The Quantum Schur Transform: I. Efficient Qudit Circuits

Abstract

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.

24 pages

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• Fast-forwarding quantum evolution
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• Quantum Error Correcting Subsystem Codes From Two Classical Linear Codes
• Fast quantum algorithms for approximating some irreducible representations of groups
• Multiparameter estimation for qubit states with collective measurements: a case study
• Permutational Quantum Computing
• Symmetric functions of qubits in an unknown basis
• How a Clebsch-Gordan Transform Helps to Solve the Heisenberg Hidden Subgroup Problem
• Symmetric coupling of four spin-1/2 systems
• A quantum algorithm for the quantum Schur-Weyl transform
• Projecting onto Qubit Irreps of Young Diagrams
• Entanglement distillation using Schur-Weyl decomposition for three qubits
• A new approach to the construction of Schur-Weyl states