aiqbench analysis
This research presents efficient algorithms for anti-symmetrizing unknown mixed quantum states, offering practical advantages in streaming settings and reducing memory requirements through simple operations like controlled-SWAPs. The proposed methods serve as foundational building blocks for advanced protocols including weak Schur sampling, unitary Schur sampling, and optimal purity amplification, outperforming existing approaches in specific regimes of copy number and local dimension.
Anti-symmetrization is cheap and useful
arXiv:2610.06737v1 Announce Type: new Abstract: We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-$r$ qudit state $\rho=\sum_{i=1}^r \lambda_i |\varphi_i\rangle\!\langle\varphi_i|$, prepare a copy of the state proportional to $\Pi_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)$, with $\Pi_{\mathrm{anti}}^r$ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses $\widetilde{O}(r / \lambda_{\min})$ copies of $\rho$, where $\lambda_{\min}$ is the minimum non-zero eigenvalue of $\rho$. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses $O(r / \lambda_{\min})$ copies of $\rho$, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies $n$ and local dimension $d$, as well as in an algorithm for optimal purity amplification for qudit states.
Abstract:We consider the computational task of anti-symmetrization of an unknown mixed quantum state: Given iid copies of a rank-$r$ qudit state $\rho=\sum_{i=1}^r \lambda_i |\varphi_i\rangle\!\langle\varphi_i|$, prepare a copy of the state proportional to $\Pi_{\mathrm{anti}}^r (|\varphi_1\rangle\otimes \ldots\otimes|\varphi_r\rangle)$, with $\Pi_{\mathrm{anti}}^r$ the anti-symmetric subspace projector. We first give a simple algorithm for anti-symmetrization that uses $\widetilde{O}(r / \lambda_{\min})$ copies of $\rho$, where $\lambda_{\min}$ is the minimum non-zero eigenvalue of $\rho$. This algorithm works even in the streaming setting with small working memory, and it uses only simple quantum operations like controlled-SWAPs and constantly many layers of single-qubit gates. As an application of this streaming anti-symmetrization procedure, we give a reduction from gapped mixed-state one-way state generators to pure-state one-way state generators. Then, using tools from Schur-Weyl duality, we give and analyze a second (non-streaming) anti-symmetrization procedure that uses $O(r / \lambda_{\min})$ copies of $\rho$, which then serves as a building block in new algorithms for weak Schur sampling and unitary Schur sampling, outperforming existing approaches in some regimes of number of copies $n$ and local dimension $d$, as well as in an algorithm for optimal purity amplification for qudit states.
arXiv:2610.06737 [quant-ph]
(or arXiv:2610.06737v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.06737 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 277 words
