TE-PAI: Exact Time Evolution by Sampling Random Circuits

TL;DR

TE-PAI introduces an effectively exact quantum time-evolution estimator that samples simple random circuits built from Pauli rotations with two fixed angles, achieving unbiased expectation values without discretisation error while allowing shallow circuit depths. The core idea uses Probabilistic Angle Interpolation to replace continuous angles with three discrete options, yielding an unbiased estimator for the full time-evolution superoperator and a tunable trade-off between circuit depth and sampling overhead through a parameter $\Delta$. Numerical demonstrations on Heisenberg spin rings and small quantum chemistry problems show that TE-PAI matches deep-Trotter means with much shallower circuits, while remaining compatible with error mitigation and classical shadows; fault-tolerant resource estimates indicate substantial reductions in non-Clifford resources compared to conventional Trotterisation. The approach is poised to benefit late-NISQ and early fault-tolerant regimes, offering a practical path to quantum advantages in dynamics and spectroscopic tasks, and it can be integrated with existing randomized protocols and circuit-cutting or shadow-techniques to further enhance scalability.

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 $\pmΔ$ and $π$, 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 $Δ$ 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.

Figures (13)

• Figure 1: A single random circuit instance of TE-PAI -- by executing multiple such random circuits and post-processing their measurement outcomes, one can implement effectively exact time evolution on average via \ref{['main_statement']}. In the present example, we consider a 5-qubit Hamiltonian defined in \ref{['eq:spinring']} and a rotation angle $\Delta = \pi/2^{6} = \pi/64$. TE-PAI then uses the Pauli gates $RXX, RYY, RZZ$ and $RZ$ only with rotation angles $\pm\Delta = \pm\pi/64$ and only rarely with $\pi$ (gate highlighted by dotted rectangle) -- when the angle $\pi$ is chosen then all measurement outcomes are multiplied by a factor $-1$. The example considers a short time evolution of $T=0.05$ which is the reason for obtaining a shallow circuit with an expected number of gates $\nu_\infty \approx 25$. We note that existing compilation techniques, including ones that were specifically developed for Trotterised circuit structures 2qan, can be applied immediately to reduce the circuit depth.
• Figure 2: Expected number of gates when simulating the time evolution under the Hamiltonian in \ref{['eq:spinring']} for 14 qubits using different rotation angle settings as $\Delta = \pi/2^{\ell},\ \ell = 1, 2, \dots , 10$. While the number of gates grows linearly with the total time $T$, the slope is determined by the angle $\Delta$ -- decreasing $\Delta$ increases the circuit depth, however, can exponentially reduce the measurement overhead as we detail below.
• Figure 3: Measurement overhead for the time-dependent Trotter circuit with different $\Delta = \pi/2^{\ell},\ \ell = 6, 7, 8, 9, 10$. We consider the Hamiltonian in \ref{['eq:spinring']} for 14 qubits. We observe that the overhead grows exponentially with the total time $T$. Since $\Delta$ directly affects the exponent, a smaller $\Delta$ results in a slower exponential blowup.
• Figure 4: Trade-off between the expected number of gates $\nu_\infty$ (left axis) and measurement overhead (right axis) as a function of the rotation angle $\Delta = \pi/2^{\ell},\ \ell = 3, 6, 9, 12, 15$ for the time-dependent Hamiltonian in \ref{['eq:spinring']} for 14 qubits and $T=1$.
• Figure 5: Histogram of the number of gates in the randomly generated TE-PAI circuits for the time-dependent Hamiltonian in \ref{['eq:spinring']} for 14 qubits and $\Delta=2^{-7}\pi,\ T=1$. The expected number of gates in \ref{['thm:num_gate2']} is $\nu _{\infty} \approx 2715$ which is in good agreement with the empirical mean (black line). Furthermore, \ref{['lemma:distribution']} guarantees that the distribution is well approximated by a Gaussian distribution $\mathcal{N}(\nu_{\infty}, \nu_{\infty})$ (red line) which is in good agreement with the histogram.

Theorems & Definitions (6)

• Theorem 1
• Lemma 1
• Theorem 2
• Remark 1
• proof
• proof