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Quantum Circuits for Clebsch-Gordon and Schur duality transformations

Quantum Circuits for Clebsch-Gordon and Schur duality transformations

D. Bacon (Caltech), I. Chuang (MIT) and A. Harrow (MIT)

quant-ph/0407082 + more unpublished

quant-ph/0407082 + more unpublished

1. Motivation2. Total angular momentum (Schur) basis3. Schur transform: applications4. Schur transform: construction5. Generalization to qudits6. Generalized phase estimation

1. Motivation2. Total angular momentum (Schur) basis3. Schur transform: applications4. Schur transform: construction5. Generalization to qudits6. Generalized phase estimation

OutlineOutline

Unitary changes of basisUnitary changes of basis

Unlike classical information, quantum information is always presented in a particular basis.

Unlike classical information, quantum information is always presented in a particular basis.

A change of basis is a unitary operation.A change of basis is a unitary operation.

|2i

|1i

|3i

|20i

|10i

|30i

UCB

QuestionsQuestions

1. When can UCB be implemented efficiently?

2. What use are bases other than the standard basis?

1. When can UCB be implemented efficiently?

2. What use are bases other than the standard basis?

AnswersAnswers

1. I’ll describe a useful and physically motivated alternate basis.

2. I’ll give an efficient quantum circuit to transform from the computational basis to this alternate basis.

1. I’ll describe a useful and physically motivated alternate basis.

2. I’ll give an efficient quantum circuit to transform from the computational basis to this alternate basis.

Example 1: position/momentumExample 1: position/momentum

Position basis: |xi=|x1i |xni

Momentum basis: |p0i= x exp(2ipx/2n)|xi / 2n/2

Position basis: |xi=|x1i |xni

Momentum basis: |p0i= x exp(2ipx/2n)|xi / 2n/2

Quantum Fourier Transform: UQFT|p0i = |pi

Quantum Fourier Transform: UQFT|p0i = |pi

Angular momentum basisAngular momentum basis

States on n qubits can be (partially) labelled by total angular momentum (J) and the Z component of angular momentum (M).

States on n qubits can be (partially) labelled by total angular momentum (J) and the Z component of angular momentum (M).

Example 2: two qubitsExample 2: two qubits

However, for >2 qubits, J and M do not uniquely specify the state.

However, for >2 qubits, J and M do not uniquely specify the state.

U(2)

spin 0

spin 1

S2

antisymmetric

(sign representation)

symmetric

(trivial representation)

Example 3: three qubitsExample 3: three qubitsU(2)

spin 3/2

spin ½

S3

?

symmetric

(trivial representation)

Example 3: three qubits cont.Example 3: three qubits cont.

This is a two-dimensional irreducible representation (irrep) of S3. Call it P½,½.

This is a two-dimensional irreducible representation (irrep) of S3. Call it P½,½.

a = |0ih1| I I + I |0ih1| I + I I |0ih1|aP½,½P½,-½ and [a, S3]=0, so P½,½P½,-½.a = |0ih1| I I + I |0ih1| I + I I |0ih1|aP½,½P½,-½ and [a, S3]=0, so P½,½P½,-½.

Schur decomposition for n qubitsSchur decomposition for n qubits

Theorem (Schur): For any J and M, PJ,M is an irrep of Sn. Furthermore, PJ,MPJ,M’ for any M0, so PJ,M is determined by J up to isomorphism.

Theorem (Schur): For any J and M, PJ,M is an irrep of Sn. Furthermore, PJ,MPJ,M’ for any M0, so PJ,M is determined by J up to isomorphism.

MJ and PJ are irreps of U(2) and Sn, respectively.

MJ and PJ are irreps of U(2) and Sn, respectively.

Diagrammatic view of Schur transformDiagrammatic view of Schur transform

VVVV

VV

|i1i

|i2i

|ini

USc

h

USc

h

|Ji

|Mi

|Pi

USc

h

USc

h

= USc

h

USc

h

RJ(V)

RJ(V)

RJ()RJ()

V 2 U(2)

2 SnRJ is a U(2)-irrep

RJ is a Sn-irrep

Applications of the Schur transformApplications of the Schur transform

Universal entanglement concentration:Given |ABi n, Alice and Bob both perform the Schur transform, measure J, discard MJ and are left with a maximally entangled state in PJ equivalent to ¼ nE() EPR pairs.

Universal entanglement concentration:Given |ABi n, Alice and Bob both perform the Schur transform, measure J, discard MJ and are left with a maximally entangled state in PJ equivalent to ¼ nE() EPR pairs.

Universal data compression:Given n, perform the Schur transform, weakly measure J and the resulting state has dimension ¼ exp(nS()).

Universal data compression:Given n, perform the Schur transform, weakly measure J and the resulting state has dimension ¼ exp(nS()).

State estimation:Given n, estimate the spectrum of , or estimate , or test to see whether the state is n.

State estimation:Given n, estimate the spectrum of , or estimate , or test to see whether the state is n.

Begin with the Clebsch-Gordon transform.MJ M½ = MJ+½ © MJ-½

Begin with the Clebsch-Gordon transform.MJ M½ = MJ+½ © MJ-½

How to perform the Schur transform?How to perform the Schur transform?

