Quantum thermal state preparation for near-term quantum processors

Abstract

Preparation of quantum thermal states of many-body systems is a key computational challenge for quantum processors, with applications in physics, chemistry, and classical optimization. We provide a simple and efficient algorithm for thermal state preparation, combining engineered bath resetting and modulated system-bath coupling to derive a quantum channel approximately satisfying quantum detailed balance relations. We show that the fixed point $\hatσ$ of the channel approximates the Gibbs state as $\|\hatσ-\hatσ_β\|\sim θ^2$, where $θ$ is the system-bath coupling and $\hatσ_β\propto e^{-β\hat H_S}$. We provide extensive numerics, for the example of the 2D Quantum Ising model, confirming that the protocol successfully prepares the thermal state throughout the finite-temperature phase diagram, including near the quantum phase transition. Simulations for free-fermion systems provide further evidence for the accuracy of the protocol for large system sizes in the weak-coupling limit. Our algorithm provides a path to efficient quantum simulation of quantum-correlated states at finite temperature with current and near-term quantum processors.

Figures (6)

• Figure 1: Protocol for quantum thermal state preparation. The system density matrix evolves under the repeated map $\hat{\rho}_{n+1} = \mathcal{E}(\hat{\rho}_n)$, consisting of three stages: (i) initializing a bath of auxiliary qubits in the $\ket{0}$ state, (ii) joint unitary evolution and (iii) reset of the bath qubits. The unitary stage consists of evolution under a time-dependent unitary $\hat{U}(\tau)$, defined in Eq. (\ref{['eq:unitary_step']}), with $-M \leq \tau \leq M$, $M \propto\sqrt{\beta}$. The coupling to the bath is smoothly switched on with an interaction strength modulated by the filter function $f(\tau)$, to ensure the resulting map approximately satisfies detailed balance. A randomization step $\hat{R}$ (iv) grants an additional suppression of unwanted system coherences. Our protocol is designed with near-term analog and digital processors in mind.
• Figure 2: Single spin cooling. (a) Scaling of steady state population errors ($\zeta^{\rm pop}$) and coherences ($\zeta^{\rm coh}$) vs. system-bath coupling $\theta^2$. (b) Comparison of randomized (solid orange, purple lines) and unrandomized (solid red, blue lines) protocols, for varying reset time $T$ and $\theta=0.25$. Black dot-dashed lines show perturbative solution for coherences, Eq. (\ref{['eq:corrections']}).
• Figure 3: Evolution of energy in 2D Quantum Ising model ($3\times3$ sites), vs. number of resets. We show results for several different choices of parameters $J, \beta$, using the unrandomized protocol with $\theta=0.25$. Energies for different curves are scaled to lie between -1 (ground state) and 1 (highest excited state) for comparison. Dot-dashed lines are the corresponding thermal state values.
• Figure 4: Steady state observables in 2D Quantum Ising model. Numerical values are depicted by open points, while coloured lines are values for the thermal density matrix obtained via exact diagonalisation. (a) Total energy in steady state for fixed $J=1$ and varying $\beta$, with system sizes between $2\times 2$ and $4\times 4$. Inset: Energy for fixed $\beta=0.5$ and varying $J$. (b) Total heat capacity in steady state, for same parameters as in (a) and varying $\beta$. Error bars are due to finite sampling over trajectories. Inset: Heat capacity for fixed $\beta=0.5$ and varying $J$. (c) Upper plot: Transverse magnetisation $\langle \hat{Z}_0\rangle$ for three different temperatures and varying $J$. Lower plot: Mutual information between a single spin and the rest of the system. System size is $3\times 3$ sites.
• Figure 5: (a) Scaling of relative error in steady state energy, relative to thermal energy, vs. coupling $\theta^2$. We show four representative points in the $J/\beta$ phase diagram. Solid lines are for the unrandomized protocol, while dot-dash lines are for the randomized protocol with parameter $\lambda = 5$. (b) Scaling of trace distance between steady state and the first order Floquet Gibbs state, vs. coupling $\theta^2$. We focus on the two low-temperature points, and for $\beta=1.5$, $J=1$ show also the value when the coherence between the degenerate ferromagnetic ground states is ignored.