Algorithmic Quantum Simulations of Quantum Thermodynamics

TL;DR

Algorithmic protocols for simulating quantum thermodynamics on quantum hardware through quantum kernel function expansion (QKFE) produce the free energy as an analytic function of temperature with uniform convergence, providing a general framework for computing thermodynamic potentials on programmable quantum devices.

Abstract

Characterizing quantum phases-of-matter at finite-temperature is essential for understanding complex materials and large-scale thermodynamic phenomena. Here, we develop algorithmic protocols for simulating quantum thermodynamics on quantum hardware through quantum kernel function expansion (QKFE), producing the free energy as an analytic function of temperature with uniform convergence. These protocols are demonstrated by simulating transverse field Ising and XY models with superconducting qubits. In both analogue and digital implementations of the QKFE algorithms, we exhibit quantitative agreement of our quantum simulation experiments with the exact results. Our approach provides a general framework for computing thermodynamic potentials on programmable quantum devices, granting access to key thermodynamic properties such as entropy, heat capacity and criticality, with far-reaching implications for material design and drug development.

Figures (3)

• Figure 1: Schematic of the Quantum Kernel Function Expansion (QKFE) algorithm for simulating quantum thermodynamics. (a) The QKFE converts real-time (t) quantum dynamics into finite-temperature (T) quantum thermodynamics by computing the free energy $F(T)$. Higher order quantities such as the entropy $S(T)$ and the heat capacity $C(T)$ are subsequently obtained via analytic differentiation. (b) DOS for the 1D 10-qubit TFIM: comparing the QKFE approximation (red line) with exact diagonalization (ED, histogram). (c) The extensive scaling of the heat capacity in the 2D TFIM obtained by implementing the QKFE algorithm on the quantum processor. The shaded areas represent the error bars corresponding to one standard deviation. Notably, for the $2\times 2$ and $2 \times 3$ cases, the error bars are smaller than the linewidth at this scale, a result of extensive averaging over samples and quantum shots.
• Figure 2: Digital quantum simulation of TFIM in one- and two-dimensions. The left three panels represent the 1D chain comprising ten qubits with periodic boundary condition, and the right panels are for the two-dimensional $3 \times 4$ lattice. The moments ($f_{n,c}$) in (a) and (d) correspond to the doubled system in the virtual-copy compiling protocol. The raw and error-mitigated data are compared to the theoretical simulation results (Exact). The deviation of the error-mitigated results from exact values is barely noticeable. (b) and (e), the thermodynamic quantities extracted from the measured moments, for one- and two-dimensional TFIMs, respectively. (c) The observed symmetry between the paramagnetic phase ($g>J$) and ferromagnetic phase ($g<J$) illustrates the experimental observation of the duality in 1D TFIM. (f) The heat capacity for the 2D TFIM. It shows a sizable peak at finite temperature in agreement with the exact results, a signature of the expected phase transition at finite temperature. The sampling errors ($\Delta(\cdot)$) of moments are suppressed down to about $10^{-2}$, by averaging over a sufficiently large number of quantum shots and random unitaries. We choose $g=J$, and $g=2J$ for the one- and two-dimensional models, respectively. The Ising coupling $J$ is set as an energy unit.
• Figure 3: Hybrid digital-analogue quantum simulation of spin XY model. (a) the expansion moments for the free energy. The theoretical (Exact), raw experimental data, and error-mitigated results are provided in the same plot for a comparison. (b) and (c), the free energy and thermal entropy as derived from the experimental measurements. We find excellent agreement for both thermodynamic quantities with the exact results in the entire temperature window. In (d), we provide the heat capacity. The sampling errors ($\Delta(\cdot)$) are below $10^{-2}$. Here, we choose a $2\times 2$ lattice and set the coupling strength $J$ as the energy unit.