Symmetric quantum signal processing provides a parameterized representation of a real polynomial, which can be translated into an efficient quantum circuit for performing a wide range of computational tasks on quantum computers. For a given polynomial, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess $\Phi^0$ that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of $\Phi^0$, on which the cost function is strongly convex under suitable conditions. This explains the aforementioned success of optimization algorithms, and solves the open problem of finding phase factors using only standard double precision arithmetic operations.
翻译:对称量子信号处理提供了一个真正的多元量子信号处理的参数化表示,它可以转换成一个高效的量子电路,用于在量子计算机上完成一系列广泛的计算任务。对于一个特定的多元量子计算机,参数(所谓的阶段因子)可以通过解决优化问题获得。然而,成本函数是非碳化的,并且具有一个非常复杂的能源景观,具有众多的全球和地方微型。因此,令人惊讶的是,从一个固定的初始猜测$\Phi ⁇ 0美元(不含输入多元量子计算机的信息)开始,这个解决方案能够从实践中强有力地获得。为了调查这一现象,我们首先明确确定成本函数的所有全球微型。然后我们证明,一个特定的全球最低值(称为最大溶液)属于$\Phi ⁇ 0美元,其成本函数在适当条件下是强烈的共产值。这就是上述优化算法的成功,并解决仅使用标准的双精度计算操作寻找阶段因子这一公开的问题。