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cleartomark
%%EndFont 
%%BeginFont: CMSY6
%!PS-AdobeFont-1.1: CMSY6 1.0
%%CreationDate: 1991 Aug 15 07:21:34
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueID 5000816 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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0C87E5954608782144AC7EA7EEDBA4383BAF02C56AAE81709D32431A46D6B980
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4D6E9AB63CE4B43E42BF55BE4E324C4078642065FF7C619F9D224753D2D496B8
1FB8DF41649C463E380B148FCFDE9DC19ECEE27E5F73CE7D2307D21301190D36
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301B44EC618BC89F9E26B33FBFD77A988137B4078474E9CA360F8A09A3FCF80C
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 8 /braceleftbig put
dup 9 /bracerightbig put
dup 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 26 /braceleftbigg put
dup 27 /bracerightbigg put
dup 32 /parenleftBigg put
dup 33 /parenrightBigg put
dup 40 /braceleftBigg put
dup 50 /bracketlefttp put
dup 51 /bracketrighttp put
dup 52 /bracketleftbt put
dup 53 /bracketrightbt put
dup 54 /bracketleftex put
dup 55 /bracketrightex put
dup 80 /summationtext put
dup 88 /summationdisplay put
dup 103 /tildewidest put
dup 112 /radicalbig put
dup 115 /radicalBigg put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
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readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSS12
%!PS-AdobeFont-1.1: CMSS12 1.0
%%CreationDate: 1991 Aug 20 17:33:47
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSS12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMSS12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-62 -251 978 758}readonly def
/UniqueID 5000804 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
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%%EndFont 
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SDict begin H.S end
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SDict begin H.S end
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3893 a
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SDict begin H.S end
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SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (equation.8) cvn H.B /DEST pdfmark
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2931 y Fq(.)404 3066 y(F)-8 b(or)46 b(eac)m(h)i Fp(0)53
b Fn(\024)g Fo(i)f Fn(\024)i Fo(N)10 b Fq(,)51 b(let)c
Fk(P)1659 3081 y Fm(i)1739 3066 y Fp(=)1868 2991 y Fi(P)1973
3095 y Fm(z)s Fj(\025)p Fl(0)2120 3066 y Fn(j)p Fo(i)p
Fp(;)17 b Fo(z)t Fn(ih)p Fo(i)p Fp(;)g Fo(z)t Fn(j)47
b Fq(denote)h(the)f(pro)5 b(jection)257 3196 y(on)m(to)34
b(the)g(subspace)h(querying)f(the)g Fo(i)1659 3160 y
Fd(th)1764 3196 y Fq(oracle)e(bit.)46 b(Let)34 b Fo(\014)2470
3211 y Fm(x;i)2587 3196 y Fp(=)29 b Fn(j)p Fk(P)2782
3211 y Fm(i)2810 3196 y Fn(j)p Fo( )2905 3160 y Fm(t)2901
3221 y(x)2945 3196 y Fn(ij)k Fq(denote)h(the)257 3317
y(absolute)i(v)-5 b(alue)36 b(of)f(the)i(amplitude)d(of)i(querying)g
(the)h Fo(i)2376 3281 y Fd(th)2483 3317 y Fq(bit)e(in)g(the)i
Fo(t)22 b Fp(+)g(1)3131 3279 y Fd(th)3238 3317 y Fq(query)-8
b(,)257 3437 y(pro)m(vided)34 b(the)g(oracle)f(is)g Fo(x)p
Fq(.)46 b(Note)34 b(that)1787 3362 y Fi(P)1892 3389 y
Fm(N)1892 3466 y(i)p Fl(=0)2027 3437 y Fo(\014)2088 3401
y Fl(2)2082 3462 y Fm(x;i)2199 3437 y Fp(=)29 b(1)k Fq(for)g(an)m(y)h
(oracle)f Fo(x)p Fq(,)h(since)g(the)257 3558 y(algorithm)c(queries)35
b(one)e(of)g(the)h Fo(N)44 b Fq(bits)33 b Fo(x)1860 3573
y Fl(1)1900 3558 y Fo(;)17 b(:)g(:)g(:)f(;)h(x)2174 3573
y Fm(N)2241 3558 y Fq(,)34 b(or)f(sim)m(ulates)f(a)h(non-query)h(b)m(y)
257 3678 y(querying)e(the)h(oracle)d(with)i Fo(i)c Fp(=)f(0)p
Fq(.)43 b(The)33 b Fo(t)22 b Fp(+)g(1)2014 3640 y Fd(th)2116
3678 y Fq(query)33 b(c)m(hanges)h(the)e(inner)f(pro)s(duct)257
3798 y(b)m(y)41 b(at)f(most)f(the)h(o)m(v)m(erlap)g(b)s(et)m(w)m(een)i
(the)f(pro)5 b(jections)40 b(of)f(the)i(t)m(w)m(o)f(states)h(on)m(to)f
(the)257 3919 y(subspace)35 b(that)d(corresp)s(onds)i(to)e(indices)g
Fo(i)h Fq(on)g(whic)m(h)g Fo(x)2386 3934 y Fm(i)2447
3919 y Fq(and)f Fo(y)2684 3934 y Fm(i)2745 3919 y Fq(di\033er,)834
3946 y Fi(\014)834 4006 y(\014)834 4066 y(\014)868 4061
y Fn(h)p Fo( )974 4025 y Fm(t)970 4086 y(x)1014 4061
y Fn(j)p Fo( )1109 4025 y Fm(t)1105 4086 y(y)1146 4061
y Fn(i)22 b(\000)h(h)p Fo( )1413 4025 y Fm(t)p Fl(+1)1409
4086 y Fm(x)1532 4061 y Fn(j)p Fo( )1627 4025 y Fm(t)p
Fl(+1)1623 4086 y Fm(y)1747 4061 y Fn(i)1786 3946 y Fi(\014)1786
4006 y(\014)1786 4066 y(\014)1846 4061 y Fp(=)1950 3946
y Fi(\014)1950 4006 y(\014)1950 4066 y(\014)1983 4061
y Fn(h)p Fo( )2089 4025 y Fm(t)2085 4086 y(x)2129 4061
y Fn(j)p Fp(\()p Fk(I)e Fn(\000)i Fk(O)2415 4076 y Fm(x)2459
4061 y Fk(O)2531 4076 y Fm(y)2572 4061 y Fp(\))p Fn(j)p
Fo( )2705 4025 y Fm(t)2701 4086 y(y)2743 4061 y Fn(i)2782
3946 y Fi(\014)2782 4006 y(\014)2782 4066 y(\014)2842
4061 y Fp(=)257 4126 y
SDict begin H.S end
 257 4126 a 257 4126 a
SDict begin 14.5 H.A end
 257 4126
a 257 4126 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.11) cvn H.B /DEST pdfmark
end
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Fi(\014)1156 4306 y(\014)1156 4365 y(\014)1190 4360 y
Fp(2)1296 4266 y Fi(X)1256 4478 y Fm(i)p Fl(:)p Fm(x)1340
4488 y Fh(i)1365 4478 y Fj(6)p Fl(=)p Fm(y)1455 4488
y Fh(i)1481 4360 y Fn(h)p Fo( )1587 4319 y Fm(t)1583
4385 y(x)1627 4360 y Fn(j)p Fk(P)1717 4375 y Fm(i)1745
4360 y Fn(j)p Fo( )1840 4319 y Fm(t)1836 4385 y(y)1877
4360 y Fn(i)1916 4246 y Fi(\014)1916 4306 y(\014)1916
4365 y(\014)1977 4360 y Fn(\024)28 b Fp(2)2188 4266 y
Fi(X)2148 4478 y Fm(i)p Fl(:)p Fm(x)2232 4488 y Fh(i)2257
4478 y Fj(6)p Fl(=)p Fm(y)2347 4488 y Fh(i)2390 4360
y Fo(\014)2445 4375 y Fm(x;i)2532 4360 y Fo(\014)2587
4375 y Fm(y)r(;i)2673 4360 y Fo(:)622 b Fq(\(11\))404
4698 y(The)40 b(bigger)e(amplitudes)g(of)h(querying)h(the)g(bits)f
Fo(i)g Fq(on)g(whic)m(h)h Fo(x)2869 4713 y Fm(i)2937
4698 y Fq(and)g Fo(y)3182 4713 y Fm(i)3249 4698 y Fq(di\033er,)257
4818 y(the)33 b(larger)f(the)h(drop)f(in)g(the)h(inner)f(pro)s(duct)h
(can)g(b)s(e.)404 4953 y(De\034ne)h(an)e(auxiliary)f(v)m(ector)i
Fo(a)1591 4968 y Fm(i)1620 4953 y Fp([)p Fo(x)p Fp(])28
b(=)f Fo(\016)1903 4968 y Fm(x)1947 4953 y Fo(\014)2002
4968 y Fm(x;i)2123 4953 y Fq(and)32 b(note)h(that)808
5188 y Fm(N)768 5218 y Fi(X)782 5428 y Fm(i)p Fl(=0)928
5313 y Fo(a)979 5272 y Fl(2)979 5337 y Fm(i)1046 5313
y Fp(=)1190 5188 y Fm(N)1150 5218 y Fi(X)1165 5428 y
Fm(i)p Fl(=0)1310 5218 y Fi(X)1363 5427 y Fm(x)1471 5313
y Fo(\016)1518 5272 y Fl(2)1514 5337 y Fm(x)1558 5313
y Fo(\014)1619 5272 y Fl(2)1613 5337 y Fm(x;i)1728 5313
y Fp(=)1832 5218 y Fi(X)1884 5427 y Fm(x)1992 5313 y
Fo(\016)2039 5272 y Fl(2)2035 5337 y Fm(x)2136 5188 y(N)2096
5218 y Fi(X)2111 5428 y Fm(i)p Fl(=0)2257 5313 y Fo(\014)2318
5272 y Fl(2)2312 5337 y Fm(x;i)2427 5313 y Fp(=)2530
5218 y Fi(X)2582 5427 y Fm(x)2691 5313 y Fo(\016)2738
5272 y Fl(2)2734 5337 y Fm(x)2806 5313 y Fp(=)27 b(1)p
Fo(:)p eop end
%%Page: 10 10
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 0 0 a 257 266 a
SDict begin H.S end
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a 257 266 a
SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark
end
 257 266 a 307 x Fq(The)34
b(drop)e(in)g(the)h(total)e(w)m(eigh)m(t)i(is)f(upp)s(er)h(b)s(ounded)g
(b)m(y)716 774 y Fi(\014)716 834 y(\014)749 859 y Fo(W)855
818 y Fm(t)907 859 y Fn(\000)22 b Fo(W)1112 818 y Fm(t)p
Fl(+1)1232 774 y Fi(\014)1232 834 y(\014)1349 859 y Fp(=)1508
745 y Fi(\014)1508 804 y(\014)1508 864 y(\014)1558 765
y(X)1582 974 y Fm(x;y)1719 859 y Fp(\000[)p Fo(x;)17
b(y)t Fp(])p Fo(\016)2028 874 y Fm(x)2071 859 y Fo(\016)2114
874 y Fm(y)2156 778 y Fi(\000)2201 859 y Fn(h)p Fo( )2303
874 y Fm(x)2348 859 y Fn(j)p Fo( )2439 874 y Fm(y)2480
859 y Fn(i)22 b(\000)h(h)p Fo( )2747 818 y Fj(0)2743
884 y Fm(x)2787 859 y Fn(j)p Fo( )2882 818 y Fj(0)2878
884 y Fm(y)2919 859 y Fn(i)2958 778 y Fi(\001)3003 745
y(\014)3003 804 y(\014)3003 864 y(\014)1349 1144 y Fp(=)1508
1030 y Fi(\014)1508 1090 y(\014)1508 1149 y(\014)1542
1144 y Fp(2)1608 1050 y Fi(X)1631 1259 y Fm(x;y)1808
1050 y Fi(X)1767 1262 y Fm(i)p Fl(:)p Fm(x)1851 1272
y Fh(i)1877 1262 y Fj(6)p Fl(=)p Fm(y)1967 1272 y Fh(i)2010
1144 y Fp(\000[)p Fo(x;)17 b(y)t Fp(])p Fo(\016)2319
1159 y Fm(x)2362 1144 y Fo(\016)2405 1159 y Fm(y)2447
1144 y Fn(h)p Fo( )2549 1159 y Fm(x)2593 1144 y Fn(j)p
Fk(P)2683 1159 y Fm(i)2711 1144 y Fn(j)p Fo( )2802 1159
y Fm(y)2843 1144 y Fn(i)2882 1030 y Fi(\014)2882 1090
y(\014)2882 1149 y(\014)1348 1423 y Fn(\024)83 b Fp(2)1574
1328 y Fi(X)1597 1537 y Fm(x;y)1734 1328 y Fi(X)1794
1538 y Fm(i)1895 1423 y Fp(\000)1956 1438 y Fm(i)1984
1423 y Fp([)p Fo(x;)17 b(y)t Fp(])p Fo(\016)2232 1438
y Fm(x)2275 1423 y Fo(\016)2318 1438 y Fm(y)2382 1423
y Fn(\001)22 b Fo(\014)2487 1438 y Fm(x;i)2575 1423 y
Fo(\014)2630 1438 y Fm(y)r(;i)1349 1698 y Fp(=)83 b(2)1574
1603 y Fi(X)1634 1813 y Fm(i)1734 1698 y Fo(a)1785 1657
y Fj(\003)1785 1722 y Fm(i)1825 1698 y Fp(\000)1886 1713
y Fm(i)1914 1698 y Fo(a)1965 1713 y Fm(i)1348 1961 y
Fn(\024)g Fp(2)1574 1866 y Fi(X)1634 2076 y Fm(i)1734
1961 y Fo(\025)p Fp(\(\000)1890 1976 y Fm(i)1918 1961
y Fp(\))p Fo(a)2007 1920 y Fl(2)2007 1986 y Fm(i)1348
2224 y Fn(\024)g Fp(2)17 b(max)1653 2285 y Fm(i)1772
2224 y Fo(\025)p Fp(\(\000)1928 2239 y Fm(i)1956 2224
y Fp(\))22 b Fn(\001)2066 2129 y Fi(X)2126 2339 y Fm(i)2226
2224 y Fo(a)2277 2183 y Fl(2)2277 2249 y Fm(i)1349 2457
y Fp(=)83 b(2)17 b(max)1653 2518 y Fm(i)1772 2457 y Fo(\025)p
Fp(\(\000)1928 2472 y Fm(i)1956 2457 y Fp(\))p Fo(:)257
2749 y Fq(Here)33 b Fo(a)538 2713 y Fj(\003)538 2773
y Fm(i)610 2749 y Fq(denotes)g(the)g(transp)s(ose)g(of)f
Fo(a)1729 2764 y Fm(i)1757 2749 y Fq(.)43 b(The)34 b(\034rst)e
(inequalit)m(y)f(b)s(ounds)i(the)g(drop)f(in)257 2869
y(inner)k(pro)s(duct)f(for)g(a)g(sp)s(eci\034c)h(pair)f(and)g(follo)m
(ws)f(from)h(Equation)2833 2869 y
SDict begin H.S end
 2833 2869 a Fq(11)2930
2805 y
SDict begin H.R end
 2930 2805 a 2930 2869 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.11) cvn H.B /ANN pdfmark end
 2930 2869 a Fq(.)53 b(The)36
b(second)257 2989 y(inequalit)m(y)24 b(follo)m(ws)g(from)g(the)i(sp)s
(ectral)f(norm)f(of)h Fp(\000)p Fq(.)40 b(The)27 b(second)f(and)f
(third)g(inequal-)257 3110 y(ities)i(state)g(that)g(the)h(b)s(est)g(p)s
(ossible)f(query)h(distributes)f(the)h(amplitude)d(of)i(the)h(query)257
3230 y(according)k(to)g(the)h(largest)f(principal)f(eigen)m(v)m(ector)i
(of)f(the)h(query)h(matrices)e Fp(\000)3213 3245 y Fm(i)3241
3230 y Fq(.)161 b Fn(u)-66 b(t)257 3443 y
SDict begin H.S end
 257 3443 a
257 3443 a
SDict begin 14.5 H.A end
 257 3443 a 257 3443 a
SDict begin [ /View [/XYZ H.V] /Dest (example.2) cvn H.B /DEST pdfmark
end
 257 3443 a 65 x Ff(Example:)86
b(Ordered)56 b(Seac)m(hing)h(2.)50 b Fe(R)-5 b(eturning)49
b(to)g(our)h(example)e(of)h(or)-5 b(der)g(e)g(d)257 3628
y(se)g(ar)g(ching,)37 b(for)g Fo(N)h Fp(=)28 b(4)p Fe(,)37
