The Second Moment of Hafnians in Gaussian Boson Sampling
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909594937196544 |
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| author | Ehrenberg, Adam Iosue, Joseph T. Deshpande, Abhinav Hangleiter, Dominik Gorshkov, Alexey V. |
| author_facet | Ehrenberg, Adam Iosue, Joseph T. Deshpande, Abhinav Hangleiter, Dominik Gorshkov, Alexey V. |
| contents | Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector $2n = 80$. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13878 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Second Moment of Hafnians in Gaussian Boson Sampling Ehrenberg, Adam Iosue, Joseph T. Deshpande, Abhinav Hangleiter, Dominik Gorshkov, Alexey V. Quantum Physics Gaussian Boson Sampling is a popular method for experimental demonstrations of quantum advantage, but many subtleties remain in fully understanding its theoretical underpinnings. An important component in the theoretical arguments for approximate average-case hardness of sampling is anticoncentration, which is a second-moment property of the output probabilities. In Gaussian Boson Sampling these are given by hafnians of generalized circular orthogonal ensemble matrices. In a companion work [arXiv:2312.08433], we develop a graph-theoretic method to study these moments and use it to identify a transition in anticoncentration. In this work, we find a recursive expression for the second moment using these graph-theoretic techniques. While we have not been able to solve this recursion by hand, we are able to solve it numerically exactly, which we do up to Fock sector $2n = 80$. We further derive new analytical results about the second moment. These results allow us to pinpoint the transition in anticoncentration and furthermore yield the expected linear cross-entropy benchmarking score for an ideal (error-free) device. |
| title | The Second Moment of Hafnians in Gaussian Boson Sampling |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2403.13878 |