Resource-efficient quantum algorithm for linear systems of equations

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 (VQA) and the framework of classical shadows. The result is the Shadow Quantum Linear Solver (SQLS), a quantum algorithm solving the QLSP avoiding the need for large controlled unitaries, 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 to other notorious variational approaches to solving linear systems of equations. We test the convergence of the SQLS on a number of linear systems, and 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 2D grid for the first time using a hybrid quantum algorithm.

💡 Research Summary

This paper introduces the Shadow Quantum Linear Solver (SQLS), a novel hybrid quantum-classical algorithm designed to solve the Quantum Linear System Problem (QLSP) with enhanced resource efficiency for current Noisy Intermediate-Scale Quantum (NISQ) hardware.

The core innovation of SQLS lies in its synergistic combination of two powerful frameworks: Variational Quantum Algorithms (VQAs) and Classical Shadows. The algorithm aims to find a parameterized quantum circuit V(θ*) that prepares a quantum state |x*⟩ proportional to the solution vector of a linear system A|x⟩ = |b⟩. Unlike the famed HHL algorithm, which requires deep circuits and large controlled unitaries impractical for NISQ devices, SQLS adopts a variational approach. It minimizes a specifically designed local cost function C_L(θ) that reaches zero when A|x(θ)⟩ equals |b⟩.

The key efficiency breakthrough comes from the evaluation of this cost function. Under the assumption that the matrix A and the state preparation unitary U for |b⟩ can be decomposed into linear combinations of k-local Pauli strings (a common setting in VQAs), the cost function itself can be expressed as a weighted sum of expectation values of many Pauli observables. Estimating these expectation values directly on hardware would typically require a large number of circuit executions. SQLS circumvents this bottleneck by employing the Classical Shadows protocol. This technique allows for the simultaneous estimation of a large number of Pauli expectation values using a number of quantum circuit runs that scales logarithmically with the number of observables and polynomially with the locality k and target error. Consequently, SQLS achieves an exponential reduction in the number of circuit executions per cost function evaluation compared to prior variational approaches like the VQLS.

Theoretically, SQLS requires a number of qubits logarithmic in the system size N (n = log N) and avoids computationally expensive controlled unitaries. The algorithm’s resource scaling is polylogarithmic in N, linear in the condition number κ, and logarithmic in the inverse error 1/ε, preserving an exponential advantage over classical solvers in the noiseless limit. Numerical experiments presented in the paper demonstrate that the theoretical bounds on SQLS’s resource consumption are conservative, with practical performance being more favorable.

Finally, to showcase its practical relevance, the authors apply SQLS to a concrete physics problem: solving the discretized Laplace equation on a 2D grid. By leveraging linear algebra decomposition theorems, they successfully transform this problem into a QLSP amenable to SQLS, marking the first application of a hybrid quantum algorithm to this problem. This work positions SQLS as a promising, resource-efficient candidate for solving linear systems on near-term quantum computers and opens new avenues for applying quantum algorithms to practical scientific and engineering problems.