b(the)h(adversary)e(matrix)h(with)h(r)-5 b(esp)g(e)g(ct)37
b(to)g(the)g(or)-5 b(der)g(e)g(d)37 b(b)-5 b(asis)257
3748 y Fp(\(0001)p Fo(;)17 b Fp(0011)p Fo(;)g Fp(0111)p
Fo(;)g Fp(1111\))31 b Fe(is)j(given)g(by)1312 4191 y
Fp(\000)1373 4150 y Fd(searc)n(h)1568 4131 y Fl(\(4\))1690
4191 y Fp(=)1794 3931 y Fi(2)1794 4107 y(6)1794 4166
y(6)1794 4230 y(4)1863 4010 y Fp(0)90 b(1)2147 3971 y
Fl(1)p 2147 3987 36 4 v 2147 4044 a(2)2285 3971 y(1)p
2285 3987 V 2285 4044 a(3)1863 4130 y Fp(1)g(0)f(1)2285
4091 y Fl(1)p 2285 4107 V 2285 4165 a(2)1870 4212 y(1)p
1870 4228 V 1870 4285 a(2)2002 4251 y Fp(1)g(0)g(1)1870
4332 y Fl(1)p 1870 4348 V 1870 4405 a(3)2008 4332 y(1)p
2008 4348 V 2008 4405 a(2)2140 4371 y Fp(1)g(0)2330 3931
y Fi(3)2330 4107 y(7)2330 4166 y(7)2330 4230 y(5)2413
4191 y Fo(:)257 4628 y Fe(The)29 b(sp)-5 b(e)g(ctr)g(al)29
b(norm)g(is)g(e)-5 b(asily)29 b(se)-5 b(en)29 b(to)g(b)-5
b(e)29 b(lower)g(b)-5 b(ounde)g(d)29 b(by)g(the)h(sum)f(of)g(the)h
(entries)257 4761 y(in)25 b(the)h(\034rst)f(r)-5 b(ow,)27
b Fo(\025)p Fp(\(\000)1077 4725 y Fd(searc)n(h)1272 4706
y Fl(\(4\))1366 4761 y Fp(\))h Fn(\025)g Fp(1)q(+)1674
4722 y Fl(1)p 1674 4738 V 1674 4796 a(2)1720 4761 y Fp(+)1807
4722 y Fl(1)p 1807 4738 V 1807 4796 a(3)1853 4761 y Fe(.)41
b(In)25 b(gener)-5 b(al,)26 b Fo(\025)p Fp(\(\000)2555
4725 y Fd(searc)n(h)2750 4761 y Fp(\))f Fe(is)g(lower)g(b)-5
b(ounde)g(d)257 4882 y(by)39 b(the)g(harmonic)e(numb)-5
b(er)38 b Fo(H)1424 4897 y Fm(N)7 b Fj(\000)p Fl(1)1582
4882 y Fe(,)39 b(which)f(is)g(at)h(le)-5 b(ast)39 b Fp(ln\()p
Fo(N)10 b Fp(\))p Fe(.)57 b(The)38 b(sp)-5 b(e)g(ctr)g(al)38
b(norm)257 5002 y(of)49 b(the)g(query)g(matric)-5 b(es)48
b Fo(\025)p Fp(\(\000)1403 4966 y Fd(searc)n(h)1403 5027
y Fm(i)1597 5002 y Fp(\))h Fe(is)g(maximize)-5 b(d)47
b(when)h Fo(i)54 b Fp(=)f Fn(b)p Fo(N)r(=)p Fp(2)p Fn(c)p
Fe(,)g(in)48 b(which)257 5122 y(c)-5 b(ase)41 b(it)g(is)g(upp)-5
b(er)41 b(b)-5 b(ounde)g(d)40 b(by)i(the)f(sp)-5 b(e)g(ctr)g(al)40
b(norm)h(of)g(the)g(in\034nite)f(Hilb)-5 b(ert)42 b(matrix)257
5243 y Fp([1)p Fo(=)p Fp(\()p Fo(r)9 b Fp(+)d Fo(s)g
Fn(\000)g Fp(1\)])804 5258 y Fm(r)n(;s)p Fj(\025)p Fl(1)980
5243 y Fe(,)29 b(which)e(is)g Fo(\031)t Fe(.)43 b(W)-7
b(e)27 b(thus)h(r)-5 b(epr)g(ove)27 b(the)g(lower)g(b)-5
b(ound)27 b(of)h Fp(\(1)6 b Fn(\000)g Fo(\017)3237 5207
y Fj(0)3261 5243 y Fp(\))3309 5195 y Fl(ln)o(\()p Fm(N)h
Fl(\))p 3309 5220 177 4 v 3375 5277 a Fm(\031)257 5363
y Fe(for)35 b(or)-5 b(der)g(e)g(d)34 b(se)-5 b(ar)g(ching)34
b(in)g(given)g([)1582 5363 y
SDict begin H.S end
 1582 5363 a Fe(30)1682 5299
y
SDict begin H.R end
 1682 5299 a 1682 5363 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.hns:ordered-search) cvn H.B /ANN pdfmark end
 1682 5363 a Fe(].)p eop end
%%Page: 11 11
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 0 0 a 257 266 a
SDict begin H.S end
 257 266
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SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark
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 257 266 a 257 473 a
SDict begin H.S end
 257
473 a 257 473 a
SDict begin 14.5 H.A end
 257 473 a 257 473 a
SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark
end
 257 473 a 100 x Fr(5)156
b(Applying)53 b(the)f(sp)t(ectral)i(metho)t(d)257 792
y Fq(The)39 b(sp)s(ectral)e(metho)s(d)g(is)g(v)m(ery)i(app)s(ealing)d
(in)h(that)g(it)g(has)h(a)f(simple)f(form)m(ulation,)257
912 y(a)f(basic)f(pro)s(of,)h(and)g(giv)m(es)g(go)s(o)s(d)e(lo)m(w)m
(er)i(b)s(ounds)g(for)g(man)m(y)f(problems.)49 b(\222palek)35
b(and)257 1033 y(Szegedy)46 b([)667 1033 y
SDict begin H.S end
 667 1033 a
Fq(51)764 968 y
SDict begin H.R end
 764 968 a 764 1033 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ss:equiv) cvn H.B /ANN pdfmark end
 764 1033 a Fq(])e(sho)m(w)g(that)g
(for)f(an)m(y)h(problem,)h(the)f(b)s(est)g(lo)m(w)m(er)g(b)s(ound)g(ac)
m(hiev)-5 b(able)257 1153 y(b)m(y)39 b(the)g(sp)s(ectral)f(metho)s(d)f
(is)h(alw)m(a)m(ys)h(at)e(least)h(as)g(go)s(o)s(d)f(as)i(the)f(b)s(est)
h(lo)m(w)m(er)f(b)s(ound)257 1273 y(ac)m(hiev)-5 b(able)47
b(b)m(y)h(an)m(y)g(of)f(the)g(previously)g(published)g(adv)m(ersary)i
(metho)s(ds.)87 b(Their)257 1394 y(pro)s(of)48 b(is)g(constructiv)m(e)i
(and)e(illuminating:)71 b(giv)m(en)48 b(an)m(y)h(lo)m(w)m(er)g(b)s
(ound)g(in)e(an)m(y)j(of)257 1514 y(the)c(previously)g(published)g(adv)
m(ersary)h(metho)s(ds,)i(they)d(construct)h(an)f(adv)m(ersary)257
1635 y(matrix)31 b Fp(\000)i Fq(and)f(pro)m(v)m(e)i(it)e(ac)m(hiev)m
(es)i(the)f(same)f(lo)m(w)m(er)h(b)s(ound.)404 1755 y(The)40
b(\034rst)g(general)e(quan)m(tum)i(lo)m(w)m(er)f(b)s(ound)g(using)g
(adv)m(ersary)i(argumen)m(ts)e(w)m(as)257 1875 y(in)m(tro)s(duced)34
b(b)m(y)g(Am)m(bainis)e(in)g([)1458 1875 y
SDict begin H.S end
 1458 1875
a Fq(3)1507 1811 y
SDict begin H.R end
 1507 1811 a 1507 1875 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:lowerb) cvn H.B /ANN pdfmark end
 1507 1875
a Fq(].)46 b(As)34 b(sho)m(wn)g(in)f([)2190 1875 y
SDict begin H.S end
 2190
1875 a Fq(51)2288 1811 y
SDict begin H.R end
 2288 1811 a 2288 1875 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ss:equiv) cvn H.B /ANN pdfmark end
 2288
1875 a Fq(],)g(it)g(can)g(b)s(e)h(deriv)m(ed)g(from)e(the)257
1996 y(sp)s(ectral)37 b(metho)s(d)g(b)m(y)g(applying)f(simple)g(b)s
(ounds)h(on)g(the)h(sp)s(ectral)e(norm)h(of)f Fp(\000)h
Fq(and)257 2116 y(eac)m(h)e Fp(\000)539 2131 y Fm(i)567
2116 y Fq(.)45 b(By)34 b(de\034nition,)f(the)g(n)m(umerator)g
Fo(\025)p Fp(\(\000\))g Fq(is)g(lo)m(w)m(er-b)s(ounded)g(b)m(y)3061
2077 y Fl(1)p 3024 2093 111 4 v 3024 2150 a Fj(j)p Fm(d)p
Fj(j)3100 2132 y Fg(2)3144 2116 y Fo(d)3195 2080 y Fj(\003)3234
2116 y Fp(\000)p Fo(d)g Fq(for)257 2250 y(an)m(y)45 b(non-negativ)m(e)e
(v)m(ector)i Fo(d)p Fq(,)h(and)e(b)m(y)h(Mathias')e(lemma)e([)2581
2250 y
SDict begin H.S end
 2581 2250 a Fq(39)2679 2186 y
SDict begin H.R end
 2679 2186 a 2679
2250 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.mathias:spectral-norm) cvn H.B /ANN pdfmark end
 2679 2250 a Fq(],)47 b(the)d(denominator)257 2370
y Fo(\025)p Fp(\(\000)413 2385 y Fm(i)441 2370 y Fp(\))33
b Fq(is)f(upp)s(er-b)s(ounded)h(b)m(y)h(the)f(pro)s(duct)f(of)g(a)h(ro)
m(w-norm)e(and)i(a)f(column-norm.)257 2395 y
SDict begin H.S end
 257 2395
a 257 2395 a
SDict begin 14.5 H.A end
 257 2395 a 257 2395 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.3) cvn H.B /DEST pdfmark
end
 257 2395 a 178 x Ff(Lemma)52
b(3)h(\([)853 2573 y
SDict begin H.S end
 853 2573 a Ff(39)964 2508 y
SDict begin H.R end
 964
2508 a 964 2573 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.mathias:spectral-norm) cvn H.B /ANN pdfmark end
 964 2573 a Ff(,)1048 2573 y
SDict begin H.S end
 1048 2573
a Ff(51)1160 2508 y
SDict begin H.R end
 1160 2508 a 1160 2573 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ss:equiv) cvn H.B /ANN pdfmark end
 1160 2573
a Ff(]\).)48 b Fe(L)-5 b(et)47 b Fo(G)g Fe(b)-5 b(e)46
b(any)h(non-ne)-5 b(gative)45 b(symmetric)h(matrix)h(and)257
2693 y Fo(M)5 b(;)17 b(N)49 b Fe(non-ne)-5 b(gative)37
b(matric)-5 b(es)37 b(such)h(that)h Fo(G)33 b Fp(=)h
Fo(M)h Fn(\016)25 b Fo(N)48 b Fe(is)38 b(the)g(entrywise)g(pr)-5
b(o)g(duct)257 2813 y(of)35 b Fo(M)46 b Fe(and)34 b Fo(N)10
b Fe(.)45 b(Then)1277 3032 y Fo(\025)p Fp(\()p Fo(G)p
Fp(\))28 b Fn(\024)71 b Fp(max)1710 3070 y Fh(x;y)1630
3126 y(G)p Fg([)p Fh(x;y)r Fg(])p Fh(>)p Fg(0)1904 3032
y Fo(r)1948 3047 y Fm(x)1992 3032 y Fp(\()p Fo(M)10 b
Fp(\))28 b Fo(c)2242 3047 y Fm(y)2284 3032 y Fp(\()p
Fo(N)10 b Fp(\))p Fo(;)257 3337 y Fe(wher)-5 b(e)37 b
Fo(r)579 3352 y Fm(x)623 3337 y Fp(\()p Fo(M)10 b Fp(\))37
b Fe(is)g(the)g Fo(`)1152 3352 y Fl(2)1191 3337 y Fe(-norm)g(of)f(the)h
Fo(x)1824 3301 y Fc(th)1926 3337 y Fe(r)-5 b(ow)37 b(in)f
Fo(M)10 b Fe(,)39 b(and)d Fo(c)2641 3352 y Fm(y)2682
3337 y Fp(\()p Fo(N)10 b Fp(\))38 b Fe(is)f(the)g Fo(`)3196
3352 y Fl(2)3235 3337 y Fe(-norm)257 3457 y(of)e(the)g
Fo(y)586 3421 y Fc(th)684 3457 y Fe(c)-5 b(olumn)34 b(in)h
Fo(N)10 b Fe(.)404 3660 y Fq(Applying)30 b(these)i(t)m(w)m(o)f(b)s
(ounds,)h(w)m(e)g(obtain)e(Am)m(bainis')f(lo)m(w)m(er)i(b)s(ound)g(in)f
([)3214 3660 y
SDict begin H.S end
 3214 3660 a Fq(3)3263 3595 y
SDict begin H.R end
 3263 3595
a 3263 3660 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:lowerb) cvn H.B /ANN pdfmark end
 3263 3660 a Fq(].)43 b(W)-8 b(e)257 3780
y(refer)41 b(to)f(the)g(metho)s(d)g(as)g(an)h(un)m(w)m(eigh)m(ted)g
(adv)m(ersary)h(metho)s(d)e(since)g(it)f(considers)257
3900 y(only)31 b(t)m(w)m(o)h(t)m(yp)s(es)g(of)f(inputs:)43
b(easy)32 b(inputs)f(and)g(hard)h(inputs.)43 b(W)-8 b(e)31
b(construct)h(a)f(zero-)257 4021 y(one)42 b(v)-5 b(alued)41
b(adv)m(ersary)i(matrix)d Fp(\000)h Fq(that)h(corresp)s(onds)g(to)f(a)h
(uniform)d(distribution)257 4141 y(o)m(v)m(er)34 b(the)f(hard)f(input)h
(pairs.)257 4161 y
SDict begin H.S end
 257 4161 a 257 4161 a
SDict begin 14.5 H.A end
 257 4161 a 257
4161 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.4) cvn H.B /DEST pdfmark
end
 257 4161 a 182 x Ff(Theorem)40 b(4)f(\(Un)m(w)m(eigh)m(ted)g
(metho)s(d)g([)1933 4343 y
SDict begin H.S end
 1933 4343 a Ff(3)1989 4279
y
SDict begin H.R end
 1989 4279 a 1989 4343 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:lowerb) cvn H.B /ANN pdfmark end
 1989 4343 a Ff(]\).)k Fe(L)-5
b(et)37 b Fo(F)50 b Fe(b)-5 b(e)36 b(a)h(p)-5 b(artial)36
b(b)-5 b(o)g(ole)g(an)36 b(func-)257 4464 y(tion,)k(and)e(let)h
Fo(A)c Fn(\022)h Fo(F)1126 4428 y Fj(\000)p Fl(1)1220
4464 y Fp(\(0\))i Fe(and)g Fo(B)j Fn(\022)35 b Fo(F)1880
4428 y Fj(\000)p Fl(1)1974 4464 y Fp(\(1\))k Fe(b)-5
b(e)38 b(subsets)h(of)f(\(har)-5 b(d\))38 b(inputs.)57
b(L)-5 b(et)257 4584 y Fo(R)48 b Fn(\022)f Fo(A)29 b
Fn(\002)h Fo(B)50 b Fe(b)-5 b(e)45 b(a)g(r)-5 b(elation,)47
b(and)d(set)h Fo(R)1896 4599 y Fm(i)1971 4584 y Fp(=)h
Fn(f)p Fp(\()p Fo(x;)17 b(y)t Fp(\))45 b Fn(2)i Fo(R)g
Fp(:)g Fo(x)2778 4599 y Fm(i)2853 4584 y Fn(6)p Fp(=)f
Fo(y)3023 4599 y Fm(i)3051 4584 y Fn(g)e Fe(for)h(e)-5
b(ach)257 4704 y Fp(1)28 b Fn(\024)g Fo(i)g Fn(\024)g
Fo(n)p Fe(.)43 b(L)-5 b(et)31 b Fo(m;)17 b(m)1114 4668
y Fj(0)1168 4704 y Fe(denote)29 b(the)h(minimal)f(numb)-5
b(er)29 b(of)h(ones)f(in)h(any)g(r)-5 b(ow)30 b(and)f(any)257
4825 y(c)-5 b(olumn)39 b(in)h(r)-5 b(elation)39 b Fo(R)q
Fe(,)i(r)-5 b(esp)g(e)g(ctively,)41 b(and)e(let)h Fo(`;)17
b(`)2239 4789 y Fj(0)2301 4825 y Fe(denote)40 b(the)f(maximal)g(numb)-5
b(er)257 4945 y(of)38 b(ones)f(in)h(any)g(r)-5 b(ow)37
b(and)h(any)f(c)-5 b(olumn)38 b(in)f(any)h(of)g(the)g(r)-5
b(elations)37 b Fo(R)2894 4960 y Fm(i)2922 4945 y Fe(,)i(r)-5
b(esp)g(e)g(ctively.)257 5066 y(Then)34 b Fo(Q)588 5081
y Fl(2)628 5066 y Fp(\()p Fo(f)11 b Fp(\))27 b(=)h(\012\()1002
4980 y Fi(p)p 1102 4980 348 4 v 86 x Fo(mm)1272 5037
y Fj(0)1296 5066 y Fo(=``)1427 5037 y Fj(0)1450 5066
y Fp(\))p Fe(.)257 5268 y(Pr)-5 b(o)g(of.)49 b Fq(Let)54
b Fo(S)70 b Fp(=)65 b Fn(f)p Fp(\()p Fo(x;)17 b(y)t Fp(\))63
