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arXiv:2404.10743

An angle rounding parameter initialization technique for ma-QAOA

Anthony Wilkie, James Ostrowski, Rebekah Herrman

Published Apr 16, 2024• 0 citations
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Abstract

The multi-angle quantum approximate optimization algorithm (ma-QAOA) is a recently introduced algorithm that gives at least the same approximation ratio as the quantum approximate optimization algorithm (QAOA) and, in most cases, gives a significantly higher approximation ratio than QAOA. One drawback to ma-QAOA is that it uses significantly more classical parameters than QAOA, so the classical optimization component more complex. In this paper, we motivate a new parameter initialization strategy in which angles are initially randomly set to multiples of $π/8$ between $-π$ and $π$ and this vector is used to seed one round of BFGS. We find that this parameter initialization strategy gives average approximation ratios of $0.900$, $0.982$, and $0.997$ for $p = 1, 2, 3$ layers of ma-QAOA. This is comparable to the average approximation ratios of ma-QAOA where the optimal parameters are found using BFGS with 1 random starting seed, which are $0.900$, $0.982$, and $0.996$. We also test another parameter initialization strategy in which angles corresponding to maximal degree vertices in the graph are set to 0 while all other are randomly initialized to random multiples of $π/8$. Using this strategy, the average approximation ratios are $0.897$, $0.984$, and $0.997$.

DOI
10.48550/arxiv.2404.10743

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Multi-angle quantum approximate optimization algorithm (ma-QAOA) has been proposed as an alternative to regular QAOA for solving optimization problems using noisy intermediate scale quantum (NISQ) hardware. Here, we first implement QAOA and ma-QAOA algorithms to solve MaxCut on weighted and unweighted versions of the Erdős–Rényi (ER) and random cubic (RC) graphs of 6 and 10 nodes. Quantum circuit parameters are updated iteratively through classical feedback with the quantum circuit for forward pass modelled on Qiskit simulator without noise models. Then, the quantum circuit for the final run with updated parameters is implemented on Qiskit simulator without noise models, Qiskit simulator with experimentally benchmarked noise models (fake back-end), and also on actual experimental quantum hardware (IBM’s 127-qubit quantum processor ibm brisbane). We show that for most graph instances, ma-QAOA performs better than QAOA (in terms of approximation ratio (AR)) and needs less stages in the quantum circuit for all three aforementioned methods. Finally, we extend our results up to 14-node RC graphs while restricting ourselves to Qiskit simulation without noise (both for parameter update and final run) and show that for most instances, ma-QAOA again needs much less stages compared to QAOA to obtain reasonable AR.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
The standard quantum approximate optimization algorithm (QAOA) has a limited expressive power in shallow circuits, whereas naively increasing the number of layers leads to greater quantum circuit depth and higher noise sensitivity. To address this trade-off, based on the ideas of the multi-angle quantum approximate optimization algorithm (MA-QAOA) and the RY-layer-assisted quantum approximate optimization algorithm (RY-QAOA), this paper proposes an improved algorithm, i.e., MA-RY-QAOA, which integrates parameter multangularization and RY-assisted architecture. In MA-RY-QAOA, the problem block incorporates edge parameter multangularization together with RY extension, while the mixer block implements qubit parameter multangularization. For layer number p=1 and node size n=4~12 , we conduct 50 simulation trials on maximum cut problem respectively for random graph with edge probability of 0.8, complete graph, 2-regular graph and 4-regular graph, by using the average approximation ratio as the evaluation metric. The simulation trial results show that, across these four class graphs, the average approximation ratio of MA-RY-QAOA and MA-QAOA significantly outperforms standard QAOA, RY-QAOA, and QAOA+ for overwhelming majority of node sizes; with respect to complete graph, the average approximation ratio of MA-QAOA outperforms that of MA-RY-QAOA for all node sizes; with regard to 2-regular graph, the average approximation ratio of MA-RY-QAOA exceeds that of MA-QAOA for all node sizes; with regard to 4-regular graph, the average approximation ratio of MA-RY-QAOA takes advantage over that of MA-QAOA for overwhelming majority of node sizes; concerning random graph, the average approximation ratio of MA-RY-QAOA is superior to that of MA-QAOA for both small node sizes and large node sizes, but is inferior to that of MA-QAOA for medium node sizes. Therefore, MA-RY-QAOA provides a useful reference for the structure design of shallow-layer QAOA on noisy intermediate-scale quantum (NISQ) devices.
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To address the limited expressive capability of the standard quantum approximate optimization algorithm (QAOA) in shallow circuits and the increased noise sensitivity caused by deeper quantum circuits, an improved QAOA, termed MA-RY-QAOA, was proposed by integrating parameter multi-angleization and an RY-assisted structure. In the proposed algorithm, edge-parameter multi-angleization and RY expansion were incorporated into the problem layer, while qubit-parameter multi-angleization was introduced into the mixing layer. Simulation experiments were conducted on the Max-cut problem of random graphs, complete graphs, 2-regular graphs, and 4-regular graphs with circuit depth ( p=1 <?fx-imagestate width="7.45066643" height="2.87866688"?> <?fx-imagestate width="7.45066643" height="2.87866688"?> ) and node sizes ranging from 4 to 12. The average approximation ratio was adopted as the evaluation metric. The results showed that the average approximation ratios achieved by MA-RY-QAOA and MA-QAOA were generally superior to those obtained by the standard QAOA, RY-QAOA, and QAOA+ algorithms. In particular, MA-RY-QAOA exhibited better performance on 2-regular and 4-regular graphs and achieved higher approximation ratios for certain scales of random graphs. These findings indicate that MA-RY-QAOA improves optimization performance while maintaining a shallow circuit structure and provides a useful reference for the design of QAOA variants on noisy intermediate-scale quantum (NISQ) devices.
The quantum approximate optimization algorithm (QAOA) has emerged as a promising candidate for demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) devices. While various QAOA parameterization schemes exist, ranging from the original single-angle approach to the more expressive multi-angle quantum approximate optimization algorithm (MA-QAOA) and automorphic-angle quantum approximate optimization algorithm (AA-QAOA), each presents distinct trade-offs between expressiveness and classical optimization complexity. In this work, we introduce the kkk-interaction-angle quantum approximate optimization algorithm (kkkA-QAOA), a parameterization scheme that groups cost function terms by their kkk-body interaction order, providing a natural middle ground between parameter efficiency and solution quality. This approach is particularly well-suited for combinatorial optimization problems defined on hypergraphs, where multi-body interactions naturally arise in applications such as Boolean satisfiability and resource allocation with multi-party constraints. We benchmark kkkA-QAOA against standard single-angle quantum approximate optimization algorithm (SA-QAOA), MA-QAOA, and AA-QAOA on two problem classes: 3-uniform cyclic sign-alternating hypergraphs and random coefficient hypergraphs. Our results demonstrate that kkkA-QAOA achieves approximation ratios comparable to MA-QAOA while requiring significantly fewer function evaluations, thereby reducing quantum resource consumption.