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b(as)h(quan)m(tum)f(T)-8 b(uring)21 b(mac)m(hines.)38 b(Quan)m(tum)22 b(T)-8 b(uring)22 b(mac)m(hines)h (and)g(quan)m(tum)448 1250 y(circuits)40 b(are)h(equiv)-5 b(alen)m(t)40 b(for)g(describing)e(quan)m(tum)j(computation,)i(with)c (mo)s(dest)448 1363 y(o)m(v)m(erhead)j(for)d(one)i(mo)s(del)e(to)i(sim) m(ulate)e(the)h(other.)70 b(There)40 b(is)f(a)i(lot)f(of)g(detailed)448 1476 y(bac)m(kground)31 b(on)g(these)h(topics)f(whic)m(h)f(I)h(will)d (not)j(go)h(in)m(to)f(here.)43 b(I)31 b(suggest)h(lo)s(oking)448 1589 y(to)24 b(Nielsen)e(and)h(Ch)m(uang)f([1)q(])i(for)f(more)g (information)e(and)i(bibliographic)c(references.)448 1875 y Fy(2)135 b(Ac)l(kno)l(wledgmen)l(ts)448 2078 y Fx(This)28 b(article)g(grew)h(out)h(of)f(a)g(somewhat)h(impromptu)c(in) m(tro)s(ductory)i(talk)h(I)g(ga)m(v)m(e)i(at)448 2191 y(Dagstuhl)806 2158 y Ft(2)867 2191 y Fx(in)20 b(the)i(F)-8 b(all)21 b(of)h(2002.)39 b(I)22 b(ha)m(v)m(e)g(enjo)m(y)m(ed)h(man)m(y) e(rew)m(arding)g(encoun)m(ters)h(and)448 2304 y(discussions)j(at)j (this)f(and)g(previous)f(Dagstuhl)h(seminars,)g(and)g(I)g(wish)f(to)j (thank)e(the)448 2417 y(organizers)41 b(of)g(the)g(seminar,)h(Harry)e (Buhrman,)i(Lance)g(F)-8 b(ortno)m(w,)44 b(and)c(Thomas)448 2530 y(Thierauf,)k(for)d(in)m(viting)f(me.)76 b(Thanks)41 b(also)h(to)g(the)g(Europ)s(ean)f(Comm)m(unit)m(y)g(for)448 2643 y(pro)m(viding)32 b(\014nancial)g(assistance)i(to)g(me)g(and)f (the)g(other)h(guests.)51 b(Finally)-8 b(,)33 b(I)g(thank)448 2756 y(Lance)42 b(F)-8 b(ortno)m(w)43 b(for)e(suggesting)h(\(the)g (nigh)m(t)f(b)s(efore\))g(that)h(I)g(giv)m(e)g(a)g(talk)f(along)448 2869 y(these)31 b(lines,)e(and)h(for)g(in)m(viting)e(me)j(write)e(it)h (up)f(for)i(BEA)-8 b(TCS.)448 3155 y Fy(3)135 b(Bo)t(olean)46 b(Circuits)448 3358 y Fx(Here)25 b(is)e(a)i(quic)m(k)f(review)f(of)i (the)f(Bo)s(olean)h(circuit)d(mo)s(del.)38 b(Our)23 b(approac)m(h)h(is) g(sligh)m(tly)448 3471 y(unortho)s(do)m(x|for)k(reasons)g(that)h(ma)m 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y Ft(2)3270 4522 y Fj(\025)f Fx(0,)448 4635 y(with)32 b(equalit)m(y)h(holding)d(i\013)j Fo(u)c Fx(=)g(0.)49 b(The)33 b Fn(Hermitian)i(norm)42 b Fj(j)p Fo(u)p Fj(j)33 b Fx(of)g Fo(u)g Fx(is)3045 4557 y Fe(p)p 3136 4557 201 4 v 78 x Fj(h)p Fo(u)p Fj(j)p Fo(u)q Fj(i)g Fx(A)448 4748 y(matrix)28 b Fo(M)38 b Fx(that)29 b(preserv)m(es)f(the)g (Hermitian)f(inner)g(pro)s(duct)g(\(that)i(is,)f