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Measuring the Hall Viscosity of the Laughlin State on Noisy Quantum Computers

Ammar Kirmani, Andrew A. Allocca, Jian-Xin Zhu, Armin Rahmani, Sriram Ganeshan, Pouyan Ghaemi

Abstract

Hall viscosity is a quantized nondissipative stress response of a fractional quantum Hall (FQH) fluid to adiabatic geometric deformations. Despite strong theoretical interest, its experimental observation in the FQH state has remained elusive, making it a promising target for realization on current NISQ devices. In this work, we employ a quasi-one-dimensional model of an FQH state coupled to a background metric to probe the geometric response under a metric quench. We design and implement a quantum-circuit protocol that realizes a Hilbert-space-truncated version of the model and extracts the Hall viscosity from the geometric response encoded in the wavefunction dynamics of the device. While the truncation prevents us from accessing the fully quantized value of Hall viscosity, the hardware data nevertheless show excellent agreement with analytical and numerical predictions within this restricted regime.

Measuring the Hall Viscosity of the Laughlin State on Noisy Quantum Computers

Abstract

Hall viscosity is a quantized nondissipative stress response of a fractional quantum Hall (FQH) fluid to adiabatic geometric deformations. Despite strong theoretical interest, its experimental observation in the FQH state has remained elusive, making it a promising target for realization on current NISQ devices. In this work, we employ a quasi-one-dimensional model of an FQH state coupled to a background metric to probe the geometric response under a metric quench. We design and implement a quantum-circuit protocol that realizes a Hilbert-space-truncated version of the model and extracts the Hall viscosity from the geometric response encoded in the wavefunction dynamics of the device. While the truncation prevents us from accessing the fully quantized value of Hall viscosity, the hardware data nevertheless show excellent agreement with analytical and numerical predictions within this restricted regime.

Paper Structure

This paper contains 1 section, 9 equations, 3 figures.

Table of Contents

  1. Acknowledgments

Figures (3)

  • Figure 1: (Top) Spin-shift $S$ obtained from exact diagonalization of \ref{['eq:HYoshioka']}. The quantization of this quantity verifies that the 1d Hamiltonian and the metric parametrization of the Hall viscosity reproduce known results. (Bottom) The overlap of the torus and truncated finite cylinder wave functions.
  • Figure 2: Quantum circuit showing the unitary that creates the ground state with $g^{11}$ and off-diagonal metric $g^{12}$ on a cylinder. Figure (a) show the setup of the Hadamard test for the imaginary part. (b) Shows the decomposition of the Unitary for the first term in Eq. \ref{['eq:finite diff']}.
  • Figure 3: Spin shift $S$ evaluated on a cylinder with circumference $L_y$ for $N_e=4$. The finite difference is taken as $\Delta=0.1$. We show the simulation of the quantum algorithm (blue curve), and the quantum device results with (red points) and without (black points) post-selection. We also plot $S$ obtained from both analytic calculation (green points, proportional to \ref{['eq:TTviscosity']}) and exact diagonalization of \ref{['eq:Htrunc']}, demonstrating good agreement between theory and quantum device results.