Fermionic quantum computation

TL;DR

The paper develops a comprehensive framework for fermionic quantum computation by formalizing locality with $C^*$-algebras, defining local fermionic modes and their parity‑preserving gate sets, and establishing a robust fermion–qubit correspondence. It demonstrates universal LFM gates, and provides efficient strategies to simulate fermionic gates with qubits (and vice versa), including logarithmic and constant‑cost encodings. It further explores Majorana fermions, offering universal and measurement‑based alternatives to multi‑Majorana gates, and introduces both a geometrically local simulation on graphs and Majorana‑based quantum codes, linking fermionic computation to error correction and stabilizer codes. These results underscore the equivalence of fermionic models with BQP and offer practical pathways for implementing fermionic computation and codes in qubit‑based architectures.

Abstract

We define a model of quantum computation with local fermionic modes (LFMs) -- sites which can be either empty or occupied by a fermion. With the standard correspondence between the Foch space of $m$ LFMs and the Hilbert space of $m$ qubits, simulation of one fermionic gate takes $O(m)$ qubit gates and vice versa. We show that using different encodings, the simulation cost can be reduced to $O(\log m)$ and a constant, respectively. Nearest-neighbors fermionic gates on a graph of bounded degree can be simulated at a constant cost. A universal set of fermionic gates is found. We also study computation with Majorana fermions which are basically halves of LFMs. Some connection to qubit quantum codes is made.