Local Quantum Circuits Hosting Free Fermions

• The paper introduces local quantum circuits that encode free fermions using customized operator mappings and ancillary qubits, enabling highly efficient simulation of fermionic dynamics.
• It details innovative circuit compression algorithms that reduce depth, preserve locality, and align with hardware connectivity constraints, thus supporting scalability for NISQ devices.
• The study also uncovers novel free-fermion realizations beyond traditional Jordan–Wigner methods, highlighting challenges in operator inversion and entanglement transitions.

A local quantum circuit hosting free fermions is a quantum information structure—typically realized as a finite-depth arrangement of unitary (and possibly nonunitary) gates with strictly local support—such that the many-body spectrum and most dynamical features are determined by underlying noninteracting (free) fermionic degrees of freedom. In these circuits, the local qubit operators encode fermionic creation and annihilation operators (either via explicit maps like Jordan–Wigner or through more intricate constructions involving nonlocal transfer matrices or ancillary qubits), enabling exact or highly efficient simulation of fermionic quantum dynamics. Realizations range from textbook matchgate circuits to recent “free fermions in disguise” constructions, and encompass both standard and exotic quantum phases, including those characterized by nontrivial topology or criticality.

1. Operator Mappings and Locality Constraints

Quantum circuits that host free fermions require a mapping from fermionic Fock space to a local qubit Hilbert space. The canonical method—the Jordan–Wigner (JW) transformation—maps fermionic modes onto qubits with nonlocal Pauli strings, enabling the simulation of free-fermion Hamiltonians like the Kitaev chain or transverse-field Ising model. Despite its efficiency in 1D systems, the JW transformation leads to increasing operator nonlocality in higher dimensions or non-planar graphs.

Recent advances have expanded the toolbox beyond JW through custom encoding schemes (Chien et al., 2020). One strategy constructs “local Majorana” operators on groups of qubits, enabling locality-preserving mappings on general interaction graphs by associating auxiliary qubits or modifications to the system graph. This approach allows mappings such as

$$\widetilde{A}_{jk} = \varepsilon_{jk} c_j^p c_k^q,\qquad \widetilde{B}_j = i^{d(j)/2} \prod_{\mathrm{incident\;edges}\;a} c_j^a$$

with $\varepsilon_{jk} = \pm1$ and $d(j)$ the degree of vertex $j$, to encode fermionic hopping and parity in the qubit system.

Quasi-local encodings, enabled by truncation of state preparation circuits (such as partial MERA) or strategic omission of system graph edges, further reduce the Pauli weight of circuit elements at the cost of strict operator locality. Such encodings are essential for efficient quantum simulation on near-term devices and for non-planar or high-connectivity systems such as those modeling the Sachdev–Ye–Kitaev (SYK) model.

Device constraints, including limited qubit connectivity (as in superconducting heavy-hexagon layouts), are addressed by matching the encoded system graph to hardware topology, further illustrating the flexibility of these mappings for practical quantum circuit design.

2. Circuit Architectures and Compression Algorithms

The architecture of local free-fermion circuits is deeply intertwined with algebraic compression methods and hardware scaling. Circuits simulating Trotterized evolution under quadratic Hamiltonians can in principle be deep, but for free-fermion models possess hidden group-theoretic or Lie-algebraic structure that allows for compression into shallow circuits with depth independent of simulation time (Camps et al., 2021, Kökcü et al., 2023).

Key to these methods is the identification of algebraic blocks—such as SU(2) rotations corresponding to local hopping or phase terms—obeying fusion (combining gates on the same site), commutation (distant gates commute), and turnover relations (permuting sequences of gates). For example,

$$R(\theta_1, \mathbf{n}_1) R(\theta_2, \mathbf{n}_2) = R(\theta_3, \mathbf{n}_3)$$

with $$\cos(\theta_3/2) = \cos(\theta_1/2)\cos(\theta_2/2) - \sin(\theta_1/2)\sin(\theta_2/2)\mathbf{n}_1\cdot\mathbf{n}_2$$ and analogous formulas for $\mathbf{n}_3$.

