We investigate notions of complexity of states in continuous many-body quantum systems. We focus on Gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. Our proposal for quantifying state complexity is based on the Fubini-Study metric. It leads to counting the number of applications of each gate (infinitesimal generator) in the transformation, subject to a state-dependent metric. We minimize the defined complexity with respect to momentum-preserving quadratic generators which form $\mathfrak{s}\mathfrak{u}(1,1)$ algebras. On the manifold of Gaussian states generated by these operations, the Fubini-Study metric factorizes into hyperbolic planes with minimal complexity circuits reducing to known geodesics. Despite working with quantum field theories far outside the regime where Einstein gravity duals exist, we find striking similarities between our results and those of holographic complexity proposals.

中文翻译：

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量子场论态复杂性的定义

我们研究了连续多体量子系统中状态复杂性的概念。我们关注于高斯态，其中包括自由量子场理论的基态及其在多尺度纠缠重归一化ansatz的连续版本中遇到的近似值。我们提出的量化状态复杂度的建议基于Fubini-Study度量。它导致对转换中每个门（无穷小生成器）的应用次数进行计数，这取决于状态相关的度量。我们将保持动量的二次生成器的定义复杂度降至最低$\mathfrak{s}\mathfrak{ü}（\mathrm{1个}，\mathrm{1个}）$代数 在这些操作生成的高斯状态的流形上，Fubini-Study度量分解为双曲平面，而复杂度最小的电路减少到已知的测地线。尽管使用的量子场论远远超出了爱因斯坦引力对偶存在的范围，但我们发现我们的结果与全息复杂性提议的结果之间有着惊人的相似之处。