Direct measurement of the quantum geometric tensor in pseudo-Hermitian systems

TL;DR

This work tackles the challenge of directly measuring the left-right quantum geometric tensor (QGT) in pseudo-Hermitian systems with real spectra. It introduces two self-contained dynamical schemes that combine adiabatic perturbation theory with generalized expectation values to extract all QGT components from time-evolved states prepared via nonadiabatic ramps. Numerical benchmarks on two q-deformed two-band models show accurate reconstruction of the QGT and, for Model I, correct Chern-number phase diagrams, illustrating the topological content captured by the LR QGT. The methods offer a practical route to extend dynamical QGT measurements from Hermitian to non-Hermitian contexts and are compatible with current quantum platforms, with potential extensions to many-body settings and Hermitian systems.

Abstract

The quantum geometric tensor (QGT) fundamentally encodes the geometry and topology of quantum states in both Hermitian and non-Hermitian regimes. While adiabatic perturbation theory links its real part (quantum metric) and imaginary part (Berry curvature) to energy fluctuations and generalized forces, respectively, in Hermitian systems, direct measurement of the QGT, which is defined using both left and right eigenstates of a non-Hermitian Hamiltonian, remains challenging. Here we develop two quantum simulation schemes to directly extract all components of the QGT in pseudo-Hermitian systems with real spectra. Each scheme independently determines the complete QGT using generalized expectation values of either the energy fluctuation operator or the generalized force operator with respect to two time-evolved states prepared through distinct nonadiabatic evolutions, thereby establishing two self-contained measurement protocols. We illustrate the validity of these schemes on two $q$-deformed two-band models: one with nontrivial topology and the other with a nonvanishing off-diagonal quantum metric. Numerical simulations demonstrate that, for suitably chosen nonadiabatic ramp velocities, both schemes achieve high-fidelity agreement with theoretical predictions for measuring the QGT in both models and successfully capture the topological phase transition of the first model using Chern numbers calculated from Berry curvatures. For larger velocities, the generalized force scheme yields greater accuracy for the real part of the QGT, while the energy fluctuation scheme better captures its imaginary part. This work establishes a framework for extending dynamical measurement schemes from Hermitian to pseudo-Hermitian systems with real spectra.

Figures (5)

• Figure 1: QGT components for model I, extracted using (a) the energy fluctuation scheme and (b) the generalized force scheme. Analytical results are shown as lines, while numerical data are represented by markers. Specifically, $Q_{\theta\theta}$ is shown with solid lines and circles, $Q_{\phi\phi}$ with dashed lines and triangles, $\mathrm{Re}[Q_{\phi\theta}]$ with dashed-dotted lines and diamonds, and $\mathrm{Im}[Q_{\phi\theta}]$ with dotted lines and thin diamonds. The parameters are set to $\Omega_1/2\pi = 10\,\mathrm{MHz}$, $\Delta_1/2\pi = 15\,\mathrm{MHz}$, $\Delta_2 = 0$, and $\phi^0 = 0$.
• Figure 2: Chern number phase diagram for model I. The Chern numbers are calculated by integrating Berry curvatures extracted using the energy fluctuation scheme (diamonds) and the generalized force scheme (circles), and are compared with analytical results (solid line). The parameters are fixed at $\Omega_1/2\pi = 10\,\mathrm{MHz}$ and $\Delta_1/2\pi = 15\,\mathrm{MHz}$, while $\Delta_2/2\pi$ is varied from $0$ to $30\,\mathrm{MHz}$, corresponding to $\Delta_2/\Delta_1$ ranging from $0$ to $2$.
• Figure 3: QGT components for model II, extracted using (a) the energy fluctuation scheme and (b) the generalized force scheme. Analytical results are shown as lines, while numerical data are represented by markers, following the same style conventions as in Fig. \ref{['fig: pH_spin_QGT']}, with subscripts adjusted for the present model. The parameters are set to $B/2\pi = 10\,\mathrm{MHz}$ and $y^0 = \pi/2$.
• Figure 4: Dependence of measured QGT components on ramp velocity $v$ for both measurement schemes. Left panels: $Q_{\theta\theta}$, $Q_{\phi\theta}$, and $Q_{\phi\phi}$ (top to bottom) for model I at $(\theta^0, \phi^0) = (\pi/2, 0)$ with $\Omega_1/2\pi = 10\,\mathrm{MHz}$, $\Delta_1/2\pi = 15\,\mathrm{MHz}$, and $\Delta_2 = 0$. Right panels: $Q_{xx}$, $Q_{yx}$, and $Q_{yy}$ (top to bottom) for model II at $(x^0, y^0) = (\pi, \pi/2)$ with $B/2\pi = 10\,\mathrm{MHz}$. Numerical results from the energy fluctuation scheme ($\Delta^2 H$) and the generalized force scheme ($f_\mu$) are distinguished by line styles as indicated in the legend. Horizontal dashed lines denote theoretical values. Circles indicate the critical ramp velocities at which the absolute deviation between numerically extracted and theoretical values first exceeds $0.005$ for model I or $0.01$ for model II (see Table \ref{['table: critical velocities']} in Appendix \ref{['sec: error']}).
• Figure 5: Quantum circuit for measuring the generalized expectation value $\langle\!\langle O \rangle\!\rangle = \frac{\langle \psi_1 | O | \psi_2 \rangle}{\langle \psi_1 | \psi_2 \rangle}$ via a controlled-swap operation. The ancilla qubit (top line) controls a swap operation between two $n$-qubit registers storing arbitrary states $|\psi_1\rangle$ and $|\psi_2\rangle$. Measurement of $\sigma_{x(y)}$ on the ancilla yields the real (imaginary) part of $\langle \psi_1 | O | \psi_2 \rangle \langle \psi_2 | \psi_1 \rangle$, corresponding to the numerator of the generalized expectation value. The denominator $|\langle \psi_1 | \psi_2 \rangle|^2$ is obtained by setting $O$ as the identity operator $I$ and measuring $\sigma_x$ on the ancilla.