Quantum Circuits for Clebsch-GordAn and Schur duality transformations

Quantum Circuits for Clebsch-GordAn 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quant-ph/0407082 + more unpublished

quant-ph/0407082 + more unpublished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. Connections to Fourier transform on Sn

1. Motivation2. Total angular momentum (Schur) basis3. Schur transform: applications4. Schur transform: construction5. Generalization to qudits6. Connections to Fourier transform on Sn

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?

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

Example 2: quantumerror-correcting codesExample 2: quantum

error-correcting codes

In the computational basis:|i1i … |ini

Errors act independently.

In the encoded basis: |encoded datai |syndromei

Correctable errors act on the syndrome.

In the computational basis:|i1i … |ini

Errors act independently.

In the encoded basis: |encoded datai |syndromei

Correctable errors act on the syndrome.

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 3: two qubitsExample 3: 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)

(C2) 2 (M1 Ptrivial) © (M0 Psign)

Example 4: three qubitsExample 4: three qubitsU(2)

spin 3/2

spin ½

S3

?

symmetric

(trivial representation)

Example 4: three qubits cont.Example 4: 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 2 or 3 qubitsSchur decomposition for 2 or 3 qubits

Summarizing:(C2) 3 M3/2 © M½ © M½

(M3/2 Ptrivial) © (M½ P½)

P½,½ P½,-½ P½

Summarizing:(C2) 3 M3/2 © M½ © M½

(M3/2 Ptrivial) © (M½ P½)

P½,½ P½,-½ P½

In hindsight, this looks similar to:(C2) 2 M1 © M0

(M1 Ptrivial) © (M0 Psign)

In hindsight, this looks similar to:(C2) 2 M1 © M0

(M1 Ptrivial) © (M0 Psign)

Schur decomposition for n qubitsSchur decomposition for n qubits

Theorem (Schur): A similar decomposition exists for n qudits.

Theorem (Schur): A similar decomposition exists for n qudits.

Diagrammatic view of Schur transformDiagrammatic view of Schur transform

uuuu

uu

|i1i

|i2i

|ini

USc

h

USc

h

|Ji or |i

|Mi

|Pi

USc

h

USc

h

= USc

h

USc

h

R(u)

R(u)R()R()

u 2 U(d)

2 SnR is a U(d)-irrep

R 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 , discard M and are left with a maximally entangled state in P equivalent to ¼ nE() EPR pairs.

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

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

Universal data compression:Given n, perform the Schur transform, weakly measure 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 on qubits.

MJ M½ MJ+½ © MJ-½

Begin with the Clebsch-Gordon transform on qubits.

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

ancilla 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(u)

RJ(u)

uu= UCGUCG

RJ0(u)

RJ0(u)

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…

n 1 2 2 3 3 4 4 4

J ½ 1 0 3/2 ½ 2 1 0

Schur duality for n quditsSchur duality for n qudits

Example:

d=2

Example:

d=2

But what is PJ?But what is PJ?

S1

J=½

1

S3

J=½

J=3/2

3

S2

J=1

J=0

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

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

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 .

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:

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 add one box to .

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 add one box to .

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

© ©

+

+

U(1) CG

2x2 reduced Wigner

add a box to

1) Sn QFT ! Schur transform: Generalized Phase Estimation-Only permits measurement in Schur basis, not full Schur transform.-Similar to [abelian QFT ! phase estimation].

1) Sn QFT ! Schur transform: Generalized Phase Estimation-Only permits measurement in Schur basis, not full Schur transform.-Similar to [abelian QFT ! phase estimation].

Connections to the QFT on SnConnections to the QFT on Sn

2) Schur transform ! Sn QFT-Just embed C[Sn] in (Cn) n and do the Schur transform-Based on Howe duality

2) Schur transform ! Sn QFT-Just embed C[Sn] in (Cn) n and do the Schur transform-Based on Howe duality

UQFTUQFT

CPyCPy

|i1i

|i2i

|ini

|p1i

|i

|p2i

|i|i

UQFTy

UQFTy

CPCP

Generalized phase estimationGeneralized phase estimation

UQFTy

UQFTy

|triviali

|i|i

UQFTUQFT

|triviali

|i|i

|pi|i

Generalized phase estimation:interpretation as Sn CG transform

Generalized phase estimation:interpretation as Sn CG transform

|i

|mi

|pi

|i

|pi

|p0i

|i

|mi

|p0i

|i

|pi

|p0i

|i

|mi

|pi

|i

|pi

|p0i

Sn

CG

Sn

CG

|i

|mi

|p0i

|i

|pi

|p0i

L()

L()C[Sn

] CP

CP

L()

L()C

P

CP

=

Using GPE to measure MUsing GPE to measure M

-Measure weight (a.k.a. “type”): the # of 1’s, # of 2’s, …, # of d’s.

-For each c = 1, …, d- Find the m positions in the states |1i,…,|ci.- Do GPE on these m positions and extract an irrep label c.

-This gives a chain of irreps 1,2,…,d=.

-Measure weight (a.k.a. “type”): the # of 1’s, # of 2’s, …, # of d’s.

-For each c = 1, …, d- Find the m positions in the states |1i,…,|ci.- Do GPE on these m positions and extract an irrep label c.

-This gives a chain of irreps 1,2,…,d=.

Performing this coherently requires O(nd) iterations of GPE, or by

looking more carefully at the S_n Fourier transform, we can use only

O(d) times the running time of GPE.

Performing this coherently requires O(nd) iterations of GPE, or by

looking more carefully at the S_n Fourier transform, we can use only

O(d) times the running time of GPE.

This is useful for many tasks in quantum information theory. Now what?

This is useful for many tasks in quantum information theory. Now what?

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|,M,PiThe generalized phase estimation algorithm allowsmeasurement of in time poly(n) + O(n¢log(d))or ,M,P in time d¢poly(n) + O(nd ¢ log(d)).

|i1,…,ini!|i1,…,ini|,M,PiThe generalized phase estimation algorithm allowsmeasurement of in time poly(n) + O(n¢log(d))or ,M,P in time d¢poly(n) + O(nd ¢ log(d)).