1. 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and Schur duality transformations D. Bacon (Caltech), I. Chuang (MIT) and A. Harrow (MIT) quant-ph/0407082 + more unpublished quant-ph/0407082 + more unpublished

2. 1. Motivation 2. Total angular momentum (Schur) basis 3. Schur transform: applications 4. Schur transform: construction 5. Generalization to qudits 6. Connections to Fourier transform on S n 1. Motivation 2. Total angular momentum (Schur) basis 3. Schur transform: applications 4. Schur transform: construction 5. Generalization to qudits 6. Connections to Fourier transform on S n Outline

3. 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

4. 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?

5. 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

6. Example 2: quantum error-correcting codes Example 2: quantum error-correcting codes In the computational basis: |i 1 i ­ … ­ |i n i Errors act independently. In the encoded basis: |encoded data i ­ |syndrome i Correctable errors act on the syndrome. In the computational basis: |i 1 i ­ … ­ |i n i Errors act independently. In the encoded basis: |encoded data i ­ |syndrome i Correctable errors act on the syndrome.

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

8. Example 3: 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) ( C 2 ) ­ 2  ( M 1 ­ P trivial ) © ( M 0 ­ P sign )

9. Example 4: three qubits U(2) spin 3/2 spin ½ S3S3 ? symmetric (trivial representation)

10. Example 4: 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.

11. Schur decomposition for 2 or 3 qubits Summarizing: ( C 2 ) ­ 3  M 3/2 © M ½ © M ½  ( M 3/2 ­ P trivial ) © ( M ½ ­ P ½ ) P ½,½  P ½,-½  P ½ Summarizing: ( C 2 ) ­ 3  M 3/2 © M ½ © M ½  ( M 3/2 ­ P trivial ) © ( M ½ ­ P ½ ) P ½,½  P ½,-½  P ½ In hindsight, this looks similar to: ( C 2 ) ­ 2  M 1 © M 0  ( M 1 ­ P trivial ) © ( M 0 ­ P sign ) In hindsight, this looks similar to: ( C 2 ) ­ 2  M 1 © M 0  ( M 1 ­ P trivial ) © ( M 0 ­ P sign )

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

13. Diagrammatic view of Schur transform u u u u u u   |i 1 i |i 2 i |i n i U Sch |J i or | i |M i |P i U Sch = R (u) R (  ) u 2 U(d)  2 S n R is a U(d)-irrep R is a S n -irrep

14. Applications of the Schur transform Universal entanglement concentration: Given |  AB i ­ 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 |  AB i ­ 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.

15. Begin with the Clebsch-Gordon transform on qubits. M J ­ M ½  M J+½ © M J-½ Begin with the Clebsch-Gordon transform on qubits. 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.

16. + + Implementing the CG transform

17. ancilla bits Doing the controlled rotation

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

19. 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

20. 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…

21. n12233444 J ½10 3/2 ½210 Schur duality for n qudits Example: d=2 Example: d=2

22. But what is P J ? S1S1 J=½ 1 S3S3 J=3/2 3 S2S2 J=1 J=0 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

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

24. 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

25. A subgroup-adapted basis for M 1 U(1) 1 2 2 U(2) 3 3 3 3 U(3) 4 U(4)

26. 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) ­  ©© + + U(1) CG 2x2 reduced Wigner add a box to

27. 1) S n QFT ! Schur transform: Generalized Phase Estimation - Only permits measurement in Schur basis, not full Schur transform. -Similar to [abelian QFT ! phase estimation]. 1) S n 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 S n 2) Schur transform ! S n QFT -Just embed C [S n ] in ( C n ) ­ n and do the Schur transform -Based on Howe duality 2) Schur transform ! S n QFT -Just embed C [S n ] in ( C n ) ­ n and do the Schur transform -Based on Howe duality

28. U QFT CPyCPy CPyCPy |i 1 i |i 2 i |i n i |p 1 i | i |p 2 i | i |i|i U QFT y CPCP CPCP Generalized phase estimation U QFT y |trivial i |i|i |i|i U QFT |trivial i |i|i |i|i |p i |i|i

29. Generalized phase estimation: interpretation as S n CG transform Generalized phase estimation: interpretation as S n CG transform | i |m i |p i |i|i |p  i |p  0 i | i |m i |p 0 i | i |p i |p 0 i | i |m i |p i |i|i |p  i |p  0 i S n CG S n CG | i |m i |p 0 i | i |p i |p  0 i  L(  ) C [S n ] CPCP CPCP L(  ) CPCP CPCP   =

30. Using 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 |1 i,…,|c i. - 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 |1 i,…,|c i. - 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.

31. This is useful for many tasks in quantum information theory. Now what? 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 |,M,P i The generalized phase estimation algorithm allows measurement of in time poly(n) + O (n ¢ log(d)) or,M,P in time d ¢ poly(n) + O (nd ¢ log(d)). |i 1,…,i n i! |i 1,…,i n i |,M,P i The generalized phase estimation algorithm allows measurement of in time poly(n) + O (n ¢ log(d)) or,M,P in time d ¢ poly(n) + O (nd ¢ log(d)).