量子电路学习输出分布的平均情况复杂度

IF 5.1 2区 物理与天体物理 Q1 PHYSICS, MULTIDISCIPLINARY
Quantum Pub Date : 2025-10-13 DOI:10.22331/q-2025-10-13-1883
Alexander Nietner, Marios Ioannou, Ryan Sweke, Richard Kueng, Jens Eisert, Marcel Hinsche, Jonas Haferkamp
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引用次数: 0

摘要

在这项工作中,我们证明了在统计查询模型中学习砖砌随机量子电路的输出分布是平均情况下的困难。该学习模型作为一种抽象的计算模型被广泛应用于大多数通用学习算法。特别是,对于深度为$n$量子位$d$的砖制随机量子电路,我们展示了三个主要结果:—在超对数电路深度$d=\omega(\log(n))$,任何学习算法都需要超多项式多次查询才能在随机绘制的实例上获得恒定的成功概率。-存在$d=O(n)$,这样任何学习算法都需要$\Omega(2^n)$查询才能在随机抽取的实例上达到$O(2^{-n})$的成功概率。在无限电路深度$d\to\infty$下,任何学习算法都需要$2^{2^{\Omega(n)}}$多次查询才能在随机抽取的实例上达到$2^{-2^{\Omega(n)}}$的成功概率。作为独立兴趣的辅助结果,我们证明了砖砌随机量子电路的输出分布在总变异距离上不断远离任何固定分布,概率为$1-O(2^{-n})$,这证实了Aaronson和Chen猜想的一个变体。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the average-case complexity of learning output distributions of quantum circuits
In this work, we show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms. In particular, for brickwork random quantum circuits on $n$ qubits of depth $d$, we show three main results:
– At super logarithmic circuit depth $d=\omega(\log(n))$, any learning algorithm requires super polynomially many queries to achieve a constant probability of success over the randomly drawn instance.
– There exists a $d=O(n)$, such that any learning algorithm requires $\Omega(2^n)$ queries to achieve a $O(2^{-n})$ probability of success over the randomly drawn instance.
– At infinite circuit depth $d\to\infty$, any learning algorithm requires $2^{2^{\Omega(n)}}$ many queries to achieve a $2^{-2^{\Omega(n)}}$ probability of success over the randomly drawn instance.
As an auxiliary result of independent interest, we show that the output distribution of a brickwork random quantum circuit is constantly far from any fixed distribution in total variation distance with probability $1-O(2^{-n})$, which confirms a variant of a conjecture by Aaronson and Chen.
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来源期刊
Quantum
Quantum Physics and Astronomy-Physics and Astronomy (miscellaneous)
CiteScore
9.20
自引率
10.90%
发文量
241
审稿时长
16 weeks
期刊介绍: Quantum is an open-access peer-reviewed journal for quantum science and related fields. Quantum is non-profit and community-run: an effort by researchers and for researchers to make science more open and publishing more transparent and efficient.
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