Quantum Circuits for Clebsch-Gordon and Schur duality transformations D. Bacon (Caltech), I. Chuang (MIT) and A. Harrow (MIT)
Outline 1. Motivation 2. Total angular momentum (Schur) basis 3. Schur transform: applications 4. Schur transform: construction 5. Generalized phase estimation
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
Questions When can UCB be implemented efficiently? What use are bases other than the standard basis?
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.
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
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 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.
? Example 3: three qubits S3 U(2)3 spin ½ symmetric (trivial representation) U(2)3 spin 3/2 spin ½
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 P½,½@P½,-½.
Schur decomposition for n qubits Theorem (Schur): For any J and M, PJ,M is an irrep of Sn. Furthermore, PJ,M@PJ,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.
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
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.
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.
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
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
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 PJ@PJ0.
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 irreps @ standard tableaux @ Gelfand-Zetlin basis
Generalized phase estimation |Ji |0i |Ji UQFT UQFTy |p1i |p2i Up Upy |i1i |i2i |ini
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?