# Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit

## TL;DR

This paper experimentally measures the quantum metric tensor using superconducting qubits, introduces two measurement methods, and explores a topological phase transition characterized by the Euler number instead of the Chern number.

## Contribution

It presents two novel methods for directly measuring the quantum metric tensor in quantum states using superconducting circuits.

## Key findings

- Successful measurement of the quantum metric tensor.
- Observation of a topological phase transition characterized by the Euler number.
- Demonstration of new techniques to explore quantum state topology.

## Abstract

Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum-metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system, which is characterized by the Euler characteristic number instead of the Chern number. The work opens up a unique approach to explore the topology of quantum states with the QGT.