b(:)h(\()p Fo(x;)17 b(y)t Fp(\))64 b Fn(2)h Fo(R)38 b
Fn(_)f Fp(\()p Fo(y)t(;)17 b(x)p Fp(\))64 b Fn(2)g Fo(R)q
Fn(g)54 b Fq(b)s(e)h(a)f(symmetrized)257 5388 y(v)m(ersion)e(of)e
Fo(R)q Fq(.)98 b(De\034ne)52 b(a)e(column)g(v)m(ector)i
Fo(d)e Fq(from)g(the)h(relation)d Fo(S)57 b Fq(b)m(y)52
b(setting)p eop end
%%Page: 12 12
TeXDict begin 12 11 bop 0 0 a
SDict begin /product where{pop product(Distiller)search{pop pop pop
version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto
closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show
grestore}if}{pop}ifelse}{pop}ifelse}if end
 0 0 a 257 266 a
SDict begin H.S end
 257 266
a 257 266 a
SDict begin H.R end
 257 266 a 257 266 a
SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark
end
 257 266 a 307 x Fo(d)308
588 y Fm(x)380 573 y Fp(=)483 488 y Fi(p)p 583 488 703
4 v 85 x Fn(jf)p Fo(y)30 b Fp(:)e(\()p Fo(x;)17 b(y)t
Fp(\))27 b Fn(2)h Fo(S)6 b Fn(gj)o Fq(,)28 b(and)e(an)g(adv)m(ersary)i
(matrix)d Fp(\000)h Fq(b)m(y)i(setting)e Fp(\000[)p Fo(x;)17
b(y)t Fp(])27 b(=)3393 534 y Fl(1)p 3336 550 150 4 v
3336 607 a Fm(d)3372 615 y Fh(x)3411 607 y Fm(d)3447
615 y Fh(y)257 716 y Fq(if)55 b(and)h(only)f(if)f Fp(\()p
Fo(x;)17 b(y)t Fp(\))66 b Fn(2)h Fo(S)6 b Fq(.)113 b(Then)56
b Fo(\025)p Fp(\(\000\))67 b Fn(\025)2294 676 y Fl(1)p
2257 693 111 4 v 2257 750 a Fj(j)p Fm(d)p Fj(j)2333 731
y Fg(2)2377 716 y Fo(d)2428 679 y Fj(\003)2467 716 y
Fp(\000)p Fo(d)f Fp(=)h(1)p Fq(.)112 b(F)-8 b(or)55 b(eac)m(h)i(of)257
857 y(the)47 b(matrices)e Fp(\000)904 872 y Fm(i)932
857 y Fq(,)k(w)m(e)e(apply)f(Lemma)1808 857 y
SDict begin H.S end
 1808 857
a Fq(3)1857 793 y
SDict begin H.R end
 1857 793 a 1857 857 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3) cvn H.B /ANN pdfmark end
 1857 857 a 46
w Fq(with)f Fo(M)10 b Fp([)p Fo(x;)17 b(y)t Fp(])51 b(=)f
Fo(N)10 b Fp([)p Fo(y)t(;)17 b(x)p Fp(])51 b(=)3124 818
y Fl(1)p 3104 834 76 4 v 3104 891 a Fm(d)3140 899 y Fh(x)3235
857 y Fq(if)45 b(and)257 1001 y(only)51 b(if)f Fp(\()p
Fo(x;)17 b(y)t Fp(\))59 b Fn(2)g Fo(S)6 b Fq(.)99 b(F)-8
b(or)51 b(ev)m(ery)i Fo(x)60 b Fn(2)f Fo(A)p Fq(,)d Fo(r)2111
1016 y Fm(x)2155 1001 y Fp(\()p Fo(M)10 b Fp(\))60 b
Fn(\024)2532 916 y Fi(p)p 2632 916 185 4 v 85 x Fo(`=d)2773
972 y Fl(2)2773 1026 y Fm(x)2875 1001 y Fn(\024)3012
916 y Fi(p)p 3112 916 176 4 v 85 x Fo(`=m)51 b Fq(and)257
1133 y Fo(c)299 1148 y Fm(y)341 1133 y Fp(\()p Fo(N)10
b Fp(\))32 b Fn(\024)646 1048 y Fi(p)p 746 1048 206 4
v 85 x Fo(`)787 1104 y Fj(0)810 1133 y Fo(=d)910 1104
y Fl(2)910 1157 y Fm(y)983 1133 y Fn(\024)1092 1048 y
Fi(p)p 1191 1048 222 4 v 1191 1133 a Fo(`)1232 1104 y
Fj(0)1256 1133 y Fo(=m)1390 1104 y Fj(0)1413 1133 y Fq(.)51
b(F)-8 b(or)34 b(ev)m(ery)i Fo(x)d Fn(2)f Fo(B)5 b Fq(,)35
b(the)h(inequalities)d(are)i(sw)m(app)s(ed.)257 1265
y(By)e(Lemma)759 1265 y
SDict begin H.S end
 759 1265 a Fq(3)807 1200 y
SDict begin H.R end
 807
1200 a 807 1265 a
SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3) cvn H.B /ANN pdfmark end
 807 1265 a Fq(,)g Fo(\025)p Fp(\(\000)1023
1280 y Fm(i)1051 1265 y Fp(\))27 b Fn(\024)i Fp(max)1403
1280 y Fm(x;y)r Fl(:\000)1564 1290 y Fh(i)1590 1280 y
Fl([)p Fm(x;y)r Fl(])p Fm(>)p Fl(0)1837 1265 y Fo(r)1881
1280 y Fm(x)1924 1265 y Fp(\()p Fo(M)10 b Fp(\))p Fo(c)2146
1280 y Fm(y)2188 1265 y Fp(\()p Fo(N)g Fp(\))28 b Fn(\024)2486
1179 y Fi(p)p 2585 1179 348 4 v 2585 1265 a Fo(``)2667
1236 y Fj(0)2690 1265 y Fo(=mm)2909 1236 y Fj(0)2933
1265 y Fq(.)469 b Fn(u)-66 b(t)404 1459 y Fq(The)36 b(un)m(w)m(eigh)m
(ted)g(adv)m(ersary)h(metho)s(d)d(is)h(v)m(ery)h(simple)e(to)g(apply)h
(as)g(it)f(requires)257 1579 y(only)j(to)f(sp)s(ecify)i(a)e(set)i
Fo(R)g Fq(of)e(hard)h(input)g(pairs.)56 b(It)37 b(giv)m(es)g(tigh)m(t)f
(lo)m(w)m(er)h(b)s(ounds)h(for)257 1700 y(man)m(y)g(computational)d
(problems,)j(including)e(in)m(v)m(erting)h(a)g(p)s(erm)m(utation)f([)
3147 1700 y
SDict begin H.S end
 3147 1700 a Fq(3)3196 1635 y
SDict begin H.R end
 3196 1635 a
3196 1700 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:lowerb) cvn H.B /ANN pdfmark end
 3196 1700 a Fq(],)j(com-)257 1820 y(puting)29
b(an)m(y)h(symmetric)f(function)g(and)g(coun)m(ting)g([)2203
1820 y
SDict begin H.S end
 2203 1820 a Fq(42)2301 1756 y
SDict begin H.R end
 2301 1756 a 2301
1820 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.nw:median) cvn H.B /ANN pdfmark end
 2301 1820 a Fq(,)2357 1820 y
SDict begin H.S end
 2357 1820 a Fq(10)2455
1756 y
SDict begin H.R end
 2455 1756 a 2455 1820 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bcwz:error) cvn H.B /ANN pdfmark end
 2455 1820 a Fq(,)2511
1820 y
SDict begin H.S end
 2511 1820 a Fq(14)2609 1756 y
SDict begin H.R end
 2609 1756 a 2609
1820 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bhmt:count) cvn H.B /ANN pdfmark end
 2609 1820 a Fq(],)h(constan)m(t-lev)m(el)g(and-)257
1940 y(or)40 b(trees)g([)651 1940 y
SDict begin H.S end
 651 1940 a Fq(3)700
1876 y
SDict begin H.R end
 700 1876 a 700 1940 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:lowerb) cvn H.B /ANN pdfmark end
 700 1940 a Fq(,)767 1940
y
SDict begin H.S end
 767 1940 a Fq(29)864 1876 y
SDict begin H.R end
 864 1876 a 864 1940 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.hmw:berror-search) cvn H.B /ANN pdfmark end
 864
1940 a Fq(],)i(and)e(v)-5 b(arious)39 b(graph)g(problems)g([)2233
1940 y
SDict begin H.S end
 2233 1940 a Fq(21)2331 1876 y
SDict begin H.R end
 2331 1876 a 2331
1940 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.dhhm:graphs) cvn H.B /ANN pdfmark end
 2331 1940 a Fq(].)65 b(F)-8 b(or)38 b(some)i(computational)257
2061 y(problems,)e(the)g(hardness)g(do)s(es)g(ho)m(w)m(ev)m(er)h(not)e
(necessarily)h(rely)f(only)f(on)h(a)g(few)h(se-)257 2181
y(lected)43 b(hard)g(instances,)j(but)d(rather)g(on)g(more)f(global)e
(prop)s(erties)j(of)f(the)h(inputs.)257 2301 y(Applying)f(the)g(un)m(w)
m(eigh)m(ted)i(metho)s(d)d(on)h(ordered)h(searc)m(hing)f(w)m(ould)g
(for)g(instance)257 2422 y(only)e(yield)g(a)g(lo)m(w)m(er)g(b)s(ound)g
(of)g(a)g(constan)m(t.)68 b(In)40 b(these)i(cases,)h(w)m(e)f(ma)m(y)e
(apply)g(the)257 2542 y(follo)m(wing)21 b(w)m(eigh)m(ted)j(v)-5
b(arian)m(t)22 b(of)g(the)i(metho)s(d,)g(due)g(to)f(Am)m(bainis)f([)
2749 2542 y
SDict begin H.S end
 2749 2542 a Fq(4)2797 2478 y
SDict begin H.R end
 2797 2478 a
2797 2542 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:degree-vs-qc) cvn H.B /ANN pdfmark end
 2797 2542 a Fq(])i(and)f(Zhang)f([)3343 2542
y
SDict begin H.S end
 3343 2542 a Fq(57)3441 2478 y
SDict begin H.R end
 3441 2478 a 3441 2542
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.zhang:ambainis) cvn H.B /ANN pdfmark end
 3441 2542 a Fq(].)257 2567 y
SDict begin H.S end
 257 2567 a 257 2567 a
SDict begin 14.5 H.A end

257 2567 a 257 2567 a
SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark
end
 257 2567 a 176 x Ff(Theorem)30
b(5)g(\(W)-9 b(eigh)m(ted)29 b(metho)s(d)h([)1779 2743
y
SDict begin H.S end
 1779 2743 a Ff(4)1835 2678 y
SDict begin H.R end
 1835 2678 a 1835 2743
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:degree-vs-qc) cvn H.B /ANN pdfmark end
 1835 2743 a Ff(,)1896 2743 y
SDict begin H.S end
 1896 2743 a Ff(57)2008
2678 y
SDict begin H.R end
 2008 2678 a 2008 2743 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.zhang:ambainis) cvn H.B /ANN pdfmark end
 2008 2743 a Ff(]\).)37
b Fe(L)-5 b(et)29 b Fo(F)42 b Fp(:)27 b Fo(S)34 b Fn(!)27
b(f)p Fp(0)p Fo(;)17 b Fp(1)p Fn(g)2935 2707 y Fm(m)3030
2743 y Fe(b)-5 b(e)29 b(a)g(p)-5 b(artial)257 2863 y(function.)45
b(L)-5 b(et)35 b Fo(w)s(;)17 b(w)1039 2827 y Fj(0)1096
2863 y Fe(denote)34 b(a)g(weight)h(scheme)e(as)i(fol)5
b(lows:)403 3064 y Fn(\017)48 b Fe(Every)h(p)-5 b(air)47
b Fp(\()p Fo(x;)17 b(y)t Fp(\))52 b Fn(2)g Fo(S)1470
3028 y Fl(2)1558 3064 y Fe(is)c(assigne)-5 b(d)46 b(a)i(non-ne)-5
b(gative)47 b(weight)g Fo(w)s Fp(\()p Fo(x;)17 b(y)t
Fp(\))51 b(=)501 3184 y Fo(w)s Fp(\()p Fo(y)t(;)17 b(x)p
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b(y)t Fp(\))26 b(=)i(0)34 b Fe(whenever)g Fo(F)14 b Fp(\()p
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Fe(.)403 3387 y Fn(\017)48 b Fe(Every)53 b(triple)g Fp(\()p
Fo(x;)17 b(y)t(;)g(i)p Fp(\))60 b Fn(2)h Fo(S)1631 3351
y Fl(2)1706 3387 y Fn(\002)36 b Fp([)p Fo(N)10 b Fp(])53
b Fe(is)g(assigne)-5 b(d)51 b(a)i(non-ne)-5 b(gative)51
b(weight)501 3507 y Fo(w)574 3471 y Fj(0)597 3507 y Fp(\()p
Fo(x;)17 b(y)t(;)g(i)p Fp(\))24 b Fe(that)i(satis\034es)e
Fo(w)1544 3471 y Fj(0)1567 3507 y Fp(\()p Fo(x;)17 b(y)t(;)g(i)p
Fp(\))27 b(=)g(0)e Fe(whenever)f Fo(x)2545 3522 y Fm(i)2602
3507 y Fp(=)j Fo(y)2753 3522 y Fm(i)2806 3507 y Fe(or)e
Fo(F)14 b Fp(\()p Fo(x)p Fp(\))28 b(=)g Fo(F)14 b Fp(\()p
Fo(y)t Fp(\))p Fe(,)501 3627 y(and)34 b Fo(w)763 3591
y Fj(0)786 3627 y Fp(\()p Fo(x;)17 b(y)t(;)g(i)p Fp(\))p
Fo(w)1163 3591 y Fj(0)1185 3627 y Fp(\()p Fo(y)t(;)g(x;)g(i)p
Fp(\))27 b Fn(\025)h Fo(w)1694 3591 y Fl(2)1733 3627
y Fp(\()p Fo(x;)17 b(y)t Fp(\))34 b Fe(for)g(al)5 b(l)35
b Fo(x;)17 b(y)t(;)g(i)34 b Fe(such)h(that)g Fo(x)3027
3642 y Fm(i)3083 3627 y Fn(6)p Fp(=)28 b Fo(y)3235 3642
y Fm(i)3262 3627 y Fe(.)257 3828 y(Then)1048 4024 y Fo(Q)1125
4039 y Fl(2)1165 4024 y Fp(\()p Fo(F)14 b Fp(\))27 b(=)h(\012)1519
3853 y Fi( )1683 4024 y Fp(min)1700 4073 y Fh(x;y)r(;i)1635
4129 y(w)q Fg(\()p Fh(x;y)r Fg(\))p Fh(>)p Fg(0)1681
4183 y Fh(x)1716 4199 y(i)1742 4183 y Fb(6)p Fg(=)p Fh(y)1821
4199 y(i)1930 3844 y Fi(s)p 2030 3844 534 4 v 2060 3956
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Fo(y)t Fp(\))p 2040 4001 514 4 v 2040 4092 a Fo(v)t Fp(\()p
Fo(x;)17 b(i)p Fp(\))p Fo(v)t Fp(\()p Fo(y)t(;)g(i)p
Fp(\))2598 3853 y Fi(!)2677 4024 y Fo(;)257 4345 y Fe(wher)-5
b(e)36 b Fo(w)s(t)p Fp(\()p Fo(x)p Fp(\))30 b(=)909 4270
y Fi(P)1015 4374 y Fm(y)1073 4345 y Fo(w)s Fp(\()p Fo(x;)17
b(y)t Fp(\))34 b Fe(and)i Fo(v)t Fp(\()p Fo(x;)17 b(i)p
Fp(\))30 b(=)1994 4270 y Fi(P)2099 4374 y Fm(y)2157 4345
y Fo(w)2230 4309 y Fj(0)2253 4345 y Fp(\()p Fo(x;)17
b(y)t(;)g(i)p Fp(\))35 b Fe(for)h(al)5 b(l)36 b Fo(x)31
b Fn(2)f Fo(S)43 b Fe(and)35 b Fo(i)c Fn(2)257 4465 y
Fp([)p Fo(N)10 b Fp(])p Fe(.)404 4666 y Fq(A)m(t)34 b(\034rst)g
(glance,)g(the)g(w)m(eigh)m(ted)h(metho)s(d)e(ma)m(y)g(lo)s(ok)g
(rather)h(complicated,)e(b)s(oth)257 4786 y(in)h(its)f(form)m(ulation)f
(and)i(use,)h(though)f(it)f(is)h(not.)45 b(W)-8 b(e)34
b(\034rst)f(assign)g(w)m(eigh)m(ts)h(to)f(pairs)257 4907
y Fp(\()p Fo(x;)17 b(y)t Fp(\))47 b Fq(of)f(inputs)h(for)g(whic)m(h)h
Fo(F)14 b Fp(\()p Fo(x)p Fp(\))52 b Fn(6)p Fp(=)h Fo(F)14
b Fp(\()p Fo(y)t Fp(\))p Fq(,)49 b(as)e(in)g(the)g(sp)s(ectral)g(metho)
s(d.)87 b(W)-8 b(e)257 5027 y(require)37 b(the)f(w)m(eigh)m(ts)h(to)f
(b)s(e)g(symmetric)f(so)h(that)g(they)h(represen)m(t)h(the)e
(di\036cult)m(y)g(in)257 5147 y(distinguishing)31 b Fe(b)-5
b(etwe)g(en)34 b Fo(x)h Fe(and)f Fo(y)t Fq(.)404 5268
y(W)-8 b(e)31 b(then)g(afterw)m(ards)h(assign)e(w)m(eigh)m(ts)i
Fo(w)1971 5232 y Fj(0)1993 5268 y Fp(\()p Fo(x;)17 b(y)t(;)g(i)p