Fj(h)p Fo(M)10 b(u)p Fj(j)p Fo(M)g(v)t Fj(i)25 b Fx(=)448 4861 y Fj(h)q Fo(u)p Fj(j)p Fo(v)s Fj(i)42 b Fx(for)g(all)e Fo(u;)15 b(v)s Fx(\))43 b(is)d(called)h Fn(unitary)p Fx(.)76 b(The)41 b Fn(adjoint)51 b Fx(of)42 b(a)g(matrix)f Fo(M)10 b Fx(,)45 b(written)448 4974 y Fo(M)546 4941 y Fh(y)582 4974 y Fx(,)29 b(is)f(the)i(conjugate)g(transp)s(ose)e(of)i Fo(M)10 b Fx(;)29 b(that)h(is,)f(the)g(\()p Fo(i;)15 b(j)5 b Fx(\)th)31 b(en)m(try)e(of)g Fo(M)3156 4941 y Fh(y)3221 4974 y Fx(is)f(the)p eop %%Page: 16 16 16 15 bop 448 573 a Fx(complex)29 b(conjugate)g(of)g(the)g(\()p Fo(j;)15 b(i)p Fx(\)th)30 b(en)m(try)f(of)g Fo(M)10 b Fx(.)40 b(It)28 b(is)g(easy)h(to)g(see)h(that)f(a)g(matrix)448 686 y Fo(M)37 b Fx(is)25 b(unitary)g(if)h(and)f(only)h(if)f Fo(M)10 b(M)1700 653 y Fh(y)1761 686 y Fx(=)25 b Fo(M)1955 653 y Fh(y)1991 686 y Fo(M)35 b Fx(=)25 b Fo(I)7 b Fx(.)40 b(This)24 b(is)i(in)f(close)h(analogy)h(with)448 799 y(real)j(orthogonal)h(matrices;)g(in)e(fact,)j(a)f(real)f(matrix)f(is)h (unitary)f(if)g(and)h(only)g(if)f(it)h(is)448 912 y(orthogonal.)40 b(This)24 b(means)h(that)i(the)e(real-amplitudes)f(mo)s(del)g(of)i (Section)g(5)g(em)m(b)s(eds)448 1024 y(nicely)h(in)g(the)h(presen)m(t)g (mo)s(del,)g(simply)e(b)m(y)i(restricting)f(the)h(amplitudes)e(to)j(b)s (e)e(real.)589 1137 y(The)33 b(computational)g(basis)f(is)h(as)g(b)s (efore,)h(but)e(allo)m(wing)g(complex)h(co)s(e\016cien)m(ts)448 1250 y(means)21 b(that)h(the)f(space)h Fj(H)f Fx(is)f(no)m(w)h(iden)m (ti\014ed)f(with)f Fg(C)2291 1217 y Ft(2)2326 1194 y Ff(n)2379 1250 y Fx(.)37 b(Quan)m(tum)21 b(gates)h(no)m(w)f(m)m(ust)448 1363 y(corresp)s(ond)40 b(to)h(unitary)e(transformations.)70 b(As)40 b(previously)-8 b(,)42 b(a)e(quan)m(tum)g(circuit)448 1476 y(starts)35 b(in)e(a)h(basis)f(state,)j(whic)m(h)d(has)h(unit)f (Hermitian)f(norm.)51 b(The)34 b(unitary)f(gates)448 1589 y(preserv)m(e)22 b(the)g(norm)f(of)h(the)g(state,)j(so)d(that)g (the)g(\014nal)e(state)2501 1521 y Fe(P)2597 1616 y Fi(x)p Fh(2f)p Ft(0)p Fi(;)p Ft(1)p Fh(g)2845 1587 y Ff(n)2906 1589 y Fo(a)2954 1603 y Fi(x)2998 1589 y Fj(j)q Fo(x)p Fj(i)i Fx(satis\014es)448 1657 y Fe(P)544 1752 y Fi(x)603 1726 y Fj(j)q Fo(a)677 1740 y Fi(x)720 1726 y Fj(j)746 1684 y Ft(2)814 1726 y Fx(=)29 b(1.)48 b(W)-8 b(e)33 b(therefore)g(in)m(terpret)f Fj(j)q Fo(a)2027 1740 y Fi(x)2070 1726 y Fj(j)2096 1684 y Ft(2)2168 1726 y Fx(as)h(the)f (probabilit)m(y)e(that)k(the)e(\014nal)448 1838 