Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.
Solving linear systems is at the foundation of many algorithms. Recently, quantum linear system algorithms (QLSAs) have attracted great attention since they converge to a solution exponentially faster than classical algorithms in terms of the problem dimension. However, low-complexity circuit implementations of the oracles assumed in these QLSAs constitute the major bottleneck for practical quantum speed-up in solving linear systems. In this work, we focus on the application of QLSAs for linear systems that are expressed as a low rank tensor sums, which arise in solving discretized PDEs. Previous works uses modified Krylov subspace methods to solve such linear systems with a per-iteration complexity being polylogarithmic of the dimension but with no guarantees on the total convergence cost. We propose a quantum algorithm based on the recent advances on adiabatic-inspired QLSA and perform a detailed analysis of the circuit depth of its implementation. We rigorously show that the total complexity of our implementation is polylogarithmic in the dimension, which is comparable to the per-iteration complexity of the classical heuristic methods.
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.
Finding the solution to linear systems is at the heart of many applications in science and technology. Over the years, a number of algorithms have been proposed to solve this problem on a digital quantum device, yet most of these are too demanding to be applied to the current noisy hardware. In this work, an original algorithmic procedure to solve the quantum linear system problem (QLSP) is presented, which combines ideas from variational quantum algorithms and the framework of classical shadows. The result is the shadow quantum linear solver (SQLS), a quantum algorithm solving the QLSP while avoiding the need for large controlled unitaries and requiring a number of qubits that is logarithmic in the system size. In particular, our heuristics show an exponential advantage of the SQLS in circuit execution per cost function evaluation when compared with other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and the results highlight how the theoretical bounds on the number of resources used by the SQLS are conservative. Finally, we apply this algorithm to a physical problem of practical relevance by leveraging decomposition theorems from linear algebra to solve the discretized Laplace equation in a two-dimensional grid.
Linear equations play a pivotal role in many areas of science and engineering, making efficient solutions to linear systems highly desirable. The development of quantum algorithms for solving linear systems has been a significant breakthrough, with such algorithms rapidly becoming some of the most influential in quantum computing. Subsequent advances have focused on improving the efficiency of quantum linear solvers and extending their core techniques to address various challenges, such as least-squares data fitting. In this article, we introduce a new quantum algorithm for solving linear systems based on the gradient descent method. Inspired by the vector/density state formalism, we represent a point, or vector, as a density state-like entity, enabling us to leverage recent advancements in quantum algorithmic frameworks, such as block encoding, to directly manipulate operators and carry out the gradient descent method in a quantum setting. The operator corresponding to the intermediate solution is updated iteratively, with a provable guarantee of convergence. The quantum state representing the final solution to the linear system can then be extracted by further measurement. We provide a detailed complexity analysis and compare our approach with existing methods, demonstrating significant improvement in certain aspects.
Solving linear systems of equations is central to many engineering and
scientific fields. Several quantum algorithms have been proposed for linear
systems, where the goal is to prepare ∣x⟩ such that A∣x⟩∝∣b⟩. While these algorithms are promising, the time horizon for their
implementation is long due to the required quantum circuit depth. In this work,
we propose a variational hybrid quantum-classical algorithm for solving linear
systems, with the aim of reducing the circuit depth and doing much of the
computation classically. We propose a cost function based on the overlap
between ∣b⟩ and A∣x⟩, and we derive an operational meaning for
this cost in terms of the solution precision ϵ. We also introduce a
quantum circuit to estimate this cost, while showing that this cost cannot be
efficiently estimated classically. Using Rigetti's quantum computer, we
successfully implement our algorithm up to a problem size of 32×32.
Furthermore, we numerically find that the complexity of our algorithm scales
efficiently in both 1/ϵ and κ, with κ the condition
number of A. Our algorithm provides a heuristic for quantum linear systems
that could make this application more near term.
Quantum computing has the potential to speed up some optimization methods. One can use quantum computers to solve linear systems via quantum linear system algorithms (QLSAs). QLSAs can be used as a subroutine for algorithms that require solving linear systems, such as the dual logarithmic barrier method (DLBM) for solving linear optimization (LO) problems. In this paper, we use a QLSA to solve the linear systems arising in each iteration of the DLBM. To use the QLSA in a hybrid setting, we read out quantum states via a tomography procedure that introduces considerable error and noise. Thus, this paper first proposes an inexact feasible variant of DLBM for LO problems and then extends it to a quantum version. Our quantum approach has quadratic convergence toward the central path with inexact directions, and we show that this method has the best known [Formula: see text] iteration complexity, where n is the number of variables, [Formula: see text] is the initial duality gap, and [Formula: see text] is the accuracy. We further use iterative refinement to improve the complexity dependence on accuracy and condition number. For LO problems with quadratically more constraints than variables, the quantum complexity of our method has a sublinear dependence on dimension. Funding: This research was supported by Defense Advanced Research Projects Agency [Grant W911NF2010022] and the National Science Foundation [Grant 2128527].
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Padé approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix exponentials. The proposed algorithm approximates the matrix exponential by the diagonal Padé approximation, which is then encoded into a large, block-sparse linear system and solved via quantum linear system algorithms (QLSA). The detailed quantum circuit is given based on quantum oracle access to the matrix, the inhomogeneous term, and the initial state. The complexity of the proposed algorithm is analyzed. Compared to the method based on Taylor approximation, which approximates the matrix exponential using a k -th order Taylor series, the proposed algorithm improves the approximation order k from two perspectives: 1) the explicit complexity dependency on k is improved, and 2) a smaller k suffices for the same precision. Numerical experiments demonstrate the advantages of the proposed algorithm comparing to other related algorithms.