Fp(\))30 b Fq(that)h(represen)m(t)h(the)f(di\036cult)m(y)257
5388 y(in)26 b(distinguishing)e Fo(x)30 b Fe(fr)-5 b(om)28
b Fo(y)k Fe(by)e(querying)f(index)f Fo(i)p Fe(.)41 b
Fq(The)28 b(harder)e(it)g(is)g(to)g(distinguish)p eop
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark
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b Fq(from)e Fo(y)k Fq(b)m(y)e(index)g Fo(i)p Fq(,)h(compared)d(to)h
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Fo(i)p Fq(,)i(the)f(more)257 693 y(w)m(eigh)m(t)33 b(w)m(e)h(put)f(on)f
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Fp(\()p Fo(y)t(;)17 b(x;)g(i)p Fp(\))p Fq(,)31 b(and)i(visa-v)m(ersa.)
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Fo(t)p Fp(\()p Fo(x;)17 b(y)t(;)g(i)p Fp(\))39 b(=)h
Fo(w)2029 777 y Fj(0)2052 814 y Fp(\()p Fo(x;)17 b(y)t(;)g(i)p
Fp(\))p Fo(=w)2478 777 y Fj(0)2500 814 y Fp(\()p Fo(y)t(;)g(x;)g(i)p
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(pairs)257 1175 y Fp(\()p Fo(u;)17 b(y)t Fp(\))p Fq(,)44
b(b)m(y)g(querying)g(index)f Fo(i)p Fq(.)75 b(If)43 b(w)m(e,)k(b)m(y)d
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(and)f(otherwise)g(small.)68 b(Consider)257 1416 y(w)m(e)41
b(query)h(an)e(index)g Fo(i)g Fq(for)g(whic)m(h)g Fo(x)1671
1431 y Fm(i)1740 1416 y Fn(6)p Fp(=)g Fo(y)1904 1431
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y Fm(i)2487 1656 y Fq(and)e Fo(F)14 b Fp(\()p Fo(x)p
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(pair)f Fp(\()p Fo(u;)17 b(y)t Fp(\))35 b Fq(for)257
1897 y(whic)m(h)46 b Fo(u)605 1912 y Fm(i)682 1897 y
Fn(6)p Fp(=)j Fo(y)855 1912 y Fm(i)928 1897 y Fq(and)d
Fo(F)14 b Fp(\()p Fo(u)p Fp(\))48 b Fn(6)p Fp(=)i Fo(F)14
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s(ose)257 2138 y Fo(t)p Fp(\()p Fo(x;)17 b(y)t(;)g(i)p
Fp(\))p Fq(.)45 b(Ha)m(ving)33 b(th)m(us)h(c)m(hosen)h
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b(w)m(e)h(set)g Fo(w)2299 2102 y Fj(0)2322 2138 y Fp(\()p
Fo(x;)17 b(y)t(;)g(i)p Fp(\))28 b(=)h Fo(w)s Fp(\()p
Fo(x;)17 b(y)t Fp(\))3059 2053 y Fi(p)p 3157 2053 339
4 v 3157 2138 a Fo(t)p Fp(\()p Fo(x;)g(y)t(;)g(i)p Fp(\))257
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2751 y(strong)44 b(lo)m(w)m(er)g(b)s(ounds)h(for)e(read-once)h(form)m
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SDict begin H.S end
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SDict begin H.R end
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2687 a 2287 2751 a
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/Dest (cite.bs:q-read-once) cvn H.B /ANN pdfmark end
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3392 2751 y
SDict begin H.S end
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SDict begin H.R end
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3441 2751 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:degree-vs-qc) cvn H.B /ANN pdfmark end
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b([)724 2872 y
SDict begin H.S end
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SDict begin H.R end
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773 2872 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.s:local) cvn H.B /ANN pdfmark end
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2872 y
SDict begin H.S end
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SDict begin H.R end
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2872 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ss:local) cvn H.B /ANN pdfmark end
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y
SDict begin H.S end
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SDict begin H.R end
 2594 2807 a 2594 2872
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.zhang:local) cvn H.B /ANN pdfmark end
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y(men)m(ts)25 b(to)e(pro)m(v)m(e)i(lo)m(w)m(er)f(b)s(ounds)g(for)f(lo)s
(cal)f(searc)m(h,)27 b(a)c(distributed)h(v)m(ersion)g(of)f(Gro)m(v)m
(er's)257 3112 y(problem.)39 b(\222palek)23 b(and)g(Szegedy)h(pro)m(v)m
(e)g(in)e([)1898 3112 y
SDict begin H.S end
 1898 3112 a Fq(51)1996 3048 y
SDict begin H.R end

1996 3048 a 1996 3112 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ss:equiv) cvn H.B /ANN pdfmark end
 1996 3112 a Fq(])h(that)f(the)h(w)m(eigh)m(ted)
h(metho)s(d)e(is)g(equiv-)257 3233 y(alen)m(t)31 b(to)g(the)h(sp)s
(ectral)f(metho)s(d\026an)m(y)g(lo)m(w)m(er)h(b)s(ound)f(that)h(can)f
(b)s(e)h(ac)m(hiev)m(ed)h(b)m(y)f(one)257 3353 y(of)44
b(the)h(t)m(w)m(o)f(metho)s(ds)g(can)h(also)e(b)s(e)h(sho)m(wn)i(b)m(y)
f(the)f(other.)78 b(Their)45 b(pro)s(of)e(is)g(con-)257
3473 y(structiv)m(e)31 b(and)e(giv)m(es)h(simple)e(expressions)j(for)e
(con)m(v)m(erting)h(one)f(in)m(to)g(the)h(other.)42 b(The)257
3594 y(main)37 b(w)m(eigh)m(ts)i Fo(w)s Fp(\()p Fo(x;)17
b(y)t Fp(\))37 b Fq(are)h(the)h(co)s(e\036cien)m(ts)g(of)f(the)h(w)m
(eigh)m(t)g(function)e Fo(W)3137 3558 y Fm(t)3205 3594
y Fq(for)h(the)257 3714 y(input)i(pair)g Fp(\()p Fo(x;)17
b(y)t Fp(\))p Fq(,)41 b(that)f(is,)i Fo(w)s Fp(\()p Fo(x;)17
b(y)t Fp(\))40 b(=)h(\000[)p Fo(x;)17 b(y)t Fp(])p Fo(\016)2149
3729 y Fm(x)2192 3714 y Fo(\016)2235 3729 y Fm(y)2277
3714 y Fq(,)42 b(and)f(the)g(secondary)g(w)m(eigh)m(ts)257
3835 y Fo(w)330 3798 y Fj(0)353 3835 y Fp(\()p Fo(x;)17
b(y)t(;)g(i)p Fp(\))32 b Fq(follo)m(w)e(from)i(Mathias')g(lemma)e([)
1946 3835 y
SDict begin H.S end
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SDict begin H.R end
 2044 3770 a
2044 3835 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.mathias:spectral-norm) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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3835 a Fq(3)2538 3770 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (theorem.3) cvn H.B /ANN pdfmark end
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3835 a Fq(\).)257 3860 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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a 257 3860 a
SDict begin [ /View [/XYZ H.V] /Dest (example.3) cvn H.B /DEST pdfmark
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b(Seac)m(hing)h(3.)49 b Fe(T)-7 b(o)46 b(apply)g(the)h(weighte)-5
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4114 y Fd(searc)n(h)2842 4150 y Fp([)p Fo(x;)17 b(y)t
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b(for)g(cho)-5 b(osing)43 b(otherwise.)257 4391 y(Now,)c(c)-5
b(onsider)37 b Fo(t)p Fp(\()p Fo(x;)17 b(y)t(;)g(i)p
Fp(\))38 b Fe(with)g Fo(F)14 b Fp(\()p Fo(x)p Fp(\))33
b Fn(\024)h Fo(i)h(<)e(F)14 b Fp(\()p Fo(y)t Fp(\))37
b Fe(so)h(that)g Fo(x)2649 4406 y Fm(i)2712 4391 y Fn(6)p
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b(querying)257 4511 y(index)32 b Fo(i)p Fe(,)h(we)f(also)g(le)-5
b(arn)32 b(to)h(distinguish)f(b)-5 b(etwe)g(en)32 b Fo(x)h
Fe(and)f Fo(z)37 b Fe(for)c(e)-5 b(ach)32 b(of)g(the)g
Fo(F)14 b Fp(\()p Fo(y)t Fp(\))i Fn(\000)i Fo(i)257 4632
y Fe(inputs)36 b Fo(z)k Fe(with)35 b Fo(i)29 b(<)f(F)14
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(b)-5 b(etwe)g(en)35 b Fo(u)f Fe(and)h Fo(y)257 4752
y Fe(for)f(e)-5 b(ach)33 b(of)h(the)g Fo(i)20 b Fn(\000)h
Fo(F)14 b Fp(\()p Fo(x)p Fp(\))20 b(+)f(1)34 b Fe(inputs)g
Fo(u)f Fe(with)h Fo(F)14 b Fp(\()p Fo(x)p Fp(\))28 b
Fn(\024)g Fo(F)14 b Fp(\()p Fo(u)p Fp(\))27 b Fn(\024)h
Fo(i)p Fe(.)45 b(W)-7 b(e)33 b(thus)h(cho)-5 b(ose)257
4872 y(to)35 b(set)1324 5023 y Fo(t)p Fp(\()p Fo(x;)17
b(y)t(;)g(i)p Fp(\))27 b(=)1806 4955 y Fn(j)p Fo(F)14
b Fp(\()p Fo(y)t Fp(\))21 b Fn(\000)h Fo(i)p Fn(j)g Fp(+)g(1)p
1804 5000 588 4 v 1804 5091 a Fn(j)p Fo(F)14 b Fp(\()p
Fo(x)p Fp(\))22 b Fn(\000)g Fo(i)p Fn(j)g Fp(+)g(1)2401
5023 y Fo(:)257 5243 y Fe(Plugging)56 b(these)g(values)g(into)g(the)g
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3665 y
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3665 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.lls:formulas) cvn H.B /ANN pdfmark end
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257 3925 y
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4106 y
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4106 a
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4042 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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814 y Fo(x)43 b Fq(are)f(one)g(\(so)g Fn(j)p Fo(x)p Fn(j)i
Fp(=)f Fo(k)s Fq(\),)i(and)d(the)g(goal)f(is)g(to)h(\034nd)g(all)e(of)i
(these)h Fo(k)i Fq(indices.)72 b(W)-8 b(e)257 934 y(sa)m(y)37
b(an)f(algorithm)d Fk(A)k Fq(succeeds)h(if)d(it)g(outputs)i(a)f(subset)
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Fo(k)j Fq(and)d Fo(S)257 1054 y Fq(con)m(tains)28 b(all)e(indices)h
Fo(i)h Fn(2)g Fp([)p Fo(N)10 b Fp(])28 b Fq(for)f(whic)m(h)h
Fo(x)1879 1069 y Fm(i)1936 1054 y Fp(=)f(1)p Fq(.)42
b(Th)m(us,)30 b(b)m(y)f(de\034nition,)e(it)g(fails)f(ev)m(en)257
1175 y(if)34 b(it)g(outputs)i(all)d(but)i(one)g(of)g(the)g
Fo(k)k Fq(indices.)50 b(The)36 b Fo(k)s Fq(-fold)e(searc)m(h)i(problem)
e(can)h(b)s(e)257 1295 y(solv)m(ed)27 b(in)e Fo(O)s Fp(\()767
1210 y Fn(p)p 850 1210 113 4 v 85 x Fo(k)s(n)p Fp(\))h
Fq(queries,)i(essen)m(tially)e(b)m(y)h(sequen)m(tially)e(running)h(Gro)
m(v)m(er's)h(searc)m(h)257 1416 y(algorithm)e Fo(k)31
b Fq(times.)41 b(Klauc)m(k,)29 b(\222palek,)g(and)f(de)h(W)-8
b(olf)26 b([)2328 1416 y
SDict begin H.S end
 2328 1416 a Fq(35)2426 1351
y
SDict begin H.R end
 2426 1351 a 2426 1416 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ksw:dpt) cvn H.B /ANN pdfmark end
 2426 1416 a Fq(])i(sho)m(w)h(that)f(if)e(the)
j(n)m(um)m(b)s(er)257 1536 y(of)g(queries)g(is)g(less)g(than)f
Fo(\017)1222 1451 y Fn(p)p 1306 1451 113 4 v 1306 1536
a Fo(k)s(n)h Fq(for)f(some)h(constan)m(t)g Fo(\017)p
Fq(,)h(then)g(the)f(success)i(probabilit)m(y)257 1656
y(of)40 b Fk(A)h Fq(is)f(exp)s(onen)m(tially)f(small)f(in)i
Fo(k)s Fq(.)67 b(They)42 b(th)m(us)g(pro)m(v)m(e)f(a)f(strong)h(direct)
f(pro)s(duct)257 1777 y(theorem)26 b(for)g(the)g Fo(k)s
Fq(-fold)f(searc)m(h)i(problem.)40 b(One)27 b(of)f(the)g(main)e(elemen)
m(ts)j(of)f(the)g(pro)s(of)257 1897 y(is)32 b(the)h(p)s(olynomial)c
(metho)s(d)j(whic)m(h)h(w)m(e)h(discuss)g(in)d(the)i(next)h(section.)
404 2023 y(In)c(v)m(ery)i(recen)m(t)g(w)m(ork,)g(Am)m(bainis)c([)1745
2023 y
SDict begin H.S end
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SDict begin H.R end
 1794 1959 a 1794
2023 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:sdp) cvn H.B /ANN pdfmark end
 1794 2023 a Fq(])j(prop)s(oses)g(an)f(extension)h(of)f(the)h
(adv)m(ersary)257 2143 y(metho)s(d)45 b(and)g(uses)h(it)e(to)g(repro)m
(v)m(e)j(the)e(strong)g(direct)f(pro)s(duct)h(theorem)g(of)f([)3343
2143 y
SDict begin H.S end
 3343 2143 a Fq(35)3441 2079 y
SDict begin H.R end
 3441 2079 a 3441
2143 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ksw:dpt) cvn H.B /ANN pdfmark end
 3441 2143 a Fq(].)257 2264 y(Though)d(the)g(follo)m(wing)c(v)m