y(state)g(of)f(the)f (circuit)f(is)h Fj(j)p Fo(x)p Fj(i)p Fx(.)589 1951 y(Do)s(es)37 b(this)e(giv)m(e)i(a)f(more)g(p)s(o)m(w)m(erful)f(mo)s(del)g(than)h (the)g(one)g(in)f(Section)h(5)h(using)448 2064 y(real)e(amplitudes?)52 b(No,)37 b(not)e(really)-8 b(.)54 b(Both)35 b(de\014ne)g(the)g(same)g (complexit)m(y)g(classes.)448 2177 y(In)26 b(fact)h(one)f(can)h(easily) e(transform)h(a)g(quan)m(tum)g(circuit)f(with)f(complex)i(amplitudes) 448 2290 y(in)m(to)35 b(an)g(equiv)-5 b(alen)m(t)34 b(quan)m(tum)h (circuit)f(with)f(real)i(amplitudes)e(at)i(the)g(exp)s(ense)g(of)448 2403 y(including)24 b(one)k(extra)g(ancilla)e(and)h(adding)f(one)i(to)g (the)g(arit)m(y)f(of)h(some)g(of)g(the)f(gates.)589 2516 y(If)e Fo(M)35 b Fx(is)24 b(an)m(y)h Fo(k)12 b Fj(\002)d Fo(`)26 b Fx(complex)f(matrix)f(\(this)g(includes)f(ro)m(w)i(and)f (column)g(v)m(ectors\),)448 2629 y(w)m(e)37 b(transform)e(it)h(in)m(to) g(a)h(2)p Fo(k)28 b Fj(\002)23 b Fx(2)p Fo(`)37 b Fx(real)f(matrix)f Fo(\032)p Fx(\()p Fo(M)10 b Fx(\))37 b(as)g(follo)m(ws:)51 b(replace)36 b(ev)m(ery)448 2742 y(en)m(try)31 b Fo(x)20 b Fx(+)g Fo(y)s(i)30 b Fx(of)h Fo(M)40 b Fx(b)m(y)31 b(the)f(2)21 b Fj(\002)f Fx(2)31 b(real)f(matrix)1706 2866 y Fe(\024)1796 2938 y Fo(x)83 b Fj(\000)p Fo(y)1800 3051 y(y)152 b(x)2091 2866 y Fe(\025)2154 2995 y Fo(:)448 3248 y Fx(W)-8 b(e)32 b(ha)m(v)m(e)f(the)g(follo)m(wing)e(facts:)585 3435 y Fj(\017)46 b Fo(\032)p Fx(\()p Fo(M)10 b(N)g Fx(\))39 b(=)f Fo(\032)p Fx(\()p Fo(M)10 b Fx(\))p Fo(\032)p Fx(\()p Fo(N)g Fx(\),)42 b(and)37 b Fo(\032)p Fx(\()p Fo(M)1958 3449 y Ft(1)2024 3435 y Fx(+)25 b Fo(aM)2256 3449 y Ft(2)2296 3435 y Fx(\))39 b(=)f Fo(\032)p Fx(\()p Fo(M)2649 3449 y Ft(1)2689 3435 y Fx(\))26 b(+)f Fo(a\032)p Fx(\()p Fo(M)3064 3449 y Ft(2)3104 3435 y Fx(\),)41 b(where)676 3548 y Fo(a)25 b Fj(2)g Fg(C)17 b Fx(,)37 b(and)30 b Fo(M)10 b Fx(,)30 b Fo(M)1374 3562 y Ft(1)1414 3548 y Fx(,)h Fo(M)1558 3562 y Ft(2)1597 3548 y Fx(,)g(and)f Fo(N)40 b Fx(ha)m(v)m(e)31 b(an)m(y)g(appropriate)f(dimensions.)585 3736 y Fj(\017)46 b Fo(\032)p Fx(\()p Fo(M)856 3703 y Fh(y)892 3736 y Fx(\))25 b(=)g Fo(\032)p Fx(\()p Fo(M)10 b Fx(\))1263 3703 y Ft(t)585 3923 y Fj(\017)46 b Fo(M)51 b Fx(is)41 b(unitary)g(if)f(and)h(only)g(if)g Fo(\032)p Fx(\()p Fo(M)10 b Fx(\))43 b(is)d(orthogonal.)76 b(This)40 b(follo)m(ws)h(from)676 4036 y(item)30 b(2.)585 4224 y Fj(\017)46 b Fo(\032)p Fx(\()p Fo(I)7 b Fx(\))35 b(=)f Fo(I)7 b Fx(.)57 b(Here)36 b(the)g(second)g