1. Quantum Circuits for Clebsch-Gordon and Schur duality transformations
D. Bacon (Caltech), I. Chuang (MIT) and A. Harrow (MIT)

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

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.
UCB
|10i
|1i
|2i
|3i
|30i
|20i

4. Questions When can UCB be implemented efficiently?
What use are bases other than the standard basis?

5. Answers
I’ll describe a useful and physically motivated alternate basis.
I’ll give an efficient quantum circuit to transform from the computational basis to this alternate basis.

6. Example 1: position/momentum
Position basis: |xi=|x1i­L­|xni
Momentum basis: |p0i= åx exp(2pipx/2n)|xi / 2n/2
Quantum Fourier Transform: UQFT|p0i = |pi

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 2: two qubits S2 U(2)­2
antisymmetric
(sign representation)
symmetric
(trivial representation)
U(2)­2
spin 0
spin 1
However, for >2 qubits, J and M do not uniquely specify the state.

9. ? Example 3: three qubits S3 U(2)­3 spin ½ symmetric
(trivial representation)
U(2)­3
spin 3/2
spin ½

10. Example 3: three qubits cont.
This is a two-dimensional irreducible representation (irrep) of S3. Call it P½,½.
(|0ih1|)­3P½,½=P½,-½ and (|0ih1|)­3 commutes with S3, so

11. Schur decomposition for n qubits
Theorem (Schur): For any J and M, PJ,M is an irrep of Sn. Furthermore, for any M0, so PJ,M is determined by J up to isomorphism.
MJ and PJ are irreps of U(2) and Sn, respectively.

12. Diagrammatic view of Schur transform
|i1i
USch
|Ji
|i2i
|Mi
|Pi
V 2 U(2)
p 2 Sn
|ini
RJ is a U(2)-irrep
RJ is a Sn-irrep p V
USch
USch V =
RJ(V)
RJ(p) V

13. Applications of the Schur transform
Universal entanglement concentration:
Given |yABi­n, 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(y) EPR pairs.
Universal data compression:
Given r­n, perform the Schur transform, weakly measure J and the resulting state has dimension ¼ exp(nS(r)).
State estimation:
Given r­n, estimate the spectrum of r, or estimate r, or test to see whether the state is s­n.

14. How to perform the Schur transform?
Begin with the Clebsch-Gordon transform.
MJ ­ M½ = MJ+½ © MJ-½ M0
Why can UCG be implemented efficiently?
1. Conditioned on J and M, UCG is two-dimensional.
2. CCG can be efficiently classically computed.

15. Diagrammatic view of CG transform
UCG
|Ji
|Ji MJ
|Mi
|J0i
MJ+½ © MJ-½ M½
|Si
|M0i
UCG
UCG =
RJ(V)
RJ0(V) V

16. Schur transform = iterated CG
(C2)­n
UCG
|½i
|J1i
|i1i
UCG
|J2i
|J2i
|M2i
|i2i
|J3i
|M3i
|i3i
|Jn-1i
UCG
|Jn-1i
|Jni
|Mn-1i
|ini
|Mi

17. Almost there… 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: (follows from Schur duality theorem)
Since U(2)­n acts irreducibly on MJ, then Schur duality implies that

18. But what is PJ? S3
J=½
J=3/2 3 S1 S6
J=3
J=2
J=1
J=0 6 S4
J=2
J=1
J=0 4 S2 S5
J=5/2
J=3/2
J=½ 5
J=1
J=½
J=0 1 2
paths of standard Gelfand-Zetlin basis

19. Generalized phase estimation
|Ji
|0i
|Ji
UQFT
UQFTy
|p1i
|p2i Up
Upy
|i1i
|i2i
|ini

20. Summary
|i1,…,ini!|J,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|Ji
The generalized phase estimation algorithm allows measurement of J in time poly(n) + O(n¢log(d)).
This is useful for many tasks in quantum information theory. Can you find more?