Models of Quantum Complexity Growth

Brandão, Fernando G. S. L.; Chemissany, Wissam; Hunter-Jones, Nicholas; Kueng, Richard; Preskill, John

doi:10.1103/prxquantum.2.030316

Cited by 73 publications

(72 citation statements)

References 58 publications

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“…A number of works have suggested that certain measures of quantum complexity may have a semiclassical bulk dual [5,12,39,40]. Notions of state complexity [41][42][43][44][45][46][47][48][49] or unitary complexity of time evolution [1,[50][51][52][53][54][55][56] have been intriguingly linked to measures of "size" of the bulk universe at the given boundary time, e.g. the volume of the bulk maximal volume slice [12,31,57] anchored at that time, or the gravitational action in the corresponding Wheeler-de Witt diamond [40].…”

Section: K-complexity and The Black Hole Interiormentioning

confidence: 99%

Random matrix theory for complexity growth and black hole interiors

Kar

Lamprou

Rozali

et al. 2022

J. High Energ. Phys.

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We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy — a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.

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Section: K-complexity and The Black Hole Interiormentioning

confidence: 99%

Random matrix theory for complexity growth and black hole interiors

Kar

Lamprou

Rozali

et al. 2022

J. High Energ. Phys.

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show abstract

“…At the same time it is expected that the complexity of the quantum state continues to grow for a time that is exponential in the system size [34]. Indeed, this can be shown rigorously for a strong notion of complexity and dynamics given by random quantum circuits [35], which are often expected to capture the qualitative behaviour of chaotic Hamiltonian timeevolution (other complexity measures are directly connected to the typicality as measured by quantum state k-designs [36]). In other words, we expect that the time-dependent quantum state |Ψ(t) continues to evolve towards a more typical state (in the sense of the Haar measure) for very long time.…”

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confidence: 99%

High-temperature thermalization implies the emergence of quantum state designs

Wilming¹,

Roth²

2022

Preprint

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“…Namely, if the average-case distance between a pair of quantum objects on N qubit systems is large, then they can be (statistically) distinguished almost perfectly using a randomized protocol with just a few implementations of local random circuits of depth O(N ). We observe that such behavior takes place in two scenarios related to those recently analyzed in the context of so-called Quantum Algorithmic Measurement [26] and complexity growth of quantum circuits [7]: (i) distinguishing Haar random N qubit pure state from maximally mixed state and (ii) distinguishing N qubit Haar random unitary from maximally depolarizing channel. Furthermore, we apply our findings to understand the effects of noise on quantum advantage proposals based on random circuit sampling [27,28].…”

mentioning

confidence: 75%

“…Application 2: Strong complexity of quantum states and unitaries. The above two examples also have interesting consequences for the notion of a strong state and unitary complexity investigated in [7]. There, the authors defined complexity C δ of N -qubit pure state ψ (resp.…”

mentioning

confidence: 99%

“…While it is natural to consider the optimal protocols when one wishes to distinguish between two objects, alas, in reality, such protocols might be not practical. For example, in general, they require high-depth, complicated quantum circuits [7]. From a complementary perspective, quantum distances are often used to compare an ideal implementation (of a state, measurement, or channel) with its noisy experimental version.…”

mentioning

confidence: 99%

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Operational Quantum Average-Case Distances

Maciejewski¹,

Puchała²,

Oszmaniec³

2021

Preprint

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We introduce operational distance measures between quantum states, measurements, and channels based on their average-case distinguishability. To this end, we analyze the average Total Variation Distance (TVD) between statistics of quantum protocols in which quantum objects are intertwined with random circuits and subsequently measured in the computational basis. We show that for circuits forming approximate 4-designs, the average TVDs can be approximated by simple explicit functions of the underlying objects, which we call average-case distances. The so-defined distances capture average-case distinguishability via moderate-depth random quantum circuits and satisfy many natural properties. We apply them to analyze effects of noise in quantum advantage experiments and in the context of efficient discrimination of high-dimensional quantum states and channels without quantum memory. Furthermore, based on analytical and numerical examples, we argue that average-case distances are better suited for assessing the quality of NISQ devices than conventional distance measures such as trace distance and the diamond norm.Introduction. In the era of Noisy Intermediate Scale Quantum (NISQ) devices [1], it is instrumental to have figures of merit that quantify how close two quantum protocols are. The distance measures commonly used for this purpose, for example, in the context of quantum error-correction [2], such as trace distance or diamond norm, have an operational interpretation in terms of optimal statistical distinguishability between two quantum states, measurements, or channels [3][4][5][6]. While it is natural to consider the optimal protocols when one wishes to distinguish between two objects, alas, in reality, such protocols might be not practical. For example, in general, they require high-depth, complicated quantum circuits [7]. From a complementary perspective, quantum distances are often used to compare an ideal implementation (of a state, measurement, or channel) with its noisy experimental version. In this context, using the distances based on optimal distinguishability gives information about the worst-case performance of a device in question. This may be impractical as well -it is not expected that the performance of typical experiments on a quantum device will be comparable to the worst-case scenario.

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Models of Quantum Complexity Growth

Cited by 73 publications

References 58 publications

Random matrix theory for complexity growth and black hole interiors

Random matrix theory for complexity growth and black hole interiors

High-temperature thermalization implies the emergence of quantum state designs

Operational Quantum Average-Case Distances

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