A fault-tolerant variational quantum algorithm with limited T-depth

We propose a variational quantum eigensolver (VQE) algorithm that uses a fault-tolerant (FT) gate-set, and is hence suitable for implementation on a future error-corrected quantum computer. VQE quantum circuits are typically designed for near-term, noisy quantum devices and have continuously parameterized rotation gates as the central building block. On the other hand, an FT quantum computer (FTQC) can only implement a discrete set of logical gates, such as the so-called Clifford+T gates. We show that the energy minimization of VQE can be performed with such an FT discrete gate-set, where we use the Ross-Selinger algorithm to transpile the continuous rotation gates to the error-correctable Clifford+T gate-set. We find that there is no loss of convergence when compared to the one of parameterized circuits if an adaptive accuracy of the transpilation is used in the VQE optimization. State preparation with VQE requires only a moderate number of T-gates, depending on the system size and transpilation accuracy. We demonstrate these properties on emulators for two prototypical spin models with up to 16 qubits. This is a promising result for the integration of VQE and more generally variational algorithms in the emerging FT setting, where they can form building blocks of the general quantum algorithms that will become accessible in an FTQC