Fo(I)43 b Fx(is)35 b(of)h(course)g(bigger)f(than)h(the)g(\014rst.)56 b(This)676 4337 y(follo)m(ws)29 b(from)h(item)g(1.)448 4524 y(If)c Fo(u)h Fx(is)f(column)f(v)m(ector)j(in)d Fg(C)1447 4491 y Fi(m)1519 4524 y Fx(,)j(then)e Fo(\032)p Fx(\()p Fo(u)p Fx(\))h(is)f(tec)m(hnically)g(a)h(2)p Fo(m)13 b Fj(\002)g Fx(2)26 b(matrix.)39 b(There)448 4637 y(are)27 b(only)f(2)p Fo(m)h Fx(real)g(degrees)g(of)g(freedom)f (in)g Fo(u)p Fx(,)h(ho)m(w)m(ev)m(er,)i(so)e(w)m(e)h(can)f(iden)m(tify) e Fo(u)h Fx(with)448 4750 y(a)31 b(v)m(ector)g(in)e Fg(R)962 4717 y Ft(2)p Fi(m)1070 4750 y Fx(.)40 b(The)30 b(real)f(dimension)f (is)h(t)m(wice)h(the)g(complex)g(dimension.)38 b(Since)448 4863 y(adding)e(a)i(new)e(qubit)g(to)i(a)g(set)f(of)h(registers)f (doubles)e(the)i(dimension)e(of)i Fj(H)q Fx(,)i(this)448 4976 y(suggests)26 b(that)f(w)m(e)g(can)g(sim)m(ulate)f(a)i(circuit)d (with)h(complex)g(amplitudes)f(b)m(y)h(a)h(circuit)p eop %%Page: 17 17 17 16 bop 448 573 a Fx(with)27 b(real)g(amplitudes)f(and)h(one)h (additional)e(ancilla,)h(and)g(an)m(y)h(gates)h(with)e(nonreal)448 686 y(en)m(tries)f(are)g(sim)m(ulated)f(b)m(y)h(gates)h(that)g(in)m (teract)g(with)d(this)h(ancilla.)38 b(All)25 b(this)g(indeed)448 799 y(w)m(orks)30 b(using)e(the)i Fo(\032)g Fx(transformation)f(ab)s(o) m(v)m(e.)41 b(The)29 b Fo(T)13 b Fx(-gate)32 b(de\014ned)c(in)g (Section)i(5)g(is)448 912 y(actually)g Fo(\032)h Fx(applied)d(to)j(the) f(one-qubit)g(gate)i(with)d(matrix)1677 1038 y Fe(\024)1766 1108 y Fx(1)153 b(0)1766 1225 y(0)84 b Fo(e)1937 1192 y Fi(i\031)r(=)p Ft(4)2120 1038 y Fe(\025)2183 1166 y Fo(;)448 1421 y Fx(whic)m(h)32 b(is)h(kind)f(of)h(\\conditional)f (phase)h(shift")g(gate.)51 b(The)33 b(circuit)f(sim)m(ulating)g(the)448 1534 y(T)-8 b(o\013oli)30 b(gate)i(in)d(Section)h(5)g(w)m(as)h(deriv)m (ed)e(from)h(a)g(w)m(ell-kno)m(wn)f(complex-amplitude)448 1647 y(quan)m(tum)h(circuit)f(\(see)j([1],)f(for)f(example\).)448 1933 y Fy(References)448 2136 y Fx([1])47 b(M.)33 b(A.)g(Nielsen)e(and) h(I.)h(L.)f(Ch)m(uang.)47 b Fn(Quantum)35 b(Computation)i(and)f (Quantum)590 2249 y(Information)p Fx(.)43 b(Cam)m(bridge)29 b(Univ)m(ersit)m(y)h(Press,)g(2000.)448 2437 y([2])47 b(Y.)27 b(Shi.)34 b(Both)28 b(To\013oli)e(and)h(con)m(trolled-NOT)g (need)g(little)f(help)g(to)i(do)f(univ)m(ersal)590 2550 y(quan)m(tum)j(computation.)40 b(Unpublished,)27 b(2002,)32 b(quan)m(t-ph/0205115.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF