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arXiv:2610.38707

Quenched Thermodynamics from Ancilla-Assisted Imaginary-Time Evolution

Ibukunoluwa Adisa, Daniele Cuomo

quant-phSubmitted Oct 28, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Quenched thermodynamics combines the equilibrium physics of fixed coupling configurations with a prescribed distribution of disorder. Problems related to quenched thermodynamics are generally intractable to approximate. We provide a quantum algorithm to calculate quenched disorder averages for the transverse-field Ising model, whose coupling configurations are organized into gauge classes. We show how to estimate these averages by preparing approximate Gibbs states through ancilla-assisted imaginary-time evolution. At each measurement, the ancilla record identifies the gauge class to which the energy estimate contributes. We validate our framework through an open $3\times3$ lattice. A one-step circuit resolves all sixteen flux classes, and a finite-shot calculation estimates the quenched energy for independently distributed bond signs. Thermodynamic integration extends the reconstruction to a quenched free-energy estimate, while comparison with exact diagonalization identifies the finite-step approximation as the dominant source of error. This use of the measurement record provides access to the thermodynamics of frustrated spin systems with a disorder distribution chosen independently of the circuit's sampling frequencies.

AI Summary

The paper asks how ancilla-assisted imaginary-time evolution can estimate quenched thermodynamics of a transverse-field Ising model with a prescribed bond-disorder distribution. It retains both outcomes of each ancilla measurement: the record sets the signs of Hamiltonian terms, while products of bond signs around independent cycles identify gauge-equivalent configurations. Class-conditioned energy estimates are then combined with disorder weights chosen in classical postprocessing, rather than with the circuit’s temperature-dependent sampling frequencies. Each elementary conditional map is exactly proportional to $e^{-s\beta c_jP_j/2}$, using an analytically specified interaction angle. A single ordered layer approximates the Gibbs state for noncommuting terms. Numerical validation uses an ideal circuit simulator and exact diagonalization on an open $3\times3$ lattice with nine spins and sixteen flux classes. For independent positive bonds with probability $0.8$, at $\beta=0.75$ the Gibbs, exact-circuit, and finite-shot mean energies are respectively $-7.351629$, $-7.384441$, and $-7.374307\pm0.013585$, using 131,072 shots across two measurement bases. Thermodynamic integration gives a circuit free-energy estimate biased by $+0.176702$ at $\beta=1.4$; quadrature corrections are approximately $2\times10^{-5}$. The stated limitations are finite-step thermal-state error, exponentially many flux classes, and sampling costs when circuit and target distributions overlap poorly. Reusing data for different disorder priors requires sufficient sampling support. Deeper circuits must preserve fixed coupling signs across steps; unconstrained measurement histories do not generally represent static disorder. The free-energy sweep uses exact circuit expectations with projected sampling uncertainties, not finite-shot measurements at every temperature.

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20 pages, 9 figures

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Quenched thermodynamics combines the equilibrium physics of fixed coupling configurations with a prescribed distribution of disorder. Problems related to quenched thermodynamics are generally intractable to approximate. We provide a quantum algorithm to calculate quenched disorder averages for the transverse-field Ising model, whose coupling configurations are organized into gauge classes. We show how to estimate these averages by preparing approximate Gibbs states through ancilla-assisted imaginary-time evolution. At each measurement, the ancilla record identifies the gauge class to which the energy estimate contributes.
We validate our framework through an open 3×33\times33×3 lattice. A one-step circuit resolves all sixteen flux classes, and a finite-shot calculation estimates the quenched energy for independently distributed bond signs. Thermodynamic integration extends the reconstruction to a quenched free-energy estimate, while comparison with exact diagonalization identifies the finite-step approximation as the dominant source of error. This use of the measurement record provides access to the thermodynamics of frustrated spin systems with a disorder distribution chosen independently of the circuit's sampling frequencies.
The paper asks how ancilla-assisted imaginary-time evolution can estimate quenched thermodynamics of a transverse-field Ising model with a prescribed bond-disorder distribution. It retains both outcomes of each ancilla measurement: the record sets the signs of Hamiltonian terms, while products of bond signs around independent cycles identify gauge-equivalent configurations. Class-conditioned energy estimates are then combined with disorder weights chosen in classical postprocessing, rather than with the circuit’s temperature-dependent sampling frequencies.
Each elementary conditional map is exactly proportional to e−sβcjPj/2e^{-s\beta c_jP_j/2}e−sβcj​Pj​/2, using an analytically specified interaction angle. A single ordered layer approximates the Gibbs state for noncommuting terms. Numerical validation uses an ideal circuit simulator and exact diagonalization on an open 3×33\times33×3 lattice with nine spins and sixteen flux classes. For independent positive bonds with probability 0.80.80.8, at β=0.75\beta=0.75β=0.75 the Gibbs, exact-circuit, and finite-shot mean energies are respectively −7.351629-7.351629−7.351629, −7.384441-7.384441−7.384441, and −7.374307±0.013585-7.374307\pm0.013585−7.374307±0.013585, using 131,072 shots across two measurement bases. Thermodynamic integration gives a circuit free-energy estimate biased by +0.176702+0.176702+0.176702 at β=1.4\beta=1.4β=1.4; quadrature corrections are approximately 2×10−52\times10^{-5}2×10−5.
The stated limitations are finite-step thermal-state error, exponentially many flux classes, and sampling costs when circuit and target distributions overlap poorly. Reusing data for different disorder priors requires sufficient sampling support. Deeper circuits must preserve fixed coupling signs across steps; unconstrained measurement histories do not generally represent static disorder. The free-energy sweep uses exact circuit expectations with projected sampling uncertainties, not finite-shot measurements at every temperature.
We introduce a quantum-circuit algorithm for performing imaginary-time evolution on infinite one-dimensional lattice systems. The method uses a parameterized quantum circuit to represent a uniform matrix product state ansatz. We derive the ITE algorithm using the time-dependent variational principle and employ the quantum Lanczos algorithm to improve the ground-state energy estimate. As a benchmark, we simulate the transverse-field Ising model using both classical simulators and IBM Quantum devices. Our analysis includes a statistical study of the distributions of the cost function and energy density obtained from quantum measurements, illustrating the effects of finite-sampling noise on convergence.
Finite-Temperature Dynamical Phase Diagram of the 2+12+12+1D Quantum Ising Model
Mapping finite-temperature dynamical phase diagrams of quantum many-body models is a necessary step towards establishing a framework of far-from-equilibrium quantum many-body universality. However, this is quite difficult due, in part, to the severe challenges in representing the volume-law entanglement that is generated under nonequilibrium dynamics at finite temperatures. Here, we address these challenges with an efficient equilibrium quantum Monte Carlo (QMC) framework for computing the finite-temperature dynamical phase diagram. Our method uses energy conservation and the self-thermalizing properties of ergodic quantum systems to determine observables at late times after a quantum quench. We use this technique to chart the dynamical phase diagram of the 2+12+12+1D quantum Ising model generated by quenches of the transverse field in initial thermal states. Our approach allows us to track the evolution of dynamical phases as a function of both the initial temperature and transverse field. Surprisingly, we identify quenches in the ordered phase that cool the system as well as an interval of initial temperatures where it is possible to quench from the paramagnetic (PM) to ferromagnetic (FM) phases. Our method gives access to dynamical properties without explicitly simulating unitary time evolution, and is immediately applicable to other lattice geometries and interacting many-body systems. Finally, we propose a quantum simulation experiment on state-of-the-art digital quantum hardware to directly probe the predicted dynamical phases and their real-time formation.
