Efficient learning of ground & thermal states within phases of matter

• URL: http://arxiv.org/abs/2301.12946v2
• Date: Fri, 5 May 2023 20:15:16 GMT
• Title: Efficient learning of ground & thermal states within phases of matter
• Authors: Emilio Onorati, Cambyse Rouz\'e, Daniel Stilck Fran\c{c}a, James D. Watson
• Abstract summary: We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter.
• Score: 1.1470070927586014
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; and (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we wish to minimise the number of samples we use to learn these properties to a given precision. For the first task, we develop new techniques to learn parameterisations of classes of systems, including quantum Gibbs states of non-commuting Hamiltonians with exponential decay of correlations and the approximate Markov property. We show it is possible to infer the expectation values of all extensive properties of the state from a number of copies that not only scales polylogarithmically with the system size, but polynomially in the observable's locality -- an exponential improvement. This set of properties includes expected values of quasi-local observables and entropies. For the second task, we develop efficient algorithms for learning observables in a phase of matter of a quantum system. By exploiting the locality of the Hamiltonian, we show that $M$ local observables can be learned with probability $1-\delta$ to precision $\epsilon$ with using only $N=O\big(\log\big(\frac{M}{\delta}\big)e^{polylog(\epsilon^{-1})}\big)$ samples -- an exponential improvement on the precision over previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponential decay of correlations. In addition, our sample complexity applies to the worse case setting whereas previous results only applied on average. Furthermore, we develop tools of independent interest, such as robust shadow tomography algorithms, Gibbs approximations to ground states, and generalisations of transportation cost inequalities for Gibbs states.

Related papers

• Predicting adaptively chosen observables in quantum systems [1.1562071835482224]
  This work investigates the adaptive setting for three classes of observables: local, Pauli, and bounded-Frobenius-norm observables. We prove that $Omega(sqrtM)$ samples of an arbitrarily large unknown quantum state are necessary to predict expectation values of $M$ adaptively chosen local and Pauli observables.
  arXiv  Detail & Related papers  (2024-10-20T20:45:10Z)
• Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems [77.88054335119074]
  We use FNOs to model the evolution of random quantum spin systems. We apply FNOs to a compact set of Hamiltonian observables instead of the entire $2n$ quantum wavefunction.
  arXiv  Detail & Related papers  (2024-09-05T07:18:09Z)
• Predicting Ground State Properties: Constant Sample Complexity and Deep Learning Algorithms [48.869199703062606]
  A fundamental problem in quantum many-body physics is that of finding ground states of local Hamiltonians. We introduce two approaches that achieve a constant sample complexity, independent of system size $n$, for learning ground state properties.
  arXiv  Detail & Related papers  (2024-05-28T18:00:32Z)
• Efficient Learning of Long-Range and Equivariant Quantum Systems [9.427635404752936]
  We consider a fundamental task in quantum many-body physics - finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. We extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems.
  arXiv  Detail & Related papers  (2023-12-28T13:42:59Z)
• Estimating properties of a quantum state by importance-sampled operator shadows [1.3854792306663213]
  We provide a simple method for estimating the expectation value of observables with an unknown quantum state. The time complexity to construct the data structure is $2O(k)$ for $k$-local observables, similar to the post-processing time of classical shadows.
  arXiv  Detail & Related papers  (2023-05-16T11:56:32Z)
• Exponentially improved efficient machine learning for quantum many-body states with provable guarantees [0.0]
  We provide theoretical guarantees for efficient learning of quantum many-body states and their properties, with model-independent applications. Our results provide theoretical guarantees for efficient learning of quantum many-body states and their properties, with model-independent applications.
  arXiv  Detail & Related papers  (2023-04-10T02:22:36Z)
• Neural network enhanced measurement efficiency for molecular groundstates [63.36515347329037]
  We adapt common neural network models to learn complex groundstate wavefunctions for several molecular qubit Hamiltonians. We find that using a neural network model provides a robust improvement over using single-copy measurement outcomes alone to reconstruct observables.
  arXiv  Detail & Related papers  (2022-06-30T17:45:05Z)
• Probing finite-temperature observables in quantum simulators of spin systems with short-time dynamics [62.997667081978825]
  We show how finite-temperature observables can be obtained with an algorithm motivated from the Jarzynski equality. We show that a finite temperature phase transition in the long-range transverse field Ising model can be characterized in trapped ion quantum simulators.
  arXiv  Detail & Related papers  (2022-06-03T18:00:02Z)
• Average-case Speedup for Product Formulas [69.68937033275746]
  Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states. Our results open doors to the study of quantum algorithms in the average case.
  arXiv  Detail & Related papers  (2021-11-09T18:49:48Z)
• Learning quantum many-body systems from a few copies [1.5229257192293197]
  Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. We identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols.
  arXiv  Detail & Related papers  (2021-07-07T16:21:51Z)
• Gaussian Process States: A data-driven representation of quantum many-body physics [59.7232780552418]
  We present a novel, non-parametric form for compactly representing entangled many-body quantum states. The state is found to be highly compact, systematically improvable and efficient to sample. It is also proven to be a universal approximator' for quantum states, able to capture any entangled many-body state with increasing data set size.
  arXiv  Detail & Related papers  (2020-02-27T15:54:44Z)

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.