Advantages of multistage quantum walks over QAOA

TL;DR

This work compares multistage continuous-time quantum walks (MSQW) and the quantum approximate optimization algorithm (QAOA) for Ising-encoded optimization, using heuristic parameter choices to ensure fair resource accounting. Through analytical derivations and spin-glass simulations, it demonstrates that MSQW more accurately approximates quantum annealing and achieves lower final energies and higher ground-state probabilities than QAOA for the same resources, with advantages already evident at few stages. Numerically, MSQW shows robust performance on Sherrington-Kirkpatrick instances, including high success probabilities (up to ~0.8) for two-stage runs with time-averaging, and streamlined parameterizations reduce classical optimization overhead. The results suggest that exploiting the native, simultaneous action of driver and problem Hamiltonians on current hardware can outperform gate-based QAOA in optimization tasks, motivating further exploration of MSQW across more problems and parameterizations.

Abstract

Methods to find the solution state for optimization problems encoded into Ising Hamiltonians are a very active area of current research. In this work we compare the quantum approximate optimization algorithm (QAOA) with multi-stage quantum walks (MSQW). Both can be used as variational quantum algorithms, where the control parameters are optimized classically. A fair comparison requires both quantum and classical resources to be assessed. Alternatively, parameters can be chosen heuristically, as we do in this work, providing a simpler setting for comparisons. Using both numerical and analytical methods, we obtain evidence that MSQW outperforms QAOA, using equivalent resources. We also show numerically for random spin glass ground state problems that MSQW performs well even for few stages and heuristic parameters, with no classical optimization.

Figures (6)

• Figure 1: Quantum walk and QAOA average energies and success probabilities for a single stage. For quantum walk, $\gamma$ and $t$ are used to parameterise the single stage protocol. For QAOA, $\alpha$ is the time for which the driver is active, and $\beta$ is the time for which the problem Hamiltonian is active. All plots are for for the 10 qubit SK spin glass instance with ID: aaaufeflfwqdwhthcrcnynopihzciv. Energy and success probability plots share the same color scale to emphasise the performance difference between the two protocols.
• Figure 2: Comparison of final problem Hamiltonian energy expectations (a) and success probabilities (b) for single stage QW and QAOA. Symbols represent the first $100$ ten qubit problems in the dataset from Callison2019_dataset. Optimal parameters were found using a $20$ by $20$ grid search with an evenly-spaced grid with $0 \le \gamma \le 4$ and $0 \le t \le 6$ for QW and $0 \le \alpha \le \pi/2$ and $0 \le \beta \le \pi/2$ for QAOA. Dashed lines are a guide to the eye showing what equal performance would look like.
• Figure 3: Quantum walk and QAOA average energies and success probabilities for two stages. For quantum walk, $\gamma$ and $t$ are used to parameterise the single stage protocol, which induce an analogous parametrization for QAOA in terms of these parameters (see equation \ref{['eq:param_QAOA']}). After time-averaging, both protocols are parameterised in terms of $\gamma_1$ and $\gamma_2$ as described in the main text, and averaged over 2,000 samples with independent random runtimes for each stage in the range $0.1-0.5$. All plots are for for the $5$ qubit SK spin glass instance with ID: aaavmaiqiolnplcovmzxjazkyvyayz. Energy and success probability plots share the same color scale to emphasise the performance difference between the two protocols.
• Figure 4: Values of $\alpha/t$ (blue, circles) and $\beta/t$ (gold, squares) for different number of stages $p$ and initial $\gamma$ values. At each stage $\gamma_{j+1}/\gamma_j= 1-\Delta\gamma$ using the parameterisation given in equation \ref{['eq:param_QAOA']}. In figure (a) the filled symbols shows the approximately optimal schedule for $p=5$ with an initial $\gamma$ value of $3$ and $\Delta \gamma=0.2$ (see figure \ref{['fig:5_stage_QAOA_prob']}). The unfilled symbols show what it would have been for the approximately optimal values of quantum walks $\gamma=4$ and $\delta \gamma=0.2$. Figure (b) shows that for certain parameters, $p=200$ and $\Delta \gamma=0.3$ with an initial $\gamma$ of $20$ in this case, a schedule which strongly resembles an annealing schedule can be obtained.
• Figure 5: Quantum walk and QAOA average energies and success probabilities for five stage protocol. For quantum walk, $\gamma$ and $t$ are used to parameterise the single stage protocol. Both are parameterised in terms of $\gamma$ and $\Delta \gamma$ as described in the main text, and averaged over 2,000 samples with independent random runtimes for each stage in the range $0.1-0.5$. All plots are for for the $5$ qubit SK spin glass instance with ID: aaavmaiqiolnplcovmzxjazkyvyayz. Energy and success probability plots share the same color scale to emphasise the performance difference between the two protocols.