Quantum algorithms for solving linear systems of equations have generated excitement because of the potential speed-ups involved and the importance of solving linear equations in many applications. However, applying these algorithms can be challenging. The Harrow-Hassidim-Lloyd algorithm and improvements thereof require complex subroutines suitable for fault-tolerant hardware such as Hamiltonian simulation, making it ill-suited to current hardware. Variational algorithms, on the other hand, involve expensive optimization loops, which can be prone to barren plateaus and local optima. We describe a quantum algorithm for solving linear systems of equations that avoids these problems. Our algorithm is based on the Woodbury identity, which analytically describes the inverse of a matrix that is a low-rank modification of another (easily-invertible) matrix. This approach only utilizes basic quantum subroutines like the Hadamard test or the swap test, so it is well-suited to current hardware. There is no optimization loop, so barren plateaus and local optima do not present a problem. The low-rank aspect of the identity enables us to efficiently transfer information to and from the quantum computer. This approach can produce accurate results on current hardware. As evidence of this, we estimate an inner product involving the solution of a system of more than 16 million equations with 2% error using IBM's Auckland quantum computer. To our knowledge, no system of equations this large has previously been solved to this level of accuracy on a quantum computer.
Quantum algorithms for solving linear systems of equations have generated excitement because of the potential speed-ups involved and the importance of solving linear equations in many applications. However, applying these algorithms can be challenging. The Harrow-Hassidim-Lloyd algorithm and improvements thereof require complex subroutines suitable for fault-tolerant hardware such as Hamiltonian simulation, making it ill-suited to current hardware. Variational algorithms, on the other hand, involve expensive optimization loops, which can be prone to barren plateaus and local optima. We describe a quantum algorithm for solving linear systems of equations that avoids these problems. Our algorithm is based on the Woodbury identity, which analytically describes the inverse of a matrix that is a low-rank modification of another (easily-invertible) matrix. This approach only utilizes basic quantum subroutines like the Hadamard test or the swap test, so it is well-suited to current hardware. There is no optimization loop, so barren plateaus and local optima do not present a problem. The low-rank aspect of the identity enables us to efficiently transfer information to and from the quantum computer. This approach can produce accurate results on current hardware. As evidence of this, we estimate an inner product involving the solution of a system of more than 16 million equations with 2% error using IBM's Auckland quantum computer. To our knowledge, no system of equations this large has previously been solved to this level of accuracy on a quantum computer.
Quantum computation offers a promising alternative to classical computing methods in many areas of numerical science, with algorithms that make use of the unique way in which quantum computers store and manipulate data often achieving dramatic improvements in performance over their classical counterparts. The potential efficiency of quantum computers is particularly important for numerical simulations, where the capabilities of classical computing systems are often insufficient for the analysis of real-world problems. In this work, we study problems involving the solution of matrix equations, for which there currently exists no efficient, general quantum procedure. We develop a generalization of the Harrow/Hassidim/Lloyd algorithm by providing an alternative unitary for eigenphase estimation. This unitary, which we have adopted from research in the area of quantum walks, has the advantage of being well defined for any arbitrary matrix equation, thereby allowing the solution procedure to be directly implemented on quantum hardware for any well-conditioned system. The procedure is most useful for sparse matrix equations, as it allows for the inverse of a matrix to be applied with O(Nnzlog(N)) complexity, where N is the number of unknowns, and Nnz is the total number of nonzero elements in the system matrix. This efficiency is independent of the matrix structure, and hence the quantum procedure can outperform classical methods for many common system types. We show this using the example of sparse approximate inverse (SPAI) preconditioning, which involves the application of matrix inverses for matrices with Nnz=O(N).
