Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform

• URL: http://arxiv.org/abs/2305.04069v3
• Date: Mon, 12 Feb 2024 04:01:22 GMT
• Title: Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform
• Authors: Adam Wills, Sergii Strelchuk
• Abstract summary: We present a transparent algorithm for implementing the qubit quantum Schur transform. We study the associated Schur states, which consist of qubits coupled via Clebsch-Gordan coefficients. It is shown that Wigner 6-j symbols and SU(N) Clebsch-Gordan coefficients naturally fit our framework.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The quantum Schur transform is a fundamental building block that maps the computational basis to a coupled basis consisting of irreducible representations of the unitary and symmetric groups. Equivalently, it may be regarded as a change of basis from the computational basis to a simultaneous spin eigenbasis of Permutational Quantum Computing (PQC) [Quantum Inf. Comput., 10, 470-497 (2010)]. By adopting the latter perspective, we present a transparent algorithm for implementing the qubit quantum Schur transform which uses $O(\log(n))$ ancillas and can be decomposed into a sequence of $O(n^3\log(n)\log(\frac{n}{\epsilon}))$ Clifford + T gates, where $\epsilon$ is the accuracy of the algorithm in terms of the trace norm. We discuss the necessity for some applications of implementing this operation as a unitary rather than an isometry, as is often presented. By studying the associated Schur states, which consist of qubits coupled via Clebsch-Gordan coefficients, we introduce the notion of generally coupled quantum states. We present six conditions, which in different combinations ensure the efficient preparation of these states on a quantum computer or their classical simulability (in the sense of computational tractability). It is shown that Wigner 6-j symbols and SU(N) Clebsch-Gordan coefficients naturally fit our framework. Finally, we investigate unitary transformations which preserve the class of computationally tractable states.

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