Quasi-Adiabatic Processing of Thermal States

TL;DR

This work introduces Quasi-Adiabatic Thermal Evolution (QATE), a finite-time unitary protocol that starts from a Gibbs state and probes thermal properties of the final Hamiltonian without requiring exponential runtimes. The authors define rigorous benchmarks—diagonality in the energy basis, $\\Delta E_{QATE}$, and energy variance—alongside off-diagonal measures COD and BOD to quantify adiabaticity violations and ETH-consistency. Analytic results for the translationally invariant TFIM show polynomial convergence in system size and evolution time, while numerical studies in free-fermion and non-integrable models reveal robust, similar scaling behaviors and practical insights, such as the detrimental effect of initial-state degeneracy and the relative insensitivity to crossing zero-temperature phase transitions. The findings indicate that, with appropriate initial states and ramp designs, QATE can recover thermal observables and offers a viable near-term approach for thermal-state preparation and quantum simulations at finite temperature, aided by isospectral constructions and Gaussian DOS insights. Overall, the work advances a concrete, scalable framework for finite-temperature adiabatic processing aligned with ETH expectations and quantum simulation capabilities.

Abstract

We investigate the performance of an adiabatic evolution protocol when initialized from a Gibbs state at finite temperature. Specifically, we identify the diagonality of the final state in the energy eigenbasis, as well as the difference in energy and in energy variance with respect to the ideal adiabatic limit as key benchmarks for success and introduce metrics to quantify the off-diagonal contributions. Provided these benchmarks converge to their ideal adiabatic values, we argue that thermal expectation values of observables can be recovered, in accordance with the eigenstate thermalization hypothesis. For the transverse-field Ising model, we analytically establish that these benchmarks converge polynomially in both the quasi-adiabatic evolution time $T$ and system size. We perform numerical studies on non-integrable systems and find close quantitative agreement for the off-diagonality metrics, along with qualitatively similar behavior in the energy convergence.

Figures (12)

• Figure 1: Graphical illustration of the QATE. The Gibbs state minimizes the energy at fixed von Neumann entropy $S[\rho] = -\Tr(\rho \ln(\rho))$, so the curve of initial states (red) lies above that of final states (blue). The adiabatic evolution is unitary and conserves the von Neumann entropy, corresponding to vertical motion in the diagram. This corresponds to isentropic cooling. While $E_{\mathrm{G}}$ represents the minimal energy at a given entropy, the preservation of the initial spectrum constrains the minimum attainable energy to $E_{\mathrm{min}}$, given by state $\rho_{\mathrm{min}}$. Running the protocol for a finite time will prepare $\rho_{\mathrm{QATE}}$ with energy $E_{\mathrm{QATE}}\geq E_{\mathrm{min}}$. We denote $\rho_{\mathrm{E}}$ and $\rho_{\mathrm{G}}$ as the Gibbs states at the same energy and entropy as $\rho_{\mathrm{QATE}}$, respectively.
• Figure 2: QATE for TFIM, evolving from $g = 1.1 \to 1.5$ at $\beta = 1$. (a) COD dependence on the evolution time $T$ for various system sizes $N$ along with a $1/T^2$ trendline (black dashed line). Inset: COD scaling with system size $N$, fitted linearly (black dashed line). Square markers in the inset correspond to the square markers in the main plot. (b) Dependence of the energy error normalized by the system size $N$, $\Delta E_{\mathrm{QATE}}/N$, on $T$ for various $N$ along with a $1/T^2$ trendline (black dashed line). Inset: $\Delta E_{\mathrm{min}}$ for different system sizes, fitted linearly (black dashed line). Square markers in the inset correspond to square markers in the main plot. (c) BOD dependence on the eigenstate energy difference for various $T$. Results are shown for $N = 1000$ and filter width $\delta = 0.04$. Inset: BOD scaling with $T$, along with $1/T^2$ (black long-dash) and $1/T^4$ (black short-dash) trendlines. Square and circle markers in the inset correspond to the respective markers in the main plot.
• Figure 3: QATE from the isospectral $H_{\mathrm{Z}}$ to $H_{\mathrm{TFI}}$ with $J = 1$, $g = 1.5$, and $\beta = 1$. (a) COD dependence on the evolution time $T$ for various system sizes $N$. Blue dashed line displays a polynomial fit for $N=128$ and $T\geq250$. Inset: system size scaling fitted at selected points ($T=500$, square markers in the inset correspond to the square markers in the main plot), with a polynomial fit (black dashed line). (b) $\Delta E_{\mathrm{QATE}} /N$ dependence on $T$. Blue dashed line displays a polynomial fit for $N=128$ and $T\geq250$. Inset: system size scaling fitted at selected points ($T=250$, square markers in the inset correspond to the square markers in the main plot), with a polynomial fit (black dashed line). (c) BOD dependence on the energy eigenvalue difference $\abs{E_i - E_j}$. Results are shown for $N = 256$ and filter width $\delta = 0.01$. Inset: polynomial fit of selected points at the tails (circle markers in the inset correspond to the circle markers in the main plot).
• Figure 4: Ising model with mixed field evolved from $(J, h, g) = (1, 0.0, 1.05)\xrightarrow{}(1,0.5,1.05)$ at $\beta=1$. (a) Dependence of COD on the evolution time $T$ for various system sizes $N$. Inset: $\Delta E_{\mathrm{min}}$ for different system sizes, along with a linear fit. (b) Dependence of energy error normalized with the system size $\Delta E_{\mathrm{QATE}}/N$ on $T$ for various system sizes $N$. Inset: Dependence of the relative variance of the $\rho_{\mathrm{QATE}}$ with respect to $\rho_{\mathrm{min}}$ on $T$. (c) BOD dependence on the energy eigenvalue difference $\abs{E_i - E_j}$ for various $T$. Inset: polynomial fit of selected points (black dashed line) at the tails (circle markers in the inset correspond to the circle markers in the main text).
• Figure 5: Ising model with mixed fields evolved from $(J, h, g) = (1, 0.0, 1.05)\xrightarrow{}(1, 0.5, 1.05)$ at $\beta = 1$. Dependence of the 1-norm distance between the reduced density matrices $\rho_{\mathrm{QATE}}$ and $\rho_{\mathrm{min}}$ on the three central sites as a function of QATE time $T$ along with a $T^{-1}$ trendline (black dashed line). The inset shows the 1-norm between $\rho_{\mathrm{min}}$ and the Gibbs state at the same entropy, $\rho_{\mathrm{G}}$ (circles), and at the same energy, $\rho_{\mathrm{E}}$ (squares), as a function of system size $N$. (See \ref{['fig:approach']} for a visualization of $\rho_{\mathrm{G}}$ and $\rho_{\mathrm{E}}$).