Boltzmann Sampling of Frustrated J1 - J2 Ising Models with Programmable Quantum Annealers

TL;DR

The paper investigates the capacity of D-Wave analog quantum annealers to sample the Boltzmann distribution of a frustrated J1-J2 Ising model (the 1D ANNNI model) by mapping 12-spin chains onto hardware graphs and varying annealing times and energy scales. It uses a direct spin-to-qubit embedding and fits an effective inverse temperature $\\beta$ to minimize the total variation distance between observed samples and the target Boltzmann distribution, reporting TVD as low as $0.0003$ at low temperature for highly frustrated cases (e.g., $J_2=1$). Sampling quality degrades near the critical frustration $J_2=0.5$, and performance differences emerge between Pegasus and Zephyr hardware graphs. The results support the potential of current analog quantum computers as thermodynamic samplers for highly frustrated systems, while leaving open questions about scalability, size-independence of optimal parameters, and competitiveness against classical Monte Carlo at larger system sizes.

Abstract

One of the surprising, and potentially very useful, capabilities of analog quantum computers, such as D-Wave quantum annealers, is sampling from the Boltzmann, or Gibbs, distribution defined by a classical Hamiltonian. In this study, we thoroughly examine the ability of D-Wave quantum annealers to sample from the Boltzmann distribution defined of a canonical type of competing magnetic frustration $J_1$-$J_2$ model; the ANNNI (axial next-nearest-neighbor Ising) model. Boltzmann sampling error rate is quantified for standard linear-ramp anneals ranging from $5$ nanosecond annealing times up to $2000$ microseconds on two different D-Wave quantum annealing processors. Interestingly, we find some analog hardware parameters which result in a very high accuracy (down to a TVD of $0.0003$) and low temperature sampling (down to $β=32.2$) in a frustrated region of the ANNNI model magnetic phase diagram. This bolsters the viability of current analog quantum computers for thermodynamic sampling applications of highly frustrated magnetic spin systems.

Figures (14)

• Figure 1: $12$-spin ANNNI model rendering (left), where red edges are ferromagnetic coupling and blue edges are antiferromagnetic coupling. This model is then sampled on the D-Wave QPU hardware; representative sampled spin configurations are shown on the right. Cyan nodes are spin down $\downarrow$ and red nodes are spin up $\uparrow$. These samples are from $5$ nanosecond annealing times, with frustration parameter coupling of $J_2=0.5$ (bottom row) and $J_2=1$ (top row). All of these spin configurations were measured on Advantage2_prototype1.4.
• Figure 2: Minimum error rate (TVD), from all evaluated annealing times, Boltzmann sampling as a function of the J coupling energy scale programmed on the analog hardware (x-axis), for the Advantage_system4.1 D-Wave quantum annealing processor. The bottom plot shows the value of the minimum error rate, TVD, found across all evaluating annealing times, and the top plot shows the corresponding $\beta$ value at which the QPU samples a Boltzmann distribution at that given error rate. Each separate line denotes a different ANNNI model frustration parameter $J_2$.
• Figure 3: Minimum error rate (TVD), from all evaluated annealing times, Boltzmann sampling as a function of the J coupling energy scale programmed on the analog hardware (x-axis), for the Advantage2_system1.4 D-Wave quantum annealing processor. The bottom plot shows the value of the minimum error rate, TVD, found across all evaluating annealing times, and the top plot shows the corresponding $\beta$ value at which the QPU samples a Boltzmann distribution at that given error rate. Each separate line denotes a different ANNNI model frustration parameter $J_2$.
• Figure 4: Error rate (TVD) as a function of inverse temperature $\beta$, across the entire spectrum of evaluated annealing times (color coded by the log-scale heatmap below the sub-plots), for the ANNNI frustration parameter $J_2=0.5$ run on the Advantage_system4.1 processor. Each sub-plot corresponds to a different analog hardware energy scale, denoted in the title of each sub-plot. Error bars are shown on datapoints where there were many best-fitted $\beta$ values that spanned at least a range of $0.1$. The annealing time which had the absolute lowest error rate within each sub-plot is notated on the plot with the exact annealing time, and the resulting TVD.
• Figure 5: Error rate (TVD) as a function of inverse temperature $\beta$, across the entire spectrum of evaluated annealing times (color coded by the log-scale heatmap below the sub-plots), for the ANNNI frustration parameter $J_2=0.5$ run on the Advantage2_system1.4 processor. Each sub-plot corresponds to a different analog hardware energy scale, denoted in the title of each sub-plot. Error bars are shown on datapoints where there were many best-fitted $\beta$ values that spanned at least a range of $0.1$. The annealing time which had the absolute lowest error rate within each sub-plot is notated on the plot with the exact annealing time, and the resulting TVD.