Quantum simulation and ground state preparation for the honeycomb Kitaev model

Bespalova, Tatiana A.; Kyriienko, Oleksandr

doi:10.48550/arxiv.2109.13883

Cited by 3 publications

(3 citation statements)

References 107 publications

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“…We choose antiferromagnetic coupling and set h z /J = 0.2. Specifically, we simulate nonequilibrium effects by performing time evolution of M z = j Z j /N for N = 12 qubits on a lattice with periodic boundary conditions [76], starting from the uniform initial state. The choice of quantum dataset with strong magnetic correlations may be especially suitable for kernel-based regression, given recent advances in learning from experiments [77].…”

Section: Resultsmentioning

confidence: 99%

Quantum Kernel Methods for Solving Differential Equations

Paine¹,

Elfving²,

Kyriienko³

2022

Preprint

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We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits (DQC) approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.

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Section: Resultsmentioning

confidence: 99%

Quantum Kernel Methods for Solving Differential Equations

Paine¹,

Elfving²,

Kyriienko³

2022

Preprint

Self Cite

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show abstract

“…Another clear extension is to consider different lattice topologies, where in the case of triangular lattices, one can build on work that studies circuits to prepare Kitaev honeycomb ground states (such as in Ref. [43]). We also foresee extensions that aim at reducing the number of required auxiliary qubits.…”

Section: Discussionmentioning

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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“…We note that a similar state preparation scheme, although not employing the Floquet stabilization of the non-abelian phase nor experimental co-design, has been explored in Ref. [39].…”

Section: B Variational State Preparationmentioning

confidence: 99%

Non-Abelian Floquet Spin Liquids in a Digital Rydberg Simulator

Kalinowski¹,

Maskara²,

Lukin³

2022

Preprint

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Understanding topological matter is an outstanding challenge across several disciplines of physical science. Programmable quantum simulators have emerged as a powerful approach to studying such systems. While quantum spin liquids of paradigmatic toric code type have recently been realized in the laboratory, controlled exploration of topological phases with non-abelian excitations remains an open problem. We introduce and analyze a new approach to simulating topological matter based on periodic driving. Specifically, we describe a model for a so-called Floquet spin liquid, obtained through a periodic sequence of parallel quantum gate operations that effectively simulates the Hamiltonian of the non-abelian spin liquid in Kitaev's honeycomb model. We show that this approach, including the toolbox for preparation, control, and readout of topological states, can be efficiently implemented in state-of-the-art experimental platforms. One specific implementation scheme is based on Rydberg atom arrays and utilizes recently demonstrated coherent qubit transport combined with controlled-phase gate operations. We describe methods for probing the non-abelian excitations, and the associated Majorana zero modes, and simulate possible fusion and braiding experiments. Our analysis demonstrates the potential of programmable quantum simulators for exploring topological phases of matter. Extensions including simulation of Kitaev materials and lattice gauge theories are also discussed.

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Quantum simulation and ground state preparation for the honeycomb Kitaev model

Cited by 3 publications

References 107 publications

Quantum Kernel Methods for Solving Differential Equations

Quantum Kernel Methods for Solving Differential Equations

Quantum circuits for solving local fermion-to-qubit mappings

Non-Abelian Floquet Spin Liquids in a Digital Rydberg Simulator

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