(ery)42 b(brief)d(description)h(of)f(the)i(pro)s(of)e(do)s(es)i(not)f
(giv)m(e)g(full)257 2384 y(justice)26 b(to)e(the)i(metho)s(d,)g(w)m(e)g
(hop)s(e)f(it)f(con)m(v)m(eys)k(some)d(of)g(the)g(in)m(tuition)e(on)i
(whic)m(h)h([)3329 2384 y
SDict begin H.S end
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y
SDict begin H.R end
 3378 2320 a 3378 2384 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:sdp) cvn H.B /ANN pdfmark end
 3378 2384 a Fq(])f(is)257 2504
y(based.)45 b(The)34 b(algorithm)29 b(runs)34 b(on)e(a)h(uniform)d(sup)
s(erp)s(osition)i(of)g(all)f(inputs.)44 b(During)257
2625 y(the)29 b(computation,)e(the)i(input)f(register)g(gets)g(en)m
(tangled)h(with)e(the)i(w)m(orkspace)h(of)e(the)257 2745
y(algorithm)i(due)k(to)f(the)h(queries)g(to)f(the)g(oracle.)45
b(W)-8 b(e)34 b(trace)f(out)g(the)h(w)m(orkspace)h(and)257
2866 y(examine)f(the)h(eigenspaces)g(of)f(the)h(densit)m(y)g(matrix)e
(of)g(the)i(input)f(register.)48 b(Due)36 b(to)257 2986
y(symmetries,)i(there)g(are)f(exactly)h Fo(k)28 b Fp(+)d(1)37
b Fq(eigenspaces,)j(indexed)e(b)m(y)g(the)f(n)m(um)m(b)s(er)h(of)257
3106 y(ones)31 b(the)f(algorithm)d(\020kno)m(ws\021)38
b(at)30 b(that)f(stage)h(of)g(the)g(algorithm.)39 b(In)30
b(the)h(b)s(eginning,)257 3227 y(all)d(amplitude)g(is)i(in)f(the)i
Fp(0)1270 3191 y Fd(th)1370 3227 y Fq(eigenspace.)44
b(One)30 b(query)i(can)e(only)f(mo)m(v)m(e)i(little)c(ampli-)257
3347 y(tude)37 b(from)d(the)i Fo(i)920 3311 y Fd(th)1027
3347 y Fq(eigenspace)h(to)e(the)h Fo(i)22 b Fp(+)g(1)2003
3308 y Fd(th)2110 3347 y Fq(eigenspace.)54 b(If)35 b(the)h(algorithm)d
(has)257 3467 y(a)d(go)s(o)s(d)f(success)k(probabilit)m(y)-8
b(,)28 b(the)j(quan)m(tum)f(amplitude)e(of)i(high)f(eigenspaces)i(m)m
(ust)257 3588 y(b)s(e)i(signi\034can)m(t,)e(since)i(the)f(algorithm)d
(m)m(ust)j(\020kno)m(w\021)41 b(most)31 b(of)h(the)h
Fo(k)i Fq(indices,)d(whic)m(h)257 3708 y(implies)e(a)j(lo)m(w)m(er)f(b)
s(ound)h(on)g(the)f(query)i(complexit)m(y)-8 b(.)257
3906 y
SDict begin H.S end
 257 3906 a 257 3906 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section.7) cvn H.B /DEST pdfmark
end
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3906 a 169 x Fr(7)156 b(P)l(olynomial)53 b(lo)l(w)l(er)g(b)t(ounds)257
4305 y Fq(There)33 b(are)f(essen)m(tially)g(t)m(w)m(o)g(di\033eren)m(t)
g(metho)s(ds)g(kno)m(wn)h(for)e(pro)m(ving)h(lo)m(w)m(er)g(b)s(ounds)
257 4425 y(on)43 b(quan)m(tum)g(computations.)73 b(The)43
b(historically)d(\034rst)j(metho)s(d)f(is)h(the)g(adv)m(ersary)257
4545 y(metho)s(d)33 b(w)m(e)h(discuss)h(ab)s(o)m(v)m(e.)46
b(It)33 b(w)m(as)h(in)m(tro)s(duced)f(in)g(1994)f(b)m(y)i(Bennett,)h
(Bernstein,)257 4666 y(Brassard,)30 b(and)e(V)-8 b(azirani,)27
b(and)g(published)h(in)f(1997)g(in)g(the)h(SIAM)h(Journal)e(on)g(Com-)
257 4786 y(puting,)e(in)d(a)h(sp)s(ecial)g(section)g(that)g(con)m
(tains)h(some)f(of)g(the)g(most)g(outstanding)g(pap)s(ers)257
4907 y(on)39 b(quan)m(tum)h(computing.)61 b(The)41 b(second)f(metho)s
(d)e(w)m(as)i(in)m(tro)s(duced)f(shortly)g(after,)257
5027 y(in)e(1998,)g(b)m(y)h(Beals,)g(Buhrman,)g(Clev)m(e,)h(Mosca,)g
(and)f(de)f(W)-8 b(olf)36 b([)2756 5027 y
SDict begin H.S end
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Fq(9)2805 4963 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bbcmw:polynomialsj) cvn H.B /ANN pdfmark end
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Fq(],)i(and)g(implicitly)257 5147 y(used)32 b(b)m(y)f(F)-8
b(ortno)m(w)31 b(and)f(Rogers)h(in)e([)1638 5147 y
SDict begin H.S end
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5147 a Fq(25)1736 5083 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.fr:limits) cvn H.B /ANN pdfmark end
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5147 a Fq(].)43 b(Their)30 b(approac)m(h)h(is)f(algebraic)f(and)i
(follo)m(ws)257 5268 y(earlier)25 b(v)m(ery)i(successful)g(w)m(ork)g
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(\(see)k(for)257 5388 y(instance)i(Beigel's)e(1993)g(surv)m(ey)k([)1538
5388 y
SDict begin H.S end
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SDict begin H.R end
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5388 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.beigel:survey) cvn H.B /ANN pdfmark end
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5388 y
SDict begin H.S end
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SDict begin H.R end
 2887 5324 a 2887
5388 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.regan:survey) cvn H.B /ANN pdfmark end
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SDict begin H.S end
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a 257 266 a
SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.18) cvn H.B /DEST pdfmark
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b(that)h(an)m(y)h(partial)d(b)s(o)s(olean)h(function)h
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y
SDict begin H.S end
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SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark
end
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Fp(1)p Fn(g)36 b Fe(b)-5 b(e)38 b(a)f(p)-5 b(artial)37
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Fq(The)37 b(crux)f(in)g([)975 1708 y
SDict begin H.S end
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1644 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bbcmw:polynomialsj) cvn H.B /ANN pdfmark end
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Fa(A)2388 1828 y Fq(that)k(represen)m(ts)i(or)e(appro)m(x-)257
1949 y(imates)i Fo(F)14 b Fq(.)257 1949 y
SDict begin H.S end
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257 1949 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.10) cvn H.B /DEST pdfmark
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b(10)g(\([)966 2155 y
SDict begin H.S end
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SDict begin H.R end
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2090 a 1022 2155 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bbcmw:polynomialsj) cvn H.B /ANN pdfmark end
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b Fk(A)h Fe(b)-5 b(e)38 b(a)g(quantum)h(algorithm)f(that)h(c)-5
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b(c)g(eptanc)g(e)257 2636 y(pr)g(ob)g(ability)35 b(of)f
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3084 y Fq(as)525 2963 y
SDict begin H.S end
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SDict begin 14.5 H.A end
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a 525 2963 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.13) cvn H.B /DEST pdfmark
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3221 y Fm(i)1801 3206 y Fp(\))p Fn(j)p Fo(i;)17 b(b)p
Fp(;)g Fo(z)t Fn(i)22 b Fp(+)g Fo(x)2292 3221 y Fm(i)2321
3206 y Fn(j)p Fo(i;)17 b(b)22 b Fn(\010)h Fp(1;)17 b
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b(Supp)s(ose)45 b(w)m(e)g(apply)f Fk(O)2435 3466 y Fj(0)2435
3527 y Fm(x)2523 3502 y Fq(on)f(some)h(sup)s(erp)s(osition)257
3548 y Fi(P)363 3652 y Fm(i;b;z)512 3623 y Fo(\013)574
3638 y Fm(i;b;z)708 3623 y Fn(j)p Fo(i;)17 b(b)p Fp(;)g
Fo(z)t Fn(i)27 b Fq(where)h(eac)m(h)g(amplitude)d Fo(\013)2020
3638 y Fm(i;b;z)2180 3623 y Fq(is)h(an)h Fo(N)10 b Fq(-v)-5
b(ariate)26 b(complex-v)-5 b(alued)257 3743 y(p)s(olynomial)36
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b(Then,)43 b(b)m(y)d(Eq.)2539 3743 y
SDict begin H.S end
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3679 y
SDict begin H.R end
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SDict begin [ /Color [1 0 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (equation.13) cvn H.B /ANN pdfmark end
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(state)257 3789 y Fi(P)363 3893 y Fm(i;b;z)512 3864 y
Fo(\014)567 3879 y Fm(i;b;z)701 3864 y Fn(j)p Fo(i;)17
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b(complex-v)-5 b(alued)28 b(p)s(olynomial)e(in)i Fo(x)i
Fq(of)f(degree)i(at)e(most)g Fo(j)22 b Fp(+)16 b(1)p
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b(p)s(olynomial)c(in)j Fo(x)h Fq(of)g(degree)h(at)e(most)g
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Fp(2)p Fo(T)14 b Fq(.)43 b(Theorem)2776 4706 y
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a Fq(10)2874 4642 y
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a 32 w Fq(follo)m(ws.)207 b Fn(u)-66 b(t)404 4907 y Fq(The)37
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Fo(F)14 b Fp(\()p Fo(x)p Fp(\))34 b Fq(for)g(all)e Fo(x)e
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SDict begin H.R end
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SDict begin H.S end
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4420 a Fq(1)2930 4356 y
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4420 a Fq(])h(in)f(whic)m(h)h(he)257 4541 y(pro)m(v)m(es)g(a)d(lo)m(w)m
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SDict begin H.S end
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h(on)e(subsets)j(b)m(y)e(Brassard,)g(H\370y)m(er,)h(and)257
1656 y(T)-8 b(app)33 b([)536 1656 y
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1592 y
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/Dest (cite.bht:collision) cvn H.B /ANN pdfmark end
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y(The)26 b(b)s(est)f(lo)m(w)m(er)g(b)s(ound)g(for)g(the)g(collision)d
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1897 y(the)33 b(adv)m(ersary)i(metho)s(d)c(is)i(only)f(a)g(constan)m
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y(P)m(ortier)34 b([)615 2138 y
SDict begin H.S end
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y
SDict begin H.R end
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/Dest (cite.knp:simon) cvn H.B /ANN pdfmark end
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b(metho)s(d)k(to)g(pro)m(v)m(e)i(a)e(lo)m(w)m(er)g(b)s(ound)h(of)f
Fp(\012\(log)18 b Fo(N)10 b Fp(\))257 2258 y Fq(for)27
b(Simon's)g(problem)f([)1161 2258 y
SDict begin H.S end
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2194 y
SDict begin H.R end
 1259 2194 a 1259 2258 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.simon) cvn H.B /ANN pdfmark end
 1259 2258 a Fq(],)i(whic)m(h)g(is)f
(tigh)m(t)g([)1966 2258 y
SDict begin H.S end
 1966 2258 a Fq(48)2064 2194
y
SDict begin H.R end
 2064 2194 a 2064 2258 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.simon) cvn H.B /ANN pdfmark end
 2064 2258 a Fq(,)2118 2258
y
SDict begin H.S end
 2118 2258 a Fq(12)2216 2194 y
SDict begin H.R end
 2216 2194 a 2216 2258
a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bh:simon) cvn H.B /ANN pdfmark end
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2379 y(b)s(o)s(olean)k(function)h(ha)m(ving)g(prop)s(erties)h(related)f
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2499 y(this)d(problem,)f(the)h(b)s(est)g(lo)m(w)m(er)f(b)s(ound)h(that)
f(can)h(b)s(e)g(obtained)f(using)g(the)h(adv)m(ersary)257
