““Both Toffoli and CNOT Both Toffoli and CNOT need little help to do need little help to do

universal QC”universal QC”

(following a (following a paperpaper by the by the same title by Yaoyun Shi)same title by Yaoyun Shi)

AbstractAbstract

• Well known fact: Well known fact: – {CNOT,S} is universal when S is an irrational {CNOT,S} is universal when S is an irrational

one qubit rotationone qubit rotation

• Less well known fact:Less well known fact:– S really only needs to not square to something S really only needs to not square to something

classicalclassical

• Another less well known fact:Another less well known fact:– {Toffoli, Hadamard} is universal{Toffoli, Hadamard} is universal

The AgendaThe Agenda

• BackgroundBackground– Completeness vs. UniversalityCompleteness vs. Universality– Kitaev-Solovay TheoremKitaev-Solovay Theorem– Another result by KitaevAnother result by Kitaev

• Completeness (existence) proofsCompleteness (existence) proofs

• Completeness: an explicit constructionCompleteness: an explicit construction

• ConclusionConclusion

UniversalityUniversality

• A (real) gate library G is universal ifA (real) gate library G is universal if– it can approximate any unitary it can approximate any unitary

(orthogonal) operator if constant inputs (orthogonal) operator if constant inputs from the computational basis are from the computational basis are allowedallowed

– for example, a TOFFOLI gate can for example, a TOFFOLI gate can approximate a CNOT gate in this senseapproximate a CNOT gate in this sense

CompletenessCompleteness

• A gate library G is complete ifA gate library G is complete if– it can approximate any unitary operator it can approximate any unitary operator

in U(2in U(2kk) for some k) for some k– no extra wires or constant inputs no extra wires or constant inputs

allowedallowed

• Completeness => UniversalityCompleteness => Universality

Why completeness?Why completeness?

• The Kitaev-Solovay Theorem:The Kitaev-Solovay Theorem:

• Any complete gate library can Any complete gate library can efficiently approximate any 1 qubit efficiently approximate any 1 qubit unitary operatorunitary operator– specifically, one can get within specifically, one can get within εε in in

polylog(1/polylog(1/εε) gates) gates

Another theorem of KitaevAnother theorem of Kitaev

• Suppose:Suppose:– MM is a (real) Hilbert space of dimension is a (real) Hilbert space of dimension

> 2> 2– is a unit vectoris a unit vector– H H SO( SO(M M ) is the stabilizer of span() is the stabilizer of span())– v v O( O(M M ), ), not an eigenvector of v not an eigenvector of v

• Then: Then: – the subgroup generated by Hthe subgroup generated by H v v-1-1Hv is Hv is

dense in SO(dense in SO(M M ) )

The AgendaThe Agenda

• BackgroundBackground

• Completeness (existence) proofsCompleteness (existence) proofs– CNOTs and RotationsCNOTs and Rotations– Eigenvectors & EigenvaluesEigenvectors & Eigenvalues– Who’s DenseWho’s Dense

• Completeness: an explicit constructionCompleteness: an explicit construction

• ConclusionConclusion

A CNOT and a rotationA CNOT and a rotation

• Fix an arbitrary one qubit rotation S Fix an arbitrary one qubit rotation S about an angle about an angle θθ– if if θθ//ππ is irrational, we know from general is irrational, we know from general

theory that {CNOT, S} is completetheory that {CNOT, S} is complete

• So, suppose So, suppose θθ is a rational multiple of is a rational multiple of pipi

A CNOT and a rotationA CNOT and a rotation

• Finally, suppose SFinally, suppose S22 does not have does not have both both 00 and and 11 as eigenvectors as eigenvectors– a theorem of Gottesman-Knill implies a theorem of Gottesman-Knill implies

that:that:•for an S failing this condition, any {S, CNOT} for an S failing this condition, any {S, CNOT}

circuit may be efficiently simulated by a circuit may be efficiently simulated by a classical computerclassical computer

– thus, such an S is not universal for QCthus, such an S is not universal for QC

• Then {S, CNOT} is complete.Then {S, CNOT} is complete.

A sketch of the proof:A sketch of the proof:

• Let U be the operator be computed byLet U be the operator be computed by

• Apply the Kitaev lemma several timesApply the Kitaev lemma several times– Q.E.D.Q.E.D.

S

S S

S

Eigenvectors & EigenvaluesEigenvectors & Eigenvalues

• Calculating U’s eigenvalues gives them asCalculating U’s eigenvalues gives them as– 1, 1, e1, 1, eii, e, e-i-i

– is incommensurable with piis incommensurable with pi

• Let Let ii be the orthonormal eigenvectors be the orthonormal eigenvectors

– U restricted to span(span(11, , 22) is the identity) is the identity

– U restricted to span(U restricted to span(33, , 44):=H):=H11 is a rotation is a rotation through the angle through the angle

Who’s DenseWho’s Dense

• U generates a dense subgroup of HU generates a dense subgroup of H11

• Call SO(span(Call SO(span(22, , 33, , 44)) H)) H22

– HH11 H H22 is the stabilizer of span( is the stabilizer of span(22))

– one CNOT, Cone CNOT, C11 fixes fixes 11, and moves , and moves span(span(22))

Who’s DenseWho’s Dense

• The Kitaev lemma applies: {U, CThe Kitaev lemma applies: {U, C11} } generates a dense subset of Hgenerates a dense subset of H22

• A similar argument shows {U, CA similar argument shows {U, C11, C, C22} } generates a dense subset of SO(4)generates a dense subset of SO(4)

