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.

中文翻译：

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费米子量子元胞自动机和广义矩阵乘积幺正

我们研究了费米子一维 (1D) 链的矩阵乘积酉算子 (MPU)。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，(ii) 并非所有费米子量子元胞自动机 (QCA) 都可以表示为费米子 MPU。然后我们介绍后者的自然概括，通过允许额外的运算符作用于其辅助空间而获得。我们描述了一类这样的具有局部性的广义 MPU，并表明，在附加惰性辅助费米自由度之前，该系列的任何代表都是费米 QCA，反之亦然。最后，我们证明了广义 MPU 的索引定理，恢复了最近推导出的费米子 QCA 一维分类。