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Nature Communications

Efficient learning of ground and thermal states within phases of matter

Cambyse Rouzé, Daniel Stilck França, Emilio Onorati, James D. Watson

Published Sep 5, 2024• 22 citations
Publisher

Abstract

We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we present sample-efficient ways to learn these properties to high precision. For the first task, we develop techniques to learn parameterisations of classes of systems, including quantum Gibbs states for classes of non-commuting Hamiltonians. We then give methods to sample-efficiently infer expectation values of extensive properties of the state, including quasi-local observables and entropies. For the second task, we exploit the locality of Hamiltonians to show that M local observables can be learned with probability 1 − δ and precision ε using $$N={\mathcal{O}}\left(\log \left(\frac{M}{\delta }\right){e}^{{\rm{polylog}}({\varepsilon }^{-1})}\right)$$ samples — exponentially improving previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponentially decaying correlations. Scalable characterisation of quantum states can be achieved by leveraging on simplifications valid for specific classes of states. Here, the authors show how to combine the strengths of shadows and many-body tomography for all states exhibiting exponential decay of correlations and the approximate Markov property.

DOI
10.1038/s41467-024-51439-x

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We consider two related tasks: (a) estimating a parameterisation of a given Gibbs state and expectation values of Lipschitz observables on this state; (b) learning the expectation values of local observables within a thermal or quantum phase of matter. In both cases, we present sample-efficient ways to learn these properties to high precision. For the first task, we develop techniques to learn parameterisations of classes of systems, including quantum Gibbs states for classes of non-commuting Hamiltonians. We then give methods to sample-efficiently infer expectation values of extensive properties of the state, including quasi-local observables and entropies. For the second task, we exploit the locality of Hamiltonians to show that M local observables can be learned with probability 1 − δ and precision ε using N=O(log⁡(Mδ)epolylog(ε−1))N={\mathcal{O}}\left(\log \left(\frac{M}{\delta }\right){e}^{{\rm{polylog}}({\varepsilon }^{-1})}\right)N=O(log(δM​)epolylog(ε−1)) samples — exponentially improving previous bounds. Our results apply to both families of ground states of Hamiltonians displaying local topological quantum order, and thermal phases of matter with exponentially decaying correlations. Scalable characterisation of quantum states can be achieved by leveraging on simplifications valid for specific classes of states. Here, the authors show how to combine the strengths of shadows and many-body tomography for all states exhibiting exponential decay of correlations and the approximate Markov property.
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 MMM local observables can be learned with probability 1−δ1-δ1−δ to precision εεε with using only N=O(log⁡(Mδ)epolylog(ε−1))N=O\big(\log\big(\frac{M}δ\big)e^{polylog(ε^{-1})}\big)N=O(log(δM​)epolylog(ε−1)) 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.
Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.
The task of learning the Hamiltonian interactions governing a quantum system, given samples of its thermal (or 'Gibbs') states, is a foundational question at the intersection of quantum learning theory and many-body physics. In this paper, we draw connections to the quantum Gibbs sampling literature to introduce a natural learning algorithm we call the stationarity test: which simply "guesses" the Hamiltonian, and measures the rate-of-change of local observables, under the associated detailed-balanced quantum Markov chain [CKG23]. We leverage the stationarity test to address the following applications: 1. To give the first learning algorithm for the underlying interaction graph, i.e. structure learning, of lattice Hamiltonians at all temperatures, given copies of their Gibbs states. 2. To give the first learning algorithm for the coefficients of a lattice Hamiltonian, given only copies of its thermal metastable states, modeled as the "local minima" of the free energy. 3. In addition, we present a refinement to the recent area law for thermal metastable states [BCV25], which holds in the thermodynamic limit. Our learning algorithms are rigorous, time-efficient, and nearly sample-optimal in system size and accuracy. At a technical level, our arguments are based on new approximate locality and convexity properties for the quantum Fisher information of these Gibbs sampling algorithms.
The combination of quantum many-body and machine learning techniques has recently proved to be a fertile ground for new developments in quantum computing. Several works have shown that it is possible to classically efficiently predict the expectation values of local observables on all states within a phase of matter using a machine learning algorithm after learning from data obtained from other states in the same phase. However, existing results are restricted to phases of matter such as ground states of gapped Hamiltonians and Gibbs states that exhibit exponential decay of correlations. In this work, we drop this requirement and show how it is possible to learn local expectation values for all states in a phase, where we adopt the Lindbladian phase definition by Coser \& Pérez-García [Coser \& Pérez-García, Quantum 3, 174 (2019)], which defines states to be in the same phase if we can drive one to other rapidly with a local Lindbladian. This definition encompasses the better-known Hamiltonian definition of phase of matter for gapped ground state phases, and further applies to any family of states connected by short unitary circuits, as well as non-equilibrium phases of matter, and those stable under external dissipative interactions. Under this definition, we show that N=O(log⁡(n/δ)2polylog(1/ε))N = O(\log(n/δ)2^{polylog(1/ε)})N=O(log(n/δ)2polylog(1/ε)) samples suffice to learn local expectation values within a phase for a system with nnn qubits, to error εεε with failure probability δδδ. This sample complexity is comparable to previous results on learning gapped and thermal phases, and it encompasses previous results of this nature in a unified way. Furthermore, we also show that we can learn families of states which go beyond the Lindbladian definition of phase, and we derive bounds on the sample complexity which are dependent on the mixing time between states under a Lindbladian evolution.
Estimating thermal expectation values of observables is a fundamental task in quantum physics, quantum chemistry, and materials science. While recent quantum algorithms have enabled efficient quantum preparation of thermal states, observable estimation via sampling remains costly: a straightforward implementation separates successive measurements by a full mixing time in order to ensure samples are approximately independent. In this work, we show that the sampling cost can be substantially reduced by using a single Gibbs-sampling trajectory. After a single burn-in period, we interleave coherent measurements that satisfy detailed balance with respect to the target Gibbs state. The efficiency of this approach rests on the fact that, in many settings, the autocorrelation time can be significantly shorter than the mixing time. For energy estimation (and more generally for observables commuting with the Hamiltonian), we implement the required measurements using Gaussian-filtered quantum phase estimation with only logarithmic overhead. We also introduce a weighted operator Fourier transform technique to mitigate measurement-induced disturbance for general observables.
We present an efficient protocol for estimating MMM observables to precision ε\varepsilonε from only O(log⁡(M)/ε2)\mathcal{O}(\log (M)/\varepsilon^2)O(log(M)/ε2) copies of a Gibbs state ρ∝exp⁡(−βH)ρ\propto\exp(-βH)ρ∝exp(−βH), given access to its Hamiltonian HHH. The protocol uses single-copy, nonadaptive measurements and a total Hamiltonian simulation time of O~(βM/ε2)\widetilde{\mathcal{O}}(βM/\varepsilon^2)O(βM/ε2); we show that the sample complexity is optimal in a black-box setting. The key idea of our protocol is a new interpretation of quantum Gibbs samplers as detailed-balanced measurement channels: measurements that preserve the Gibbs state when outcomes are marginalized, and thus enable Gibbs samples to be reused for measuring subsequent observables. Consequently, shadow tomography of thermal states admits a general efficient algorithm when controlled Hamiltonian simulation is available, substantially boosting the data acquisition rate in quantum thermal simulation workflows.