In this thesis, we study the properties of the transverse field Ising (TFI) chain in several non-equilibrium setups. One such setup is finite-time quantum quench—quench in which a Hamiltonian parameter is continuously changed during a finite-time, as opposed to the usual instantaneous change in the sudden quenches. A different type of quantum quench, which does not involve changing a Hamiltonian parameter, can be realised by the so-called partitioning protocol. In this protocol, two independent chains are prepared at different temperatures and then coupled into a single system and evolved as such. Another possibility to take the system out of equilibrium is to introduce a driving—for example, by suddenly and continuously changing a Hamiltonian parameter over long times. We discuss the general treatment of each of these situations, and apply them to several quench and driving protocols in the TFI chain. We analyse the behaviour of several observables, such as the total energy, spin correlation functions and Loschmidt echo, during and after quenches and during driving. While we choose the specific protocols to display the dependence of observables on the protocol features, we also investigate the universal behaviours of the observables in the late-time limit. Isolated quenched TFI chains do not relax at late times, however, local observables in these systems do—they relax to a special stationary state described by the generalised Gibbs ensemble (GGE). A similar situation happens in the case of periodic driving of the TFI chain—the late time state of the driven TFI chain, at least as far as the local observables are concerned, it is a periodic steady state described by the periodic Gibbs ensemble (PGE). In this thesis, we observe the relaxation behaviour for various quench and driving protocols and observables. In addition to studying statistical ensembles in the context of relaxation of out-of-equilibrium systems, we describe a machine learning algorithm based on the generalised Gibbs ensemble. We first describe a traditional machine learning algorithm, the Boltzmann machine, and its quantum version. Based on our knowledge of the relaxation of the out-of-equilibrium TFI chain, we modify the existing Boltzmann machine algorithm to include the many GGE charges and test it on the commonly used MNIST dataset. Rather than learning a complicated energy function and using the Boltzmann distribution, we then use the relatively simple TFI Hamiltonian and learn the effective temperatures of the GGE and find reasonably low error rates, while learning a comparatively low number of parameters.
Imperfections in correlated materials can alter their ground state as well as finite-temperature properties in significant ways. Here, we develop a method based on numerical linked-cluster expansions for calculating exact finite-temperature properties of disordered lattice models directly in the thermodynamic limit. We show that a continuous distribution for disordered parameters can be achieved using a set of carefully chosen discrete modes in the distribution, which allows for the averaging of properties over all disorder realizations. We benchmark our results for thermodynamic properties of the square-lattice Ising and quantum Heisenberg models with bond disorder against Monte Carlo simulations and study them as the strength of disorder changes. We also apply the method to the disordered Heisenberg model on the frustrated checkerboard lattice, which is closely connected to Sr2Cu(Te0.5W0.5)O6{\mathrm{Sr}}_{2}\mathrm{Cu}({\mathrm{Te}}_{0.5}{\mathrm{W}}_{0.5}){\mathrm{O}}_{6}Sr2​Cu(Te0.5​W0.5​)O6​. Our method can be used to study finite-temperature properties of other disordered quantum lattice models, including those for interacting lattice fermions.
The transverse field Ising model of N fully connected spins provides a mathematical accessible, yet non-trivial, system to investigate the non-thermal dynamics after a sudden quantum quench. The focus in this thesis is three-fold. First, in a 1/N expansion the dynamics of the mean magnetization within its variance is obtained analytically in the large N limit. The variance constitutes a leading order correction to the mean field limit at infinite N. While mean field theories are one of the most accessible approximations to deal with the quantum complexity, its validity in time out of equilibrium has to be checked on a case by case basis. By studying the dynamics of the variance, we see that the mean field approximation can break down at surprisingly early times scaling with the square root of N. The underlying mechanism in the fully connected Ising model is identified as a dephasing effect between nearby effective orbits. The dynamics of the magnetization variance discriminates four qualitatively different regions in the dynamical phase diagram that are indistinguishable by just looking at the mean magnetization. Second, bipartite entanglement between disjoint groups of the spins in the fully connected Ising model is studied. In the large N limit the entanglement Hamiltonian is shown to be a harmonic oscillator. A quantitative relation between the angular frequency of the oscillator and a squeezing measure of the spin state is derived. Explicit expressions for the Renyi entanglement entropies are given. In contrast to the variance, which vanishes in the thermodynamic limit, the entanglement entropy saturate to an N-independent positive value. The dynamics of the entanglement Hamiltonian and the entanglement entropies is discussed. Linear growth as a function of time is linked to the unstable fixed point and the homoclinic orbit of a pitchfork bifurcation, while logarithmic increase is explained due to a dephasing mechanism of nearby effective orbits. A generalization to other models that permit a semiclassical effective description (on a possibly higher dimensional phase space) is given. As a byproduct the connection between entangled Gaussian states and the symplectic eigenvalues of its reduced covariance is derived. An upper bound on the von Neumann entanglement entropy of any bipartite pure state is given in terms of the symplectic properties of its reduced covariance. Third, the existence of dynamical phase transitions (DPTs) in the fully connected Ising model is argued by two different approaches. DPTs are defined as non-analyticities in the time-dependence of the return probability rate function in the thermodynamic limit. Like entanglement entropy, the behavior of the return rate function is an observable-independent quantity used to characterize non-equilibrium states. A numerically based connection between the DPT and the analytic property of the infinite time averaged order parameter as a function of quench strength is discussed. The concept of Feynman history states is used to devise the notion of history entanglement. An information theoretic interpretation of the history entanglement entropy as the precision of a quantum clock follows from Holevo’s bound. The dependence of this entanglement entropy on the observation time constitutes an operator-independent characterization of the quantum dynamics in closed systems. For observation times comparable to the Heisenberg time, the history entanglement becomes sensitive to the statistics of the energy gaps, and thus distinguishes integrable and non-integrable dynamics. This is demonstrated for two examples, a random matrix ensemble, and a one-dimensional hardcore boson model with integrability breaking next nearest neighbor interaction.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.