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<105 fermionic modes can be performed with roughly O(1) and O(N2) 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.
Highlights
The physics of interacting electrons in the presence of external fields predicts the properties of many materials as well as the dynamics of most chemical reactions
We show that practical simulations of the Hubbard model and plane wave electronic structure models can be performed with roughly O(1) and O(N 2) T complexities until the models are so large (e.g., N > 105 orbitals) that only one Trotter step is required to obtain target precision, or other assumptions of our analysis break down
We present an improved bound on the second-order Trotter error, and on the ground state energy shift, in Appendix E; this bound does not require the product of the evolution time and norm of the Hamiltonian to be small
Summary
The physics of interacting electrons in the presence of external fields predicts the properties of many materials as well as the dynamics of most chemical reactions. Allowing the target error to grow with system size significantly ameliorates the poor scaling of low-order Trotter methods with respect to time and precision In this context, we show that practical simulations of the Hubbard model and plane wave electronic structure models can be performed with roughly O(1) and O(N 2) T complexities until the models are so large (e.g., N > 105 orbitals) that only one Trotter step is required to obtain target precision, or other assumptions of our analysis break down. Use split-operator or fermionic swap network Trotter step for evolution within phase estimation (Section 2); Optimize Trotter step circuits to maximize simultaneous rotations for Hamming weight phasing (Section 2.1); Bound Trotter errors and maximum shift in eigenphases (Section 3.1); Determine precision target—either relative to total energy, or absolute (Section 3.2); Numerically minimize number of costly fault-tolerant gates (Section 3.2); Estimate surface code resource requirements (Section 3.3); End Result: Fault-tolerant costs of estimating system ground state energy (Section 3.3, Section 4)
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