2619 y(metho)s(d)j(is)g(a)h(constan)m(t.)404 2740 y(In)45
b(con)m(trast,)50 b(for)45 b(an)m(y)h Fe(total)56 b Fq(b)s(o)s(olean)44
b(function)g Fo(F)64 b Fp(:)49 b Fn(f)p Fp(0)p Fo(;)17
b Fp(1)p Fn(g)2778 2704 y Fm(N)2894 2740 y Fn(!)49 b(f)p
Fp(0)p Fo(;)17 b Fp(1)p Fn(g)p Fq(,)48 b(the)257 2860
y(adv)m(ersary)e(and)e(p)s(olynomial)c(metho)s(d)k(are)g(b)s(oth)f(p)s
(olynomially)d(related)k(to)f(blo)s(c)m(k)257 2980 y(sensitivit)m(y)-8
b(,)1013 3103 y Fi(p)p 1113 3103 343 4 v 90 x Fq(bs)q
Fp(\()p Fo(F)14 b Fp(\))p Fo(=)p Fp(6)26 b Fn(\024)1590
3179 y Fi(g)1589 3193 y Fp(deg)q(\()p Fo(F)14 b Fp(\))27
b Fn(\024)i Fp(deg)q(\()p Fo(F)14 b Fp(\))28 b Fn(\024)g
Fq(bs)2547 3151 y Fl(3)2587 3193 y Fp(\()p Fo(F)14 b
Fp(\))3322 3048 y
SDict begin H.S end
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SDict begin 17.5 H.A end
 3322 3048 a
3322 3048 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.14) cvn H.B /DEST pdfmark
end
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1350 3270 246 4 v 1350 3361 a Fq(bs)q Fp(\()p Fo(F)g
Fp(\))27 b Fn(\024)i Fq(A)m(dv)q Fp(\()p Fo(F)14 b Fp(\))27
b Fn(\024)h Fq(bs)2283 3318 y Fl(2)2322 3361 y Fp(\()p
Fo(F)14 b Fp(\))p Fo(:)3322 3215 y
SDict begin H.S end
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a
SDict begin 17.5 H.A end
 3322 3215 a 3322 3215 a
SDict begin [ /View [/XYZ H.V] /Dest (equation.15) cvn H.B /DEST pdfmark
end
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3582 y(It)33 b(follo)m(ws)f(from)f([)941 3582 y
SDict begin H.S end
 941 3582
a Fq(19)1039 3518 y
SDict begin H.R end
 1039 3518 a 1039 3582 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.buhrmanwolf:dectreesurvey) cvn H.B /ANN pdfmark end
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a Fq(])h(that)h Fp(deg)q(\()p Fo(F)14 b Fp(\))28 b Fn(\024)g
Fq(bs)1836 3540 y Fl(3)1876 3582 y Fp(\()p Fo(F)14 b
Fp(\))p Fq(,)32 b(and)h(from)e(Nisan)h(and)h(Szegedy)i([)3371
3582 y
SDict begin H.S end
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SDict begin H.R end
 3468 3518 a 3468
3582 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ns:poly) cvn H.B /ANN pdfmark end
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3689 y Fi(g)527 3703 y Fp(deg)q(\()p Fo(F)14 b Fp(\))827
3667 y Fl(2)910 3703 y Fn(\025)44 b Fq(bs)q Fp(\()p Fo(F)14
b Fp(\))p Fq(.)71 b(Buhrman)41 b(and)h(de)g(W)-8 b(olf)41
b([)2421 3703 y
SDict begin H.S end
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SDict begin H.R end
 2519 3639
a 2519 3703 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.buhrmanwolf:dectreesurvey) cvn H.B /ANN pdfmark end
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3823 y(surv)m(ey)35 b(of)d(these)i(and)f(other)f(complexit)m(y)g
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y(adv)m(ersary)33 b(metho)s(d)e(for)g(some)g(total)e(b)s(o)s(olean)h
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SDict begin H.S end
 2642 4064 a Fq(4)2690
4000 y
SDict begin H.R end
 2690 4000 a 2690 4064 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ambainis:degree-vs-qc) cvn H.B /ANN pdfmark end
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(es)h(a)257 4184 y(b)s(o)s(olean)36 b(function)h Fo(F)50
b Fp(:)36 b Fn(f)p Fp(0)p Fo(;)17 b Fp(1)p Fn(g)1430
4148 y Fl(4)1505 4184 y Fn(!)36 b(f)p Fp(0)p Fo(;)17
b Fp(1)p Fn(g)36 b Fq(on)h(four)g(bits,)i(whic)m(h)f(can)g(b)s(e)f
(describ)s(ed)257 4305 y(as)e(\020the)f(four)g(input)g(bits)g(are)h
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SDict begin H.S end
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4240 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.lls:formulas) cvn H.B /ANN pdfmark end
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Fp(deg)q(\()p Fo(F)14 b Fp(\))31 b(=)f(2)k Fq(and)h(for)f(whic)m(h)257
4425 y(there)27 b(exists)g(an)f(adv)m(ersary)h(matrix)e
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Fo(=)17 b Fp(max)2915 4440 y Fm(i)2960 4425 y Fo(\025)p
Fp(\(\000)3116 4389 y Fm(F)3116 4450 y(i)3174 4425 y
Fp(\))28 b(=)f(2)p Fo(:)p Fp(5)p Fq(.)257 4545 y(W)-8
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4545 y Fp(=)f Fo(F)i Fn(\016)257 4666 y Fp(\()p Fo(F)s(;)17
b(F)s(;)g(F)s(;)g(F)d Fp(\))26 b(:)i Fn(f)p Fp(0)p Fo(;)17
b Fp(1)p Fn(g)1063 4630 y Fl(16)1164 4666 y Fn(!)28 b(f)p
Fp(0)p Fo(;)17 b Fp(1)p Fn(g)29 b Fq(de\034ned)k(on)e(16)f(bits)h(for)f
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Fo(\025)p Fp(\(\000)832 4750 y Fm(F)877 4759 y Fg(2)916
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4786 y Fo(\025)p Fp(\(\000)1401 4743 y Fm(F)1446 4752
y Fg(2)1401 4812 y Fm(i)1484 4786 y Fp(\))28 b(=)g(2)p
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b Fo(N)51 b Fp(=)40 b(4)1968 4870 y Fm(n)2055 4907 y
Fq(bits)g(of)g(degree)h Fp(deg)r(\()p Fo(F)14 b Fp(\))40
b(=)h(2)3190 4870 y Fm(n)3236 4907 y Fq(,)i(with)257
5027 y(sp)s(ectral)35 b(lo)m(w)m(er)g(b)s(ound)g Fp(2)p
Fo(:)p Fp(5)1311 4991 y Fm(n)1389 5027 y Fp(=)d(deg)q(\()p
Fo(F)14 b Fp(\))1797 4991 y Fl(1)p Fm(:)p Fl(32)p Fm(:::)1985
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b(The)257 5147 y(th)m(us)c(constructed)g(function)e Fo(F)58
b Fq(is)44 b(an)h(example)f(of)g(an)g(iterated)h(function)f(of)g(lo)m
(w)257 5268 y(degree)26 b(and)g(high)e(quan)m(tum)h(query)i(complexit)m
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5388 y(gap)38 b(b)s(et)m(w)m(een)j(the)e(p)s(olynomial)c(metho)s(d)i
(and)i(the)g(adv)m(ersary)h(metho)s(d)e(for)g(a)g(total)p
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SDict begin H.R end
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SDict begin [ /View [/XYZ H.V] /Dest (page.21) cvn H.B /DEST pdfmark
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SDict begin H.S end
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y
SDict begin H.R end
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/Dest (cite.ambainis:degree-vs-qc) cvn H.B /ANN pdfmark end
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b(olf,)47 b(and)f(Zalk)-5 b(a)44 b([)2408 1295 y
SDict begin H.S end
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1295 a Fq(10)2505 1231 y
SDict begin H.R end
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/Dest (cite.bcwz:error) cvn H.B /ANN pdfmark end
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1295 a Fq(])i(sho)m(w)h(that)e(an)m(y)h(quan-)257 1416
y(tum)36 b(algorithm)e(for)i(Gro)m(v)m(er's)h(problem)f(that)g
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SDict begin H.S end
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3398 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bnrw:robust) cvn H.B /ANN pdfmark end
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(of)g(suc)m(h)i(p)s(olynomials,)257 3582 y(and)d(refer)g(to)f(them)g
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SDict begin H.S end
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a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (theorem.12) cvn H.B /DEST pdfmark
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b(12)h(\(Robust)f(p)s(olynomials)f([)2077 3763 y
SDict begin H.S end
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3763 a Ff(15)2189 3698 y
SDict begin H.R end
 2189 3698 a 2189 3763 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bnrw:robust) cvn H.B /ANN pdfmark end
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3763 a Ff(]\).)d Fe(A)n(n)f(appr)-5 b(oximate)46 b Fo(N)10
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3965 y Fl(1)p 1647 3981 V 1647 4038 a(3)1720 4004 y Fe(for)g(every)f
Fo(i)h Fp(=)g(1)p Fo(;)17 b(:)g(:)g(:)e(;)i(M)10 b Fe(.)43
b(The)35 b Fq(robust)25 b(degree)257 4124 y Fe(of)31
b(a)h(b)-5 b(o)g(ole)g(an)30 b(function)h Fo(F)41 b Fp(:)28
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Fe(,)31 b(denote)-5 b(d)30 b Fp(rdeg)r(\()p Fo(F)14 b
Fp(\))p Fe(,)31 b(is)h(the)f(minimal)f(de)-5 b(gr)g(e)g(e)257
4244 y(of)35 b(a)g(r)-5 b(obust)35 b(p)-5 b(olynomial)33
b(appr)-5 b(oximating)33 b Fo(F)14 b Fe(.)404 4425 y
Fq(Robust)45 b(p)s(olynomials)d(comp)s(ose)i(b)m(y)i(de\034nition.)80
b(Buhrman)44 b(et)i(al.)d([)3116 4425 y
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4361 y
SDict begin H.R end
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SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.bnrw:robust) cvn H.B /ANN pdfmark end
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4545 y(that)30 b(the)g(robust)g(degree)h(of)f(an)m(y)g(total)e
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b Fp(1)p Fn(g)2534 4509 y Fm(N)2627 4545 y Fn(!)28 b(f)p
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Fo(N)10 b Fp(\))29 b Fq(b)m(y)257 4666 y(giving)21 b(a)h(classical)f
(algorithm)e(that)j(uses)i(a)e(quan)m(tum)h(subroutine)f(for)g(Gro)m(v)
m(er's)i(prob-)257 4786 y(lem)34 b([)471 4786 y
SDict begin H.S end
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a Fq(27)568 4722 y
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/Dest (cite.grover:search) cvn H.B /ANN pdfmark end
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(and)f(de)g(W)-8 b(olf)33 b([)3343 4786 y
SDict begin H.S end
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Fq(29)3441 4722 y
SDict begin H.R end
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/Dest (cite.hmw:berror-search) cvn H.B /ANN pdfmark end
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Fq(].)257 4907 y(Buhrman)h(et)h(al.)e([)971 4907 y
SDict begin H.S end
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4907 a Fq(15)1069 4842 y
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/Dest (cite.bnrw:robust) cvn H.B /ANN pdfmark end
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Fi(g)2367 4907 y Fp(deg)q(\()p Fo(F)14 b Fp(\))j(log)2827
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5388 y Fp(deg)q(\()p Fo(F)d Fp(\)\))p Fq(.)93 b(It)49
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SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.hmw:berror-search) cvn H.B /ANN pdfmark end
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2299 y Fn(p)p 1743 2299 V 84 x Fo(N)43 b Fq(in)m(tegers)32
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SDict begin H.S end
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a 2300 2383 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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SDict begin H.S end
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257 2584 a
SDict begin 14.5 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (section.9) cvn H.B /DEST pdfmark
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4184 y
SDict begin H.S end
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SDict begin H.R end
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4184 a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
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(d)e(that)257 4305 y(exactly)j(c)m(haracterizes)h(the)f(quan)m(tum)f
