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arXiv:2207.03404

The Quantum Approximate Optimization Algorithm performance with low entanglement and high circuit depth

Rishi Sreedhar, Pontus Vikstål, Marika Svensson, Andreas Ask, Göran Johansson, Laura García-Álvarez

Published Jul 7, 2022• 8 citations
arXivHTMLPDF

Abstract

Variational quantum algorithms constitute one of the most widespread methods for using current noisy quantum computers. However, it is unknown if these heuristic algorithms provide any quantum-computational speedup, although we cannot simulate them classically for intermediate sizes. Since entanglement lies at the core of quantum computing power, we investigate its role in these heuristic methods for solving optimization problems. In particular, we use matrix product states to simulate the quantum approximate optimization algorithm with reduced bond dimensions $D$, a parameter bounding the system entanglement. Moreover, we restrict the simulation further by deterministically sampling solutions. We conclude that entanglement plays a minor role in the MaxCut and Exact Cover 3 problems studied here since the simulated algorithm analysis, with up to $60$ qubits and $p=100$ algorithm layers, shows that it provides solutions for bond dimension $D \approx 10$ and depth $p \approx 30$. Additionally, we study the classical optimization loop in the approximated algorithm simulation with $12$ qubits and depth up to $p=4$ and show that the approximated optimal parameters with low entanglement approach the exact ones.

