The Quantum Esscher Transform
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866910907689336832 |
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| author | Qiu, Yixian Koor, Kelvin Rebentrost, Patrick |
| author_facet | Qiu, Yixian Koor, Kelvin Rebentrost, Patrick |
| contents | The Esscher Transform is a tool of broad utility in various domains of applied probability. It provides the solution to a constrained minimum relative entropy optimization problem. In this work, we study the generalization of the Esscher Transform to the quantum setting. We examine a relative entropy minimization problem for a quantum density operator, potentially of wide relevance in quantum information theory. The resulting solution form motivates us to define the \textit{quantum} Esscher Transform, which subsumes the classical Esscher Transform as a special case. Envisioning potential applications of the quantum Esscher Transform, we also discuss its implementation on fault-tolerant quantum computers. Our algorithm is based on the modern techniques of block-encoding and quantum singular value transformation (QSVT). We show that given block-encoded inputs, our algorithm outputs a subnormalized block-encoding of the quantum Esscher transform within accuracy $ε$ in $\tilde O(κd \log^2 1/ε)$ queries to the inputs, where $κ$ is the condition number of the input density operator and $d$ is the number of constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07561 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Quantum Esscher Transform Qiu, Yixian Koor, Kelvin Rebentrost, Patrick Quantum Physics The Esscher Transform is a tool of broad utility in various domains of applied probability. It provides the solution to a constrained minimum relative entropy optimization problem. In this work, we study the generalization of the Esscher Transform to the quantum setting. We examine a relative entropy minimization problem for a quantum density operator, potentially of wide relevance in quantum information theory. The resulting solution form motivates us to define the \textit{quantum} Esscher Transform, which subsumes the classical Esscher Transform as a special case. Envisioning potential applications of the quantum Esscher Transform, we also discuss its implementation on fault-tolerant quantum computers. Our algorithm is based on the modern techniques of block-encoding and quantum singular value transformation (QSVT). We show that given block-encoded inputs, our algorithm outputs a subnormalized block-encoding of the quantum Esscher transform within accuracy $ε$ in $\tilde O(κd \log^2 1/ε)$ queries to the inputs, where $κ$ is the condition number of the input density operator and $d$ is the number of constraints. |
| title | The Quantum Esscher Transform |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2401.07561 |