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 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.