Filtering variational quantum algorithms for combinatorial optimization

Abstract

Current gate-based quantum computers have the potential to provide a computational advantage if algorithms use quantum hardware efficiently. To make combinatorial optimization more efficient, we introduce the Filtering Variational Quantum Eigensolver (F-VQE) which utilizes filtering operators to achieve faster and more reliable convergence to the optimal solution. Additionally we explore the use of causal cones to reduce the number of qubits required on a quantum computer. Using random weighted MaxCut problems, we numerically analyze our methods and show that they perform better than the original VQE algorithm and the Quantum Approximate Optimization Algorithm (QAOA). We also demonstrate the experimental feasibility of our algorithms on a Honeywell trapped-ion quantum processor.

A case study of variational quantum algorithms for a job shop scheduling problem

“…This confirms the fast convergence of this algorithm first observed for the weighted MaxCut problem in Ref. [13]. Another advantage of F-VQE compared to Var-QITE is that F-VQE does not require inversion of the-typically ill-conditioned-matrix A in Eq.…”

“…Ideally, we would like an algorithm to return the ground state with a frequency P ψ (gs) ≈ 1, which implies small average energy ψ ≈ 0. The converse is not true because a superposition of low-energy excited states |ψ can exhibit a small average energy ψ ≈ 0 but small overlap with the ground state P ψ (gs) ≈ 0 [13].…”

“…F-VQE is a generalization of VQE with faster and more reliable convergence to the optimal solution [13]. The algorithm uses filtering operators to modify the energy landscape at each optimization step.…”

“…and equivalently for F 2 ψ . Our implementation of F-VQE adapts the parameter τ dynamically at each optimization step to keep the gradient norm of the objective close to some large, fixed value (see [13] for details).…”

“…We selected the popular quantum approximate optimization algorithm (QAOA) [10] and the variational quantum eigensolver (VQE) [11] as well as the less well studied variational quantum imaginary time evolution algorithm (VarQITE) [12] and the filtering variational quantum eigensolver (F-VQE) [13] recently introduced by some of the present authors. Despite its promising properties, such as supporting a form of quantum advantage, [14][15][16] and considerable progress with regards to its experimental realization [17], in general the QAOA ansatz requires circuit depths that are challenging for current quantum hardware.…”