Compression to fixed-depth circuits is achieved through recursive application of these relations, resulting in circuit architectures with linear (or better) scaling in qubit number and fixed depth versus simulation time—crucial for NISQ hardware.

Handling long-range hopping and arbitrary graph structures is addressed by encoding fermionic swap chains (FSWAPs), so long-range terms are implemented as compressed sequences of nearest-neighbor gates, preserving the scalability and locality of the approach even for arbitrarily connected or two-dimensional lattices.

3. Novel Free-Fermion Realizations Beyond Jordan–Wigner

Recent work has uncovered extensive classes of systems—“free fermions in disguise” (FFD)—in which the Hamiltonian spectrum matches that of noninteracting fermions, yet there exists no local Jordan–Wigner transformation mapping the circuit to bilinear fermion operators (Fendley et al., 2023, Pozsgay et al., 5 Feb 2024, Vernier et al., 21 Jul 2025, Szász-Schagrin et al., 26 Sep 2025). In these circuits, spin chain Hamiltonians are constructed using abstract generators (e.g., $h_j = Z_{j-2} Z_{j-1} X_j$) satisfying specific algebras (FFD algebra) that enable the spectrum to be encoded via transfer matrix techniques: $$T(u) = \sum_{r=0}^{r_{\max}} (-u)^r \overline{Q}^{(r)}$$ Obeying an inversion relation $T(u)T(-u) = P(u^2)$ (for a system-specific polynomial $P$), the roots $u_k$ define mode energies $\epsilon_k = 1/u_k$, and the raising/lowering operators

$$\Psi_{\pm k} = \frac{1}{N_k} T(\mp u_k)\chi T(\pm u_k)$$

satisfy canonical anticommutation relations and diagonalize the Floquet (single time-step evolution) operator or Hamiltonian.

Crucially, such spectra exhibit exponentially degenerate eigenspaces; the Hilbert space factorizes as $\mathcal{H} = \mathcal{H}_F \otimes \mathcal{H}_D$, the latter supporting “ancillary free fermions” and symmetry operators that resolve this degeneracy (Vernier et al., 21 Jul 2025). These constructions require non-local transfer matrices (implemented as MPOs) for both diagonalization and efficient classical simulation of dynamics.

The absence of a JW mapping means that local spin operators are generally highly nonlocal and nonlinear in the hidden fermions, complicating the calculation of spin correlation functions. Nonetheless, carefully constructed observables (e.g., edge or boundary terms) admit bilinear expansions enabling efficient simulation of local quench or dynamical protocols (Vona et al., 31 May 2024).

4. Circuit Floquet Dynamics and Classical Simulability

For both standard and FFD circuits, the Floquet operator (one period of the driven circuit dynamics) often determines the stroboscopic evolution and encodes the free-fermion spectrum. In circuits built from FFD elements and recursively defined via transfer matrices, the spectrum is determined via explicit recursion, with the roots of an associated polynomial specifying the single-particle (fermionic) energies.

Classical simulation is efficient for observables that can be written as sums or products of the hidden fermions, particularly when the initial state is a product state. The contraction of the necessary (MPO) tensors and the root-finding for the eigenmodes scale polynomially in system size, enabling simulation of large systems (demonstrated up to $\sim$150 sites in practice) (Szász-Schagrin et al., 26 Sep 2025).

Nonunitary open circuit generalizations, constructed as Trotterized dissipative Hubbard chains with complex (imaginary) interaction, yield CPTP maps (Kraus operators) that remain exactly solvable (traceable) in the presence of local dephasing, but break integrability if interactions or dephasing are altered (Sá et al., 2020). The spectral statistics of such circuits can be classified by the complex spacing ratio technique, revealing transitions from Poisson (integrable) to Ginibre universality (chaotic).