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4425 y(out)i(also)f(to)h(b)s(e)g(its)f(w)m(eakness:)49
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f(can)h(also)f(b)s(e)h(sho)m(wn)i(b)m(y)f(one)f(of)f(the)h(other)h(t)m
(w)m(o)257 4666 y(metho)s(ds.)k(In)30 b(a)g(v)m(ery)i(recen)m(t)g(w)m
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SDict begin H.S end
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4602 y
SDict begin H.R end
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/Dest (cite.ambainis:sdp) cvn H.B /ANN pdfmark end
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(ersary)h(metho)s(d)257 4786 y(and)40 b(uses)g(it)e(to)h(repro)m(v)m(e)
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SDict begin H.S end
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SDict begin H.R end
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a
SDict begin [ /Color [0 1 0] /H /I /Border [0 0 12] /Subtype /Link
/Dest (cite.ksw:dpt) cvn H.B /ANN pdfmark end
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b(et)g(al.)f([)910 5027 y
SDict begin H.S end
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y
SDict begin H.R end
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a 812 1420 a
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y
SDict begin H.S end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.bcwz:error) cvn H.B /DEST
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SDict begin H.S end
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1589 a 501 1720 a Ft([19])49 b(H.)36 b(Buhrman)i(and)f(R.)f(de)h(W)-8
b(olf.)59 b(Complexit)m(y)37 b(measures)g(and)g(decision)h(tree)690
1833 y(complexit)m(y:)73 b(A)46 b(surv)m(ey)-8 b(.)88
b Fs(The)-5 b(or)g(etic)g(al)48 b(Computer)g(Scienc)-5
b(e)p Ft(,)51 b(288\(1\):21\02543,)690 1946 y(2002.)257
1949 y
SDict begin H.S end
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SDict begin 13.6 H.A end
 257 1949 a 257 1949 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.cdnt:inner) cvn H.B /DEST
pdfmark end
 257
1949 a 501 2103 a Ft([20])e(R.)33 b(Clev)m(e,)i(W.)e(v)-5
b(an)34 b(Dam,)g(M.)g(Nielsen,)g(and)g(A.)f(T)-8 b(app.)51
b(Quan)m(tum)35 b(en)m(tangle-)690 2216 y(men)m(t)29
b(and)f(the)h(comm)m(unication)h(complexit)m(y)f(of)f(the)g(inner)h
(pro)s(duct)f(function.)690 2328 y(In)41 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g
(dings)46 b(of)c(the)h(1st)f(NASA)g(QCQC)g(c)-5 b(onfer)g(enc)g(e)p
Ft(,)46 b(Lecture)d(Notes)e(in)690 2441 y(Computer)32
b(Science)f(1509,)g(pages)g(61\02574,)g(1998.)257 2462
y
SDict begin H.S end
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SDict begin 13.6 H.A end
 257 2462 a 257 2462 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.dhhm:graphs) cvn H.B /DEST
pdfmark end
 257 2462
a 501 2598 a Ft([21])49 b(Ch.)40 b(D\374rr,)j(M.)d(Heiligman,)j(P)-8
b(.)41 b(H\370y)m(er,)h(and)f(M.)e(Mhalla.)70 b(Quan)m(tum)41
b(query)690 2711 y(complexit)m(y)24 b(of)f(some)g(graph)g(problems.)30
b(In)22 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)29 b(of)c(31st)h
(International)690 2824 y(Col)5 b(lo)-5 b(quium)35 b(on)e(A)n(utomata,)
h(L)-5 b(anguages)36 b(and)e(Pr)-5 b(o)g(gr)g(amming)p
Ft(,)34 b(Lecture)f(Notes)690 2937 y(in)e(Computer)h(Science)f(3142,)g
(pages)g(481\025493,)g(2004.)257 2958 y
SDict begin H.S end
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2958 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.dj:rapid) cvn H.B /DEST
pdfmark end
 257 2958 a 501 3093 a Ft([22])49
b(D.)d(Deutsc)m(h)g(and)g(R.)f(Jozsa.)85 b(Rapid)46 b(solution)g(of)f
(problems)h(b)m(y)g(quan)m(tum)690 3206 y(computation.)c
Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)36 b(of)c(the)h(R)-5
b(oyal)32 b(So)-5 b(ciety,)32 b(L)-5 b(ondon)p Ft(,)30
b(A439:553\025558,)690 3319 y(1992.)257 3322 y
SDict begin H.S end
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a 257 3322 a
SDict begin 13.6 H.A end
 257 3322 a 257 3322 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.ehk:subgroup) cvn H.B /DEST
pdfmark end
 257 3322 a 501 3476
a Ft([23])49 b(M.)24 b(Ettinger)j(and)e(P)-8 b(.)25 b(H\370y)m(er,)h
(and)f(E.)g(Knill.)32 b(The)25 b(quan)m(tum)h(query)e(complexit)m(y)690
3589 y(of)30 b(the)h(hidden)h(subgroup)f(problem)h(is)e(p)s(olynomial.)
41 b Fs(Information)31 b(Pr)-5 b(o)g(c)g(essing)690 3702
y(L)g(etters)p Ft(,)31 b(91\(1\):43-48,)h(2004.)257 3728
y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.fggs99:search) cvn H.B /DEST
pdfmark end
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a 501 3858 a Ft([24])49 b(E.)29 b(F)-8 b(arhi,)30 b(J.)f(Goldstone,)g
(S.)g(Gutmann,)h(and)f(M.)f(Sipser.)37 b Fs(Invariant)30
b(quantum)690 3971 y(algorithms)j(for)f(insertion)f(into)h(an)g(or)-5
b(der)g(e)g(d)35 b(list)p Ft(.)40 b(quan)m(t-ph/9901059,)33
b(1999.)257 3997 y
SDict begin H.S end
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SDict begin 13.6 H.A end
 257 3997 a 257
3997 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.fr:limits) cvn H.B /DEST
pdfmark end
 257 3997 a 501 4128 a Ft([25])49 b(L.)28 b(F)-8
b(ortno)m(w)29 b(and)f(J.)g(D.)f(Rogers.)35 b(Complexit)m(y)29
b(limitations)f(on)g(quan)m(tum)g(com-)690 4241 y(putation.)78
b Fs(Journal)44 b(of)f(Computer)g(and)g(System)g(Scienc)-5
b(es)p Ft(,)47 b(59\(2\):240\025252,)690 4354 y(1999.)257
4357 y
SDict begin H.S end
 257 4357 a 257 4357 a
SDict begin 13.6 H.A end
 257 4357 a 257 4357 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.gasarch:pir) cvn H.B /DEST
pdfmark end
 257
4357 a 501 4510 a Ft([26])i(W.)38 b(Gasarc)m(h.)66 b(A)37
b(surv)m(ey)i(on)f(priv)-5 b(ate)39 b(information)h(retriev)-5
b(al.)65 b(The)39 b(compu-)690 4623 y(tational)34 b(complexit)m(y)g
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b(TCS)p Ft(,)31 b(82:72\025107,)690 4736 y(2004.)257
4739 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.grover:search) cvn H.B /DEST
pdfmark end
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4739 a 501 4893 a Ft([27])49 b(L.)23 b(K.)g(Gro)m(v)m(er.)29
b(A)23 b(fast)f(quan)m(tum)i(mec)m(hanical)h(algorithm)f(for)e
(database)i(searc)m(h.)690 5006 y(In)g Fs(Pr)-5 b(o)g(c)g(e)g(e)g
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b(ory)27 b(of)g(Computing)p Ft(,)e(pages)690 5119 y(212\025219,)32
b(1996.)257 5139 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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5139 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.gross:2005) cvn H.B /DEST
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 257 5139 a 501 5275 a Ft([28])49 b(D.)26 b(Gross.)32
b Fs(The)d(F)-7 b(utur)i(e)29 b(of)f(Physics.)33 b Ft(CERN)25
b(Collo)s(quium,)j(Jan)m(uary)f(26,)g(2005.)690 5388
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SDict begin H.R end
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SDict begin H.S end
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473 a 257 473 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.hmw:berror-search) cvn H.B
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a Ft([29])49 b(P)-8 b(.)26 b(H\370y)m(er,)g(M.)e(Mosca,)h(and)g(R.)f
(de)h(W)-8 b(olf.)31 b(Quan)m(tum)26 b(searc)m(h)g(on)e(b)s
(ounded-error)690 686 y(inputs.)50 b(In)33 b Fs(Pr)-5
b(o)g(c)g(e)g(e)g(dings)38 b(of)d(30th)g(International)f(Col)5
b(lo)-5 b(quium)36 b(on)e(A)n(utomata,)690 799 y(L)-5
b(anguages)31 b(and)f(Pr)-5 b(o)g(gr)g(amming)p Ft(,)29
b(Lecture)g(Notes)e(in)g(Computer)i(Science)f(2719,)690
912 y(pages)j(291\025299,)g(2003.)257 932 y
SDict begin H.S end
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257 932 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.hns:ordered-search) cvn
H.B /DEST pdfmark end
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b(P)-8 b(.)41 b(H\370y)m(er,)h(J.)f(Neerb)s(ek,)h(and)e(Y.)f(Shi.)69
b(Quan)m(tum)42 b(complexities)f(of)e(ordered)690 1181
y(searc)m(hing,)h(sorting,)f(and)e(elemen)m(t)i(distinctness.)60
b Fs(A)n(lgorithmic)-5 b(a)p Ft(,)38 b(34\(4\):429\025)690
1294 y(448,)31 b(2002.)257 1315 y
SDict begin H.S end
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a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.holevo:comm) cvn H.B /DEST
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b(A.)37 b(S.)g(Holev)m(o.)61 b(Bounds)38 b(for)e(the)i(quan)m(tit)m(y)g
(of)f(information)h(transmitted)h(b)m(y)690 1564 y(a)i(quan)m(tum)h
(comm)m(unication)h(c)m(hannel.)73 b Fs(Pr)-5 b(oblemy)43
b(Per)-5 b(e)g(dachi)44 b(Informatsii)p Ft(,)690 1676
y(9\(3\):3\02511,)33 b(1973.)42 b(English)32 b(translation)g(is)e
Fs(Pr)-5 b(oblems)34 b(in)e(Information)f(T)-7 b(r)i(ans-)690
1789 y(mission)p Ft(,)29 b(9:177\025183,)j(1973.)257
1810 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.hs:compose) cvn H.B /DEST
pdfmark end
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1810 a 501 1946 a Ft([32])49 b(P)-8 b(.)24 b(H\370y)m(er)e(and)h(R.)f
(\222palek.)28 b Fs(Tight)d(adversary)h(b)-5 b(ounds)26
b(for)f(c)-5 b(omp)g(osite)27 b(functions)p Ft(.)690
2059 y(quan)m(t-ph/0509067,)34 b(2005.)257 2085 y
SDict begin H.S end
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2085 a 257 2085 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.jlb:search) cvn H.B /DEST
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2216 a Ft([33])49 b(M.)34 b(B.)g(Jacok)m(es,)i(A.)e(J.)h(Landahl,)h
(and)f(E.)f(Bro)s(ok)m(es.)52 b Fs(A)n(n)35 b(impr)-5
b(ove)g(d)37 b(quantum)690 2328 y(algorithm)c(for)f(se)-5
b(ar)g(ching)34 b(an)e(or)-5 b(der)g(e)g(d)35 b(list)p
Ft(.)40 b(Man)m(uscript,)31 b(2005.)257 2349 y
SDict begin H.S end
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a 257 2349 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.knp:simon) cvn H.B /DEST
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a Ft([34])49 b(P)-8 b(.)36 b(K)m(oiran,)h(V.)e(Nesme)f(and)h(N.)g(P)m
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b(In)32 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)38 b(of)c(32nd)g(Interna-)
690 2711 y(tional)g(Col)5 b(lo)-5 b(quium)34 b(on)f(A)n(utomata,)h(L)-5
b(anguages)35 b(and)f(Pr)-5 b(o)g(gr)g(amming)p Ft(,)34
b(Lecture)690 2824 y(Notes)c(in)h(Computer)h(Science)f(3580,)g(pages)g
(1287\0251298,)h(2005.)257 2845 y
SDict begin H.S end
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a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.ksw:dpt) cvn H.B /DEST pdfmark
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b(H.)36 b(Klauc)m(k,)i(R.)d(\222palek,)j(and)e(R.)g(de)g(W)-8
b(olf.)57 b(Quan)m(tum)37 b(and)g(classical)f(strong)690
3093 y(direct)e(pro)s(duct)f(theorems)h(and)f(optimal)h(time-space)g
(tradeo\033s.)49 b(In)32 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(d-)690
3206 y(ings)27 b(of)g(the)h(45th)g(IEEE)f(Symp)-5 b(osium)27
b(on)g(F)-7 b(oundations)27 b(of)g(Computer)h(Scienc)-5
