Modular Parity Quantum Approximate Optimization

Abstract

The parity transformation encodes spin models in the low-energy subspace of a larger Hilbertspace with constraints on a planar lattice. Applying the Quantum Approximate Optimization Algorithm (QAOA), the constraints can either be enforced explicitly, by energy penalties, or implicitly, by restricting the dynamics to the low-energy subspace via the driver Hamiltonian. While the explicit approach allows for parallelization with a system-size-independent circuit depth, the implicit approach shows better QAOA performance. Here we combine the two approaches in order to improve the QAOA performance while keeping the circuit parallelizable. In particular, we introduce a modular parallelization method that partitions the circuit into clusters of subcircuits with fixed maximal circuit depth, relevant for scaling up to large system sizes.

Universal Parity Quantum Computing

“…The introduced decoding scheme can be used to decode the output of optimization algorithms run in the parity scheme as for example proposed in Refs. [41][42][43], in order to further work with the resulting quantum states. With its availability of non-local and multi-qubit operations and the intrinsic error-correction capability, we expect our work to be a step towards the next generation of quantum computers.…”

Applications of Universal Parity Quantum Computation

“…In combination with a partial parallelization of CNOT chains (see Appendix of Ref. [31]), this leads to a minimal circuit depth for any single-qubit unitary of…”

“…3 (leaving out the R z rotations and replacing the physical Hadamard gates by parameterized R x rotations). Such implementations and similar variants have been analyzed thoroughly in recent literature [31,41,42]. Typically, an additional Hamiltonian containing parity constraints is added to give an energy penalty to states which are not in the code space.…”