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Solving Computationally Hard Problems Using Quantum Computing

Angadh Singh

Published Jan 1, 2024• 0 citations
OpenAlex

Abstract

Some problems are computationally solvable but to complexity of nature of problem, they take a long time to solve. Quantum computers (QC) offer the ability to solve these problems in reasonable time using principles of quantum mechanics. One such hard problem is MaxCut, with various applications like portfolio optimization, drug discovery, and inventory management. This poster dives into solving these problems on QC with introduction to how problems are encoded mathematically (QUBO); converted into quantum language (Hamiltonian) and solved using quantum algorithm (QAOA).The goal of research was to run MaxCut problem on different variants of a quantum algorithms to analyze results.

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Some problems are computationally solvable but to complexity of nature of problem, they take a long time to solve. Quantum computers (QC) offer the ability to solve these problems in reasonable time using principles of quantum mechanics. One such hard problem is MaxCut, with various applications like portfolio optimization, drug discovery, and inventory management. This poster dives into solving these problems on QC with introduction to how problems are encoded mathematically (QUBO); converted into quantum language (Hamiltonian) and solved using quantum algorithm (QAOA).The goal of research was to run MaxCut problem on different variants of a quantum algorithms to analyze results.
Quantum computing has the potential to improve financial portfolio optimization by addressing scalability and computational complexity issues. This article explores the application of quantum algorithms to portfolio optimization. It begins by discussing the limitations of classical optimization methods and introduces the basics of quantum computing. Two key quantum algorithms, quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA), are presented in detail. These algorithms are applied to solve the Quadratic Unconstrained Binary Optimization (QUBO) formulation of the portfolio optimization problem. The article provides a high-level quantum algorithm, along with its pseudo-code Python implementation. The potential computational speedup of quantum algorithms is analyzed, highlighting the theoretical quadratic speedup over classical methods. However, the article also acknowledges the challenges and limitations currently facing quantum computing. It also concludes by emphasizing the promising future of quantum computing in finance and encourages further research to unlock the full potential of quantum technologies in portfolio optimization and other complex financial problems.
Optimization problems in finance, physics, and computer science are typically very hard to tackle in classical computing; quantum computing could help speed up computations and provide efficient methods for tackling large problems. Typically, to treat a problem with a quantum computer, the optimal solution is cast as the ground state of a diagonal Hamiltonian. We develop a method, called imaginary-time evolution block encoding (ITE-BE), based on a recent imaginary-time algorithm, which requires no variational parameter optimization, as all parameters can be derived analytically from the target Hamiltonian. We also demonstrate that our method can be successfully combined with other quantum algorithms such as the quantum approximate optimization algorithm (QAOA). For illustration, here we study the MaxCut problem. We find that the QAOA ansatz increases the postselection success of ITE-BE, and shallow QAOA circuits, when boosted with ITE-BE, achieve better performance than deeper QAOA circuits. For the special case of the transverse initial state, we adapt our block-encoding scheme to allow for a deterministic application of the first layer of the circuit.
Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions can be assembled to form a single Polynomial Binary Unconstrained Optimization instance with fewer variables. This technique can be applied to any QUBO instance and leads to either an all-classical or a hybrid quantum-classical approach. When quantum heuristics like the Quantum Approximate Optimization Algorithm (QAOA) are used, our proposal leads to a double advantage: a substantial reduction of quantum resources, specifically an average of ~42% fewer qubits to solve MaxCut on random 3-regular graphs, together with an improvement in the quality of the approximate solutions reached.
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.
The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Because of the Z2{\mathbb{Z}}_{2}Z2​ symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. In view of this, we propose QAOA-in-QAOA (QAOA2{\mathrm{QAOA}}^{2}QAOA2) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of QAOA2{\mathrm{QAOA}}^{2}QAOA2 is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, QAOA2{\mathrm{QAOA}}^{2}QAOA2 attains a competitive or even better performance over classical algorithms when the node count is around 200020002000. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems.