Distributed Quantum Inner Product Estimation with Structured Random Circuits
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908801356005376 |
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| author | Zheng, Congcong Wang, Kun Yu, Xutao Xu, Ping Zhang, Zaichen |
| author_facet | Zheng, Congcong Wang, Kun Yu, Xutao Xu, Ping Zhang, Zaichen |
| contents | Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. Existing rigorous sample complexity analyses are limited to unitary $4$-designs, which pose significant practical challenges for near-term quantum devices. This work addresses this challenge by exploring DIPE with structured random circuits. We first establish that DIPE with an arbitrary unitary $2$-design ensemble achieves an average sample complexity of $\mathcal{O}(\sqrt{2^n})$, where $n$ is the number of qubits. We then analyze ensembles below unitary $2$-designs -- specifically, the brickwork and local unitary $2$-design ensembles -- demonstrating average sample complexities of $\mathcal{O}(\sqrt{2.18^n})$ and $\mathcal{O}(\sqrt{2.5^n})$, respectively. Furthermore, we analyze the state-dependent sample complexity. For brickwork ensembles, we develop a tensor network approach to compute the asymptotic state-dependent sample complexity, showing that it converges to $\mathcal{O}(\sqrt{2.18^n})$ as the circuit depth increases. Remarkably, we find that DIPE with the global Clifford ensemble requires $Θ(\sqrt{2^n})$ copies, matching the performance of unitary $4$-designs. For both local and global Clifford ensembles, we find that the efficiency can be further enhanced by the nonstabilizerness of states. Additionally, for approximate unitary $4$-designs, the performance exponentially approaches that of exact unitary $4$-designs as the circuit depth increases. Our results provide theoretically guaranteed methods for implementing DIPE with experimentally feasible unitary ensembles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19574 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distributed Quantum Inner Product Estimation with Structured Random Circuits Zheng, Congcong Wang, Kun Yu, Xutao Xu, Ping Zhang, Zaichen Quantum Physics Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. Existing rigorous sample complexity analyses are limited to unitary $4$-designs, which pose significant practical challenges for near-term quantum devices. This work addresses this challenge by exploring DIPE with structured random circuits. We first establish that DIPE with an arbitrary unitary $2$-design ensemble achieves an average sample complexity of $\mathcal{O}(\sqrt{2^n})$, where $n$ is the number of qubits. We then analyze ensembles below unitary $2$-designs -- specifically, the brickwork and local unitary $2$-design ensembles -- demonstrating average sample complexities of $\mathcal{O}(\sqrt{2.18^n})$ and $\mathcal{O}(\sqrt{2.5^n})$, respectively. Furthermore, we analyze the state-dependent sample complexity. For brickwork ensembles, we develop a tensor network approach to compute the asymptotic state-dependent sample complexity, showing that it converges to $\mathcal{O}(\sqrt{2.18^n})$ as the circuit depth increases. Remarkably, we find that DIPE with the global Clifford ensemble requires $Θ(\sqrt{2^n})$ copies, matching the performance of unitary $4$-designs. For both local and global Clifford ensembles, we find that the efficiency can be further enhanced by the nonstabilizerness of states. Additionally, for approximate unitary $4$-designs, the performance exponentially approaches that of exact unitary $4$-designs as the circuit depth increases. Our results provide theoretically guaranteed methods for implementing DIPE with experimentally feasible unitary ensembles. |
| title | Distributed Quantum Inner Product Estimation with Structured Random Circuits |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2506.19574 |