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Quantum Circuits for Clebsch- Gordon and Schur duality transformations D. Bacon (Caltech), I. Chuang (MIT) and A. Harrow (MIT) quant-ph/ more unpublished quant-ph/ more unpublished

1. Motivation 2. Total angular momentum (Schur) basis 3. Schur transform: applications 4. Schur transform: construction 5. Generalization to qudits 6. Generalized phase estimation 1. Motivation 2. Total angular momentum (Schur) basis 3. Schur transform: applications 4. Schur transform: construction 5. Generalization to qudits 6. Generalized phase estimation Outline

Unitary changes of basis Unlike classical information, quantum information is always presented in a particular basis. A change of basis is a unitary operation. |2 i |1 i |3 i |2 0 i |1 0 i |3 0 i U CB

Questions 1.When can U CB be implemented efficiently? 2.What use are bases other than the standard basis? 1.When can U CB be implemented efficiently? 2.What use are bases other than the standard basis?

Answers 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/momentum Position basis: |x i =|x 1 i­  ­ |x n i Momentum basis: |p 0 i =  x exp(2  ipx/2 n ) | x i / 2 n/2 Position basis: |x i =|x 1 i­  ­ |x n i Momentum basis: |p 0 i =  x exp(2  ipx/2 n ) | x i / 2 n/2 Quantum Fourier Transform:U QFT |p 0 i = |p i

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

Example 2: two qubits However, for >2 qubits, J and M do not uniquely specify the state. U(2) spin 0 spin 1 S2S2 antisymmetric (sign representation) symmetric (trivial representation)

Example 3: three qubits U(2) spin 3/2 spin ½ S3S3 ? symmetric (trivial representation)

Example 3: three qubits cont. This is a two-dimensional irreducible representation (irrep) of S 3. Call it P ½,½. a = |0 ih 1| ­ I ­ I + I ­ |0 ih 1| ­ I + I ­ I ­ |0 ih 1| a P ½,½  P ½,-½ and [a, S 3 ]=0, so P ½,½  P ½,-½. a = |0 ih 1| ­ I ­ I + I ­ |0 ih 1| ­ I + I ­ I ­ |0 ih 1| a P ½,½  P ½,-½ and [a, S 3 ]=0, so P ½,½  P ½,-½.

Schur decomposition for n qubits Theorem (Schur): For any J and M, P J,M is an irrep of S n. Furthermore, P J,M  P J,M’ for any M 0, so P J,M is determined by J up to isomorphism. M J and P J are irreps of U(2) and S n, respectively.

Diagrammatic view of Schur transform V V V V V V   |i 1 i |i 2 i |i n i U Sch |J i |M i |P i U Sch = R J (V) RJ()RJ() RJ()RJ() V 2 U(2)  2 S n R J is a U(2)-irrep R J is a S n -irrep

Applications of the Schur transform Universal entanglement concentration: Given |  AB i ­ n, Alice and Bob both perform the Schur transform, measure J, discard M J and are left with a maximally entangled state in P J equivalent to ¼ nE(  ) EPR pairs. Universal entanglement concentration: Given |  AB i ­ n, Alice and Bob both perform the Schur transform, measure J, discard M J and are left with a maximally entangled state in P J 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. M J ­ M ½ = M J+½ © M J-½ Begin with the Clebsch-Gordon transform. M J ­ M ½ = M J+½ © M J-½ How to perform the Schur transform? Why can U CG be implemented efficiently? 1. Conditioned on J and M, U CG is two-dimensional. 2. C CG can be efficiently classically computed. Why can U CG be implemented efficiently? 1. Conditioned on J and M, U CG is two-dimensional. 2. C CG can be efficiently classically computed.

+ + Implementing the CG transform

garbage bits Doing the controlled rotation

Diagrammatic view of CG transform U CG |M i |J i |S i |J i |J 0 i |M 0 i U CG R J (V) V V = U CG R J 0 (V) MJMJ M½M½ M J+½ © M J-½

Schur transform = iterated CG U CG |i 1 i |½ i |i 2 i |i n i |J 1 i |J 2 i |M 2 i |i 3 i U CG |J 2 i |J 3 i |M 3 i |J n-1 i |M n-1 i U CG |J n-1 i |J n i |M i (C2)­n(C2)­n

Q: What do we do with |J 1 …J n-1 i ? A: Declare victory! Let P J 0 = Span{|J 1 …J n-1 i : J 1,…,J n-1 is a valid path to J} Proof: Since U(2) acts appropriately on M J and trivially on P J 0, Schur duality implies that P J  P J 0 under S n. Q: What do we do with |J 1 …J n-1 i ? A: Declare victory! Let P J 0 = Span{|J 1 …J n-1 i : J 1,…,J n-1 is a valid path to J} Proof: Since U(2) acts appropriately on M J and trivially on P J 0, Schur duality implies that P J  P J 0 under S n. Almost there…

But what is P J ? S1S1 S2S2 J=½ J=1 J=0 1 S3S3 J=½ J=3/2 3 2 S4S4 J=2 J=1 J=0 4 S5S5 J=5/2 J=3/2 J=½ 5 S6S6 J=3 J=2 J=1 J=0 6 paths of irreps  standard tableaux  Gelfand-Zetlin basis

n J ½10 3/2 ½210 Irreps of U(d) and S n are labelled by partitions of n into 6 d parts, i.e. ( 1,…, d ) such that d = n. Let M be a U(d) irrep and P a S n irrep. Then: Irreps of U(d) and S n are labelled by partitions of n into 6 d parts, i.e. ( 1,…, d ) such that d = n. Let M be a U(d) irrep and P a S n irrep. Then: Schur duality for n qudits Example: d=2 Example: d=2

U(d) irreps U(1) irreps are labelled by integers n:  n (x) = x n 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 .

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 weights Example: d=2, = (2J, n-2J) Highest weight state is |M=J i. Annihilated by  + and acted on by Example: d=2, = (2J, n-2J) Highest weight state is |M=J i. Annihilated by  + and acted on by

A subgroup-adapted basis for M 1 U(1) U(2) 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. Clebsch-Gordon series for U(d) ­  ©©

U QFT UU UU |i 1 i |i 2 i |i n i |p 1 i | i |p 2 i | i |i|i U QFT y UyUy UyUy Generalized phase estimation U QFT y |trivial i |i|i |i|i U QFT |trivial i |i|i |i|i

This is useful for many tasks in quantum information theory. Can you find more? Summary |i 1,…,i n i! |,M,P i : The Schur transform maps the angular momentum basis of ( C d ) ­ n into the computational basis in time n ¢ poly(d). |i 1,…,i n i! |,M,P i : The Schur transform maps the angular momentum basis of ( C d ) ­ n into the computational basis in time n ¢ poly(d). |i 1,…,i n i! |i 1,…,i n i | i The generalized phase estimation algorithm allows measurement of in time poly(n) + O (n ¢ log(d)). |i 1,…,i n i! |i 1,…,i n i | i The generalized phase estimation algorithm allows measurement of in time poly(n) + O (n ¢ log(d)).