Minimal constraints in the parity formulation of optimization problems

Abstract

As a means to solve optimization problems using quantum computers, the problem is typically recast into an Ising spin model whose ground-state is the solution of the optimization problem. An alternative to the Ising formulation is the Lechner–Hauke–Zoller model, which has the form of a lattice gauge model with nearest neighbor four-body constraints. Here we introduce a method to find the minimal strength of the constraints which are required to conserve the correct ground-state. Based on this, we derive upper and lower bounds for the minimal constraints strengths. We find that, depending on the problem class, the exponent ranges from constant α = 0 to quadratic α = 2 scaling with the number of logical qubits.

Scalable set of reversible parity gates for integer factorization

“…build on programmable local fields J ij σ ij and problem-independent penalty terms acting on plaquettes of neighboring spins. If the energy penalty c penalty is strong enough, it separates the logical subspace from the rest, allowing a one-to-one correspondence with the logical model 40 .…”

“…An even number of parity changes gives rise to constraint Introducing penalty terms for sufficiently many independent constraints, an Ising Hamiltonian H Ising = ∑ i < j J i j σ i σ j is implemented through the Hamiltonian build on programmable local fields J i j σ i j and problem-independent penalty terms acting on plaquettes of neighboring spins. If the energy penalty c penalty is strong enough, it separates the logical subspace from the rest, allowing a one-to-one correspondence with the logical model 40 .…”

Error Mitigation for Quantum Approximate Optimization

“…. , Ω p ) are found using repeated measurements of the state in the computational basis to estimate and optimize the objective function (7) where the penalty strength c is a positive constant introduced to penalize invalid states and should be larger than H p 's lowest energy gap [57]. The objective function in Eq.…”

Modular Parity Quantum Approximate Optimization

“…Fig. 1) and a constraint strength c l > 0 [30]. Here, l i are labels of physical qubits with the property that all problem-spin indices involved in constraint Ĉl appear an even amount of times across all the l i .…”