概率和量子计算的通用代数描述
A common algebraic description for probabilistic and quantum computations
摘要：
我们研究了问题SFT（无和公式部分迹）的计算复杂性：给定复域（C，+，.）的子半环上的张量公式F加一个正整数k，在所有输入都是L2-范数1和保范数方阵的列向量，输出矩阵是列向量的限制下，判定$F\dagg{F}$的k-部分迹是否优于1/2.矩阵的k-部分迹是其最低k个对角元素之和。我们还考虑了这个问题的承诺版本，其中1/2阈值是一个孤立的分割点。我们展示了如何将量子或可逆门阵列编码成满足上述条件的张量公式，反之亦然;我们使用这个来证明SFT的承诺版本对于半环（Q^+，+，.）上的公式的BPP类是完备的。在正有理数中，对于在域（Q，+，.）上定义的公式的情况下的BQP，而对于P，在布尔半环上定义的公式的情况下，都是在对数空间一致约简下。这表明，概率和量子多项式时间计算机之间的区别可能最终在于，在后一种情况下，并行计算之间存在相消干涉的可能性。
英文摘要：
We study the computational complexity of the problem SFT (Sum-free Formula partial Trace): given a tensor formula F over a subsemiring of the complex field (C,+,.) plus a positive integer k, under the restrictions that all inputs are column vectors of L2-norm 1 and norm-preserving square matrices, and that the output matrix is a column vector, decide whether the k-partial trace of $F\dagg{F}$ is superior to 1/2. The k-partial trace of a matrix is the sum of its lowermost k diagonal elements. We also consider the promise version of this problem, where the 1/2 threshold is an isolated cutpoint. We show how to encode a quantum or reversible gate array into a tensor formula which satisfies the above conditions, and vice-versa; we use this to show that the promise version of SFT is complete for the class BPP for formulas over the semiring (Q^+,+,.) of the positive rational numbers, for BQP in the case of formulas defined over the field (Q,+,.), and for P in the case of formulas defined over the Boolean semiring, all under logspace-uniform reducibility. This suggests that the difference between probabilistic and quantum polynomial-time computers may ultimately lie in the possibility, in the latter case, of having destructive interference between computations occuring in parallel.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
