Designs via Free Probability

• URL: http://arxiv.org/abs/2308.06200v3
• Date: Wed, 28 Aug 2024 10:04:08 GMT
• Title: Designs via Free Probability
• Authors: Michele Fava, Jorge Kurchan, Silvia Pappalardi,
• Abstract summary: Unitary Designs have become a vital tool for investigating pseudorandomness since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relation to quantum chaotic evolution and in particular to the Eigenstate Thermalization Hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and $k$-designs through Free Probability theory.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Unitary Designs have become a vital tool for investigating pseudorandomness since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relation to quantum chaotic evolution and in particular to the Eigenstate Thermalization Hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and $k$-designs through Free Probability theory. First, by introducing the more general notion of $k$-freeness, we show that it can be used as an alternative probe to designs. In turn, free probability theory comes with several tools, useful for instance for the calculation of mixed moments or the so-called $k$-fold quantum channels. Our second result is the connection to quantum dynamics. Quantum ergodicity, and correspondingly ETH, apply to a restricted class of physical observables, as already discussed in the literature. In this spirit, we show that unitary evolution with generic Hamiltonians always leads to freeness at sufficiently long times, but only when the operators considered are restricted within the ETH class. Our results provide a direct link between unitary designs, quantum chaos and the Eigenstate Thermalization Hypothesis, and shed new light on the universality of late-time quantum dynamics.

Related papers

• Efficient Quantum Pseudorandomness from Hamiltonian Phase States [41.94295877935867]
  We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem. We show that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We show that our assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives.
  arXiv  Detail & Related papers  (2024-10-10T16:10:10Z)
• Quantum walks as thermalizations, with application to fullerene graphs [0.0]
  We show a quantum walk can be interpreted as an equilibration of a kind investigated in the literature on thermalization in unitarily evolving quantum systems. In particular, quantum walks over fullerene graphs constitute a counter-example to the hypothesis that subsystems equilibrate to the Gibbs state.
  arXiv  Detail & Related papers  (2023-04-04T06:50:48Z)
• Measurement and Probability in Relativistic Quantum Mechanics [0.0]
  This paper provides a relativistic model of measurement, in which the state of the universe is decomposed into decoherent histories of measurements recorded within it. The wave functions that we actually use for such experiments are local reductions of very coarse-grained superpositions of universal eigenstates.
  arXiv  Detail & Related papers  (2022-09-26T04:21:52Z)
• Quantum security and theory of decoherence [0.0]
  We investigate how the standard cryptographic assumption of shielded laboratory, stating that data generated by a secure quantum device remain private unless explicitly published, is disturbed by the einselection mechanism of quantum Darwinism. We derive a trade-off relation between eavesdropper's guessing probability $P_guess$ and the collective decoherence factor $Gamma.
  arXiv  Detail & Related papers  (2022-05-25T17:25:28Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• Probabilistic imaginary-time evolution by using forward and backward real-time evolution with a single ancilla: first-quantized eigensolver of quantum chemistry for ground states [0.0]
  Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. We propose a new approach of PITE which requires only a single ancillary qubit. We discuss the application of our approach to quantum chemistry by focusing on the scaling of computational cost.
  arXiv  Detail & Related papers  (2021-11-24T12:54:27Z)
• Quantum Causal Inference in the Presence of Hidden Common Causes: an Entropic Approach [34.77250498401055]
  We put forth a new theoretical framework for merging quantum information science and causal inference by exploiting entropic principles. We apply our proposed framework to an experimentally relevant scenario of identifying message senders on quantum noisy links. This approach can lay the foundations of identifying originators of malicious activity on future multi-node quantum networks.
  arXiv  Detail & Related papers  (2021-04-24T22:45:50Z)
• Probing the limits of quantum theory with quantum information at subnuclear scales [0.13844779265721088]
  We propose a new theoretical framework of Q-data tests. It recognises the established validity of quantum theory, but allows for more general -- 'post-quantum' -- scenarios in certain physical regimes.
  arXiv  Detail & Related papers  (2021-03-22T16:47:39Z)
• Preparing random states and benchmarking with many-body quantum chaos [48.044162981804526]
  We show how to predict and experimentally observe the emergence of random state ensembles naturally under time-independent Hamiltonian dynamics. The observed random ensembles emerge from projective measurements and are intimately linked to universal correlations built up between subsystems of a larger quantum system. Our work has implications for understanding randomness in quantum dynamics, and enables applications of this concept in a wider context.
  arXiv  Detail & Related papers  (2021-03-05T08:32:43Z)
• From a quantum theory to a classical one [117.44028458220427]
  We present and discuss a formal approach for describing the quantum to classical crossover. The method was originally introduced by L. Yaffe in 1982 for tackling large-$N$ quantum field theories.
  arXiv  Detail & Related papers  (2020-04-01T09:16:38Z)
• Probing the Universality of Topological Defect Formation in a Quantum Annealer: Kibble-Zurek Mechanism and Beyond [46.39654665163597]
  We report on experimental tests of topological defect formation via the one-dimensional transverse-field Ising model. We find that the quantum simulator results can indeed be explained by the KZM for open-system quantum dynamics with phase-flip errors. This implies that the theoretical predictions of the generalized KZM theory, which assumes isolation from the environment, applies beyond its original scope to an open system.
  arXiv  Detail & Related papers  (2020-01-31T02:55:35Z)

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.