Variational Quantum Simulation of Valence-Bond Solids

Daniel Huerga

Stewart Blusson Quantum Matter Institute, University of British Columbia, Vancouver V6T 1Z4, BC, Canada
Department of Physical Chemistry, University of the Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain

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Abstract

We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational optimization to the U(1) subspace. We benchmark the method against the $J1-J2$ Heisenberg model on the square lattice and uncover its phase diagram, which hosts long-range ordered Neel and columnar anti-ferromagnetic phases, as well as an intermediate valence-bond solid phase characterized by a periodic pattern of 2×2 strongly-correlated plaquettes. Our results show that the convergence of the algorithm is guided by the onset of long-range order, opening a promising route to synthetically realize frustrated quantum magnets and their quantum phase transition to paramagnetic valence-bond solids with currently developed superconducting circuit devices.

Variational quantum algorithms (VQA), generically characterized by a feedback loop between a quantum device and a classical optimizer, are at the center of current research for their potentiality in providing first useful applications of noisy intermediate scale quantum (NISQ) devices in problems ranging machine learning and quantum simulation. However, various roadblocks have been identified in their optimization, potentially hindering any applicability of VQA. Quantum simulation of two-dimensional (2D) frustrated quantum magnets offers a natural arena for benchmark and development of VQA, for they pose a challenge to state-of-the-art numerical techniques and at the same time host a plethora of phases with implications for quantum computation.

Here, we present a VQA to simulate 2D frustrated quantum magnets in the thermodynamic limit. Building upon the cluster-Gutzwiller ansatz of hierarchical mean-field theory (HMFT), a parameterized quantum circuit provides the wave function of the cluster, while information of the infinite lattice is provided through a mean-field embedding. Benchmark numerical simulations of this \textit{quantum-assisted} (Q-) HMFT on the paradigmatic J1-J2 Heisenberg antiferromagnet on the square lattice show that the convergence of the algorithm is pushed by the onset of long-range order, opening a promising route for quantum simulation of 2D quantum magnets and their quantum phase transitions to valence bond solid phases with current superconducting circuit technology.

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[2] Rasmus Berg Jensen, Simon Panyella Pedersen, and Nikolaj Thomas Zinner, "Dynamical quantum phase transitions in a noisy lattice gauge theory", Physical Review B 105 22, 224309 (2022).

[3] Verena Feulner and Michael J. Hartmann, "Variational quantum eigensolver ansatz for the J<SUB>1</SUB>-J<SUB>2</SUB> -model", Physical Review B 106 14, 144426 (2022).

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