Short Proofs of Linear Growth of Quantum Circuit Complexity

Li, Zhi

doi:10.48550/arxiv.2205.05668

Cited by 1 publication

(1 citation statement)

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“…1 More recently, Refs. [34,35] used algebraicgeometric techniques to prove a long-time linear growth of the complexity of random circuits for qubit systems, but only for the exact complexity, i.e. demanding that the target unitary be implemented exactly by quantum circuits capable of implementing arbitrary two qubit gates.…”

Section: Introductionmentioning

confidence: 99%

Saturation and recurrence of quantum complexity in random quantum circuits

Oszmaniec¹,

Horodecki²,

Hunter-Jones³

2022

Preprint

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Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing-in studying the dynamics of quantum many-body systems and the long-time properties of AdS black holes. In this context Brown and Susskind [1] conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly-exponential times. In this work we prove the saturation and recurrence of the complexity of quantum states and unitaries in a model of chaotic time-evolution based on random quantum circuits, in which a local random unitary transformation is applied to the system at every time step. Importantly, our findings hold for quite general random circuit models, irrespective of the gate set and geometry of qubit interactions. Our results advance an understanding of the long-time behaviour of chaotic quantum systems and could shed light on the physics of black hole interiors. From a technical perspective our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs.

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Section: Introductionmentioning

confidence: 99%