This paper builds on the idea of simulating stabiliser circuits through transformations of {quadratic form expansions}. This is a representation of a quantum state which specifies a formula for the expansion in the standard basis, describing real and imaginary relative phases using a degree-2 polynomial over the integers. We show how, with deft management of the quadratic form expansion representation, we may simulate individual stabiliser operations in $\mathcal{O}(n^2)$ time matching the overall complexity of other simulation techniques [1,2,3]. Our techniques provide economies of scale in the time to simulate simultaneous measurements of all (or nearly all) qubits in the standard basis. Our techniques also allow single-qubit measurements with deterministic outcomes to be simulated in constant time. We also describe throughout how these bounds may be tightened when the expansion of the state in the standard basis has relatively few terms (has low `rank'), or can be specified by sparse matrices. Specifically, this allows us to simulate a `local' stabiliser syndrome measurement in time $\mathcal{O}(n)$, for a stabiliser code subject to Pauli noise --- matching what is possible using techniques developed by Gidney [4] without the need to store which operations have thus far been simulated.

中文翻译：

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使用二次形式展开的快速稳定器仿真

本文建立在通过{二次形式展开}的变换来模拟稳定器电路的想法之上。这是一个量子态的表示，它指定了标准基中的展开公式，使用整数上的 2 次多项式描述实部和虚部的相对相位。我们展示了如何通过对二次形式展开表示的灵巧管理，我们可以在 $\mathcal{O}(n^2)$ 时间内模拟单个稳定器操作，以匹配其他模拟技术的整体复杂性 [1,2,3]。我们的技术在时间上提供了规模经济，以在标准基础上模拟所有（或几乎所有）量子位的同时测量。我们的技术还允许在恒定时间内模拟具有确定性结果的单量子比特测量。当标准基础中的状态扩展具有相对较少的项（具有低“等级”）或可以由稀疏矩阵指定时，我们还自始至终描述了如何收紧这些界限。具体来说，这使我们能够在时间 $\mathcal{O}(n)$ 中模拟“本地”稳定器综合症测量，用于受泡利噪声影响的稳定器代码 --- 匹配使用 Gidney [4] 开发的技术可能实现的结果无需存储迄今为止模拟了哪些操作。