Scaling Whole-Chip QAOA for Higher-Order Ising Spin Glass Models on Heavy-Hex Graphs

TL;DR

It is shown that the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution, and the best quantum processors find lower energy solutions up to p = 2 or p = 3, and find mean energies that are about a factor of two off from the noise-free distribution.

Abstract

We show through numerical simulation that the Quantum Approximate Optimization Algorithm (QAOA) for higher-order, random-coefficient, heavy-hex compatible spin glass Ising models has strong parameter concentration across problem sizes from $16$ up to $127$ qubits for $p=1$ up to $p=5$, which allows for straight-forward transfer learning of QAOA angles on instance sizes where exhaustive grid-search is prohibitive even for $p>1$. We use Matrix Product State (MPS) simulation at different bond dimensions to obtain confidence in these results, and we obtain the optimal solutions to these combinatorial optimization problems using CPLEX. In order to assess the ability of current noisy quantum hardware to exploit such parameter concentration, we execute short-depth QAOA circuits (with a CNOT depth of 6 per $p$, resulting in circuits which contain $1420$ two qubit gates for $127$ qubit $p=5$ QAOA) on $100$ higher-order (cubic term) Ising models on IBM quantum superconducting processors with $16, 27, 127$ qubits using QAOA angles learned from a single $16$-qubit instance. We show that (i) the best quantum processors generally find lower energy solutions up to $p=3$ for 27 qubit systems and up to $p=2$ for 127 qubit systems and are overcome by noise at higher values of $p$, (ii) the best quantum processors find mean energies that are about a factor of two off from the noise-free numerical simulation results. Additional insights from our experiments are that large performance differences exist among different quantum processors even of the same generation and that dynamical decoupling significantly improve performance for some, but decrease performance for other quantum processors. Lastly we show $p=1$ QAOA angle mean energy landscapes computed using up to a $414$ qubit quantum computer, showing that the mean QAOA energy landscapes remain very similar as the problem size changes.

Figures (20)

• Figure 1: Examples of higher order Ising models that are hardware-compatible with a heavy-hex hardware graph: (left) A $27$-qubit device: Nodes correspond to linear terms, edges to quadratic terms, and hyperedges encircling three neighboring nodes to cubic terms. Ising coefficients of $-1$ and $+1$ are depicted in red and green, respectively. (right) A $16$-qubit device: Illustrating the terminology of Equation \ref{['equation:problem_instance']}, we have $W = \{2,4,5,10,11,13\}$, with the remaining nodes in $V_2$ being $V_2 \setminus W = \{0,6,9,15\}$. For node $4 \in W$, we have neighbors $\{n_1(l), n_2(l)\} = \{1,7\} \subset V_3$. (bottom) A $127$-qubit device: Higher order Ising model comprised of $127$ linear, $144$ quadratic and $71$ cubic terms.
• Figure 2: From Refs.pelofske2023qavsqaoapelofske2023short: QAOA circuit description for heavy-hex graph compatible higher order Ising models of arbitrary size. The graph is bipartite and has an arbitrary 3-edge-coloring given by Kőnig's line coloring theorem. (left) 3-edge-coloring and bipartite grey-shading of the nodes. Adjacent purple lines denote the cubic terms terms. (right) Any quadratic term (colored edge) gives rise to a combination of two CNOTs and a Rz-rotation in the phase separator, giving a CNOT depth of 6 due to the degree-3 nodes. When targeting the degree-2 nodes with the CNOT gates, these constructions can be nested to implement the cubic terms with just one additional Rz-rotation.
• Figure 3: Error analysis performed for MPS simulations. Here we show error in the energy $|E_{\chi} - E_{2048}|$ as a function of bond dimension $\chi$ for different values of $p$. Solid black lines represent the error in the energy averaged over one hundred instances of $H_C$. All computed errors, for all instances, are contained within gray areas around black lines. The gray areas are relatively small, especially at large $\chi$. This indicates that different instances of $H_C$ result in similar errors. Errors are small; they are all below $0.1$, which is observed in the hardest case of $p=5$. Simulations with $p<5$ incur smaller errors.
• Figure 4: Classical simulations of mean energies demonstrating (noiseless) concentration of QAOA parameters. We simulate 100 random instances for each circuit size using fixed QAOA angles (trained on a single 16 qubit instance): (left) The angles for $1\leq p \leq5$ are used to execute QAOA on 100 random 16-qubit higher-order heavy-hex instances, (center) The same angles are used for 100 random $27$-qubit instances, (right) MPS simulation with bond dimension $\chi=2048$ is used for 100 random $127$-qubit instances. For growing circuit size $16,27,127$, for every random higher-order Ising model, as $p$ increases the mean energy strictly improves, showing that parameter transfer succeeds in a noiseless setting. In each plot, also the mean energy across the instance ensemble is plotted as a dashed black line.
• Figure 5: MPS simulation sample distributions, using a bond dimension of $\chi=2048$ for $127$-qubit QAOA circuits sampling eight of the higher-order Ising model problem instances. For each $p$, a total of $8192$ samples are computed. The ground state energy is marked in all plots with a dashed vertical black line, and the minimum energy found within each $p$ energy distribution is marked with a vertical solid line. In addition to the means of the distributions improving as a function of $p$ (shown for all $100$ problem instances in Figure \ref{['fig:parameter_transfer_scaling_plots']}), we see here that the minimum energy sampled also improves as $p$ increases. Notably, none of the distributions sampled the optimal energy, although the minimum energies from $p=4$ and $p=5$ are close to the optimal energy. These distributions show the ideal QAOA sampling capabilities, using the classical simulation method of MPS, if the quantum computation was noiseless.