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arXiv:2402.02984 · Journal of Physics A Mathematical and Theoretical

Quantum circuits with free fermions in disguise

Kohei Fukai, Balázs Pozsgay

Published Apr 15, 2025• 10 citations
arXivHTMLPDF

Abstract

Abstract Recently multiple families of spin chain models were found, which have a free fermionic spectrum, even though they are not solvable by a Jordan–Wigner transformation. Instead, the free fermions emerge as a result of a rather intricate construction. In this work we consider the quantum circuit formulation of the problem. We construct circuits using local unitary gates built from the terms in the local Hamiltonians of the respective models, and ask the question: which circuit geometries (sequence of gates) lead to a free fermionic spectrum? Our main example is the 4-fermion model of Fendley, where we construct free fermionic circuits with various geometries. In certain cases we prove the free fermionic nature, while for other geometries we confirm it numerically. Surprisingly, we find that many standard brickwork circuits are not free fermionic, but we identify certain symmetric constructions which are. We also consider a recent generalization of the 4-fermion model and obtain the factorization of its transfer matrix, and subsequently derive a free-fermionic circuit for this case as well.

DOI
10.1088/1751-8121/adcd18

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Abstract Recently multiple families of spin chain models were found, which have a free fermionic spectrum, even though they are not solvable by a Jordan–Wigner transformation. Instead, the free fermions emerge as a result of a rather intricate construction. In this work we consider the quantum circuit formulation of the problem. We construct circuits using local unitary gates built from the terms in the local Hamiltonians of the respective models, and ask the question: which circuit geometries (sequence of gates) lead to a free fermionic spectrum? Our main example is the 4-fermion model of Fendley, where we construct free fermionic circuits with various geometries. In certain cases we prove the free fermionic nature, while for other geometries we confirm it numerically. Surprisingly, we find that many standard brickwork circuits are not free fermionic, but we identify certain symmetric constructions which are. We also consider a recent generalization of the 4-fermion model and obtain the factorization of its transfer matrix, and subsequently derive a free-fermionic circuit for this case as well.
Abstract We provide a systematic construction for local quantum circuits hosting free fermions in disguise (FFD), both with staircase and brickwork architectures. Similar to the original Hamiltonian model introduced by Fendley, these circuits are defined by the fact that the Floquet operator corresponding to a single time step can not be diagonalized by means of any Jordan–Wigner transformation, but still displays a free-fermionic spectrum. Our construction makes use of suitable non-local transfer matrices commuting with the Floquet operator, allowing us to establish the free fermionic spectrum. We also study the dynamics of these circuits after they are initialized in arbitrary product states, proving that the evolution of certain local observables can be simulated efficiently on classical computers. Our work proves recent conjectures in the literature and raises new questions on the classical simulability of FFD.
The applications of free fermions in statistical physics and quantum computation have long been well established. In particular, the Jordan–Wigner (JW) transformation can map certain spin-1/21/21/2 chains to free fermionic systems, revealing their complete solvability. On the other hand, free fermionic systems are also classically simulable and therefore provide useful benchmarks for experimental implementations of quantum computers. Recently, Free Fermions in Disguise (FFD) have been introduced: these are new spin-1/21/21/2 chain models that can be mapped to free fermions, but not through the standard Jordan–Wigner transformation. In this thesis, we address the problem of constructing quantum circuits from Free Fermions in Disguise. This is challenging because not all circuits built from these models remain free fermionic, in contrast to the JW-diagonalizable case. We present a systematic approach to demonstrate that some circuits previously proposed in the literature are indeed free fermionic. We then study the circuit dynamics of certain observables expressible in terms of fermionic operators for small system sizes, performing an exact-diagonalization check comparing the spin and fermionic evolutions. These results raise new questions about the classical simulability of free fermions in disguise, which appears to be a more subtle issue than in standard JW-diagonalizable models.
The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such “free fermions in disguise” models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.
Abstract Free fermions in disguise (FFD) Hamiltonians describe spin chains which can be mapped to free fermions, but not via a Jordan-Wigner transformation. Although the mapping gives access to the full Hamiltonian spectrum, the computation of spin correlation functions is generally hard. Indeed, the dictionary between states in the spin and free-fermion Hilbert spaces is highly non-trivial, due to the non-linear and non-local nature of the mapping, as well as the exponential degeneracy of the Hamiltonian eigenspaces. In this work, we provide a series of results characterizing the Hilbert space associated to FFD Hamiltonians. We focus on the original model introduced by Paul Fendley and show that the corresponding Hilbert space admits the exact factorization H = H F ⊗ H D , where H F hosts the fermionic operators, while H D accounts for the exponential degeneracy of the energy eigenspaces. By constructing a family of spin operators generating the operator algebra supported on H D , we further show that H D = H F ′ ⊗ H D ~ , where H F ′ hosts ancillary free fermions in disguise, while H D ~ is generated by the common eigenstates of an extensive set of commuting Pauli strings. Our construction allows us to fully resolve the exponential degeneracy of all Hamiltonian eigenspaces and is expected to have implications for the computation of spin correlation functions, both in and out of equilibrium.
We consider quantum spin chains with a hidden free fermionic structure, distinct from the Jordan-Wigner transformation and its generalizations. We express selected local operators with the hidden fermions. This way, we can exactly solve the real-time dynamics in various physical scenarios, including the computation of selected dynamical two-point functions, in continuous or discrete time. In the latter case, we build a quantum circuit that can be implemented on a quantum computer. With this, we extend the family of classically simulable quantum many-body processes.
Recently, a number of spin chain models have been discovered that are solvable via hidden free-fermionic structures, going beyond the Jordan-Wigner paradigm. However, all examples in the literature displayed degeneracies that grow exponentially with the volume and that are homogeneous in the spectrum (identical degeneracies for all energy levels). In this note we present a model that can be solved by ``free fermions in disguise'' (FFD), such that the spectrum is free from exponential degeneracies for generic coupling constants. The model can be seen as a particular perturbation of two Ising chains. Alternatively, it can be realized as an interpolation between a standard Jordan-Wigner solvable chain and the original FFD model of Fendley. We used ChatGPT Pro 5.4 and 5.5 as a research assistant; in the Supplemental Material we provide details about the collaboration between the AI and the human author.