费米量子元胞自动机与广义矩阵积么正
Fermionic quantum cellular automata and generalized matrix product unitaries
摘要：
研究一维费米子链的矩阵积酉算子。与一维量子系统的情况形成鲜明对比，我们表明，（i）费米子MPU不一定具有严格的因果锥和（ii）不是所有的费米子量子元胞自动机（QCA）可以表示为费米子MPU。然后，我们介绍了后者的一个自然的推广，通过允许一个额外的运营商作用于他们的辅助空间。我们刻画了一个家庭这样的广义MPU的局部保持，并表明，附加惰性辅助费米自由度，这个家庭的任何代表是一个费米QCA，反之亦然。最后，我们证明了一个指数定理的广义MPU，恢复最近推导的分类费米子QCA在一维。作为我们分析的一个技术工具，我们还介绍了一个费米子矩阵乘积态的分级标准形，证明了它的唯一性相似变换。
英文摘要：
We study matrix product unitary operators (MPUs) for fermionic one-dimensional (1D) chains. In stark contrast with the case of 1D qudit systems, we show that (i) fermionic MPUs do not necessarily feature a strict causal cone and (ii) not all fermionic Quantum Cellular Automata (QCA) can be represented as fermionic MPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic degrees of freedom, any representative of this family is a fermionic QCA and viceversa. Finally, we prove an index theorem for generalized MPUs, recovering the recently derived classification of fermionic QCA in one dimension. As a technical tool for our analysis, we also introduce a graded canonical form for fermionic matrix product states, proving its uniqueness up to similarity transformations.
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