Abstract In this work we investigate a binned version of quantum phase estimation (QPE) set out by Somma (2019 New J. Phys. 21 123025) and known as the quantum eigenvalue estimation problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, Clinton et al (2021 Nat. Commun. 12 1–10), can improve the performance of QEEP in the noisy intermediate scale quantum (NISQ) regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in Clinton et al (2021 Nat. Commun. 12 1–10). Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, we estimate that the depolarizing single qubit error rate required to implement a minimum two bin example of QEEP—with a 5 × 5 Fermi-Hubbard model and up to a precision of 10 % —can be reduced from 10−7 to 10−5. We explore possible modifications to this protocol and propose an application, which we dub randomized quantum eigenvalue estimation problem (rQEEP). rQEEP outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians.
Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision
ε in Heisenberg-limited time
T=Θ(1/ε). Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling
T=O(ε1log(δ−1)) in both the precision
ε and failure probability
δ. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.
In this work we investigate a binned version of Quantum Phase Estimation (QPE) set out by [Somma 2019] and known as the Quantum Eigenvalue Estimation Problem (QEEP). Specifically, we determine whether the circuit decomposition techniques we set out in previous work, [Clinton et al 2020], can improve the performance of QEEP in the NISQ regime. To this end we adopt a physically motivated abstraction of NISQ device capabilities as in [Clinton et al 2020]. Within this framework, we find that our techniques reduce the threshold at which it becomes possible to perform the minimum two-bin instance of this algorithm by an order of magnitude. This is for the specific example of a two dimensional spin Fermi-Hubbard model. For example, given a
10% acceptable error on a
3×3 spin Fermi-Hubbard model, with a depolarizing noise rate of
10−6, we find that the phase estimation protocol of [Somma 2019] could be performed with a bin width of approximately
1/9 the total spectral range at the circuit depth where traditional gate synthesis methods would yield a bin width that covers the entire spectral range. We explore possible modifications to this protocol and propose an application, which we call Randomized Quantum Eigenvalue Estimation Problem (rQeep). rQeep outputs estimates on the fraction of eigenvalues which lie within randomly chosen bins and upper bounds the total deviation of these estimates from the true values. One use case we envision for this algorithm is resolving density of states features of local Hamiltonians, such as spectral gaps.