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arXiv:2106.02812

Optimizing Ansatz Design in QAOA for Max-cut

Ritajit Majumdar, Dhiraj Madan, Debasmita Bhoumik, Dhinakaran Vinayagamurthy, Shesha Raghunathan, Susmita Sur‐Kolay

Published Jun 5, 2021• 18 citations
arXivHTMLPDF

Abstract

Quantum Approximate Optimization Algorithm (QAOA) is studied primarily to find approximate solutions to combinatorial optimization problems. For a graph with $n$ vertices and $m$ edges, a depth $p$ QAOA for the Max-cut problem requires $2\cdot m \cdot p$ CNOT gates. CNOT is one of the primary sources of error in modern quantum computers. In this paper, we propose two hardware independent methods to reduce the number of CNOT gates in the circuit. First, we present a method based on Edge Coloring of the input graph that minimizes the the number of cycles (termed as depth of the circuit), and reduces upto $\lfloor \frac{n}{2} \rfloor$ CNOT gates. Next, we depict another method based on Depth First Search (DFS) on the input graph that reduces $n-1$ CNOT gates, but increases depth of the circuit moderately. We analytically derive the condition for which the reduction in CNOT gates overshadows this increase in depth, and the error probability of the circuit is still lowered. We show that all IBM Quantum Hardware satisfy this condition. We simulate these two methods for graphs of various sparsity with the \textit{ibmq\_manhattan} noise model, and show that the DFS based method outperforms the edge coloring based method, which in turn, outperforms the traditional QAOA circuit in terms of reduction in the number of CNOT gates, and hence the probability of error of the circuit.

