Quantum Finite Temperature Lanczos Method

Abstract

The computation of thermal properties of quantum many-body systems is a central challenge in our understanding of quantum mechanics. We introduce the Quantum Finite Temperature Lanczos Method (QFTLM), which extends the finite-temperature Lanczos method to quantum computers by combining real-time quantum Krylov methods with efficient preparation of typical states for trace estimation. This approach enables the computation of thermal expectation values while avoiding the exponential scaling inherent to classical exact simulation techniques. Numerical experiments on the transverse-field Ising model show that QFTLM can reproduce thermal observables over a wide temperature range. We further analyze the influence of Krylov dimension, number of trace-estimator states, and Trotter error, and show that suitable regularization is essential for robustness in noisy settings. These results establish QFTLM as a promising framework for finite-temperature quantum simulation.

Figures (5)

• Figure 1: Thermal expectation values of the energy (a) and heat capacity (b) as a function of the temperature for $L=14$ spins in the TFIM (see \ref{['eq:tfim']}). We compare the results obtained with QFTLM for varying Krylov dimensions $D$ with the exact solution (in black). The real-time Krylov basis is approximated with 4 first-order Trotter steps and the expectation values are computed by averaging over $K=40$ quantum Hutchinson states. In the low-temperature limit, choosing a smaller Krylov subspace leads to significant deviations from the exact solutions. For high temperatures, all simulations qualitatively align with the ground truth.
• Figure 2: Propagation of Trotter error through the QFTLM protocol. In panel (a), we plot the relative error of the thermal energies as a function of temperature for QFTLM simulations with varying number of Trotter steps used to build the real-time Krylov basis. The Krylov dimension is fixed to $D=120$ and we average over $K=40$ quantum Hutchinson states. At low temperatures, the Trotter error is clearly noticeable whereas at high temperatures other sources dominate the overall error on the thermal energy. In panel (b), we explicitly plot the scaling of the relative energy error as a function of the number of Trotter steps in the low-temperature limit ($T=0.025$, indicated by the gray dashed line in the left plot). For a large enough Krylov dimension $D$, the error scales roughly as $O(\delta t^2)$ (indicated by the black dotted line), where $\delta t$ denotes the Trotter time step.
• Figure 3: Scaling of the QFTLM error with system size $L$. We plot the relative error in the energy expectation values at three temperatures, $T=0.01, 1,$ and $100$, in panels (a), (b), and (c), respectively, as a function of the number of spins $L$, for a varying number of quantum Hutchinson states $K$. The Krylov subspace dimension is fixed to $D=20$ and the number of Trotter steps to 4. At low temperatures, the error at fixed Krylov dimension increases with the system size. Conversely, at high temperatures, we observe a decrease in the error as the number of spins increases.
• Figure 4: Effect of noise on the error of thermal energies obtained through QFTLM. Gaussian noise with varying standard deviation is added to the measured Gram matrix $S$ and projected unitary $U$. In panel (a), the time evolution is performed exactly without Trotterization. In this case, the noise is the dominant source of error at low temperatures whereas for sufficiently small standard deviations the error saturates at high temperatures. In panel (b), we perform the time evolution by splitting $e^{-iH\Delta t}$ into 4 Trotter steps. Now, the Trotter error dominates at low temperatures for small enough noise levels. Colorbar ticks indicate the simulated noise levels.
• Figure 5: Importance of appropriate regularization for noisy simulations. We plot the absolute energy error averaged over all temperatures (between $10^{-2}$ and $10^2$) as a function of the Tikhonov shift $\eta$ chosen to regularize the Gram matrix (Line 4 in \ref{['algo:regularization']}). We find that for each noise level there is a threshold (roughly a factor of 100 above the noise level) defining the minimal $\eta$ above which we obtain reasonable errors in the energy. Below that threshold, the error spikes up to $O(1)$, whereas above the threshold the error remains low up to $\eta \gtrsim 10^{-4}$. Colorbar ticks indicate the simulated noise levels.