Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916120261296128 |
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| author | Wills, Adam Strelchuk, Sergii |
| author_facet | Wills, Adam Strelchuk, Sergii |
| contents | 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 $O(n^3\log(n)\log(\frac{n}ε))$ Clifford + T gates, where $ε$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_04069 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform Wills, Adam Strelchuk, Sergii Quantum Physics 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 $O(n^3\log(n)\log(\frac{n}ε))$ Clifford + T gates, where $ε$ 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. |
| title | Generalised Coupling and An Elementary Algorithm for the Quantum Schur Transform |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2305.04069 |