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

Chen, Yuan; Xu, Yijia

doi:10.48550/arxiv.2201.05153

Cited by 6 publications

(11 citation statements)

References 25 publications

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“…[9,10]) where JWT were carried out explicitly, recent techniques [7,14,20] take a different approach where one first defines bosonic operators from the fermionic ones, which can then be mapped directly onto quantum spin operators without the need to order the fermions. The equivalence of these bosonization procedures in 2D was proven by Chen et al [15].…”

Section: Introductionmentioning

confidence: 89%

“…As shown by Ref. [15], other bosonisation procedures are equivalent in 2D. In order to reduce the number of degrees of freedom, the auxiliary system is subject to a Gauss-law constraint of the form…”

Section: Bosonizationmentioning

confidence: 99%

“…In recent years, we have witnessed a renewed interest in methods for simulating fermionic systems through a set of local qubit gates [11][12][13][14][15]. This more recent theoretical ac-tivity is again driven by two main motives.…”

Section: Introductionmentioning

confidence: 99%

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Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Nys¹,

Carleo²

2022

Preprint

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Through the introduction of auxiliary fermions, or an enlarged spin space, one can map local fermion Hamiltonians onto local spin Hamiltonians, at the expense of introducing a set of additional constraints. We present a variational Monte-Carlo framework to study fermionic systems through higher-dimensional (>1D) Jordan-Wigner transformations. We provide exact solutions to the parity and Gauss-law constraints that are encountered in bosonization procedures. We study the spinless Fermi-Hubbard (t-V ) model in 2D and demonstrate how both the ground state and the low-energy excitation spectra can be retrieved in combination with neural network quantum state ansatze.

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Section: Introductionmentioning

confidence: 89%

Section: Bosonizationmentioning

confidence: 99%

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Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Nys¹,

Carleo²

2022

Preprint

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“…In recent years, we have witnessed a strong renewed interest in methods for simulating fermionic systems through a set of local qubit gates [5,[13][14][15][16][17][18][19]. Compared to earlier methods (such as Refs.…”

Section: Introductionmentioning

confidence: 99%

Quantum circuits for solving local fermion-to-qubit mappings

Nys¹,

Carleo²

2022

Preprint

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Local Hamiltonians of fermionic systems on a lattice can be mapped onto local qubit Hamiltonians. Maintaining the locality of the operators comes at the expense of increasing the Hilbert space with auxiliary degrees of freedom. In order to retrieve the lower-dimensional physical Hilbert space that represents fermionic degrees of freedom, one must satisfy a set of constraints. In this work, we introduce quantum circuits that exactly satisfy these stringent constraints. We demonstrate how maintaining locality allows one to carry out a Trotterized time-evolution with constant circuit depth per time step. Our construction is particularly advantageous to simulate the time evolution operator of fermionic systems in d>1 dimensions. We also discuss how these families of circuits can be used as variational quantum states, focusing on two approaches: a first one based on general constantfermion-number gates, and a second one based on the Hamiltonian variational ansatz where the eigenstates are represented by parametrized time-evolution operators. We apply our methods to the problem of finding the ground state and time-evolved states of the spinless 2D Fermi-Hubbard model.

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“…Mathematically, this corresponds to this string operator defining an anomalous 1-form symmetry[80,81]. This is key to being able to 'fermionize'[29,[82][83][84][85][86][87] the system and solve it using free fermions (Sec. IV B).…”

mentioning

confidence: 99%

Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids

Verresen¹,

Vishwanath²

2022

Preprint

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The exploration of quantum spin liquids (QSLs) has been guided by different approaches including the resonating valence bond (RVB) picture, deconfined lattice gauge theories and the Kitaev model. More recently, a spin liquid ground state was numerically established on the ruby lattice, inspired by the Rydberg blockade mechanism. Here we unify these varied approaches in a single parent Hamiltonian, in which local fluctuations of anyons stabilize deconfinement. The parent Hamiltonian is defined on kagomé triangles-each hosting four RVB-like states-and includes only Ising interactions and single-site transverse fields. In the weak-field limit, the ruby spin liquid and exactly soluble kagomé dimer models are recovered, while the strong-field limit reduces to the Kitaev honeycomb model, thereby unifying three seemingly different approaches to QSLs. We similarly obtain the chiral Yao-Kivelson model, honeycomb toric code and a new spin-1 quadrupolar Kitaev model. The last is shown to be in a QSL phase by a non-local mapping to the kagomé Ising antiferromagnet. We demonstrate various applications of our framework, including (a) an adiabatic deformation of the ruby lattice model to the exactly soluble kagomé dimer model, conclusively establishing the QSL phase in the former; and (b) demystifying the dynamical protocol for measuring off-diagonal strings in the Rydberg implementation of the ruby lattice spin liquid. More generally, we find an intimate connection between Kitaev couplings and the repulsive interactions used for emergent dimer models. For instance, we show how a spin-1/2 XXZ model on the ruby lattice encodes a Kitaev honeycomb model, providing a new route toward realizing the latter in cold-atom or solid-state systems. CONTENTSI. Introduction 1 II. Model 3 A. Emergent dimer model in a tensor product Hilbert space 3 B. Hamiltonian 6 C. Brief summary of the phase diagram(s) 8 III. Quantum liquids from e-anyon fluctuations 9 A. Weak-field limit: dimer liquid stabilizer model 9 B. Arbitrary field: duality to spin-1/2 transverse field Ising model on the kagomé lattice 9 C. Strong-field limit: spin-1 quadrupolar Kitaev model 10 IV. Quantum liquids from f -anyon fluctuations 11 A. Weak-field limit: distinct dimer liquid SETs 11 B. Exact solution for arbitrary field: Majorana Fermi surfaces and chiral non-Abelian Ising topological order 12 C. Strong-field limit: spin-1/2 Kitaev model 13 V. Applications 13 A. Adiabatically connecting ruby lattice spin liquid to solvable stabilizer Hamiltonian 13 B. The Hadamard transformation for a dimer model 14 C. A natural family of spin-1 Kitaev models 15 D. Kagomé dimer model as honeycomb toric code 16 VI. Local rewriting of spin-3/2 model as two-body spin-1/2 'grandparent' models 17 A. Ruby lattice 17 B. Star lattice 18 VII. Conclusions and future directions 19 Acknowledgments 21 References 21 A. Spin-3/2 Hilbert space 25 1. Algebra: Majoranas and bosonic operators 25 2. Z 3 isomorphisms 26 3. Z 2 Hadamard 26 4. Reduction to spin-1/2 26 5. Matrix representation 27 6. Connection t...

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

Cited by 6 publications

References 25 publications

Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Variational solutions to fermion-to-qubit mappings in two spatial dimensions

Quantum circuits for solving local fermion-to-qubit mappings

Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids

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