Extending matchgates into universal quantum computation

TL;DR

This paper proves that any nonmatchgate parity-preserving unitary can extend matchgates into universal quantum computation, broadening the understanding of their computational capabilities in fermionic systems.

Contribution

It generalizes previous results by identifying the key invariant enabling matchgates to achieve universality with any nonmatchgate parity-preserving unitary.

Findings

Any nonmatchgate parity-preserving unitary extends matchgates to universal quantum computation.

Identification of the local invariant responsible for this extension.

Discussion of implications for fermionic systems and quantum computing.

Abstract

Matchgates are a family of two-qubit gates associated with noninteracting fermions. They are classically simulatable if acting only on nearest neighbors, but become universal for quantum computation if we relax this restriction or use SWAP gates [Jozsa and Miyake, Proc. R. Soc. A 464, 3089 (2008)]. We generalize this result by proving that any nonmatchgate parity-preserving unitary is capable of extending the computational power of matchgates into universal quantum computation. We identify the single local invariant of parity-preserving unitaries responsible for this, and discuss related results in the context of fermionic systems.