Spectral properties and coding transitions of Haar-random quantum codes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917544779055104 |
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| author | Sommers, Grace M. Jacoby, J. Alexander Weinstein, Zack Huse, David A. Gopalakrishnan, Sarang |
| author_facet | Sommers, Grace M. Jacoby, J. Alexander Weinstein, Zack Huse, David A. Gopalakrishnan, Sarang |
| contents | A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random $\textit{stabilizer}$ codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher $\textit{detection}$ threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral properties and coding transitions of Haar-random quantum codes Sommers, Grace M. Jacoby, J. Alexander Weinstein, Zack Huse, David A. Gopalakrishnan, Sarang Quantum Physics Statistical Mechanics A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random $\textit{stabilizer}$ codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher $\textit{detection}$ threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound. |
| title | Spectral properties and coding transitions of Haar-random quantum codes |
| topic | Quantum Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2510.07396 |