AI 中文总结
本文研究峰值电路生成的优化景观，发现现有优化器衰减率与Aaronson等人的指数关系不符，L-BFGS-B表现更优，深度极限下无多项式参数族可击败2^-n，景观连通但可达范围缩小。
AI 中文摘要
峰值电路是一类随机量子电路，其测量返回某一比特串的概率远高于随机概率，是可验证量子优势的候选实现路径，而瓶颈在于经典生成。Aaronson和Zhang固定一个随机电路，附加一个深度为其一半的可训练砖墙结构，通过梯度下降优化以提升某一选定输出串的概率。他们的方法在与规模相关的上限处陷入平台，将此归因于贫瘠高原。仍存在一个二分问题：要么优化器未充分收敛，存在更优算法可达到更高概率；要么不存在高效方法。本文绘制了决定该问题答案的优化景观。在量子比特数n=8至16的范围内，每个规模测试18个实例，在固定且收敛的预算下，未发现Aaronson和Zhang拟合的固定底数指数关系与优化器可达范围匹配：收敛时，每量子比特的衰减率从n=16时的1.16陡增至1.295（仅基于n=8至14的数据，p值分别为0.011和0.025），使得他们对n=50的估计不成立；在n=16时，他们报告的结果高于我们在所有测量预算下的结果。有一个优化器优于我们的方法：在n=16的3个实例上，L-BFGS-B的结果比收敛的Adam高3.9±1.6%。该差值在其注册规则下推翻了我们的难度猜想，但未改变衰减率：所有测量的优化器每量子比特都会损失1.3倍的因子。贫瘠高原确实存在，但无法解释该衰减率：可达范围远高于Haar下界2^-n，且精确的二阶振幅数据与深度无关，而可达范围与深度相关。解也不聚集在该下界附近：它们不相关但可通过高于2^-n 10^2至10^3倍的路径连接，这些路径的下界从端点的0.73降至0.23。路径搜索是单侧的，因此近优解的聚集问题仍未解决。在深度极限下，我们证明不存在多项式规模参数族平均上能击败多项式规模的2^-n。最终结论为：优化景观是连通的，但可达范围在缩小。
英文摘要
Peaked circuits are random quantum circuits whose measurement returns one bitstring far more often than chance. The probability of that string is the peakedness, and a verifier who knows the string checks the device in a few shots. Peaked circuits are a candidate route to verifiable quantum advantage, and classical generation is the bottleneck. Aaronson and Zhang train a variational circuit by gradient descent, and the peakedness they reach decays exponentially with the number of qubits. They state that either their optimizer stalls or no efficient generation method exists. Here we show that no bare fixed base fits the decay we measure and that neither the barren plateau nor fragmentation of the solution set explains it. We measure the optimization landscape on eighteen instances per size, from eight to sixteen qubits, under a protocol whose falsifiers were fixed in advance. The decay steepens as the qubit number grows, leaving extrapolations to larger devices unsupported. A better optimizer exists, but it gains a few percent and the peakedness it reaches decays at nearly the same rate. Aaronson and Zhang attribute the difficulty to a barren plateau. The plateau is present, but the second-order amplitude data are depth-independent while the attained peakedness is not. Independent runs land on uncorrelated solutions, yet paths connecting them stay well above the scale of a random state, so fragmentation fails at that scale. In the deep limit we take the scrambled state Haar-random. There we prove that no search over a polynomial-parameter Lipschitz family, exhaustive search included, beats the scale of a random state by more than a poly(n) factor on average. Both branches of the Aaronson-Zhang alternative therefore stay open. A generation method must beat a preregistered baseline, and a hardness argument must accommodate a solution set connected at the scale of a random state.
Comments35 pages, 6 figures, 2 tables. Analysis code, logs and data archived on Zenodo (doi:10.5281/zenodo.22121530, doi:10.5281/zenodo.22033295); see the Data and code availability section
