Abstract
Characterizing the quantum complexity of local random quantum circuits is a very deep problem with implications to the seemingly disparate fields of quantum information theory, quantum many-body physics and high energy physics. While our theoretical understanding of these systems has progressed in recent years, numerical approaches for studying these models remains severely limited. In this paper, we discuss a special class of numerically tractable quantum circuits, known as quantum automaton circuits, which may be particularly well suited for this task. These are circuits which preserve the computational basis, yet can produce highly entangled output wave functions. Using ideas from quantum complexity theory, especially those concerning unitary designs, we argue that automaton wave functions have high quantum state complexity. We look at a wide variety of metrics, including measurements of the output bit-string distribution and characterization of the generalized entanglement properties of the quantum state, and find that automaton wave functions closely approximate the behavior of fully Haar random states. In addition to this, we identify the generalized out-of-time ordered 2k-point correlation functions as a particularly useful probe of complexity in automaton circuits. Using these correlators, we are able to numerically study the growth of complexity well beyond the scrambling time for very large systems. As a result, we are able to present evidence of a linear growth of design complexity in local quantum circuits, consistent with conjectures from quantum information theory.
Highlights
Understanding the evolution of a quantum wave functions from a simple initial state to a generic vector in an exponentially large Hilbert space is a notoriously difficult problem in modern theoretical physics
We go beyond this and show that, when acting on initial product states not in the computational basis, automaton circuits produce highly entangled wave functions in which the quantum state complexity grows with circuit depth in the same way as in universal local random circuits
We study in detail the quantum state complexity of wave functions that are the output of local automaton circuits
Summary
Understanding the evolution of a quantum wave functions from a simple initial state to a generic vector in an exponentially large Hilbert space is a notoriously difficult problem in modern theoretical physics. [20], it was realized that the operator entanglement and OTO correlator properties of such circuits appear to give results that are identical to that of a generic chaotic dynamics We go beyond this and show that, when acting on initial product states not in the computational basis, automaton circuits produce highly entangled wave functions in which the quantum state complexity grows with circuit depth in the same way as in universal local random circuits. The generalized k-point OTO correlation functions can describe the growth of quantum state complexity beyond the scrambling time [11] According to this metric, complexity in automaton circuits appears to grow in the same way as in generic Haar random circuits. V we summarize our results and discuss potential applications of this work
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