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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.18420 |
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| _version_ | 1866918331279212544 |
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| author | Chen, Yiyuan Helsen, Jonas Ozols, Maris |
| author_facet | Chen, Yiyuan Helsen, Jonas Ozols, Maris |
| contents | The Sachdev--Ye--Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie--Trotter--Suzuki formulas. Building on recent results by Chen and Brandão (arXiv:2111.05324), we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models. For the $k$-local SYK model on $n$ Majorana fermions, at time $t$, our gate complexity estimates for the first-order Lie--Trotter--Suzuki formula scales with $\tilde{\mathcal{O}}(n^{k+\frac{5}{2}}t^2)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+3}t^2)$ for odd $k$, and the gate complexity of simulations using higher-order formulas scales with $\tilde{\mathcal{O}}(n^{k+\frac{1}{2}}t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+1}t)$ for odd $k$. Given that the SYK model has $Θ(n^k)$ terms, these estimates are close to optimal. These gate complexities can be further improved upon in the context of simulating the time evolution of an arbitrary fixed input state $|ψ\rangle$, leading to a $\mathcal{O}(n^2)$-reduction in gate complexity for first-order and $\mathcal{O}(\sqrt{n})$-reduction for higher-order formulas. Regarding the sparse SYK model, which is a simplified variant of the SYK model obtained by deleting all but a $Θ(n)$ fraction of the terms in a uniformly i.i.d. manner, our average gate complexity estimates for higher-order formulas scale as $\tilde{\mathcal{O}}(n^{1+\frac{1}{2}} t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{2} t)$ for odd $k$. Similar to the full SYK model, we obtain a $\mathcal{O}(\sqrt{n})$-reduction simulating the time evolution of an arbitrary fixed input state $|ψ\rangle$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18420 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trotter error and gate complexity of the SYK and sparse SYK models Chen, Yiyuan Helsen, Jonas Ozols, Maris Quantum Physics Mathematical Physics The Sachdev--Ye--Kitaev (SYK) model is a prominent model of strongly interacting fermions that serves as a toy model of quantum gravity and black hole physics. In this work, we study the Trotter error and gate complexity of the quantum simulation of the SYK model using Lie--Trotter--Suzuki formulas. Building on recent results by Chen and Brandão (arXiv:2111.05324), we derive bounds on the first- and higher-order Trotter error of the SYK model, and subsequently find near-optimal gate complexities for simulating these models. For the $k$-local SYK model on $n$ Majorana fermions, at time $t$, our gate complexity estimates for the first-order Lie--Trotter--Suzuki formula scales with $\tilde{\mathcal{O}}(n^{k+\frac{5}{2}}t^2)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+3}t^2)$ for odd $k$, and the gate complexity of simulations using higher-order formulas scales with $\tilde{\mathcal{O}}(n^{k+\frac{1}{2}}t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{k+1}t)$ for odd $k$. Given that the SYK model has $Θ(n^k)$ terms, these estimates are close to optimal. These gate complexities can be further improved upon in the context of simulating the time evolution of an arbitrary fixed input state $|ψ\rangle$, leading to a $\mathcal{O}(n^2)$-reduction in gate complexity for first-order and $\mathcal{O}(\sqrt{n})$-reduction for higher-order formulas. Regarding the sparse SYK model, which is a simplified variant of the SYK model obtained by deleting all but a $Θ(n)$ fraction of the terms in a uniformly i.i.d. manner, our average gate complexity estimates for higher-order formulas scale as $\tilde{\mathcal{O}}(n^{1+\frac{1}{2}} t)$ for even $k$ and $\tilde{\mathcal{O}}(n^{2} t)$ for odd $k$. Similar to the full SYK model, we obtain a $\mathcal{O}(\sqrt{n})$-reduction simulating the time evolution of an arbitrary fixed input state $|ψ\rangle$. |
| title | Trotter error and gate complexity of the SYK and sparse SYK models |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2502.18420 |