Abstract
Quantum phase estimation (QPE) is the workhorse behind any quantum algorithm and a promising method for determining ground state energies of strongly correlated quantum systems. Low-cost QPE techniques make use of circuits which only use a single ancilla qubit, requiring classical post-processing to extract eigenvalue details of the system. We investigate choices for phase estimation for a unitary matrix with low-depth noise-free or noisy circuits, varying both the phase estimation circuits themselves as well as the classical post-processing to determine the eigenvalue phases. We work in the scenario when the input state is not an eigenstate of the unitary matrix. We develop a new post-processing technique to extract eigenvalues from phase estimation data based on a classical time-series (or frequency) analysis and contrast this to an analysis via Bayesian methods. We calculate the variance in estimating single eigenvalues via the time-series analysis analytically, finding that it scales to first order in the number of experiments performed, and to first or second order (depending on the experiment design) in the circuit depth. Numerical simulations confirm this scaling for both estimators. We attempt to compensate for the noise with both classical post-processing techniques, finding good results in the presence of depolarizing noise, but smaller improvements in 9-qubit circuit-level simulations of superconducting qubits aimed at resolving the electronic ground state of a H4-molecule.
Highlights
Even though our paper focuses on quantum phase estimation where the phases corresponds to eigenvalues of a unitary matrix, our post-processing techniques may be applicable to multi-parameter estimation problems in quantum optical settings
We have presented and studied the performance of two estimators for quantum phase estimation at low K for different experiment protocols, different systems, and under simplistic and realistic noise conditions
We observe scaling laws for our time-series estimator; we find it first-order sensitive to the overlap A0 between starting state and ground state, second-order sensitive to the gap between the ground state and the nearest eigenstates, and second-order sensitive to the coherence time of the system
Summary
It is known that any problem efficiently solvable on a quantum computer can be formulated as eigenvalue sampling of a Hamiltonian or eigenvalue sampling of a sparse unitary matrix [1]. We ask what the spectral-resolving power of such phase estimation circuits is, both in terms of the number of applications of the controlled-U circuit in a single experiment, and the number of times the experiment is repeated Such repeated phase estimation experiments require classical post-processing of measurement outcomes, and we study two such algorithms for doing this. Even though our paper focuses on quantum phase estimation where the phases corresponds to eigenvalues of a unitary matrix, our post-processing techniques may be applicable to multi-parameter estimation problems in quantum optical settings. In these settings the focus is on determining an optical phase-shift [13,14,15] through an interferometric set-up. There is experimental work on (silicon) quantum photonic processors [16,17,18] on multiple-eigenvalue estimation for Hamiltonians which could benefit from using the classical post-processing techniques that we develop in this paper
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