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arXiv 2201.11774quant-phmath-phmath.MP
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量子门泛集效率的可计算下界

Calculable lower bounds on the efficiency of universal sets of quantum gates

Oskar Słowik, Adam Sawicki

备注：
22 pages, 2 figures, accepted in J. Phys. A after minor changes
发表信息：
Oskar Słowik and Adam Sawicki 2023 J. Phys. A: Math. Theor. 56 115304
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摘要：

目前可用的量子计算机，所谓的噪声中间尺度量子（NISQ）设备，其特征在于相对低数量的量子比特和适度的门并行度。在这种情况下，量子纠错的实现是不可能的，并且那些设备的性能是相当适度的。特别地，可实现合理高保真度的电路的深度是有限的，并且需要电路深度的最小化。这种深度取决于计算中使用的通用门集合$\mathcal{S}$的效率，并且可以使用Solovay-Kitaev定理来限制。然而，已知对于特定的$\mathcal{S}$，可以获得$\mathcal{O}(\mathrm{log}(ε^{-1}))$形式的更好的渐近紧边界。这些边界由所谓的谱隙控制，表示为$\mathrm{gap}(\mathcal{S})$。然而，对于一般的$\mathcal{S}$，$\mathrm{gap}(\mathcal{S})$的计算是不可能的，并且在实践中，人们考虑以特定尺度$r(ε)$（表示为$\mathrm{gap}_r(\mathcal{S})$）的频谱间隙。这证明是足够的约束效率的$\mathcal{S}$提供了一个感兴趣的物理上可行的情况下，其中错误$ε$是有界的从下面。在本文中，我们推导出的$\mathrm{gap}_r(\mathcal{S})$的下界，并作为一个结果，对$d$ -维量子门$\mathcal{S}$满足一个附加条件的通用集的效率。对于一般的量子门，例如Haar随机门，自然满足该条件。我们的界限是明确的，在这个意义上，所有的参数可以确定的数值计算在现有的计算机上，至少对于小$d$。这与涉及具有模糊值的参数的$\mathrm{gap}_r(\mathcal{S})$上的已知下限形成对比。

英文摘要：

Currently available quantum computers, so called Noisy Intermediate-Scale Quantum (NISQ) devices, are characterized by relatively low number of qubits and moderate gate fidelities. In such scenario, the implementation of quantum error correction is impossible and the performance of those devices is quite modest. In particular, the depth of circuits implementable with reasonably high fidelity is limited, and the minimization of circuit depth is required. Such depths depend on the efficiency of the universal set of gates $\mathcal{S}$ used in computation, and can be bounded using the Solovay-Kitaev theorem. However, it is known that much better, asymptotically tight bounds of the form $\mathcal{O}(\mathrm{log}(ε^{-1}))$, can be obtained for specific $\mathcal{S}$. Those bounds are controlled by so called spectral gap, denoted $\mathrm{gap}(\mathcal{S})$. Yet, the computation of $\mathrm{gap}(\mathcal{S})$ is not possible for general $\mathcal{S}$ and in practice one considers spectral gap at a certain scale $r(ε)$, denoted $\mathrm{gap}_r(\mathcal{S})$. This turns out to be sufficient to bound the efficiency of $\mathcal{S}$ provided that one is interested in a physically feasible case, in which an error $ε$ is bounded from below. In this paper we derive lower bounds on $\mathrm{gap}_r(\mathcal{S})$ and, as a consequence, on the efficiency of universal sets of $d$-dimensional quantum gates $\mathcal{S}$ satisfying an additional condition. The condition is naturally met for generic quantum gates, such as e.g. Haar random gates. Our bounds are explicit in the sense that all parameters can be determined by numerical calculations on existing computers, at least for small $d$. This is in contrast with known lower bounds on $\mathrm{gap}_r(\mathcal{S})$ which involve parameters with ambiguous values.

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