TE-PAI: Exact Time Evolution by Sampling Random Circuits

TL;DR

TE-PAI introduces an exact, sampling-based quantum simulation method that achieves shallow circuit depths and reduces resource requirements, enabling efficient time evolution simulation on NISQ and early fault-tolerant quantum devices.

Contribution

It presents TE-PAI, a novel sampling technique for exact quantum time evolution that avoids discretisation errors and optimizes resource use, especially T-states, for fault-tolerant implementations.

Findings

TE-PAI achieves exact time evolution without discretisation error.

The method requires significantly fewer T-states than Trotterization.

TE-PAI offers a flexible trade-off between circuit depth and measurement overhead.

Abstract

Simulating time evolution under quantum Hamiltonians is one of the most natural applications of quantum computers. We introduce TE-PAI, which simulates time evolution exactly by sampling random quantum circuits for the purpose of estimating observable expectation values at the cost of an increased circuit repetition. The approach builds on the Probabilistic Angle Interpolation (PAI) technique and we prove that it simulates time evolution without discretisation or algorithmic error while achieving shallow circuit depths with optimal scaling that saturates the Lieb-Robinson bound. Another significant advantage of TE-PAI is that it only requires executing random circuits that consist of Pauli rotation gates of only two kinds of rotation angles $\pm\Delta$ and $\pi$, along with measurements. While TE-PAI is highly beneficial for NISQ devices, we additionally develop an optimised early fault-tolerant implementation using catalyst circuits and repeat-until-success teleportation, concluding that the approach requires orders of magnitude fewer T-states than conventional techniques, such as Trotterization -- we estimate $3 \times 10^{5}$ T states are sufficient for the fault-tolerant simulation of a $100$-qubit Heisenberg spin Hamiltonian. Furthermore, TE-PAI allows for a highly configurable trade-off between circuit depth and measurement overhead by adjusting the rotation angle $\Delta$ arbitrarily. We expect that the approach will be a major enabler in the late NISQ and early fault-tolerant periods as it can compensate circuit-depth and qubit-number limitations through an increased circuit repetition.