Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909155139256320 |
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| author | Mehraban, Saeed Tahmasbi, Mehrdad |
| author_facet | Mehraban, Saeed Tahmasbi, Mehrdad |
| contents | The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of ${|T\rangle}^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. For the "approximate" rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than $\tilde Ω(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde Ω(n^2)$ for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over $\mathbb{F}_2$, resolving a question in [Wil18]. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with $T$ gates by [LKS18]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_10277 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach Mehraban, Saeed Tahmasbi, Mehrdad Quantum Physics Computational Complexity The approximate stabilizer rank of a quantum state is the minimum number of terms in any approximate decomposition of that state into stabilizer states. Bravyi and Gosset showed that the approximate stabilizer rank of a so-called "magic" state like $|T\rangle^{\otimes n}$, up to polynomial factors, is an upper bound on the number of classical operations required to simulate an arbitrary quantum circuit with Clifford gates and $n$ number of $T$ gates. As a result, an exponential lower bound on this quantity seems inevitable. Despite this intuition, several attempts using various techniques could not lead to a better than a linear lower bound on the "exact" rank of ${|T\rangle}^{\otimes n}$, meaning the minimal size of a decomposition that exactly produces the state. For the "approximate" rank, which is more realistically related to the cost of simulating quantum circuits, no lower bound better than $\tilde Ω(\sqrt n)$ has been known. In this paper, we improve the lower bound on the approximate rank to $\tilde Ω(n^2)$ for a wide range of the approximation parameters. An immediate corollary of our result is the existence of polynomial time computable functions which require a super-linear number of terms in any decomposition into exponentials of quadratic forms over $\mathbb{F}_2$, resolving a question in [Wil18]. Our approach is based on a strong lower bound on the approximate rank of a quantum state sampled from the Haar measure, a step-by-step analysis of the approximate rank of a magic-state teleportation protocol to sample from the Haar measure, and a result about trading Clifford operations with $T$ gates by [LKS18]. |
| title | Quadratic Lower bounds on the Approximate Stabilizer Rank: A Probabilistic Approach |
| topic | Quantum Physics Computational Complexity |
| url | https://arxiv.org/abs/2305.10277 |