Partition Function Estimation Using Analog Quantum Processors

TL;DR

The paper investigates partition function estimation for Ising models using analog quantum processors (D-Wave QPUs). It develops two QA-based sampling protocols—linear-ramp annealing and iterated reverse annealing (QEMC)—to build a density-of-states histogram that yields $Z$ at arbitrary temperatures, and benchmarks them against classical Wang-Landau and Multiple Histogram Reweighting methods. Empirical results on a 25-spin random $\pm J$ Ising model show that QA-based DoS sampling can achieve comparable accuracy to classical heuristics, with ultra-fast anneals delivering very low log relative errors (e.g., $7.6\times 10^{-6}$) in modest QPU time. The work demonstrates a practical analog quantum approach to thermodynamics on near-term devices and highlights the potential for scaling and applications to more complex, frustrated systems.

Abstract

We evaluate using programmable superconducting flux qubit D-Wave quantum annealers to approximate the partition function of Ising models. We propose the use of two distinct quantum annealer sampling methods: chains of Monte Carlo-like reverse quantum anneals, and standard linear-ramp quantum annealing. The control parameters used to attenuate the quality of the simulations are the effective analog energy scale of the J coupling, the total annealing time, and for the case of reverse annealing the anneal-pause. The core estimation technique is to sample across the energy spectrum of the classical Hamiltonian of interest, and therefore obtain a density of states estimate for each energy level, which in turn can be used to compute an estimate of the partition function with some sampling error. This estimation technique is powerful because once the distribution is sampled it allows thermodynamic quantity computation at arbitrary temperatures. On a $25$ spin $\pm J$ hardware graph native Ising model we find parameter regimes of the D-Wave processors that provide comparable result quality to two standard classical Monte Carlo methods, Multiple Histogram Reweighting and Wang-Landau. Remarkably, we find that fast quench-like anneals can quickly generate ensemble distributions that are very good estimates of the true partition function of the classical Ising model; on a Pegasus graph-structured QPU we report a logarithmic relative error of $7.6 \times 10^{-6}$, from $171,000$ samples generated using $0.2$ seconds of QPU time with an anneal time of $8$ nanoseconds per sample which is interestingly within the closed system dynamics timescale of the superconducting qubits.

Figures (13)

• Figure 1: The time-dependent functions $A(s)$ and $B(s)$ used for: (a) the standard forward annealing linear-ramp sampling protocol (b) reverse quantum annealing "Monte Carlo" chains, for the three different D-Wave QPUs.
• Figure 2: An example representative reverse quantum annealing Monte Carlo chain consisting of three reverse annealing cycles. Each cycle performs a $2.5$$\mu$s ramp from $s=1$ to $s=0.5$, followed by a $15$$\mu$s pause at $s=0.5$, and then a ramp back to $s=1$. Each time the hardware reaches $s=1$, the state of all qubits is measured (timescale for the measurement are not shown in this schematic); that exact measured spin configuration is then used to initialize the subsequent simulation.
• Figure 3: A comparison of the quantum annealing sampling techniques, with an optimized set of analog hardware control parameters, (a-d) and the standard classical sampling algorithms Wang-Landau and MHR for estimating the partition function in panels e and f. The x-axis denotes the absolute total sample count that was generated by each respective algorithm, in the case of the classical algorithms this corresponds to the total number of proposed spin updates which is the primary source of compute time and complexity for these algorithms. In the case of QA sampling, the x-axis corresponds to the number of measured samples, which is distributed incrementally from each hardware parameter up to the total final sample count. Random sampling error rates are shown for reference as dashed horizontal dashed lines in the right-column plots.
• Figure 4: Linear-ramp standard quantum annealing convergence with cumulative sampling over larger energy scales. For each fixed annealing time (log scale y-axis), the partition function estimation and log relative error are computed from an energy histogram that accumulates more samples as the $J$ energy scale increases along the x-axis for the Advantage_system4.1 processor (a,b) and Advantage2_system1.9 (c,d). The colormap encodes log relative error (b,d) and the actual partition function value estimate (a,c), where the true value is marked in the legend. The blue line traced out in in panels b,d define the lowest error-rate sampling for this particular protocol, which shows that there is a clear minimum -- any additional higher $J$ energy scales result in high error rates.
• Figure 5: Linear-ramp standard quantum annealing convergence with cumulative sampling over increasing annealing times. For each fixed energy scale (log scale y-axis), the partition function estimation and log relative error are computed from an energy histogram that accumulates more samples as the annealing time increases on the x-axis. Results from the Advantage_system4.1 processor (a,b) and Advantage2_system1.9 (c,d). The blue line traced in panels b,d show that the minimum error rate region of the parameter space converges at long annealing times.