Quantum Error-Correction for Fermionic Qubits

Yifan Tang, Ingo Roth, Philippe Faist +3 more

Fermionic platforms offer compelling architectures for quantum computing, ranging from topologically protected Majorana-based qubits to fermionic cold atoms. To achieve scalability, however, they require quantum error correction. In this work, we prove that any exact and sufficiently accurate approximate fermionic quantum error correction necessarily requires non-Gaussian operations, beyond the free-fermion regime of quadratic dynamics. This is in sharp contrast to the qubit setting, where the efficiently classically simulable stabilizer operations form the standard framework for quantum error correction. Specifically, we show that the logical space of any non-trivial fermionic error-correcting code contains no pure fermionic Gaussian state, utilizing a fundamental incompatibility between fermionic error correction and Wick's theorem. We further show that the required number of bounded-weight non-Gaussian gates for unitary codeword preparation grows at least linearly with both the code distance and the number of encoded modes, revealing an intrinsic resource overhead that increases simultaneously with error-protection strength and logical capacity. Furthermore, we analyze the performance of fermionic Gaussian operations in entanglement distillation, revealing a distinction from their bosonic counterparts. Our results reveal fundamental difficulties for fermionic error correction from the perspectives of both physical implementation and classical simulation, suggesting connections to fermionic phases of matter and state preparation complexity.

Takao Aoki, Go Takahashi, Tadashi Kajiya +4 more

Sagar Vijay, Liang Fu

We introduce error-correcting codes that can correct for fermion parity-violating (quasiparticle poisoning) and parity-conserving errors in systems of complex fermions and of Majorana fermions. After establishing properties of fermion codes, we introduce a generic construction of fermion codes from weakly self-dual classical, binary error-correcting codes. We use this method to construct (i) the shortest fermion code to correct for quasiparticle poisoning errors, (ii) translationally-invariant fermion codes and (iii) other codes that correct higher-weight errors. We conclude by discussing a possible physical realizations of one such code in mesoscopic superconductors hosting Majorana zero modes.

Jesús Yaljá Montiel-Pérez, Jordi Fabián González-Contreras, Erik Zamora +1 more

Ying Li

An important approach to fault-tolerant quantum computation is protecting logical information using quantum error correction. Usually, logical information is in the form of logical qubits, which are encoded in physical qubits using quantum error correction codes. Compared with qubit quantum computation, fermionic quantum computation has advantages in quantum simulations of fermionic systems, e.g., molecules. In this paper, we show that fermionic quantum computation can be universal and fault tolerant if we encode logical Majorana fermions in physical Majorana fermions. We take a color code as an example to demonstrate the universal set of fault-tolerant operations on logical Majorana fermions, and we numerically find that the fault-tolerance threshold is about $0.8%$.