Distinguishing thermal versus quantum annealing using probability-flux signatures across interaction networks

Abstract

Simulated annealing provides a heuristic solution to combinatorial optimization problems. The cost function of a problem is mapped onto the energy function of a physical many-body system, and, by using thermal or quantum fluctuations, the system explores the state space to find the ground state, which corresponds to the optimal solution of the problem. Studies have highlighted both the similarities and differences between thermal and quantum fluctuations. Nevertheless, fundamental understanding of thermal and quantum annealing remains incomplete, making it unclear how quantum annealing outperforms thermal annealing in which problem instances. Here, we investigate the many-body dynamics of thermal and quantum annealing by examining all possible interaction networks of $\pm J$ Ising spin systems up to seven spins. Our comprehensive investigation reveals that differences between thermal and quantum annealing emerge for particular interaction networks, indicating that the structure of the energy landscape distinguishes the two dynamics. We identify the microscopic origin of these differences through probability fluxes in state space, finding that the two dynamics are broadly similar, but that quantum tunnelling produces qualitative differences. Our results provide insight into how thermal and quantum fluctuations navigate a system toward the ground state in simulated annealing, and are experimentally verifiable in atomic, molecular, and optical systems. Furthermore, these insights may improve mappings of optimization problems to Ising spin systems, yielding more accurate solutions in faster simulated annealing and thus benefiting real-world applications in industry. Our comprehensive survey of interaction networks and visualization of probability flux can help to understand, predict, and control quantum advantage in quantum annealing.

Figures (7)

• Figure 1: The success rate dependency on the interaction network structure. a, $+J$ five-spin interaction networks. The node represents spins and the edges represent interactions. The colour of nodes indicates the spin identifier. The solid green edges represent the ferromagnetic interaction $+J/N$, and the dashed red edges represent the antiferromagnetic interaction $-J/N$ (not shown here). Interaction network identifiers are shown on the top of each network. b, Time evolution of the success rate corresponding to panel a. The dark and light solid lines represent TA and QA results respectively, and the dark and light dashed lines represent the adiabatic limits of TA and QA respectively. External field is set to be $h_i = 0.5J/N$ for all $i$. The reciprocal annealing schedule is used.
• Figure 1: The speed limit for the order parameter dynamics. The transition speed of the order parameter, $| \frac{\mathrm{d}}{\mathrm{d}t} \langle m \rangle |$ or $| \frac{\mathrm{d}}{\mathrm{d}t} \langle \hat{m} \rangle |$, and their bounds for 5-spin systems with ferromagnetic interactions from Fig. \ref{['fig:fig-01']}a. In each panel, the order parameter dynamics are shown for TA (grey line) from eq. \ref{['eq:order-parameter-thermal']}, TA speed limit (grey dashed-dotted line) from eq. \ref{['eq:order-parameter-bound-thermal']}, QA (black line) from eq. \ref{['eq:order-parameter-quantum']}, and QA speed limit (black dashed-dotted line) from eq. \ref{['eq:order-parameter-bound-quantum']}.
• Figure 2: Comparing success rate difference between TA and QA. From left to right, the panels represent the results for three- to seven-spin interaction networks. From top to bottom, the panels show the results for the success rate difference between QA and TA, i.e., $p_{\text{QA}}(\infty) - p_{\text{TA}}(\infty)$, the difference between the adiabatic limits of TA and TA, i.e., $p_{\text{ATA}}(\infty) - p_{\text{TA}}(\infty)$, and the difference between the adiabatic limits of QA and QA, i.e., $p_{\text{AQA}}(\infty) - p_{\text{QA}}(\infty)$. The horizontal axis represents the number of nonzero interactions, and the vertical axis represents external longitudinal field strength. Each data point corresponds to an interaction network instance. For visualization, we add small random noise to the data points to remove the overlap of data points.
• Figure 3: Time-integrated probability fluxes of TA and QA. a, Time-integrated quantum probability fluxes $\{\Delta \mathcal{J}_\text{Q}(\bm{s}, \bm{s}^\prime)\}$. b, Time-integrated thermal probability fluxes $\{\Delta \mathcal{J}_\text{T}(\bm{s}, \bm{s}^\prime)\}$. c, Difference between quantum and thermal probability fluxes, $\{\Delta \mathcal{J}_\text{Q}(\bm{s}, \bm{s}^\prime) - \Delta \mathcal{J}_\text{T}(\bm{s}, \bm{s}^\prime)\}$. Each arrow represents the time-integrated thermal probability flux, and its width is proportional to the magnitude of the flux $|\Delta \mathcal{J}(\bm{s}, \bm{s}^\prime)|$, and direction corresponds to the sign of the flux $\mathop{\mathrm{sgn}}\limits \bm{(} \Delta \mathcal{J}(\bm{s}, \bm{s}^\prime) \bm{)}$. The colour of each arrow indicates the spin identifier of the flipped spin, cf. Fig. \ref{['fig:fig-01']}a.
• Figure 4: Probability fluxes difference between TA and QA. a, Time evolution of success rate of network 5-219. We select seven time points to investigate the time-dependency of probability flux. This panel is magnified version of corresponding panel in Fig. \ref{['fig:fig-01']}b. b, Quantum probability fluxes at each time point. c, Thermal probability fluxes at each time point. d, Difference between quantum and thermal probability fluxes at each time point. For b--d, the arrow indicates the same as Fig. \ref{['fig:fig-03']}a--c but for probability flux at each time point $\mathcal{J}(\bm{s}, \bm{s}^\prime; t)$.