With the rapid development of Quantum Computers (QC) and QC Simulators, there will be an increased demand for functioning Quantum Algorithms in the near future. Some of the most ubiquitously useful algorithms are solvers for linear systems of equations. Since the conception of the Quantum Linear Solver Algorithm (QLSA) by Harrow, Hassidim and Lloyd (HHL) in 2009, many improvements have been made, although a generic implementation for arbitrary matrices and vectors is still not available. In this thesis a variant of the HHL QLSA is studied, and the open challenges are investigated. Solutions for two of the challenges, namely the Eigenvalue Inversion subroutine and the Higher-Order Ancilla Rotation subroutine, are discussed. As part of the thesis project, these subroutines have been implemented in the QX Quantum Computer Simulator, and the subroutines are combined to form a complete Quantum Linear Solver (QLS), with the restraint that the implementation for the vector and Hamiltonian of the matrix must be provided by the user. A proof-of-concept QLS by Cao et al. is also implemented in the QX simulator, and using the implementation of the vector and Hamiltonian of Cao et al. the complete solver is tested. In the process of this thesis, a framework for basic Quantum Arithmetic is built providing three variants of Integer Adders, two variants of Integer Subtracters, one Integer Multiplier and one Integer Divider. In addition, gates not natively available in the QX simulator are implemented, and a number of improvements and extensions of algorithms presented in the literature are given, making the described algorithms function on the QX simulator and extending features.
Solving linear systems of equations is essential for many problems in science and technology, including problems in machine learning. Existing quantum algorithms have demonstrated the potential for large speedups, but the required quantum resources are not immediately available on near-term quantum devices. In this work, we study near-term quantum algorithms for linear systems of equations of the form Ax=b. We investigate the use of variational algorithms and analyze their optimization landscapes. There exist types of linear systems for which variational algorithms designed to avoid barren plateaus, such as properly-initialized imaginary time evolution and adiabatic-inspired optimization, suffer from a different plateau problem. To circumvent this issue, we design near-term algorithms based on a core idea: the classical combination of variational quantum states (CQS). We exhibit several provable guarantees for these algorithms, supported by the representation of the linear system on a so-called Ansatz tree. The CQS approach and the Ansatz tree also admit the systematic application of heuristic approaches, including a gradient-based search. We have conducted numerical experiments solving linear systems as large as 2300×2300 by considering cases where we can simulate the quantum algorithm efficiently on a classical computer. These experiments demonstrate the algorithms' ability to scale to system sizes within reach in near-term quantum devices of about 100-300 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.
Solving systems of linear equations is a key subroutine in many quantum algorithms. In the last 15 years, many quantum linear solvers (QLS) have been developed, competing to achieve the best asymptotic worst-case complexity. Most QLS assume fault-tolerant quantum computers, so they cannot yet be benchmarked on real hardware. Because an algorithm with better asymptotic scaling can underperform on instances of practical interest, the question of which of these algorithms is the most promising remains open. In this work, we implement a method to partially address this question. We consider four well-known QLS algorithms which directly implement an approximate matrix inversion function: the Harrow-Hassidim-Lloyd algorithm, two algorithms utilizing a linear combination of unitaries, and one utilizing the quantum singular value transformation (QSVT). These methods, known as functional QLS, share nearly identical assumptions about the problem setup and oracle access. Their computational cost is dominated by query calls to a matrix oracle encoding the problem one wants to solve. We provide formulas to count the number of queries needed to solve specific problem instances; these can be used to benchmark the algorithms on real-life instances, under benevolent, idealized assumptions, without requiring access to quantum hardware to implement the procedure. We select three data sets: random generated instances that obey the assumptions of functional QLS, linear systems from simplex iterations on MIPLIB, and Poisson equations. Our methods can be easily extended to other data sets and provide a high-level guide to evaluate the performance of a QLS algorithm. In particular, our work shows that HHL underperforms in comparison to the other methods across all data sets, often by orders of magnitude, while the Chebyshev-approximation based methods typically have the best performance.
The Harrow-Hassidim-Lloyd (HHL) quantum algorithm for sampling from the solution of a linear system provides an exponential speed-up over its classical counterpart. The problem of solving a system of linear equations has a wide scope of applications, and thus HHL constitutes an important algorithmic primitive. In these notes, we present the HHL algorithm and its improved versions in detail, including explanations of the constituent sub- routines. More specifically, we discuss various quantum subroutines such as quantum phase estimation and amplitude amplification, as well as the important question of loading data into a quantum computer, via quantum RAM. The improvements to the original algorithm exploit variable-time amplitude amplification as well as a method for implementing linear combinations of unitary operations (LCUs) based on a decomposition of the operators using Fourier and Chebyshev series. Finally, we discuss a linear solver based on the quantum singular value estimation (QSVE) subroutine.