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.

Classical microprocessors operate on irreversible gates, that, when combined with , half-adder and full-adder operations, execute complex tasks such as multiplication of integers. We introduce parity versions of all components of a multiplication circuit. The parity gates are reversible quantum gates based on the recently introduced parity transformation and build on ground-space encoding of the corresponding gate logic. Using a quantum optimization heuristic, e.g., an adiabatic quantum computing protocol, allows one to quantum mechanically reverse the process of multiplication and thus factor integers, which has applications in cryptography. Our parity approach builds on nearest-neighbor constraints equipped with local fields, able to encode the logic of a binary multiplication circuit in a modular and scalable way.

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