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Scientific Reports

Range dependent Hamiltonian algorithms for numerical QUBO formulation

Hyunju Lee, Kyungtaek Jun

Published Mar 14, 2025• 1 citation
Publisher

Abstract

With the advent and development of quantum computers, various quantum algorithms that can solve linear equations and eigenvalues faster than classical computers have been developed. In particular, a hybrid solver provided by D-Wave's Leap quantum cloud service can utilize up to two million variables. Using this technology, quadratic unconstrained binary optimization (QUBO) models have been proposed for linear systems, eigenvalue problems, RSA cryptosystems, and computed tomography (CT) image reconstructions. Generally, QUBO formulation is obtained through simple arithmetic operations, which offers great potential for future development with the progress of quantum computers. A common method here was to binarize the variables and match them to multiple qubits. To achieve the accuracy of 64 bits per variable, 64 logical qubits must be used. Finding the global minimum energy in quantum optimization becomes more difficult as more logical qubits are used; thus, a quantum parallel computing algorithm that can create and compute multiple QUBO models is introduced here. This new algorithm divides the entire domain each variable can have into multiple subranges to generate QUBO models. This paper demonstrates the superior performance of this new algorithm particularly when utilizing an algorithm for binary variables.

DOI
10.1038/s41598-025-93552-x

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With the advent and development of quantum computers, various quantum algorithms that can solve linear equations and eigenvalues faster than classical computers have been developed. In particular, a hybrid solver provided by D-Wave's Leap quantum cloud service can utilize up to two million variables. Using this technology, quadratic unconstrained binary optimization (QUBO) models have been proposed for linear systems, eigenvalue problems, RSA cryptosystems, and computed tomography (CT) image reconstructions. Generally, QUBO formulation is obtained through simple arithmetic operations, which offers great potential for future development with the progress of quantum computers. A common method here was to binarize the variables and match them to multiple qubits. To achieve the accuracy of 64 bits per variable, 64 logical qubits must be used. Finding the global minimum energy in quantum optimization becomes more difficult as more logical qubits are used; thus, a quantum parallel computing algorithm that can create and compute multiple QUBO models is introduced here. This new algorithm divides the entire domain each variable can have into multiple subranges to generate QUBO models. This paper demonstrates the superior performance of this new algorithm particularly when utilizing an algorithm for binary variables.
In the era of quantum computing, the emergence of quantum computers and subsequent advancements have led to the development of various quantum algorithms capable of solving linear equations and eigenvalues, surpassing the pace of classical computers. Notably, the hybrid solver provided by the D-wave system can leverage up to two million variables. By exploiting this technology, quantum optimization models based on quadratic unconstrained binary optimization (QUBO) have been proposed for applications, such as linear systems, eigenvalue problems, RSA cryptosystems, and CT image reconstruction. The formulation of QUBO typically involves straightforward arithmetic operations, presenting significant potential for future advancements as quantum computers continue to evolve. A prevalent approach in these developments is the binarization of variables and their mapping to multiple qubits. These methods increase the required number of qubits as the range and precision of each variable increase. Determining the optimal value of a QUBO model becomes more challenging as the number of qubits increases. Furthermore, the accuracies of the existing Qiskit simulator, D-Wave system simulator, and hybrid solver are limited to two decimal places. Problems arise because the qubits yielding the optimal value for the QUBO model may not necessarily correspond to the solution of a given problem. To address these issues, we propose a new iterative algorithm. The novel algorithm sequentially progresses from the highest to the lowest exponent in binarizing each number, whereby each number is calculated using two variables, and the accuracy can be computed up to a maximum of 16 decimal places.
With the advent and development of quantum computers, various quantum algorithms that can solve linear equations and eigenvalues faster than classical computers have been developed. The Harrow-Hassidim-Lloyd algorithm is an algorithm that can solve linear equations in a gate model quantum computer. Still, it is constrained by the use of quantum RAM and the size limit of the matrix according to the total number of qubits in the quantum computer. Recently, Jun and Lee developed a QUBO model for solving linear systems and eigenvalue problems in the quantum computer. However, even though their model uses 2048 qubits, the number of qubits for variables that can be used for the problem is only 64. To solve this problem, we introduce an algorithm that can be used by dividing the size of the entire domain according to the number of qubits. We also form a QUBO model related to each subregion.
