Utilizing the framework of \mathbb{Z}_2 ℤ 2 lattice gauge theories in the context of Pauli stabilizer codes, we present methodologies for simulating fermions via qubit systems on a two-dimensional square lattice. We investigate the symplectic automorphisms of the Pauli module over the Laurent polynomial ring. This enables us to systematically increase the code distances of stabilizer codes while fixing the rate between encoded logical fermions and physical qubits. We identify a family of stabilizer codes suitable for fermion simulation, achieving code distances of d=2,3,4,5,6,7, allowing correction of any \lfloor \frac{d-1}{2} \rfloor ⌊ d − 1 2 ⌋ -qubit error. In contrast to the traditional code concatenation approach, our method can increase the code distances without decreasing the (fermionic) code rate. In particular, we explicitly show all stabilizers and logical operators for codes with code distances of d=3,4,5. We provide syndromes for all Pauli errors and invent a syndrome-matching algorithm to compute code distances numerically.
Quantum simulation of fermionic systems is a leading application of quantum computers. One promising approach is to represent fermions with qubits via fermion-to-qubit mappings. In this work, we present high-distance fermion-to-qubit stabilizer codes for simulating 2D and 3D fermionic systems. These codes achieve arbitrarily large code distances while keeping stabilizer weights constant. They also preserve locality by mapping local fermionic operators to local qubit operators at any fixed distance. Notably, our 3D construction is the first to simultaneously achieve high distance, constant stabilizer weights, and locality preservation. Our construction is based on concatenating a small-distance 2D or 3D fermion-to-qubit code with a high-distance fermionic color code. Together, these features provide a robust and scalable pathway to quantum simulation of fermionic systems.
It has previously been shown by Rajput, Roggero, and Wiebe that
Z2 Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary
t-qubit errors, where
t is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for
t≤3 (distance
d≤7).
Simulations of fermionic systems with a quantum computer are one of the leading applications in quantum computing; an elegant approach is to define a fermion-to-qubit mapping which in this work we present local fermion-to-qubit mapping for stabilizer codes with an arbitrary code distance and preserving the stabilizer weight for 2D and 3D fermionic simulations. Our 3D construction is the first to achieve high distance, constant stabilizer weights, and locality preservation all at once. This is accomplished via code concatenation of a small-distance 2D or 3D fermion-to-qubit code with a high-distance fermionic color code, resulting in a scalable pathway to simulate fermionic systems on a quantum computer.
Topological quantum field theory (TQFT) provides a unifying framework for describing topological phases of matter and for constructing quantum error-correcting codes, playing a central role across high-energy physics, condensed matter, and quantum information. A central challenge is to formulate topological order on lattices and to extract the properties of topological excitations from microscopic Hamiltonians. In this work, we construct new classes of lattice gauge theories as Pauli stabilizer models, realizing a wide range of TQFTs in general dimensions. We develop a lattice description of extended excitations and systematically determine their generalized statistics. Our main example is the (4+1)D fermionic-loop toric code, obtained by condensing the
e2m2-loop in the (4+1)D
Z4 toric code. We show that the loop excitation exhibits fermionic loop statistics: the 24-step loop-flipping process yields a phase of
−1. Our Pauli stabilizer models realize all twisted 2-form gauge theories in (4+1)D, the higher-form Dijkgraaf-Witten TQFT classified by
H5(B2G,U(1)). Beyond (4+1)D, the fermionic-loop toric codes form a family of
Z2 topological orders in arbitrary dimensions, realized as explicit Pauli stabilizer codes using
Z4 qudits. Finally, we develop a Pauli-based framework that defines generalized statistics for extended excitations in any dimension, yielding computable lattice unitary processes to detect nontrivial statistics. For example, we propose anyonic membrane statistics in (6+1)D, as well as fermionic membrane and volume statistics in arbitrary dimensions. We construct new families of
Z2 topological orders: the fermionic-membrane toric code and the fermionic-volume toric code. In addition, we demonstrate that
p-dimensional excitations in
2p+2 spatial dimensions can support anyonic
p-brane statistics for only even
p.
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.
Simulating fermionic systems on qubit-based quantum computers often demands significant computational resources due to the requirement to map fermions to qubits. Thus, designing a fault-tolerant quantum computer that operates directly with fermions offers an effective solution to this challenge. Here, we introduce a protocol for fault-tolerant fermionic quantum computation utilizing fermion-to-fermion low-density parity-check (LDPC) codes. Our method employs a fermionic LDPC memory, which transfers its state to fermionic color code processors, where logical operations are subsequently performed. We propose using odd-weight logical Majorana operators to form the code space, serving as memory for the fermionic LDPC code, and provide an algorithm to identify these logical operators. We present examples showing that the encoding rate of fermionic codes often matches that of qubit codes, while the logical failure rate can be significantly lower than the physical error rate. Furthermore, we propose two methods for performing fermionic lattice surgery to facilitate state transfer. Finally, we simulate the dynamics of a fermionic system using our protocol, illustrating effective error suppression.