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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>&lt;</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>5</mml:mn></mml:msup></mml:math> fermionic modes can be performed with roughly <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>N</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math> 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.