DOI
10.48550/arxiv.2106.02812

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Quantum Approximate Optimization Algorithm (QAOA) is studied primarily to find approximate solutions to combinatorial optimization problems. For a graph with nnn vertices and mmm edges, a depth ppp QAOA for the Max-cut problem requires 2⋅m⋅p2\cdot m \cdot p2⋅m⋅p CNOT gates. CNOT is one of the primary sources of error in modern quantum computers. In this paper, we propose two hardware independent methods to reduce the number of CNOT gates in the circuit. First, we present a method based on Edge Coloring of the input graph that minimizes the the number of cycles (termed as depth of the circuit), and reduces upto ⌊n2⌋\lfloor \frac{n}{2} \rfloor⌊2n​⌋ CNOT gates. Next, we depict another method based on Depth First Search (DFS) on the input graph that reduces n−1n-1n−1 CNOT gates, but increases depth of the circuit moderately. We analytically derive the condition for which the reduction in CNOT gates overshadows this increase in depth, and the error probability of the circuit is still lowered. We show that all IBM Quantum Hardware satisfy this condition. We simulate these two methods for graphs of various sparsity with the \textit{ibmq_manhattan} noise model, and show that the DFS based method outperforms the edge coloring based method, which in turn, outperforms the traditional QAOA circuit in terms of reduction in the number of CNOT gates, and hence the probability of error of the circuit.
While a Quantum Approximate Optimization Algorithm (QAOA) is intended to provide a quantum advantage in finding approximate solutions to combinatorial optimization problems, noise in the system is a hurdle in exploiting its full potential. Several error mitigation techniques have been studied to lessen the effect of noise on this algorithm. Recently, Majumdar et al. proposed a Depth First Search (DFS) based method to reduce n−1n-1n−1 CNOT gates in the ansatz design of QAOA for finding Max-Cut in a graph G = (V, E), |V| = n. However, this method tends to increase the depth of the circuit, making it more prone to relaxation error. The depth of the circuit is proportional to the height of the DFS tree, which can be n−1n-1n−1 in the worst case. In this paper, we propose an O(Δ⋅n2)O(Δ\cdot n^2)O(Δ⋅n2) greedy heuristic algorithm, where ΔΔΔ is the maximum degree of the graph, that finds a spanning tree of lower height, thus reducing the overall depth of the circuit while still retaining the n−1n-1n−1 reduction in the number of CNOT gates needed in the ansatz. We numerically show that this algorithm achieves nearly 10 times increase in the probability of success for each iteration of QAOA for Max-Cut. We further show that although the average depth of the circuit produced by this heuristic algorithm still grows linearly with n, our algorithm reduces the slope of the linear increase from 1 to 0.11.
Quantum Approximate Optimization Algorithm (QAOA) is a prospective candidate for providing quantum advantage in finding approximate solutions to optimization problems using near-term quantum devices. In this paper, the goal is to reduce the number of CNOT gates and the depth of quantum ansatz circuits for QAOA. First, we present a generalized QAOA formulation for any Hamiltonian that involves upto two-body interactions, as a graph. The circuit realization of the depth-p QAOA requires 2mp2mp2mp CNOT gates where the graph for the Hamiltonian has n vertices and m edges. Presently, a CNOT gate is one of the primary sources of error. We propose a graph-theoretic algorithm which can eliminate n−1n-1n−1 CNOTs from the QAOA ansatz while retaining functional equivalence with the original circuit. This improves upon previously proposed Depth First Search (DFS) method by restricting the increase in depth of the circuit, which was a drawback of the method, by ≃84.8%\simeq 84.8\%≃84.8% for Erdős-Renyi graphs. Finally, if the underlying connectivity (hardware coupling map) and the initial qubit placement for a hardware are known, then we present how to extend the greedy heuristic method in order to obtain a functionally equivalent circuit with ≃5%\simeq 5\%≃5%; fewer SWAP gates for Erdős-Renyi graphs.
Quantum computing has emerged as a promising paradigm to tackle computationally intensive problems that classical computers struggle with. In this study, we explore the application of the Edge-based quantum approximate optimization algorithm (QAOA) to the MAX-CUT problem, a well-known combinatorial optimization challenge. MAX-CUT aims to partition the vertices of a graph into two subsets such that the number of edges between the subsets is maximized. We define the edge-based MAX-CUT problem and propose a method for applying QAOA specifically tailored to this formulation. We conduct simulations using IBM’s Qiskit framework, examining both vertex-based and edge-based QAOA implementations across various graph structures. Our results highlight the comparative performance of these approaches in terms of solution quality and computational efficiency. Specifically, we analyze the impact of different graph sizes and edge densities on the complexity and CNOT gate counts of the proposed algorithms. This analysis provides insights into leveraging quantum computing for combinatorial optimization tasks, particularly focusing on the implications for practical applications and future research directions.
The max-cut problem (MCP) is a classic problem in the field of combinatorial optimization and has important applications in various fields, including statistical physics and image processing. However, except for some special cases, the MCP still encounters a non-deterministic polynomial complete problem (an NP-complete problem), and there is currently no known efficient classical algorithm that can solve it in polynomial time. The quantum approximate optimization algorithm (QAOA), as a pivotal algorithm in the noisy intermediate-scale quantum (NISQ) computing era, has shown significant potential for solving the MCP. However, due to the lack of quantum error correction, the reliability of computations in NISQ systems sharply declines as the circuit depth of the algorithm increases. Therefore, designing an efficient, shallow-depth, and low-complexity QAOA for the MCP is a critical challenge in demonstrating the advantages of quantum computing in the NISQ era.In this paper, according to the standard QAOA algorithm, we introduce Pauli Y rotation gates into the target Hamiltonian circuit for the MCP. By enhancing the flexibility of quantum trial functions and improving the efficiency of Hilbert space exploration within a single iteration, we significantly improve the performance of QAOA on the MCP.We conduct extensive numerical simulations using the MindSpore quantum platform, and compare the proposed RY-layer-assisted QAOA with standard QAOA and its existing variants, including MA-QAOA and QAOA+. The experiments are performed on various graph types, including complete graphs, 3-regular graphs, 4-regular graphs, and random graphs with edge probabilities between 0.3 and 0.5. Our results show that the RY-layer-assisted QAOA achieves higher approximation ratios in all graph types, particularly in regular and random graphs, where traditional QAOA variants are difficult to implement. Moreover, the proposed method exhibits strong robustness as the graph size increases, and can maintain high performance even for larger graphs. Importantly, the RY-layer-assisted QAOA requires fewer CNOT gates and has a lower circuit depth than the standard QAOA and its variants, making it more suitable for NISQ devices with limited coherence times and high error rates.In conclusion, the RY-layer-assisted QAOA provides a promising approach for solving MCP in the NISQ era. By improving the approximation ratio while reducing circuit complexity, this method demonstrates significant potential for practical quantum computing applications, thus paving the way for developing more efficient and reliable quantum optimization algorithms.
The quantum approximate optimization algorithm (QAOA) is a hybrid variational quantum-classical algorithm that solves combinatorial optimization problems. While there is evidence suggesting that the fixed form of the standard QAOA ansatz is not optimal, there is no systematic approach for finding better ansätze. We address this problem by developing an iterative version of QAOA that is problem-tailored, and which can also be adapted to specific hardware constraints. We simulate the algorithm on a class of Max-Cut graph problems and show that it converges much faster than the standard QAOA, while simultaneously reducing the required number of CNOT gates and optimization parameters. We provide evidence that this speedup is connected to the concept of shortcuts to adiabaticity.
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.