5. Criticality, Entanglement, and Topological Structure

Entanglement dynamics after local quenches—especially splitting/joining protocols—differs sharply between CFTs with free (or rational) and strongly-coupled (holographic) content. In 2D free-fermion CFTs, local quenches create sharply localized entanglement density peaks on the light cone and entanglement entropies with step-like or logarithmic time dependence (Shimaji et al., 2018). The entanglement density, defined as the second derivative of entanglement entropy with respect to interval endpoints, directly visualizes the propagation of entangled pairs.

Correspondences have been established between local nonunitary fermionic circuits and static problems of disordered noninteracting fermions in higher dimension, with entanglement phase transitions (from logarithmic to area laws) mapped to metal-insulator transitions in the Altland–Zirnbauer symmetry classification. In random measurement circuits or Gaussian tensor network architectures, the symmetry class is controlled by constraints on the covariance matrices, and boundary entanglement scaling and mutual information display universal conformal patterns (Jian et al., 2020).

Topological free-fermion circuits, such as those realizing Majorana chains or Chern insulators, can be constructed by frustration-free decompositions with exponential (rather than sharp) locality tail. For instance, strictly local parent Hamiltonians may not support nontrivial topology, but allowing exponentially decaying hopping (via square-root decompositions of the band Hamiltonian) realizes Chern insulators with exponentially split ground state multiplets (Sengoku et al., 2 May 2025). Edge observables and bipartite fluctuations can then be used to read out topological invariants in both circuit and spin chain realizations (Hur et al., 13 Jun 2025).

6. Quantum Algorithmic Speedup and Generalization

Quantum algorithms for free-fermion simulation exploit block-encoding strategies, with the active correlation matrix $M$ embedded as a subblock of a larger unitary acting on $m$ ancillas: $\langle 0^m|U_A|0^m\rangle = M/\alpha$ Efficient circuit constructions using oracles allow application to large, sparse, or even non-lattice graphs with scaling $\mathrm{poly}(\log N)$, representing exponential speedup over classical brute-force approaches (Stroeks et al., 6 Sep 2024). This is possible because all manipulations—time evolution, observable measurement, thermalization—can be formulated as manipulations of the compressed correlation matrix block, sidestepping explicit many-body Hilbert space manipulations.

This approach naturally generalizes to other quadratic Hamiltonians (bosons, with adaptation of occupation number normalization) and to systems with pairing, establishing the flexibility of the local circuit hosting concept.

7. Summary of Open Problems and Controversies

While substantial progress has been made in the principled construction, compression, and simulation of local circuits hosting free fermions (including FFD structures), important open questions remain:

• Operator mapping inversion: In FFD and related settings, it is highly nontrivial to find a closed-form decomposition of general local spin operators in terms of the hidden fermion and degeneracy sector operators (Vernier et al., 21 Jul 2025). This limits analytic access to correlation functions beyond specific observables or boundary terms.
• Universality of classical simulability: While many local free-fermion circuits (standard or disguised) are classically simulable for product initializations and local observables, the generic characterization of such circuits—especially in the presence of ancillary degeneracy sectors—remains an active area (Szász-Schagrin et al., 26 Sep 2025).
• Distinguishing from interacting or chaotic dynamics: Although local free-fermion circuits with rationally independent single-particle spectra can mimic Poisson-like many-body energy statistics locally, higher-order correlations remain non-Poissonian due to the tensor product structure, distinguishing integrable from truly generic chaotic dynamics (Riddell et al., 18 Apr 2024).
• Entanglement transitions and measurement-induced criticality: The mapping between unitary and nonunitary circuit entanglement phases and free disordered fermion criticality has uncovered new transitions and scaling regimes that remain to be analytically characterized for interacting generalizations (Jian et al., 2020).

The composite architecture of local quantum circuits hosting free fermions, integrating custom encoding, algebraic compression, nonlocal transfer matrix techniques, and robust algorithmic frameworks, provides an evolving platform for efficient quantum simulation, quantum phase diagnosis, and the systematic investigation of emergent many-body phenomena.