复杂度几何的多项式等价性
Polynomial Equivalence of Complexity Geometries
AI总结：
本文证明了一类广泛的量子复杂度几何度量与量子电路多项式等价，并将等价类扩展到曲率适中的度量，从而为用微分几何证明复杂度下界提供更可行的起点。
摘要：
本文证明了一类广泛的量子计算复杂度定义之间的多项式等价性。我们研究酉群上的右不变度量——通常按照 Nielsen 提出的量子复杂度定义称为“复杂度几何”——并刻画了与量子电路具有相同计算能力的度量的等价类。在这个普适类中，在一种度量下可以到达的任意酉算子，都可以在该类中的任何其他度量下被近似，其减速至多是长度和量子比特数的多项式级别，并且允许误差为逆多项式级别。我们描述了在两种可容忍的不同误差类型下的等价类：Killing 距离误差和算子范数误差。两个等价类中的所有度量都被证明具有指数直径；算子范数等价类中的所有度量也被证明给出了量子复杂度类 BQP 的另一种定义。我的结果扩展了 Nielsen 等人的结果，他们在 2006 年证明了某一特定度量与量子电路多项式等价。Nielsen 等人的度量具有极高的曲率。我证明了本文建立的大大扩展的等价类中也包含曲率适中的度量。我认为适中的曲率使这些度量更易于使用微分几何工具，因此使它们成为 Nielsen 计划中更有希望的起点，该计划旨在利用微分几何证明复杂度下界。
英文摘要：
This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group -- often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen -- and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds.
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