Learning-Driven Annealing with Adaptive Hamiltonian Modification for Solving Large-Scale Problems on Quantum Devices

TL;DR

Learning-Driven Annealing is a framework that links individual quantum annealing evolutions into a global solution strategy to mitigate hardware constraints such as short annealing times and integrated control errors and is a step towards practical quantum computation that enables today's quantum devices to compete with classical solvers.

Abstract

We present Learning-Driven Annealing (LDA), a framework that links individual quantum annealing evolutions into a global solution strategy to mitigate hardware constraints such as short annealing times and integrated control errors. Unlike other iterative methods, LDA does not tune the annealing procedure (e.g. annealing time or annealing schedule), but instead learns about the problem structure to adaptively modify the problem Hamiltonian. By deforming the instantaneous energy spectrum, LDA suppresses transitions into high-energy states and focuses the evolution into low-energy regions of the Hilbert space. We demonstrate the efficacy of LDA by developing a hybrid quantum-classical solver for large-scale spin glasses. The hybrid solver is based on a comprehensive study of the internal structure of spin glasses, outperforming other quantum and classical algorithms (e.g., reverse annealing, cyclic annealing, simulated annealing, Gurobi, Toshiba's SBM, VeloxQ and D-Wave hybrid) on 5580-qubit problem instances in both runtime and lowest energy. LDA is a step towards practical quantum computation that enables today's quantum devices to compete with classical solvers.

Figures (11)

• Figure 1: A $24$-qubit spin-glass with $N_{dom} = 3$. (a) Spin-glass graph with the biases $h_i$ as nodes and couplers $J_{ij}$ as edges. Dashed edges indicate periodic boundary conditions. The colored rectangles represent the spin domains: $\mathcal{D}_1$ (green), $\mathcal{D}_2$ (purple), and $\mathcal{D}_3$ (orange). Each domain has two distinct minima: $\alpha$ and $\overline{\alpha}$ ($\mathcal{D}_1$), $\beta$ and $\overline{\beta}$ ($\mathcal{D}_2$), $\gamma$ and $\overline{\gamma}$ ($\mathcal{D}_3$). Panels (a.1-3) show the eigenenergies $E$ of the corresponding domains as a function of the Hamming distance $d_{ham}$ to the respective ground state $\alpha, \beta, \gamma$. (b) Hypercube representation of the Hilbert space, with each axis representing the Hamming distance $d_{ham}$ from the ground state ($\alpha, \beta, \gamma$) of the respective spin domains $\mathcal{D}_k$. Color indicates the lowest eigenenergy at each point. The corners of the hypercube correspond to the $2^{N_{dom}} = 8$ local minima of the spin glass. (c) Energy landscape of the spin glass obtained by unfolding the hypercube facets. Each point is projected onto the closest facet (smallest Hamming distance), with the blue surface representing the lowest eigenenergies on these facets. The eight local minima are marked by colored diamonds at the facets' corners, with the states surrounding it being the energy valleys. Additionally, black dots indicate $2000$ samples from a quantum annealing simulation with annealing time $T = 10$, with the probability density shown on the lower orange plane. Panels (c.1-3) highlight the specific biases $h_i$ and couplers $J_{ij}$ satisfied by three of the eight local minima, with the unsatisfied terms set to $0$ (white). The color scheme matches that of panel (a).
• Figure 2: Comparison of the quanitiy $q_F(\alpha^*, \beta)$ to (a) the Edward-Anderson order parameter $q_{EA}(\alpha^*, \beta)$ and (b) the energy difference $\Delta E = E_{\beta} - E_{\alpha^*}$ for states $\beta$ w.r.t. the global minimum $\alpha^*$ of the $24$-qubit spin-glass shown in Fig. \ref{['fig:spin_glass_graph']}a. The colored diamonds mark the eight local minima (see Fig. \ref{['fig:spin_glass_graph']}c).
• Figure 3: (a) Instantaneous energy spectrum of the transverse field Ising model (see Eq. (\ref{['eq:H_QA']})) for the $24$-qubit spin-glass instance shown in Fig. \ref{['fig:spin_glass_graph']}a, using a linear annealing schedule $\Gamma(t)$. The plot presents a selection of energy levels corresponding to the $8$ valleys in the energy landscape (see Fig. \ref{['fig:spin_glass_graph']}c). For each valley, the local minimum (solid lines) is shown along with three states at Hamming distance $d_{ham} = 1$ (dashed line) and three states at $d_{ham} = 2$ (dash-dotted lines), representing the lowest-energy states along the edges of the hypercube. (b) Adiabatic ratios (see Eq. (\ref{['eq:g']})) for the states presented in panel (a). The 24-qubit eigenstates and eigenvalues in the vicinity of each local minimum were obtained numerically by solving the time-dependent Schrödinger equation (see e.g. TDSE) for reverse-annealing starting in the given states.
• Figure 4: Construction of $H_{F\mathcal{M}}$ (e, Eq. (\ref{['eq:H_F_M']})) for the $24$-qubit spin-glass instance shown in Fig. \ref{['fig:spin_glass_graph']}, with $\lambda = 0.1$. The feature Hamiltonian $H_F$ (b) is derived from four sampled states (a.1-4) using Eq. (\ref{['eq:H_F_mod']}) and the bitmask $\mathcal{M}$, with $m_i = 1$ if the sampled states agree in the $i$-th bit (this is the case for all non-white nodes shown in panel (b)). Panels (c) and (d) depict subsets of the original problem Hamiltonian, constructed using Eq. (\ref{['eq:H_P_mod']}), with (c) applying to the terms included in $H_F$, and (d) addressing the remaining biases and couplers. Panels (b-e.1) illustrate the corresponding energy landscapes of the Hamiltonians in (b-e), with the four sampled states shown as black dots in (b.1).
• Figure 5: Two iterations of the local search protocol applied to the $24$-qubit spin-glass instance shown in Fig. \ref{['fig:spin_glass_graph']}a. The protocol is executed with $Q_{\mathcal{T}} = 0.98$ and $N_{\mathcal{T}} = 5$, using (a) $\lambda = 0.2$ for the first and (b) $\lambda = 1.0$ for the second iteration. The top panels depict the energy landscape ($E$) of $H_{F\mathcal{M}}$ in the first and second iteration, respectively. Black dots represent $2000$ samples from a quantum annealing simulation ($T = 10$), with the probability density ($P$) shown on the bottom plane. Red dots indicate the subset $\mathcal{T}$. The initial state $\alpha_0$ of the first iteration (panel a) is marked as a red dot in Fig. \ref{['fig:spin_glass_graph']}c. The color scales are consistent with those in Fig. \ref{['fig:spin_glass_graph']}c. Panels (a-b.1) display the instantaneous energy spectrum of the transverse field Ising model with $H_{F\mathcal{M}}$ for each iteration. The energy levels correspond to those in Fig. \ref{['fig:energyspectrum_g_orig']}, where the color of the lines indicates the association to the eight valleys in the energy landscape of $H_P$. A linear annealing schedule $\Gamma(t)$ is used throughout as an example. Panels (a-b.2) show the adiabatic ratios $g$ (see Eq. (\ref{['eq:g']})) for the states shown in panels (a-b.1), respectively.

Theorems & Definitions (5)

• Theorem C.1
• Theorem C.1
• proof
• Theorem C.2
• proof