Certified algorithms for equilibrium states of local quantum Hamiltonians

• URL: http://arxiv.org/abs/2311.18706v2
• Date: Mon, 25 Nov 2024 14:51:49 GMT
• Title: Certified algorithms for equilibrium states of local quantum Hamiltonians
• Authors: Hamza Fawzi, Omar Fawzi, Samuel O. Scalet,
• Abstract summary: Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. We show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results.
• Score: 10.2138250640885
• License:
• Abstract: Predicting observables in equilibrium states is a central yet notoriously hard question in quantum many-body systems. In the physically relevant thermodynamic limit, certain mathematical formulations of this task have even been shown to result in undecidable problems. Using a finite-size scaling of algorithms devised for finite systems often fails due to the lack of certified convergence bounds for this limit. In this work, we design certified algorithms for computing expectation values of observables in the equilibrium states of local quantum Hamiltonians, both at zero and positive temperature. Importantly, our algorithms output rigorous lower and upper bounds on these values. This allows us to show that expectation values of local observables can be approximated in finite time, contrasting related undecidability results. When the Hamiltonian is commuting on a 2-dimensional lattice, we prove fast convergence of the hierarchy at high temperature and as a result for a desired precision $\varepsilon$, local observables can be approximated by a convex optimization program of quasi-polynomial size in $1/\varepsilon$.

Related papers

• Optimizing random local Hamiltonians by dissipation [44.99833362998488]
  We prove that a simplified quantum Gibbs sampling algorithm achieves a $Omega(frac1k)$-fraction approximation of the optimum. Our results suggest that finding low-energy states for sparsified (quasi)local spin and fermionic models is quantumly easy but classically nontrivial.
  arXiv  Detail & Related papers  (2024-11-04T20:21:16Z)
• 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)
• Robust Extraction of Thermal Observables from State Sampling and Real-Time Dynamics on Quantum Computers [49.1574468325115]
  We introduce a technique that imposes constraints on the density of states, most notably its non-negativity, and show that this way, we can reliably extract Boltzmann weights from noisy time series. Our work enables the implementation of the time-series algorithm on present-day quantum computers to study finite temperature properties of many-body quantum systems.
  arXiv  Detail & Related papers  (2023-05-30T18:00:05Z)
• Sparse random Hamiltonians are quantumly easy [105.6788971265845]
  A candidate application for quantum computers is to simulate the low-temperature properties of quantum systems. This paper shows that, for most random Hamiltonians, the maximally mixed state is a sufficiently good trial state. Phase estimation efficiently prepares states with energy arbitrarily close to the ground energy.
  arXiv  Detail & Related papers  (2023-02-07T10:57:36Z)
• Efficient learning of ground & thermal states within phases of matter [1.1470070927586014]
  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.
  arXiv  Detail & Related papers  (2023-01-30T14:39:51Z)
• Observation of partial and infinite-temperature thermalization induced by repeated measurements on a quantum hardware [62.997667081978825]
  We observe partial and infinite-temperature thermalization on a quantum superconducting processor. We show that the convergence does not tend to a completely mixed (infinite-temperature) state, but to a block-diagonal state in the observable basis.
  arXiv  Detail & Related papers  (2022-11-14T15:18:11Z)
• 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)
• Provably accurate simulation of gauge theories and bosonic systems [2.406160895492247]
  We develop methods for bounding the rate of growth of local quantum numbers. For the Hubbard-Holstein model, we compute a bound on $Lambda$ that achieves accuracy $epsilon$. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution.
  arXiv  Detail & Related papers  (2021-10-13T18:00:02Z)
• 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)
• Bounding the finite-size error of quantum many-body dynamics simulations [6.657101721138396]
  We derive rigorous upper bounds on the Finite-size error (FSE) of local observables in real time quantum dynamics simulations from a product state. Our bounds are practically useful in determining the validity of finite-size results, as we demonstrate in simulations of the one-dimensional (1D) quantum Ising and Fermi-Hubbard models.
  arXiv  Detail & Related papers  (2020-09-25T04:30:25Z)
• The role of boundary conditions in quantum computations of scattering observables [58.720142291102135]
  Quantum computing may offer the opportunity to simulate strongly-interacting field theories, such as quantum chromodynamics, with physical time evolution. As with present-day calculations, quantum computation strategies still require the restriction to a finite system size. We quantify the volume effects for various $1+1$D Minkowski-signature quantities and show that these can be a significant source of systematic uncertainty.
  arXiv  Detail & Related papers  (2020-07-01T17:43:11Z)

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.