• So, {U, CSo, {U, C11, C, C22} is complete} is complete

The AgendaThe Agenda

• BackgroundBackground

• Completeness (existence) proofsCompleteness (existence) proofs

• Completeness: an explicit constructionCompleteness: an explicit construction– Barenko’s ReductionBarenko’s Reduction– the Z gatethe Z gate– Grover’s AlgorithmGrover’s Algorithm

• ConclusionConclusion

An Explicit ConstructionAn Explicit Construction

• Recall {CNOT, S} is complete Recall {CNOT, S} is complete – when Swhen S22

doesn’t have both basis states doesn’t have both basis states as eigenvectors as eigenvectors

• It is true that {TOFFOLI, S} is It is true that {TOFFOLI, S} is completecomplete– when S doesn’t have both basis states when S doesn’t have both basis states

as eigenvectors as eigenvectors – a similar proof existsa similar proof exists

An Explicit ConstructionAn Explicit Construction

• Additionally, Shi explicitly {TOFFOLI, Additionally, Shi explicitly {TOFFOLI, S} approximates an arbitrary one S} approximates an arbitrary one qubit gatequbit gate

• By Barenko’s decomposition, this is By Barenko’s decomposition, this is sufficient to approximate an arbitrary sufficient to approximate an arbitrary unitary matrixunitary matrix

Some preliminariesSome preliminaries

• Define UDefine Utt to be rotation by the angle t to be rotation by the angle t

• Let S be the one-qubit gate in our libraryLet S be the one-qubit gate in our library– define define θθ by S = U by S = Uθθ

• Let W be the desired one qubit operator Let W be the desired one qubit operator – define define by W = U by W = U

Reduction of the problemReduction of the problem

• It suffices to approximateIt suffices to approximate– the Z gatethe Z gate

– a gate Wa gate W/2/2 s.t. W s.t. W /2 /200kk = U = U/2 /2 00 00k-1k-1

• Using these gates and the TOFFOLI, Using these gates and the TOFFOLI, one may simulate a gate Wone may simulate a gate W satisfying satisfying

– WW (( 00k-1k-1) = U) = U 00k-1k-1

The Z GateThe Z Gate

• How to use S to flip a signHow to use S to flip a sign– Suppose Suppose θθ = pi/4 = pi/4– One can use a well known trick:One can use a well known trick:

– This works because: XUThis works because: XUpi/4pi/411=-U=-Upi/4pi/411

S S†

Z

=

11 11

The Z GateThe Z Gate

• For arbitrary For arbitrary θθ, it’s more difficult, it’s more difficult– XUXUθθ11 could be anywhere relative to could be anywhere relative to

UUθθ11

The Z GateThe Z Gate

• A similar construction exists, howeverA similar construction exists, however

• UUθθ00UUθθ11 = a( = a(1111--0000) + b) + b0101 + + cc1010– swap the basis vectors swap the basis vectors 1111, , 0000– this is within sqrt(bthis is within sqrt(b22+c+c22) of a sign flip) of a sign flip– sqrt(bsqrt(b22+c+c22) < 1, so do a lot of these) < 1, so do a lot of these

The WThe W /2 /2 Gate Gate

• Want: WWant: W /2 /200kk = U = U/2 /2 00 00k-1k-1

• Idea ? Idea ?

Prelude to Grover’s Prelude to Grover’s AlgorithmAlgorithm

• Let Let 00 = = 002k2k

• Use S, CNOT, to build a T such that Use S, CNOT, to build a T such that – 00TT00 is small and positive is small and positive– define define φφ = T = T00

• Let Let 11 be the vector perpendicular to be the vector perpendicular to 00 in the plane spanned by in the plane spanned by 00 , , φφ

Using Grover’s AlgorithmUsing Grover’s Algorithm

• The system begins in the state The system begins in the state 0000 – apply Iapply ITT– the state = the state = 00φφ

• Iteratively reflect Iteratively reflect φφ about about 11 ala ala GroverGrover– want: want: φφ -> cos( -> cos(/2/2 ))11 + sin( + sin(/2/2 ))00

– state = state = 00(cos((cos(/2/2 ))11 + sin( + sin(/2/2 ))00))

Using Grover’s AlgorithmUsing Grover’s Algorithm

• Apply an appropriately conjugated 2k-Apply an appropriately conjugated 2k-cnot to flip the first bit if the remaining cnot to flip the first bit if the remaining 2k are orthogonal to 2k are orthogonal to 00 – state = state = 1111cos(cos(/2/2 ) + ) + 0000sin(sin(/2/2 ))

• Apply a controlled-TApply a controlled-T-1-1 : : 1111 -> -> 1100– state = (cos(state = (cos(/2/2 ))11 + sin( + sin(/2/2 ))00))00

The AgendaThe Agenda

• BackgroundBackground

• Completeness (existence) proofsCompleteness (existence) proofs

• Completeness: an explicit Completeness: an explicit constructionconstruction

• ConclusionConclusion

ConclusionConclusion

• The CNOT needs only a one qubit rotation The CNOT needs only a one qubit rotation whose square is nonclassical to form a whose square is nonclassical to form a complete librarycomplete library

• The Toffoli can partner with The Toffoli can partner with anyany nonclassical gate for a complete librarynonclassical gate for a complete library

• In the second case, we have an explicit In the second case, we have an explicit approximation algorithmapproximation algorithm

Questions?Questions?