Solving 1D Poisson problem with a Variational Quantum Linear Solver

TL;DR

The paper addresses solving 1D Poisson–derived tridiagonal linear systems on near-term quantum devices using the Variational Quantum Linear Solver (VQLS). It introduces two unitary decompositions of the tridiagonal matrix: a naive Pauli-string expansion and a multi-qubit gate–based approach that employs SWAP and center-switch gates to reduce the number of terms. By evaluating both decompositions within the VQLS framework and testing on both simulator and real hardware, the work demonstrates that the multi-qubit approach can substantially reduce unitary terms at the cost of deeper circuits, highlighting a trade-off between implementability on NISQ devices and circuit complexity. The results show that, despite hardware noise and sampling limits, the VQLS can recover high-fidelity solutions for small instances and offers a viable step toward applying variational linear solvers to real-world discretized PDE problems on near-term quantum hardware.

Abstract

Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.

Figures (2)

• Figure 1: Center-switch gate corresponding to the $(011,\, 100)$ permutation. Qubits are ordered from least to most significant, going from $q_0$ to $q_2$. Each one of the gates corresponds to a Toffoli gate, i.e. gates having two control qubits and one target. An empty dot means that the corresponding qubit must be in state 0 to flip the target qubit, and the analogous holds for a full dot and state 1. This figure was produced using the IBM Qiskit library Qiskit.
• Figure 2: Cost vs. number of iterations for two different tridiagonal quantum linear system problems (QLSPs). Both a simulator ('qasm_simulator') and a real quantum computer ('ibmq_athens') IBMQ were used for the solution. On the left hand side is the $2\times 2$ case. While the cost assesses to 0 in the simulated case, it does not for the experiments on real hardware. This is due to undersampling and noise, which pollute the cost evaluation. Nevertheless, as shown by the inset, the runs on real hardware all deliver a final solution with high fidelity. On the right hand side is the $4\times 4$ case, in which the decomposition of $A$ includes the SWAP gate. This time, the cost function is non-normalized to improve convergence. Considerations similar to the $2\times 2$ problem apply here too. Even in this case, the cost at convergence differs between simulated and quantum runs. However the annotations next to the graphs show that high fidelities are reached at convergence.