The emergence of quantum annealers has catalyzed the development of quantum optimization algorithms across various fields. The formulation of a quadratic unconstrained binary optimization (QUBO) model for quantum linear systems suggests that quantum optimization algorithms can play an important role in science and engineering. Recently, a QUBO model was developed by applying a quantum linear system to computed tomography (CT) image reconstruction to clearly show the internal structure of the sample. To date, ultra-clear CT images have not been reconstructed using quantum optimization algorithms owing to limitations in logical qubits. However, the quantum CT image reconstruction algorithm using a quantum linear system provides confidence in the accuracy of the CT image by leveraging the difference between the global and theoretical minimum energies. In this paper, we propose an algorithm that can use two or more QUBO models for CT image reconstruction. The proposed algorithm formulates several QUBO models from CT image reconstruction and verifies them using the hybrid solver of the D-Wave system. The proposed algorithm suggests that quantum-optimized CT algorithms can make greater progress in the future.
With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all science and engineering. The Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on gate model quantum computers, was invented and several advanced variations have been developed. For a given n by n matrix A and a vector b, we will find unconstrained binary optimization (QUBO) models for a vector x that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, we provide a Python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in the D-Wave system.
Abstract With the advent of quantum computers, many quantum computing algorithms are being developed. Solving linear systems is one of the most fundamental problems in almost all of science and engineering. Harrow-Hassidim-Lloyd algorithm, a monumental quantum algorithm for solving linear systems on the gate model quantum computers, was invented and several advanced variations have been developed. For a given square matrix A∈R(n×n) and a vector b∈R(n), we will find unconstrained binary optimization (QUBO) models for a vector x∈R(n) that satisfies Ax=b. To formulate QUBO models for a linear system solving problem, we make use of a linear least-square problem with binary representation of the solution. We validate those QUBO models on the D-Wave system and discuss the results. For a simple system, We provide a python code to calculate the matrix characterizing the relationship between the variables and to print the test code that can be used directly in D-Wave system.
Abstract Quantum computers have strict requirements for the problems that they can efficiently solve. One of the principal limiting factor for the performances of noisy intermediate-scale quantum (NISQ) devices is the number of qubits required by the running algorithm. Several combinatorial optimization problems can be solved with NISQ devices once that a corresponding quadratic unconstrained binary optimization (QUBO) form is derived. Numerous techniques have been proposed to achieve such reformulations and, depending on the method chosen, the number of binary variables required, and therefore of qubits, can vary considerably. The aim of this work is to drastically reduce the variables needed for these QUBO reformulations in order to unlock the possibility to efficiently obtain optimal solutions for a class of optimization problems with NISQ devices. This goal is achieved by introducing novel tools that allow an efficient use of slack variables, even for problems with non-linear constraints, without the need to approximate the starting problem. We divide our new techniques in two independent parts, called the iterative quadratic polynomial and the master-satellite methods. Hence, we show how to apply our techniques in case of an NP-hard optimization problem inspired by a real-world financial scenario called Max-Profit Balance Settlement . We follow by submitting several instances of this problem to two D-wave quantum annealers, comparing the performances of our novel approach with the standard methods used in these scenarios. Moreover, this study allows to appreciate several performance differences between the D-wave Advantage and next-generation Advantage2 quantum annealers. We show that the adoption of our techniques in this context allows to obtain QUBO formulations with significantly fewer slack variables, i.e. around 90% less, and D-wave annealers provide considerably higher correct solution rates, which moreover do not decrease with the input size as fast as when adopting standard techniques.