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Topological phases of matter remain a focus of interest due to their unique properties -fractionalisation, ground state degeneracy, and exotic excitations. While some of these properties can occur in systems of free fermions, their emergence is generally associated with interactions between particles. Here we quantify the role of interactions in general classes of topological states of matter in all spatial dimensions, including parafermion chains and string-net models. Using the interaction distance [Nat. Commun. 8, 14926 (2017)], we measure the distinguishability of states of these models from those of free fermions. We find that certain topological states can be exactly described by free fermions, while others saturate the maximum possible interaction distance. Our work opens the door to understanding the complexity of topological models and to applying new types of fermionisation procedures to describe their low-energy physics.

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The toric code is known to be equivalent to free fermions. This paper presents explicit local unitary transformations that map the Z2 toric and surface code -the open boundary equivalent of the toric code -to fermions. Through this construction it is shown that the surface code can be mapped to a set of free fermion modes, while the toric code requires additional fermionic symmetry operators. Finally, it is demonstrated how the anyonic statistics of these codes are encoded in the fermionic representations.PACS numbers: I. INTRODUCTIONThe toric code [1][2][3] and its open boundary version, the Z 2 surface code [4,5], have been the test-bed for numerous investigations of condensed matter phenomena as well as quantum information applications [6][7][8]. The main reasons for the popularity of the toric code are its ability to support Abelian anyons, exotic quasiparticles that can fault-tolerantly encode and manipulate quantum information, its eigenstates have nontrivial topological entanglement entropy [9], while it is exactly solvable. An important feature of this topological model is that it is relatively simple, where for example, the anyonic statistics and fusion rules emerge directly from the algebraic properties of Pauli matrices. At the same time the toric code enjoys many applications. It can be used as a fault tolerant quantum memory protecting against spurious local perturbations [10], it can perform topological quantum computation resilient against control errors [1], or it can encode more complex anyonic models such as Majorana fermions at lattice defects [11,12].The toric code has been experimentally simulated with highly entangled four-photon GHZ states [13] and the four-body interaction has been physically realised with Josephson junctions [14,15]. However, it has been argued by Wen that the Hilbert space of the toric code, in the presence of an external magnetic field contains a low energy subspace that can be described effectively by hopping fermionic excitations coupled to a Z 2 gauge field [16]. This gauge field does not introduce interactions, but encodes the exotic statistics of the excitations. Moreover, previous investigation of the toric code's ground state in terms of the interaction distance [17] showed that the system is equivalent to free fermions [18]. As this paper will show this is part of a more general result, where all eigenstates of the toric code are Gaussian states having entanglement spectra given in terms of free fermions. In addition, the energy spectrum has a similar decomposition in terms of single particle energies. Hence, it is expected that a unitary transformation exists that maps the toric code to a free fermion Hamiltonian. Nevertheless, a free fermion system can support neither anyonic statistics nor eigenstates with non-trivial topological entanglement entropy. Hence, these properties have to be encoded non-trivially in the unitary transformation that maps between these two physically different models.Previous works studying transformations of the toric code include th...