DOI
10.48550/arxiv.2207.03404

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Variational quantum algorithms constitute one of the most widespread methods for using current noisy quantum computers. However, it is unknown if these heuristic algorithms provide any quantum-computational speedup, although we cannot simulate them classically for intermediate sizes. Since entanglement lies at the core of quantum computing power, we investigate its role in these heuristic methods for solving optimization problems. In particular, we use matrix product states to simulate the quantum approximate optimization algorithm with reduced bond dimensions DDD, a parameter bounding the system entanglement. Moreover, we restrict the simulation further by deterministically sampling solutions. We conclude that entanglement plays a minor role in the MaxCut and Exact Cover 3 problems studied here since the simulated algorithm analysis, with up to 606060 qubits and p=100p=100p=100 algorithm layers, shows that it provides solutions for bond dimension D≈10D \approx 10D≈10 and depth p≈30p \approx 30p≈30. Additionally, we study the classical optimization loop in the approximated algorithm simulation with 121212 qubits and depth up to p=4p=4p=4 and show that the approximated optimal parameters with low entanglement approach the exact ones.
The role of entanglement in quantum optimization remains actively debated. To address this question, we focus on the fixed-parameter expanding-depth regime of the quantum approximate optimization algorithm (QAOA), where a compact two-parameter schedule is trained once on small instances and then applied as the problem size and circuit depth increase. To probe this regime beyond full state-vector simulation, we perform approximate matrix product state simulations for up to 50 qubits and 100 layers and quantify entanglement by the bond dimension. We observe an entangle--disentangle profile, with the peak bond dimension decreasing with depth and eventually saturating. This observation motivates an extreme approximation: projecting the state onto the product-state manifold (bond dimension one) after every two-qubit interaction. Based on this approximation, we introduce BOND-1, a quantum-inspired classical solver. Despite the drastic simplification, BOND-1 achieves cut ratios above 0.95 relative to the best known values on standard GSet MaxCut benchmarks with up to 20000 variables, and in some cases it matches those values. It achieves these results without per-instance optimization and has linear memory cost, while per-instance tuning can provide further improvement. These results show that, in this regime, a substantial fraction of the optimization power of QAOA survives even in the complete absence of entanglement. Our conclusions, however, are specific to this setting and do not imply that entanglement is unnecessary for quantum optimization in general.
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The advent of quantum computers promises exponential speed ups in the execution of various computational tasks. While their capabilities are hindered by quantum decoherence, they can be exactly simulated on classical hardware at the cost of an exponential scaling in terms of number of qubits. To circumvent this, quantum states can be represented as matrix product states (MPS), a product of tensors separated by so-called bond dimensions. Limiting bond dimensions growth approximates the state, but also limits its ability to represent entanglement. Methods based on this representation have been the most popular tool at simulating large quantum systems. But how to trust resulting approximate quantum states for such intractable systems sizes ? I propose here a method for inferring the fidelity of an approximate quantum state without direct comparison to its exact counterpart, and use it to design an ``entanglement-aware'' (EA) algorithm for both pure and mixed states. As opposed to state of the art methods which limit bond dimensions up to an arbitrary maximum value, this algorithm receives as input a fidelity, and adapts dynamically its bond dimensions to both local entanglement and noise such that the final quantum state fidelity at least reaches the input fidelity. I show that this algorithm far surpasses standard fixed bond dimension truncation schemes. In particular, a noiseless random circuit of 300 qubits and depth 75 simulated using MPS methods takes one week of computation time, while EA-MPS only needs 2 hours to reach similar quantum state fidelity.
Many quantum algorithms seek to output a specific bitstring solving the problem of interest---or a few if the solution is degenerate. It is the case for the quantum approximate optimization algorithm (QAOA) in the limit of large circuit depth, which aims to solve quadratic unconstrained binary optimization problems. Hence, the expected final state for these algorithms is either a product state or a low-entangled superposition involving a few bitstrings. What happens in between the initial NNN-qubit product state ${|0\ensuremath{\rangle}}^{\ensuremath{\bigotimes}N}$ and the final one regarding entanglement? Here, we consider the QAOA algorithm for solving the paradigmatic MaxCut problem on different types of graphs. We study the entanglement growth and spread resulting from randomized and optimized QAOA circuits and find that there is a volume-law entanglement barrier between the initial and final states. We also investigate the entanglement spectrum in connection with random matrix theory. In addition, we compare the entanglement production with a quantum annealing protocol aiming to solve the same MaxCut problems. Finally, we discuss the implications of our results for the simulation of QAOA circuits with tensor network-based methods relying on low-entanglement for efficiency, such as matrix product states.
The trading of fidelity for scale enables approximate classical simulators such as matrix product states (MPSs) to run quantum circuits beyond exact methods.A control parameter, the so-called bond dimension χ for MPSs, governs the allocated computational resources and the output fidelity.Here, we characterize the fidelity for the quantum approximate optimization algorithm by the expectation value of the cost function that it seeks to minimize and find that it follows a scaling law F ln χ N , where N is the number of qubits.With ln χ amounting to the entanglement that a MPS can encode, we show that the relevant variable for investigating the fidelity is the entanglement per qubit.Importantly, our results calibrate the classical computational power required to achieve the desired fidelity and benchmark the performance of quantum hardware in a realistic setup.For instance, we quantify the hardness of performing better classically than a noisy superconducting quantum processor by readily matching its output to the scaling function.Moreover, we relate the global fidelity to that of individual operations and establish its relationship with χ and N .We sharpen the requirements for noisy quantum computers to outperform classical techniques at running a quantum optimization algorithm in speed, size, and fidelity.
The promise of quantum computing to address complex problems requiring high computational resources has long been hindered by the intrinsic and demanding requirements of quantum hardware development. Nonetheless, the current state of quantum computing, denominated Noisy Intermediate-Scale Quantum (NISQ) era, has introduced algorithms and methods that are able to harness the computational power of current quantum computers with advantages over classical computers (referred to as quantum advantage). Achieving quantum advantage is of particular relevance for the combinatorial optimization domain, since it often implies solving an NP-Hard optimization problem. Moreover, combinatorial problems are highly relevant for practical application areas, such as operations research, or resource allocation problems. Among quantum computing methods, Variational Quantum Algorithms (VQA) have emerged as one of the strongest candidates towards reaching practical applicability of NISQ systems. This paper explores the current state and recent developments of VQAs, emphasizing their applicability to combinatorial optimization. We identify the Quantum Approximate Optimization Algorithm (QAOA) as the leading candidate for these problems. Furthermore, we implement QAOA circuits with varying depths to solve the MaxCut problem on graphs with 10 and 20 nodes, demonstrating the potential and challenges of using VQAs in practical optimization tasks. We release our code, dataset and optimized circuit parameters under https://github.com/DanielFPerez/VQA-for-MaxCut.