# Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization

## TL;DR

This paper demonstrates that optimized Trotter-Suzuki product formulas can enhance fault-tolerant quantum simulations of condensed-phase correlated electrons, achieving efficient, high-precision results with manageable qubit and gate complexities.

## Contribution

It introduces improved Trotterization techniques for quantum simulation, outperforming previous methods like linear combinations of unitaries in certain condensed-phase models.

## Key findings

- Low-order Trotter methods perform well with phase estimation.
- Simulations of models with up to 10^5 fermionic modes are feasible.
- Split-operator techniques reduce Trotter error compared to alternatives.

## Abstract

Recent work has deployed linear combinations of unitaries techniques to reduce the cost of fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve upon those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with $N < 10^5$ fermionic modes can be performed with roughly $O(1)$ and $O(N^2)$ T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.

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Source: https://tomesphere.com/paper/1902.10673