b(e)p Ft(,)690 3319 y(pages)31 b(12\02521,)g(2004.)257
3340 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.kutin:collision) cvn H.B
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3340 a 501 3476 a Ft([36])49 b(S.)37 b(Kutin.)61 b(Quan)m(tum)39
b(lo)m(w)m(er)g(b)s(ound)e(for)g(the)g(collision)h(problem)g(with)g
(small)690 3589 y(range.)j Fs(The)-5 b(ory)33 b(of)f(Computing)p
Ft(,)e(1:29\02536,)i(2005.)257 3610 y
SDict begin H.S end
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3610 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.lls:formulas) cvn H.B /DEST
pdfmark end
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b(S.)33 b(Laplan)m(te,)i(T.)d(Lee,)i(and)f(M.)f(Szegedy)-8
b(.)47 b(The)33 b(quan)m(tum)h(adv)m(ersary)e(metho)s(d)690
3858 y(and)27 b(form)m(ula)g(size)e(lo)m(w)m(er)j(b)s(ounds.)33
b(In)26 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)32 b(of)c(20th)g(IEEE)h
(Confer)-5 b(enc)g(e)690 3971 y(on)32 b(Computational)g(Complexity)p
Ft(,)e(2005.)257 3992 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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a 257 3992 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.lm:kolmogorov-lb) cvn H.B
/DEST pdfmark end
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b(Laplan)m(te)h(and)f(F.)g(Magniez.)k(Lo)m(w)m(er)e(b)s(ounds)d(for)h
(randomized)h(and)f(quan)m(tum)690 4241 y(query)31 b(complexit)m(y)i
(using)e(K)m(olmogoro)m(v)j(argumen)m(ts.)44 b(In)31
b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)37 b(of)c(19th)690
4354 y(IEEE)g(Confer)-5 b(enc)g(e)33 b(on)g(Computational)e(Complexity)
p Ft(,)g(pages)f(294\025304,)i(2004.)257 4375 y
SDict begin H.S end
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a 257 4375 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.mathias:spectral-norm) cvn
H.B /DEST pdfmark end
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a Ft([39])49 b(R.)25 b(Mathias.)33 b(The)26 b(sp)s(ectral)g(norm)h(of)e
(a)h(nonnegativ)m(e)h(matrix.)34 b Fs(Line)-5 b(ar)27
b(A)n(lgebr)-5 b(a)690 4623 y(and)32 b(its)g(Applic)-5
b(ations)p Ft(,)31 b(139:269\025284,)h(1990.)257 4644
y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.mss:triangle) cvn H.B /DEST
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a 501 4780 a Ft([40])49 b(F.)34 b(Magniez,)h(M.)e(San)m(tha,)j(and)e
(M.)f(Szegedy)-8 b(.)51 b(Quan)m(tum)35 b(algorithms)f(for)g(the)690
4893 y(triangle)43 b(problem.)74 b(In)41 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g
(dings)46 b(of)d(16th)f(A)n(CM-SIAM)f(Symp)-5 b(osium)42
b(on)690 5006 y(Discr)-5 b(ete)34 b(A)n(lgorithms)p Ft(,)c(pages)h
(1109\0251117,)h(2005.)257 5027 y
SDict begin H.S end
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a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.ns:poly) cvn H.B /DEST pdfmark
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b(N.)35 b(Nisan)g(and)h(M.)f(Szegedy)-8 b(.)56 b(On)36
b(the)f(degree)i(of)e(b)s(o)s(olean)g(functions)h(as)f(real)690
5275 y(p)s(olynomials.)81 b(In)43 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)48
b(of)c(24th)h(A)n(CM)e(Symp)-5 b(osium)44 b(on)h(The)-5
b(ory)44 b(of)690 5388 y(Computing)p Ft(,)30 b(pages)h(462\025467,)g
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.nw:median) cvn H.B /DEST
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a Ft([42])49 b(A.)26 b(Na)m(y)m(ak)h(and)g(F.)f(W)-8
b(u.)35 b(The)27 b(quan)m(tum)h(query)e(complexit)m(y)i(of)e(appro)m
(ximating)690 686 y(the)33 b(median)g(and)f(related)h(statistics.)46
b(In)31 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)38 b(of)33
b(31st)h(A)n(CM)f(Symp)-5 b(o-)690 799 y(sium)32 b(on)g(The)-5
b(ory)33 b(of)f(Computing)p Ft(,)e(pages)h(384\025393,)g(1999.)257
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SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.paturi:poly) cvn H.B /DEST
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a 501 961 a Ft([43])49 b(R.)34 b(P)m(aturi.)55 b(On)35
b(the)g(degree)g(of)g(p)s(olynomials)g(that)g(appro)m(ximate)h
(symmetric)690 1074 y(b)s(o)s(olean)22 b(functions)g(\(preliminary)i(v)
m(ersion\).)j(In)21 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)28
b(of)c(the)h(24th)g(A)n(CM)690 1186 y(Symp)-5 b(osium)32
b(on)g(The)-5 b(ory)33 b(of)f(Computing)p Ft(,)e(pages)g(468\025474,)i
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SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.regan:survey) cvn H.B /DEST
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b(P)m(olynomials)36 b(and)e(com)m(binatorial)i(de\034nitions)f(of)f
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b(ory)47 b(R)-5 b(etr)g(osp)g(e)g(ctive)49 b(II)14 b
Ft(,)45 b(Springer-V)-8 b(erlag,)52 b(pages)46 b(261\025293,)690
1574 y(1997.)257 1580 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.santha:and-or) cvn H.B /DEST
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b(On)33 b(the)h(Mon)m(te)g(Carlo)f(decision)h(tree)g(complexit)m(y)g
(of)f(read-once)690 1849 y(form)m(ulae.)42 b Fs(R)-5
b(andom)32 b(Structur)-5 b(es)34 b(and)e(A)n(lgorithms)p
Ft(,)e(6\(1\):75\02587,)i(1995.)257 1878 y
SDict begin H.S end
 257 1878 a
257 1878 a
SDict begin 13.6 H.A end
 257 1878 a 257 1878 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.science:2005) cvn H.B /DEST
pdfmark end
 257 1878 a 501 2011
a Ft([46])49 b(Ch.)35 b(Seife.)51 b(\020What)35 b(are)f(the)h(limits)f
(of)g(con)m(v)m(en)m(tional)j(computing?\021)78 b Fs(Scienc)-5
b(e)p Ft(,)690 2124 y(309\(5731\):96,)33 b(1)d(July)h(2005.)257
2153 y
SDict begin H.S end
 257 2153 a 257 2153 a
SDict begin 13.6 H.A end
 257 2153 a 257 2153 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.shi:collision) cvn H.B /DEST
pdfmark end
 257
2153 a 501 2286 a Ft([47])49 b(Y.)37 b(Shi.)61 b(Quan)m(tum)39
b(lo)m(w)m(er)g(b)s(ounds)f(for)f(the)g(collision)h(and)g(the)f(elemen)
m(t)i(dis-)690 2399 y(tinctness)e(problems.)59 b(In)36
b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)41 b(of)d(the)g(43r)-5
b(d)39 b(A)n(nnual)e(Symp)-5 b(osium)37 b(on)690 2512
y(the)c(F)-7 b(oundations)32 b(of)g(Computer)h(Scienc)-5
b(e)p Ft(,)31 b(pp.)g(513-519,)h(2002.)257 2535 y
SDict begin H.S end
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2535 a 257 2535 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.simon) cvn H.B /DEST pdfmark
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 257 2535 a 501
2674 a Ft([48])49 b(D.)35 b(R.)g(Simon.)55 b(On)35 b(the)h(p)s(o)m(w)m
(er)g(of)f(quan)m(tum)h(computation.)57 b Fs(SIAM)35
b(Journal)690 2787 y(on)d(Computing)p Ft(,)e(26\(5\):1474\0251483,)j
(1997.)257 2815 y
SDict begin H.S end
 257 2815 a 257 2815 a
SDict begin 13.6 H.A end
 257 2815 a 257
2815 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.snir:dec) cvn H.B /DEST
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b(Lo)m(w)m(er)27 b(b)s(ounds)d(on)h(probabilistic)h(decision)f(trees.)
31 b Fs(The)-5 b(or)g(etic)g(al)30 b(Com-)690 3061 y(puter)j(Scienc)-5
b(e)p Ft(,)32 b(38:69\02582,)g(1985.)257 3085 y
SDict begin H.S end
 257 3085
a 257 3085 a
SDict begin 13.6 H.A end
 257 3085 a 257 3085 a
SDict begin [ /View [/XYZ H.V] /Dest (cite.ss:local) cvn H.B /DEST
pdfmark end
 257 3085 a 501 3223
a Ft([50])49 b(M.)32 b(San)m(tha)h(and)f(M.)f(Szegedy)-8
b(.)45 b(Quan)m(tum)34 b(and)e(classical)g(query)f(complexities)690
3336 y(of)24 b(lo)s(cal)g(searc)m(h)g(are)g(p)s(olynomially)h(related.)
30 b(In)24 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)30 b(of)c(the)h(36th)f(A)
n(CM)690 3449 y(Symp)-5 b(osium)32 b(on)g(The)-5 b(ory)33
b(of)f(Computing)p Ft(,)e(pages)g(494\025501,)i(2004.)257
3473 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.ss:equiv) cvn H.B /DEST
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 257
3473 a 501 3611 a Ft([51])49 b(R.)31 b(\222palek)h(and)g(M.)f(Szegedy)
-8 b(.)45 b(All)31 b(quan)m(tum)i(adv)m(ersary)f(metho)s(ds)f(are)h
(equiv-)690 3724 y(alen)m(t.)61 b(In)36 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g
(dings)42 b(of)37 b(32nd)h(International)g(Col)5 b(lo)-5
b(quium)39 b(on)e(A)n(utomata,)690 3837 y(L)-5 b(anguages)28
b(and)e(Pr)-5 b(o)g(gr)g(amming)28 b Ft(,)d(Lecture)g(Notes)f(in)g
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(2005.)257 3973 y
SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.sw:and-or) cvn H.B /DEST
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(Wigderson.)33 b(Probabilistic)27 b(b)s(o)s(olean)f(decision)g(trees)g
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b(games)e(trees.)33 b(In)25 b Fs(Pr)-5 b(o)g(c)g(e)g(e)g(dings)32
b(of)27 b(the)h(27th)h(A)n(nnual)690 4338 y(Symp)-5 b(osium)32
b(on)g(F)-7 b(oundations)32 b(of)g(Computer)h(Scienc)-5
b(e)p Ft(,)31 b(pages)g(29\02538,)g(1986.)257 4361 y
SDict begin H.S end

257 4361 a 257 4361 a
SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.syz:graphs) cvn H.B /DEST
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 257 4361
a 501 4500 a Ft([53])49 b(X.)32 b(Sun,)h(A.)f(C.)h(Y)-8
b(ao,)33 b(and)f(S.)h(Zhang.)46 b(Graph)33 b(prop)s(erties)g(and)g
(circular)g(func-)690 4613 y(tions:)50 b(Ho)m(w)36 b(lo)m(w)g(can)f
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SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.valiant:2003) cvn H.B /DEST
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a 501 5000 a Ft([54])49 b(L.)31 b(G.)g(V)-8 b(alian)m(t.)42
b(Three)32 b(problems)f(in)g(computer)h(science.)42 b
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b(of)g(Computing)g(Machinery)p Ft(,)f(50\(1\):96\02599,)h(2003.)257
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SDict begin H.S end
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SDict begin 13.6 H.A end
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SDict begin [ /View [/XYZ H.V] /Dest (cite.dewolf:survey) cvn H.B /DEST
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5142 a 501 5275 a Ft([55])49 b(R.)23 b(de)h(W)-8 b(olf.)29
b(Quan)m(tum)c(comm)m(unication)h(and)e(complexit)m(y)-8
b(.)31 b Fs(The)-5 b(or)g(etic)g(al)28 b(Com-)690 5388
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SDict begin 13.6 H.A end
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a Ft([56])49 b(Ch.)33 b(Zalk)-5 b(a.)48 b(Gro)m(v)m(er's)34
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SDict begin H.S end
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SDict begin 13.6 H.A end
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b(S.)31 b(Zhang.)41 b(On)30 b(the)h(p)s(o)m(w)m(er)g(of)f(Am)m
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b(Lecture)g(Notes)g(in)f(Computer)i(Science)f(3142,)h(pages)f
(1238\0251250,)h(2004.)257 1080 y
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SDict begin 13.6 H.A end
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b(S.)e(Zhang.)88 b Fs(\(A)n(lmost\))47 b(tight)g(b)-5
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