Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems*

Quantum simulation within the quantum phase estimation-based family of algorithms yields exact eigenenergies for fermionic models that are otherwise intractable due to the Monte Carlo sign problem. In circuit-based quantum computation (CBQC), such algorithms repeatedly apply quantum gates to an input wavefunction to achieve time evolution for runtimes that increase exponentially with required bit precision. In measurement-based quantum computation (MBQC), time evolution is effectively driven by sequences of local measurements on an initial entangled resource state. We propose that the gate time burden in CBQC quantum simulation can be shifted to a burden on measurement precision in MBQC quantum simulation. Along these lines, we construct example single-qubit measurement patterns to implement MBQC algorithms for Kitaev and Hubbard chains. We also construct an example hybrid MBQC algorithm with a subroutine for eigenvalue estimation using an offline time series. We consider the scaling of measurements, precision costs, and tolerable errors in the MBQC algorithm. Our examples show that MBQC eigenvalue estimation yields a runtime advantage over CBQC when measurements can be performed quickly and accurately.