Conditional t-independent spectral gap for random quantum circuits and implications for t-design depths

• URL: http://arxiv.org/abs/2411.13739v1
• Date: Wed, 20 Nov 2024 22:46:10 GMT
• Title: Conditional t-independent spectral gap for random quantum circuits and implications for t-design depths
• Authors: James Allen, Daniel Belkin, Bryan K. Clark,
• Abstract summary: We establish a new bound on the spectral gap of the t-th moment of a one-dimensional brickwork architecture on N qudits. The improved spectral gaps gives large improvements to the constant factors in known results.
• Score: 0.0
• License:
• Abstract: A fundamental question is understanding the rate at which random quantum circuits converge to the Haar measure. One quantity which is important in establishing this rate is the spectral gap of a random quantum ensemble. In this work we establish a new bound on the spectral gap of the t-th moment of a one-dimensional brickwork architecture on N qudits. This bound is independent of both t and N, provided t does not exceed the qudit dimension q. We also show that the bound is nearly optimal. The improved spectral gaps gives large improvements to the constant factors in known results on the approximate t-design depths of the 1D brickwork, of generic circuit architectures, and of specially-constructed architectures which scramble in depth O(log N). We moreover show that the spectral gap gives the dominant epsilon-dependence of the t-design depth at small epsilon. Our spectral gap bound is obtained by bounding the N-site 1D brickwork architecture by the spectra of 3-site operators. We then exploit a block-triangular hierarchy and a global symmetry in these operators in order to efficiently bound them. The technical methods used are a qualitatively different approach for bounding spectral gaps and and have little in common with previous techniques.

Related papers

• Measurement-induced entanglement and complexity in random constant-depth 2D quantum circuits [0.0]
  We analyse the entanglement structure of states generated by random constant-depth quantum circuits. We prove that macroscopic long-ranged entanglement is generated above some constant critical depth in several natural classes of circuit architectures.
  arXiv  Detail & Related papers  (2024-10-30T17:33:02Z)
• Exact spectral gaps of random one-dimensional quantum circuits [0.3774866290142281]
  spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. We show that one can exactly compute the associated spectral gaps. We verify our results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random and symplectic circuits.
  arXiv  Detail & Related papers  (2024-08-20T21:23:42Z)
• Assessing the query complexity limits of quantum phase estimation using symmetry aware spectral bounds [0.0]
  computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian. We introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms.
  arXiv  Detail & Related papers  (2024-03-07T18:38:49Z)
• Spectral Algorithms on Manifolds through Diffusion [1.7227952883644062]
  We study the convergence performance of spectral algorithms in the Reproducing Kernel Space. We employ integral operator techniques to derive tight convergence upper bounds concerning generalized norms. Our study confirms that the spectral algorithms are practically significant in the broader context of high-dimensional approximation.
  arXiv  Detail & Related papers  (2024-03-06T12:43:53Z)
• Approximate t-designs in generic circuit architectures [0.0]
  Unitary t-designs are distributions on the unitary group whose first t moments appear maximally random. Previous work has established several upper bounds on the depths at which certain random quantum circuit ensembles approximate t-designs. Here we show that these bounds can be extended to any fixed architecture of Haar-random two-site gates.
  arXiv  Detail & Related papers  (2023-10-30T17:49:44Z)
• Quantum Gate Generation in Two-Level Open Quantum Systems by Coherent and Incoherent Photons Found with Gradient Search [77.34726150561087]
  We consider an environment formed by incoherent photons as a resource for controlling open quantum systems via an incoherent control. We exploit a coherent control in the Hamiltonian and an incoherent control in the dissipator which induces the time-dependent decoherence rates.
  arXiv  Detail & Related papers  (2023-02-28T07:36:02Z)
• Improved Quantum Algorithms for Fidelity Estimation [77.34726150561087]
  We develop new and efficient quantum algorithms for fidelity estimation with provable performance guarantees. Our algorithms use advanced quantum linear algebra techniques, such as the quantum singular value transformation. We prove that fidelity estimation to any non-trivial constant additive accuracy is hard in general.
  arXiv  Detail & Related papers  (2022-03-30T02:02:16Z)
• Three-fold way of entanglement dynamics in monitored quantum circuits [68.8204255655161]
  We investigate the measurement-induced entanglement transition in quantum circuits built upon Dyson's three circular ensembles. We obtain insights into the interplay between the local entanglement generation by the gates and the entanglement reduction by the measurements.
  arXiv  Detail & Related papers  (2022-01-28T17:21:15Z)
• Unified Field Theory for Deep and Recurrent Neural Networks [56.735884560668985]
  We present a unified and systematic derivation of the mean-field theory for both recurrent and deep networks. We find that convergence towards the mean-field theory is typically slower for recurrent networks than for deep networks. Our method exposes that Gaussian processes are but the lowest order of a systematic expansion in $1/n$.
  arXiv  Detail & Related papers  (2021-12-10T15:06:11Z)
• Realization of arbitrary doubly-controlled quantum phase gates [62.997667081978825]
  We introduce a high-fidelity gate set inspired by a proposal for near-term quantum advantage in optimization problems. By orchestrating coherent, multi-level control over three transmon qutrits, we synthesize a family of deterministic, continuous-angle quantum phase gates acting in the natural three-qubit computational basis.
  arXiv  Detail & Related papers  (2021-08-03T17:49:09Z)
• Hyperentanglement in structured quantum light [50.591267188664666]
  Entanglement in high-dimensional quantum systems, where one or more degrees of freedom of light are involved, offers increased information capacities and enables new quantum protocols. Here, we demonstrate a functional source of high-dimensional, noise-resilient hyperentangled states encoded in time-frequency and vector-vortex structured modes. We generate highly entangled photon pairs at telecom wavelength that we characterise via two-photon interference and quantum state tomography, achieving near-unity visibilities and fidelities.
  arXiv  Detail & Related papers  (2020-06-02T18:00:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.