Abstract Solving large-scale linear systems is an integral part of many scientific disciplines. Classical linear solvers have a polynomial time complexity and it seems impossible to further improve their performance. This fact has directed scientific research to the study of corresponding quantum algorithms, which promise even exponential acceleration compared to existing methods. In this paper, we present and analyze some of the most important related results. In particular, we present the HHL and WZP algorithms based on the eigen-decomposition of a matrix, the row and column iteration methods, as well as two hybrid algorithms that require the cooperation of a classical and a quantum computer, namely an algorithm that uses random walks and the VQLS algorithm. The latter is also examined from an experimental standpoint using the Qiskit open-source framework and a suitable quantum circuit is proposed which can enhance its performance.
Abstract Solving large-scale linear systems is an integral part of many scientific disciplines. Classical linear solvers have a polynomial time complexity and it seems impossible to further improve their performance. This fact has directed scientific research to the study of corresponding quantum algorithms, which promise even exponential acceleration compared to existing methods. In this paper, we present and analyze some of the most important related results. In particular, we present the HHL and WZP algorithms based on the eigen-decomposition of a matrix, the row and column iteration methods, as well as two hybrid algorithms that require the cooperation of a classical and a quantum computer, namely an algorithm that uses random walks and the VQLS algorithm. The latter is also examined from an experimental standpoint using the Qiskit open-source framework and a suitable quantum circuit is proposed which can enhance its performance.
We consider the quantum implementations of two classical iterative solvers for a system of linear equations, including the row (Kaczmarz) method which utilizes a row of the coefficient matrix in each iteration step, and the column (coordinate descent) method which uses a column instead. These two methods are widely applied in big data science due to their simple iteration schemes. We propose fast quantum algorithms for these two approaches by constructing efficient unitary operators in each iteration step based on the block-encoding technique. The construction is based on the unitaries to prepare the quantum states of rows or columns of the coefficient matrix. If the quantum states are efficiently prepared, for example, by qRAM, then the quantum iterative methods achieve an exponential speedup in the problem size n over the classical versions. Meanwhile the dependence on the number of steps is linear, which is the same as the classical counterparts. The complexity of the quantum iterative linear solvers is $O({\ensuremath{\kappa}}_{s}^{2}(logn)\left(log\frac{1}{\ensuremath{\epsilon}}\right))$, where ${\ensuremath{\kappa}}_{s}$ is the scaled condition number, and $\ensuremath{\epsilon}$ is the error. The quantum iterative linear solvers do not depend on Hamiltonian simulations and quantum phase estimation, and also work for the dense case.
We introduce a quantum linear system solving algorithm based on the Kaczmarz method, a widely used workhorse for large linear systems and least-squares problems that updates the solution by enforcing one equation at a time. Its simplicity and low memory cost make it a practical choice across data regression, tomographic reconstruction, and optimization. In contrast to many existing quantum linear solvers, our method does not rely on oracle access to query entries, relaxing a key practicality bottleneck. In particular, when the rank of the system of interest is sufficiently small and the rows of the matrix of interest admit an appropriate structure, we achieve circuit complexity O(ε1logm), where m is the number of variables and ε is the target precision, without dependence on the sparsity s, and could possibly be without explicit dependence on condition number κ. This shows a significant improvement over previous quantum linear solvers where the dependence on κ,s is at least linear. At the same time, when the rows have an arbitrary structure and have at most s nonzero entries, we obtain the circuit depth O(ε1logs) using extra O(s) ancilla qubits, so the depth grows only logarithmically with sparsity s. When the sparsity s grows as O(logm), then our method can achieve an exponential improvement with respect to circuit depth compared to existing quantum algorithms, while using (asymptotically) the same amount of qubits.
