Gauge freedom in measurement based quantum compiling

Quantum computers may offer multiple advantages, consisting of quantum speed-up when performing hard computation tasks, better compression and expressivity, reduced power consumption, and the capability of simulating natively elementary processes by representing their constituents by qubit. Therefore, chemical, nuclear and elementary particle reactions or Hamiltonian dynamics have been simulated consistently with the number of available qubits. The compact use of resources is of paramount importance for both simulating larger problems or, in prospect, introduce more robust quantum error correction routines to extend the depth of the circuits. Measurement-based quantum computing (MBQC) consists of a virtualization of gate model quantum computing over a limited subset of gate operation involving intermediate measurement processes as well, particularly suitable for photonic hardware. Here, we review the method recently introduced for direct compiling gate model circuit based on unitary processes over the measurement-based scheme. Surprisingly, in addition to better performances with respect to previously known methods, we find a gauge freedom which makes possible to even improve the efficiency. Such gauge invariance provides graphical rules, as happens with Feynman rules in perturbative quantum mechanics. Compared to Measurement Calculus, the ancillary qubits are reduced by 50% on QFT and 75% on QAOA algorithms.