Self-Testing Quantum Error Correcting Codes: Analyzing Computational Hardness
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929484466225152 |
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| author | Kuo, En-Jui Hsu, Li-Yi |
| author_facet | Kuo, En-Jui Hsu, Li-Yi |
| contents | We present a generalization of the tilted Bell inequality for quantum [[n,k,d]] error-correcting codes and explicitly utilize the simplest perfect code, the [[5,1,3]] code, the Steane [[7,1,3]] code, and Shor's [[9,1,3]] code, to demonstrate the self-testing property of their respective codespaces. Additionally, we establish a framework for the proof of self-testing, as detailed in \cite{baccari2020device}, which can be generalized to the codespace of CSS stabilizers. Our method provides a self-testing scheme for $\cosθ\lvert \bar{0} \rangle + \sinθ\lvert \bar{1} \rangle$, where $θ\in [0, \fracπ{2}]$, and also discusses its experimental application. We also investigate whether such property can be generalized to qudit and show one no-go theorem.
We then define a computational problem called ISSELFTEST and describe how this problem formulation can be interpreted as a statement that maximal violation of a specific Bell-type inequality can self-test a particular entanglement subspace. We also discuss the computational complexity of ISSELFTEST in comparison to other classical complexity challenges and some related open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01987 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-Testing Quantum Error Correcting Codes: Analyzing Computational Hardness Kuo, En-Jui Hsu, Li-Yi Quantum Physics We present a generalization of the tilted Bell inequality for quantum [[n,k,d]] error-correcting codes and explicitly utilize the simplest perfect code, the [[5,1,3]] code, the Steane [[7,1,3]] code, and Shor's [[9,1,3]] code, to demonstrate the self-testing property of their respective codespaces. Additionally, we establish a framework for the proof of self-testing, as detailed in \cite{baccari2020device}, which can be generalized to the codespace of CSS stabilizers. Our method provides a self-testing scheme for $\cosθ\lvert \bar{0} \rangle + \sinθ\lvert \bar{1} \rangle$, where $θ\in [0, \fracπ{2}]$, and also discusses its experimental application. We also investigate whether such property can be generalized to qudit and show one no-go theorem. We then define a computational problem called ISSELFTEST and describe how this problem formulation can be interpreted as a statement that maximal violation of a specific Bell-type inequality can self-test a particular entanglement subspace. We also discuss the computational complexity of ISSELFTEST in comparison to other classical complexity challenges and some related open problems. |
| title | Self-Testing Quantum Error Correcting Codes: Analyzing Computational Hardness |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2409.01987 |