Abstract Finding solutions to systems of linear equations is a common problem in many areas of science and engineering, with much potential for a speed up on quantum devices. While the Harrow–Hassidim–Lloyd (HHL) quantum algorithm yields up to an exponential speed up over classical algorithms in some cases, it requires a fault-tolerant quantum computer, which is unlikely to be available in the near-term. Thus, attention has turned to the investigation of quantum algorithms for noisy intermediate-scale quantum (NISQ) devices where several near-term approaches to solving systems of linear equations have been proposed. This paper focuses on the Variational Quantum Linear Solvers (VQLS), and other closely related methods and adaptions. Several contributions are made in this paper, which include: the first application of the Evolutionary Ansatz to the VQLS (EAVQLS), the first implementation of the Logical Ansatz VQLS (LAVQLS), based on the Classical Combination of Quantum States (CQS) method, a proof of principle demonstration of the CQS method on real quantum hardware and a method for the implementation of the Adiabatic Ansatz on the VQLS (AAVQLS). These approaches are implemented and contrasted. The CQS method is run with moderate success on a real quantum device. The EAVQLS and AAVQLS show promise as possible improvements to the standard VQLS algorithm once refined.
Abstract In a sense, gate-based quantum computers are just machines that apply (extremely) large matrices to input states (vectors), and so it is natural to ask whether linear algebra applications can benefit from quantum computing. In the case of solving linear systems, this question has been answered with an emphatic ‘yes’. In 2009, Aram Harrow, Avinatan Hassidim, and Seth Lloyd proposed a quantum algorithm to solve linear systems exponentially faster than the best classical counterpart. The quantum linear systems solver is ubiquitously known by the initials of its inventors, ‘HHL’. As well as detailing HHL, this chapter also explores some of the quantum machine learning applications that have been proposed based on HHL along with results showing that these algorithms are often ‘dequantised’ in the problem space of interest – i.e., there exists an efficient classical ‘quantum-inspired’ algorithm.
A large-scale quantum computer, if constructed, would break the public-key cryptography in everyday use outright. It is therefore essential to gather evidence that current post-quantum schemes truly resist quantum attack. For multivariate schemes, whose security rests on the hardness of solving polynomial systems over a finite field (the PoSSo problem), this boils down to asking how far quantum algorithms can accelerate exact solving. The standard approach consists of linearisation methods such as XL and F4, which reduce solving to linear algebra on large Macaulay matrices; performing that linear algebra with a quantum linear-systems solver such as HHL, rather than classically, is the natural route to a speed-up. The ill-conditioning of those matrices, however, undermines the expected gain. We explore an alternative approach based on symbolic homotopy and Newton–Hensel lifting, a symbolic variant of Newton iteration. A numerical quantum version of it is due to Xue, Wu and Guo; we ask whether an exact one, over a finite field, can sidestep the construction of ill-conditioned matrices altogether. Our answer is that, given the current state of the art, it cannot. The thesis assembles the classical and quantum tools needed for a comparative complexity analysis and clarifies the obstacles that stand in the way of exact solving. The evidence gathered is that the quantum paradigms surveyed here do not threaten, at their current parameters, multivariate UOV-based schemes such as MAYO; a quantum attack on them, should one exist, would have to exploit their structure directly.
In the field of quantum linear system algorithms, quantum computing has realized exponential computational advantages over classical computing. However, the focus has been on square coefficient matrices, with few quantum algorithms addressing nonsquare matrices. Towards this kind of problems defined by Ax=b where $A\ensuremath{\in}{\mathbb{R}}^{m\ifmmode\times\else\texttimes\fi{}n}$, we propose a quantum algorithm inspired by the classical multirow iteration method and provide an explicit quantum circuit based on the quantum comparator and quantum random access memory. The time complexity of our quantum multirow iteration algorithm is O(Klog2m), with K representing the number of iteration steps, which demonstrates an exponential speedup compared to the classical version.. Based on the convergence of the classical multirow iteration algorithm, we prove that our quantum algorithm converges faster than the quantum one-row iteration algorithm presented by Shao and Xiang [C. Shao and H. Xiang, Phys. Rev. A 101, 022322 (2020)]. Moreover, our algorithm places less demand on the coefficient matrix, making it suitable for solving inconsistent systems and quadratic optimization problems.