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arXiv:2005.10780 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Quantum-Enhanced Simulation-Based Optimization

Julien Gacon, Christa Zoufal, Stefan Woerner

Published Oct 1, 2020• 18 citations
arXivHTMLPDF

Abstract

In this paper, we introduce a quantum-enhanced algorithm for simulation-based optimization. Simulation-based optimization seeks to optimize an objective function that is computationally expensive to evaluate analytically, and thus, is approximated using simulation techniques such as Monte Carlo simulation. Quantum Amplitude Estimation (QAE) can achieve a quadratic speed-up over classical Monte Carlo simulation. Hence, in many cases, it can achieve a speed-up for simulation-based optimization as well. Combining QAE with ideas from quantum combinatorial optimization, we show how this can be used not only for continuous but also for discrete optimization problems. Furthermore, the algorithm is demonstrated on illustrative problems such as portfolio optimization with a Value at Risk constraint and inventory management.

DOI
10.1109/qce49297.2020.00017

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In this paper, we introduce a quantum-enhanced algorithm for simulation-based optimization. Simulation-based optimization seeks to optimize an objective function that is computationally expensive to evaluate analytically, and thus, is approximated using simulation techniques such as Monte Carlo simulation. Quantum Amplitude Estimation (QAE) can achieve a quadratic speed-up over classical Monte Carlo simulation. Hence, in many cases, it can achieve a speed-up for simulation-based optimization as well. Combining QAE with ideas from quantum combinatorial optimization, we show how this can be used not only for continuous but also for discrete optimization problems. Furthermore, the algorithm is demonstrated on illustrative problems such as portfolio optimization with a Value at Risk constraint and inventory management.
Simulation-based optimization is a widely used method to solve stochastic optimization problems. This method aims to identify an optimal solution by maximizing the expected value of the objective function. However, due to its computational complexity, the function cannot be accurately evaluated directly, hence is estimated through simulation. Exploiting the enhanced efficiency of Quantum Amplitude Estimation (QAE) compared to classical Monte Carlo simulation, it frequently outpaces classical simulation-based optimization, resulting in notable performance enhancements in various scenarios. In this work, we use a quantum-enhanced algorithm for simulation-based optimization and apply it to solve a variant of the classical Newsvendor Problem known to be NP-hard. Such problems provide the building block for supply chain management, particularly in inventory management and procurement optimization under risks and uncertainty.
We investigate the feasibility of integrating quantum algorithms as subroutines of simulation-based optimisation problems with relevance to and potential applications in mathematical finance. To this end, we conduct a thorough analysis of all systematic errors arising in the formulation of quantum Monte Carlo integration in order to better understand the resources required to encode various distributions such as a Gaussian, and to evaluate statistical quantities such as the Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR) of an asset. Finally, we study the applicability of quantum Monte Carlo integration for fundamental financial use cases in terms of simulation-based optimisations, notably Mean-Conditional-Value-at-Risk (Mean-CVaR) and (risky) Mean-Variance (Mean-Var) optimisation problems. In particular, we study the Mean-Var optimisation problem in the presence of noise on a quantum device, and benchmark a quantum error mitigation method that applies to quantum amplitude estimation -- a key subroutine of quantum Monte Carlo integration -- showcasing the utility of such an approach.
Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.
Abstract Quantum computing is increasingly discussed as a possible source of new computational methods for economics, but its relevance depends on whether economic bottlenecks match quantum algorithmic primitives. This critical review evaluates that match for computational economics. It synthesizes work on quantum amplitude estimation and quantum Monte Carlo for simulation and risk measurement; quadratic unconstrained binary optimization (QUBO), Ising formulations, quantum annealing, and the quantum approximate optimization algorithm for discrete optimization; and quantum linear-algebra and machine-learning primitives for econometrics and model solution. The review argues that there is no general-purpose quantum algorithm for economics analogous to Shor's algorithm for factoring. Credible claims are problem-class specific and depend on state preparation, output requirements, conditioning, hardware noise, embedding overhead, and comparison with strong classical and quantum-inspired baselines. The most mature economic applications are derivative pricing and financial risk, payment-system liquidity optimization, quantum Monte Carlo for stress testing, and early quantum-annealing formulations of dynamic programming. Transportation and infrastructure applications are relevant mainly as networked economic optimization problems. The article proposes an evidence ladder for quantum computational economics and a research agenda centered on reproducible benchmarks, economically meaningful objective functions, end-to-end resource accounting, and evaluation by economic value rather than hardware novelty. JEL Classification: C02; C15; C61; C63; C88; E37; G11; G17; R41
This article proposes a new, more efficient quantum algorithm for portfolio optimization and uses quantum combinatorial optimization (QCO) techniques. The recent construction developed in 2021 has sparked the field of financial portfolio optimization through the Quantum Walk Optimization Algorithm (QWOA). We study the complexity and efficiency of quantum optimization algorithms with a special interest in QWOA. The objective is to minimize the risk of investing by having a good combination of assets in the portfolio. We also focus on reducing the number of iterations yet attaining a high-quality resolution through contraction of the solution space that eases computations. We extend QWOA’s idea by constructing a newly outperforming scheme known as ’Quantum Mix Optimization Algorithm (QMOA)’. QMOA algorithm codes are provided for the implementation and simulation of numerical results. In addition, QMOA’s efficiency (better than the existing QCO algorithms) is discussed. For instance, the least QWOA number of computations required to execute the initial state equation is p > 2 while this value is p ≥ 2 in the proposed QMOA.
In the post-pandemic world, manufacturing enterprises face increasing uncertainties, especially with vulnerabilities in global supply chains. Although supply chain management has been extensively studied, the critical influence of decision-makers (DMs) in these systems remains underexplored. This study studies the inventory management problem under risk using the newsvendor model by incorporating DMs risk preferences. By employing the Quantum Monte Carlo (QMC) combined with Quantum Amplitude Estimation (QAE) algorithm, the estimation of probabilities or expectation values can be done more efficiently. This offers near-quadratic speedup compared to classical Monte Carlo methods. Our findings illuminate the intricate relationship between risk-aware decision-making and inventory management, providing essential insights for enhancing supply chain resilience and adaptability in uncertain conditions