Bosonization based on Clifford algebras and its gauge theoretic interpretation

Bochniak, Arkadiusz; Ruba, Błażej

doi:10.1007/jhep12(2020)118

Cited by 20 publications

(36 citation statements)

References 47 publications

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“…2. A potential flipper 7 for the separator G f on a white face is the product of X connecting the white face to the grey face below (Fig. 13).…”

Section: General Construction For Compact Fermion-to-qubit Mappingsmentioning

confidence: 99%

“…From both theoretical and practical points of view, mapping local fermionic operators to local spin operators in higher dimensions is an essential topic. In the last two decades, there have been many proposals of fermionto-qubit mappings for two dimensions [1][2][3][4][5][6][7][8][9] and three or arbitrary dimensions [10][11][12]. These fermion-to-qubit mappings play important roles in various topics of modern physics, such as exactly solvable models for topological phases [3,[13][14][15], fermionic quantum simulations [2,4,5,8,10], and quantum error correction [16][17][18][19][20][21].…”

mentioning

confidence: 99%

“…In particular, the exact bosonizations in Refs. [1,7,11,12] construct the toric code with fermions in arbitrary dimensions and impose gauge constraints to restrict in the subspace with emergent fermions, which provide an elegant spacetime description by the Chern-Simons and the Steenrod square topological action. The spacetime pictures for other fermion-to-qubit mappings mentioned above are not manifest.…”

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confidence: 99%

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Equivalence between fermion-to-qubit mappings in two spatial dimensions

Chen¹,

Xu²

2022

Preprint

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We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Ref. [1], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary (gLU) transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new super-compact encoding using 1.25 qubits per fermion on the square lattice, which is lower than any method in the literature. We prove the existence of fermionto-qubit mappings with qubit-fermion ratios r = 1 + 1 2k for positive integers k, where the proof utilizes the trivialness of quantum cellular automata (QCA) in two spatial dimensions. When the ratio approaches 1, the fermion-to-qubit mapping reduces to the 1d Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.

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“…2. A potential flipper 7 for the separator G f on a white face is the product of X connecting the white face to the grey face below (Fig. 13).…”

Section: General Construction For Compact Fermion-to-qubit Mappingsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Equivalence between fermion-to-qubit mappings in two spatial dimensions

Chen¹,

Xu²

2022

Preprint

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“…In [10] the existence of Symmetry Protected Topological (SPT) phases protected by higher symmetries was proposed. Another motivation to study higher gauge theories is provided by its relation with bosonization in arbitrary dimension [18][19][20]. Furthermore, certain models in string theory may be described as higher gauge theories [21].…”

Section: Jhep09(2021)068mentioning

confidence: 99%

Dynamics of a lattice 2-group gauge theory model

Bochniak

Hadasz

Korcyl

et al. 2021

J. High Energ. Phys.

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We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge.

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“…On the other hand, it turned out for a given fermionic system on a lattice, the mapping to a bosonic quantum spin system maintaining the locality does exist in general [14,15,16]. The recent developments in the highlighted papers and their sequels [17,18,19,20,21] led to various new findings, including the transformations keeping the symmetries manifest, the connection to the higher-form symmetries, the connection to earlier heuristic constructions, and the reduction of the number of auxiliary qubits. While the general construction is quite nontrivial and there are several subtleties, as an illustration I quote the quantum spin model equivalent to the standard spinful Hubbard model on the square lattice, as derived in the highlighted paper 3, in Fig.…”

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confidence: 99%

Jordan-Wigner Transformation in Higher Dimensions

2021

Journal Club for Condensed Matter Physics

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Particle statistics is one of the most fascinating aspects of quantum mechanics. While the Bose-Einstein statistics is already counter-intuitive, bosons may be understood as elementary excitations of quantized classical fields. Fermions, on the other hand, do not seem to have any classical counterpart. Formally they arise by quantizing Grassmann fields, but this may not be too helpful for intuitive understanding. Yet, fermions are allowed within principles of quantum mechanics, and indeed they are important building blocks of our universe. Not only the electrons, but also the "matter fields" in the standard model of elementary particles are fermionic. Fermi-Dirac statistics is indeed important for the stability of the matter. On the other hand, Fermi-Dirac statistics also poses computational challenge. The notorious negative sign problem in Quantum Monte Carlo simulations is a consequence of the Fermi-Dirac statistics (although the sign problem can appear also in bosonic systems). The computational difficulty of fermions has also become a renewed issue [1], as quantum computers will be based on bosonic qubits.Given the fundamental difference between bosons and fermions, it is remarkable that a mapping between a fermionic system and a bosonic system, known as Jordan-Wigner transformation [2], exists in one spatial dimension. The mapping can be defined exactly for a one-dimensional lattice, as discussed in Ref. [3]. The exact solution of the quantum S = 1/2 chain, which follows from the exact Jordan-Wigner transformation, is an important basis to understand quantum magnetism in one dimension. It is then a natural desire to extend the

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Bosonization based on Clifford algebras and its gauge theoretic interpretation

Cited by 20 publications

References 47 publications

Equivalence between fermion-to-qubit mappings in two spatial dimensions

Equivalence between fermion-to-qubit mappings in two spatial dimensions

Dynamics of a lattice 2-group gauge theory model

Jordan-Wigner Transformation in Higher Dimensions

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