Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

Haferkamp, Jonas

doi:10.48550/arxiv.2203.16571

Cited by 5 publications

(9 citation statements)

References 23 publications

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“…The purpose of this paper is to show that a wide variety of Brownian quantum many-body systems form good approximate k-designs in time linear in k. This scal-ing of the time-to-design is essentially optimal in its kdependence up to a log k factor [12,18], so our results establish that these quantum chaotic model systems are optimal generators of quantum randomness [26]. Our calculations build on a growing body of work featuring random circuits and Brownian models [20,[27][28][29][30][31][32][33][34][35][36][37][38][39][40], which have been used to establish polynomial complexity growth in specific circuit constructions. Our Brownian spin and fermion models are based on similar technical methods, but the simple mean-field nature of our models allows us to straightforwardly establish linear complexity growth in a large-N limit.…”

Section: Introductionmentioning

confidence: 67%

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Linear Growth of Circuit Complexity from Brownian Dynamics

Jian¹,

Bentsen²,

Swingle³

2022

Preprint

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We calculate the frame potential for Brownian clusters of N spins or fermions with timedependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the kth frame potential comes within of the Haar value after a time of order t ∼ kN +k log k +log −1 . Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ∼ kN . We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable.

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Section: Introductionmentioning

confidence: 67%

“…Before going further, we need to define good approximate k-designs. Given an ensemble E of unitaries acting on H, the k-th moment map ME,k acts on an operator X on H ⊗k as [39] ME,k (X) = E U ∈E U ⊗k X(U † ) ⊗k .…”

Section: Formal Definitions Of K-designsmentioning

confidence: 99%

Linear Growth of Circuit Complexity from Brownian Dynamics

Jian¹,

Bentsen²,

Swingle³

2022

Preprint

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“…Specifically, it is known that if one takes the local dimension to be large (e.g. q ≥ 6k 2 ) the spectral gap is constant and the design depth scales linearly in k. Quite recently, the exponent in Theorem 2 was improved from k 9.5 to k 4+o (1) via an improved RQC spectral gap [55].…”

Section: Designs From Local Quantum Circuitsmentioning

confidence: 99%

“…As the above theorem only holds for moments up to the square root of the dimension, we can use an exponentially small but k-independent spectral gap, proved in Refs. [19,55], to establish that exponentially deep random circuits form high degree designs.…”

Section: Designs From Local Quantum Circuitsmentioning

confidence: 99%

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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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Linear growth of circuit complexity from Brownian dynamics

Jian,

Bentsen,

Swingle

2023

J. High Energ. Phys.

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How rapidly can a many-body quantum system generate randomness? Using path integral methods, we demonstrate that Brownian quantum systems have circuit complexity that grows linearly with time. In particular, we study Brownian clusters of N spins or fermions with time-dependent all-to-all interactions, and calculate the Frame Potential to characterize complexity growth in these models. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using path integral methods. Within this framework it is straightforward to show that the kth Frame Potential comes within ϵ of the Haar value after a time of order t ~ kN + k log k + log ϵ−1. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary k-design after a time of order t ~ kN. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a k-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are analytically tractable and are directly applicable to practical applications calling for unitary k-designs.

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Random quantum circuits are approximate unitary $t$-designs in depth $O\left(nt^{5+o(1)}\right)$

Cited by 5 publications

References 23 publications

Linear Growth of Circuit Complexity from Brownian Dynamics

Linear Growth of Circuit Complexity from Brownian Dynamics

Saturation and recurrence of quantum complexity in random quantum circuits

Linear growth of circuit complexity from Brownian dynamics

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