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

Ian D. Kivlichan1,2, Craig Gidney3, Dominic W. Berry4, Nathan Wiebe5, Jarrod McClean1, Wei Sun6, Zhang Jiang1, Nicholas Rubin1, Austin Fowler3, Alán Aspuru-Guzik7,8, Hartmut Neven1, and Ryan Babbush1

1Google Research, Venice, CA 90291, USA
2Department of Physics, Harvard University, Cambridge, MA 02138, USA
3Google Research, Santa Barbara, CA 93117, USA
4Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2113, Australia
5Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA
6Google Research, Mountain View, CA 94043, USA
7Department of Chemistry, University of Toronto, Toronto, Ontario M5G 1Z8, Canada
8Department of Computer Science, University of Toronto, Toronto, Ontario M5G 1Z8, Canada

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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 ${\cal O}(1)$ and ${\cal O}(N^2)$ 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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