Dynamical Fermion-to-Qubit Mappings

• Dynamical fermion-to-qubit mappings are algorithmic techniques that convert fermionic degrees of freedom into qubit operators, preserving essential anticommutation relations.
• Methods such as Jordan–Wigner, Bravyi–Kitaev, and tensor network encodings optimize qubit usage and circuit depth, achieving significant improvements in simulation efficiency.
• Hardware-aware strategies and error-correcting approaches ensure robust state preparation and fault-tolerant quantum simulation applicable to quantum chemistry, condensed matter, and high-energy physics.

Dynamical fermion-to-qubit mappings are algorithmic and structural transformations that enable the simulation of fermionic systems on quantum computers by systematically converting fermionic degrees of freedom and their dynamics into qubit operators and circuits. These mappings are central to quantum simulation in quantum chemistry, condensed matter, and high-energy physics, directly impacting simulation algorithm depth, hardware feasibility, circuit noise, and resource requirements.

1. Theoretical Foundations and Motivating Challenges

A fermion-to-qubit mapping specifies how states and operators of a fermionic system (antisymmetric Fock space, anticommutation relations) are represented in a qubit system (tensor product Hilbert space, Pauli group algebra). The key challenge is that fermionic operators anticommute when acting on distinct modes, whereas individual qubits commute except when addressing the same site. Naively expressing a local fermionic operator as a short qubit operator fails in higher-dimensional or connected systems due to the nonlocal signature of anticommutation, especially in the presence of cycles in the interaction graph.

In one dimension on tree graphs, the Jordan–Wigner (JW) mapping provides an exact, local encoding. On general graphs with cycles or higher dimensions, exact local encodings are impossible; nonlocality must be distributed either in the mapped operators or in the encoded states, or else one must enlarge the Hilbert space with auxiliary qubits or logical constraints (Guaita, 18 Jan 2024).

The technical objectives are:

• Minimize the number of qubits per fermionic mode (space complexity).
• Minimize the locality (Pauli weight) of mapped operators (circuit depth, gate count).
• Preserve dynamical structure: entanglement classes, symmetries, gauge sectors.
• Ensure robust state preparation, error correction, and efficient implementation on available hardware.

2. Core Mapping Constructions and Their Properties

2.1. Linear Ancilla-free Mappings

Jordan–Wigner (JW) Mapping

For $N$ fermionic modes (indexed $j$), JW encodes each mode in a single qubit:

• $a_j \mapsto \left(\prod_{k<j} Z_k\right)\sigma^-_j$
• $$a_j^\dagger \mapsto \left(\prod_{k<j} Z_k\right)\sigma^+_j$$

This results in Pauli strings with operator weight $|i-j|+1$ for interactions between modes $i$ and $j$; nonlocality grows with distance (Havlíček et al., 2017, Chiew et al., 2021).

Bravyi–Kitaev (BK) and Generalized/Segmented BK

BK balances update and parity operations using classical binary tree (Fenwick tree) data structures:

• Both occupation and parity checks involve $O(\log\,N)$ qubits.
• Segmented BK partitions sites into blocks mapped as separate trees, reducing support for local interactions (e.g., in 2D models) to $O(\log\,w)$, where $w$ is lattice width (Havlíček et al., 2017).

Ordering Optimization

The ordering (enumeration) of fermionic modes crucially controls Pauli weight. By casting the mapping optimization as a minimum linear arrangement (MLA) or quadratic assignment problem (QAP), explicit cost functions (e.g., $\sum_{(i,j)\in E}|f(i)-f(j)|$) can be minimized (Chiew et al., 2021, Chiew et al., 30 Apr 2025). Patterns such as the Mitchison–Durbin enumeration minimize average Pauli weight by up to 13.9% vs. standard JW strings.

2.2. Local and Ancilla-assisted Mappings

Auxiliary Fermion (AF) Mapping

By introducing auxiliary modes for nonlocal couplings, the AF scheme localizes operator support (e.g., 4-local hopping terms in the 2D Hubbard model), trading increased qubit overhead for lower circuit depth (Havlíček et al., 2017).

Compact and Super-compact Encodings

Local stabilizer code-inspired mappings assign qubits to vertices (modes) and faces (e.g., on square lattices). These achieve qubit-to-mode ratios as low as $1.25:1$ (Chen et al., 2022), with maximum Pauli weights of 2–3 for local interactions (Derby et al., 2020). Constraints enforce loop identities and gauge invariance, yielding logical operator sets matching the fermionic algebra.

Quantum Code Encodings

Mappings based on classical code theory (e.g., binary addressing, segment codes, nonlinear codes) exploit particle number conservation or symmetry to compress into the information-theoretic minimum number of qubits, trading circuit locality for qubit count (Steudtner et al., 2017, Harrison et al., 2022, Carolan et al., 5 Oct 2024).

Tensor Network and Topological Encodings

Projected entangled pair operators (PEPO) enable unitary, locality-preserving fermion-qubit mappings on nontrivial manifolds (torus), correctly treating charge sectors and twisted boundary conditions (O'Brien et al., 11 Apr 2024). The intertwiner is built from even parity tensors that implement the duality between local fermionic and spin degrees of freedom, with ancillary qubit(s) encoding topological data.

3. Algorithmic and Hardware-Aware Optimization

3.1. Ancilla Insertion and Circuit Compression

Incrementally adding ancilla qubits to JW or similar schemes and choosing optimal subsets for operator compensation can reduce total Pauli weight by up to 67%—with only a modest hardware overhead (e.g., $\leq 10$ ancilla) (Chiew et al., 30 Apr 2025). Optimization typically involves balancing the reduction in nonlocal support for problematic Pauli strings with minimal increase in ancilla-induced overhead.

3.2. Tree-based and Hardware-tailored Mappings

The Bonsai and Hamiltonian-Adaptive Ternary Tree (HATT) algorithms construct ternary trees whose paths are mapped to Pauli strings, guaranteeing Majorana anticommutation and mapping Fock product states directly to computational basis states. HATT further adapts the mapping to the structure of the target Hamiltonian, minimizing Pauli weight and enabling circuit depth reductions of 5–20% in benchmarks (Miller et al., 2022, Liu et al., 3 Sep 2024). Tree growth is hardware-aware, e.g., aligned with connectivity in heavy-hexagon processors, and is optimized (from $O(N^4)$ to $O(N^3)$ complexity) by traversing cached descendant-ancestor maps.

4. Circuit-Level Realization and Parallelization

The physical viability of dynamical fermion-to-qubit mappings is dictated by achievable circuit depth and native gate overhead:

• Cavity-QED and multi-mode ancillae allow hardware-efficient realization of nonlocal parity strings, replacing long CNOT or CZ ladders with a single dispersively coupled many-body gate, slashing circuit depth by up to $O(N^2)$ (Zhu et al., 2017).
• On systems with square-grid or other fixed qubit connectivity, mapping schemes that localize operator support and are designed to enable high gate cancellation and parallelism (e.g., through circuit compression and XYZ operator decomposition) achieve Trotter step depths under 20, far surpassing JW-based baselines by up to 70% (Algaba et al., 2023).
• Customized path-coloring and conflict-graph techniques (distinguishing weak vs. strong coloring, with chromatic number $\chi$ dictating minimal sequential layers) further reduce simulation depth and expose more parallelism, especially on complex connectivity graphs (Bringewatt et al., 2022).

5. Entanglement Structure, SLOCC Classification, and Limitations

Operator mappings can preserve or obscure the underlying entanglement structure:

• For 2 and 3 qubits (mapped to $n$-fermion states with $2n$ modes), the embedding is surjective and one-to-one under LU and SLOCC operations, exactly preserving qubit entanglement classes (Chen et al., 2013).
• For 4 qubits ($n=4$), SLOCC injectivity is retained and generic LU injectivity holds, as proven via polynomial invariant comparison (Theorem 2), enabling rigorous SLOCC family classification (Chen et al., 2013).
• For $n>3$ the mapping is no longer surjective: extra fermionic orbits (entanglement classes) exist that are unattainable from the qubit embedding.

A fundamental limitation, proven in (Guaita, 18 Jan 2024), is that fully local qubit-encoding of fermionic operators is only possible on tree locality graphs. On two-dimensional (cyclic) lattices, any exact mapping requires either nonlocal operators or maps product states in the fermionic basis to long-range entangled qubit states, with circuit depth for preparation scaling linearly with system size.

6. Error Correction, Quantum Codes, and Fault Tolerance

Mappings based on repetition or color codes construct error-correcting codes directly for encoded fermions:

• Fermionic repetition codes map logical fermions into the ground state space of Kitaev chains, correcting phase (dephasing) errors (Schuckert et al., 13 Nov 2024).
• Fermionic color codes, via mapping weakly self-dual CSS codes, correct both phase and loss errors using only local Majorana operators.
• Logical gate sets (transversal Clifford, number-conserving, and non-number-conserving gates such as BRAID) are realized using native fermionic operations and, when interfaced with qubit color codes, enable qubit-controlled, fault-tolerant fermionic time evolution (Schuckert et al., 13 Nov 2024).
• Implementation in neutral atoms is feasible via photodissociation-based BRAID gates, Rydberg atom gates, and site-resolved tunneling operations, achieving exponential improvements in depth for the fermionic fast Fourier transform (FFFT).

7. Summary and Future Outlook

Dynamical fermion-to-qubit mappings constitute the algorithmic backbone of quantum simulation of fermionic systems, mediating between the nonlocal algebra of fermionic creation/annihilation and physically local operations on qubits. Multi-pronged advances address the tension between minimizing qubit resources and localizing operator support, using a spectrum of techniques: optimized ordering via QAP/MLA, local and super-compact stabilizer encodings, hardware-tailored tree mappings, ancilla-augmented hybrid strategies, tensor network PEPO intertwiners (capturing topology), and explicit code-theoretic encodings for particle-number constraints. Practical circuit reductions are achieved through both algorithmic and hardware-aware methods, supporting scalable and robust simulation platforms. The landscape is shaped by mathematical limitations arising from the topology of underlying locality graphs and by the necessity of error correction and modular fault tolerance in realistic hardware. Future directions focus on extending these principles to more general symmetries, higher spatial dimensions, and hybrid quantum architectures supporting both native fermionic and qubit registers.