Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms

The Schur basis on $n$ $d$-dimensional quantum systems is a generalization of the total angular momentum basis that is useful for exploiting symmetry under permutations or collective unitary rotations. We present efficient {size $\mathrm{poly}[n,d,\log (1/ϵ)]$ for accuracy $ϵ$} quantum circuits for the Schur transform, which is the change of basis between the computational and the Schur bases. Our circuits provide explicit efficient methods for solving such diverse problems as estimating the spectrum of a density operator, quantum hypothesis testing, and communicating without a shared reference frame. We thus render tractable a large series of methods for extracting resources from quantum systems and for numerous quantum information protocols.

Figure 1

Quantum circuit implementing ${\mathbf{U}}_{\mathrm{CG}}$ to convert between the $|j,m⟩|s⟩$ and $|j^{\prime },m^{\prime },p⟩$ bases, for the $d=2$ (qubit) case. Following standard conventions [12], time goes from left to right, the $|j⟩$ and $|m⟩$ wires hold multiple qubits, and $|s⟩$ is one qubit. The controlled $X$ operation ${\mathbf{C}}_X$ adds the control to the target qubits, i.e., ${\mathbf{C}}_X|s⟩|m⟩=|s⟩|m+s⟩$. The doubly controlled $R_y({\theta }_{j,m^{\prime }})$ gate implements the rotation given by Eq. (7) using the $j$ and $m^{\prime }$ qubits.

Figure 2

Quantum circuit for the Schur transformation ${\mathbf{U}}_{\mathrm{Sch}}$, transforming between $|i_1{i}_2\cdots i_n⟩$ and $|j,m,p⟩$. The fact that the $|p_1,p_2,\dots ,p_n⟩$ is a full basis is intimately related to Schur duality.