量子电路的代价集中与梯度消失的等价性：黎曼表述中的一个初等证明
Equivalence of cost concentration and gradient vanishing for quantum circuits: An elementary proof in the Riemannian formulation
AI总结：
该研究在量子电路优化的黎曼表述中，通过初等推导证明代价函数单门方差与黎曼单门梯度方差严格等价，明确梯度衰减与代价变化相伴，指出贫瘠高原问题无法用无梯度方法规避。
摘要：
量子电路的优化可能会因平均梯度幅值随系统规模增大而衰减受到阻碍。当这种衰减呈指数级时，就被称为贫瘠高原（barren plateau）问题。针对显式的电路参数化（以旋转角度表示），Arrasmith等人于2022年发表在《Quantum Sci. Technol.》第7卷045015页的研究已证明，贫瘠高原等价于代价函数差值的方差呈指数衰减。我们发现，在这类优化问题的（无参数化的）黎曼表述中，该问题尤为简明，并且我们得到了更紧的代价函数方差界。一项初等推导表明，代价函数的单门方差严格等于黎曼单门梯度方差的一半，其中我们按照均匀Haar测度对可变门进行采样。随后，代价函数及其梯度的总方差的上界均由各单门方差之和给出，反过来，总方差也构成单门方差的上界。因此，梯度衰减与代价函数变化是相伴发生的，贫瘠高原问题无法通过用无梯度优化方法替代基于梯度的方法来解决。
英文摘要：
The optimization of quantum circuits can be hampered by a decay of average gradient amplitudes with increasing system size. When the decay is exponential, this is called the barren plateau problem. Considering explicit circuit parametrizations (in terms of rotation angles), it has been shown in Arrasmith et al., Quantum Sci. Technol. 7, 045015 (2022) that barren plateaus are equivalent to an exponential decay of the variance of cost-function differences. We show that the issue is particularly simple in the (parametrization-free) Riemannian formulation of such optimization problems and obtain a tighter bound for the cost-function variance. An elementary derivation shows that the single-gate variance of the cost function is strictly equal to half the variance of the Riemannian single-gate gradient, where we sample variable gates according to the uniform Haar measure. The total variances of the cost function and its gradient are then both bounded from above by the sum of single-gate variances and, conversely, bound single-gate variances from above. So, decays of gradients and cost-function variations go hand in hand, and barren plateau problems cannot be resolved by avoiding gradient-based in favor of gradient-free optimization methods.
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