Why can UCG be implemented efficiently?1. Conditioned on J and M, UCG is two-dimensional.2. CCG can be efficiently classically computed.

Why can UCG be implemented efficiently?1. Conditioned on J and M, UCG is two-dimensional.2. CCG can be efficiently classically computed.

+

+

Implementing the CG transformImplementing the CG transform

garbage bits

Doing the controlled rotationDoing the controlled rotation

Diagrammatic view of CG transformDiagrammatic view of CG transform

UCGUCG|Mi

|Ji

|Si

|Ji

|J0i

|M0i

UCGUCG

RJ(V)

RJ(V)

VV= UCGUCG

RJ0(V)

RJ0(V)

MJ

M½

MJ+½ © MJ-½

Schur transform = iterated CGSchur transform = iterated CG

UCGUCG|i1i

|½i

|i2i

|ini

|J1i

|J2i

|M2i

|i3i

UCGUCG

|J2i

|J3i

|M3i

|Jn-1i|Mn-1i UCGUCG

|Jn-1i

|Jni

|Mi

(C2) n

Q: What do we do with |J1…Jn-1i?A: Declare victory!

Let PJ0 = Span{|J1…Jn-1i : J1,…,Jn-1 is a valid

path to J}Proof:

Since U(2) acts appropriately on MJ and trivially on PJ

0, Schur duality implies that PJPJ

0 under Sn.

Q: What do we do with |J1…Jn-1i?A: Declare victory!

Let PJ0 = Span{|J1…Jn-1i : J1,…,Jn-1 is a valid

path to J}Proof:

Since U(2) acts appropriately on MJ and trivially on PJ

0, Schur duality implies that PJPJ

0 under Sn.

Almost there…Almost there…

But what is PJ?But what is PJ?

S1 S2

J=½

J=1

J=0

1

S3

J=½

J=3/2

3

2

S4

J=2

J=1

J=0 4

S5

J=5/2

J=3/2

J=½ 5

S6

J=3

J=2

J=1

J=0 6

paths of irreps standard tableaux Gelfand-Zetlin basis

n 1 2 2 3 3 4 4 4

J ½ 1 0 3/2 ½ 2 1 0

Irreps of U(d) and Sn are labelled by partitions of n into 6d parts, i.e. (1,…,d) such that 1+...+d = n.

Let M be a U(d) irrep and P a Sn irrep. Then:

Irreps of U(d) and Sn are labelled by partitions of n into 6d parts, i.e. (1,…,d) such that 1+...+d = n.

Let M be a U(d) irrep and P a Sn irrep. Then:

Schur duality for n quditsSchur duality for n qudits

Example:

d=2

Example:

d=2

U(d) irrepsU(d) irreps

U(1) irreps are labelled by integers n: n(x) = xn

U(1) irreps are labelled by integers n: n(x) = xn

U(d) irreps are induced from irreps of the torus

T(d) has irreps labelled by integers 1,…,d:

U(d) irreps are induced from irreps of the torus

T(d) has irreps labelled by integers 1,…,d:

A vector v in a U(d) irrep has weight if T(d) acts on v according to .

A vector v in a U(d) irrep has weight if T(d) acts on v according to .

M has a unique vector |i2M that

a) has weight

b) is fixed by R(U) for U of the form:

(i.e. is annihilated by the raising operators)

M has a unique vector |i2M that

a) has weight

b) is fixed by R(U) for U of the form:

(i.e. is annihilated by the raising operators)

M via highest weightsM via highest weights

Example: d=2, = (2J, n-2J)

Highest weight state is |M=Ji. Annihilated by + and acted on by

Example: d=2, = (2J, n-2J)

Highest weight state is |M=Ji. Annihilated by + and acted on by

A subgroup-adapted basis for MA subgroup-adapted basis for M

1

U(1)

1

2

2

U(2)

3

3

3

3

U(3)

4

U(4)

To perform the CG transform in poly(d) steps: Perform the U(d-1) CG transform, a controlled d£d rotation given by the reduced Wigner coefficients and then a coherent classical computation.

To perform the CG transform in poly(d) steps: Perform the U(d-1) CG transform, a controlled d£d rotation given by the reduced Wigner coefficients and then a coherent classical computation.

Clebsch-Gordon series for U(d)Clebsch-Gordon series for U(d)

© ©

UQFTUQFT

UU

|i1i

|i2i

|ini

|p1i

|i

|p2i

|i|i

UQFTy

UQFTy

Uy

Uy

Generalized phase estimationGeneralized phase estimation

UQFTy

UQFTy

|triviali

|i|i

UQFTUQFT

|triviali

|i|i

This is useful for many tasks in quantum information theory. Can you find more?

This is useful for many tasks in quantum information theory. Can you find more?

SummarySummary

|i1,…,ini!|,M,Pi:The Schur transform maps the angular

momentum basis of (Cd) n into the computational basis in time n¢poly(d).

|i1,…,ini!|,M,Pi:The Schur transform maps the angular

momentum basis of (Cd) n into the computational basis in time n¢poly(d).

|i1,…,ini!|i1,…,ini|iThe generalized phase estimation algorithm

allows measurement of in time poly(n) + O(n¢log(d)).

|i1,…,ini!|i1,…,ini|iThe generalized phase estimation algorithm

allows measurement of in time poly(n) + O(n¢log(d)).