Measuring full counting statistics in a trapped-ion quantum simulator

TL;DR

This work demonstrates the measurement of full counting statistics ($\chi(\alpha)$) and probability distribution functions ($p(q)$) of subsystem observables in a trapped-ion quantum simulator after quenches. It leverages randomized measurements and classical shadows to extract FCS and PDFs for transverse and longitudinal magnetizations, including nonconserved observables, in Néel and tilted ferromagnet initial states. A bit-flip error model and its incorporation into FCS predictions are developed, showing that FCS robustly diagnose experimental imperfections and distinguish quantum states. The approach is broadly applicable to other quantum platforms and enables detailed probing of fluctuations, symmetry breaking, and relaxation in many-body dynamics.

Abstract

In quantum mechanics, the probability distribution function (PDF) and full counting statistics (FCS) play a fundamental role in characterizing the fluctuations of quantum observables, as they encode the complete information about these fluctuations. In this letter, we measure these two quantities in a trapped-ion quantum simulator for the transverse and longitudinal magnetization within a subsystem. We utilize the toolbox of classical shadows to postprocess the measurements performed in random bases. The measurement scheme efficiently allows access to the FCS and PDF of all possible operators on desired choices of subsystems of an extended quantum system.

Figures (11)

• Figure 1: a) Measurement protocol using classical shadows: The protocol used to measure the full counting statistics (FCS) and the probability distribution function (PDF) of an observable $O$ in a subsystem $A$ is based on collecting randomized shadows of the reduced density matrix $\rho_A$. The quantum system is prepared in the desired state $\ket{\Psi(t)}$ and measured in Haar random bases. Experimental repetitions with various random bases provide access to the FCS and PDF. (b, e) Magnetization PDFs in a quench from the Néel state: The PDF $p_x(q)$, of the longitudinal magnetization $S_A^x$, and the PDF $p_z(q)$, of the transverse magnetization $S_A^z$ at different times is shown. (c, d, f, g) Magnetization FCS in a quench from the Néel state: Real and imaginary parts of the FCS of $S_A^x$ (in (c-d)) and that of $S_A^z$ (in (f-g)) are shown. As a proof of principle, the experiment captures the expected behavior for the PDF and FCS (see main text for details). In all the panels, different colors correspond to three different times; $t=0$ms (gray), $t=1$ms (red), and $t=4$ms (blue). The dotted lines show the theoretical predictions for a perfect Néel state whereas the dashed lines are the predictions for initial state in presence of experimental errors. Symbols and error bars show experimental data. The total system size is $N=10$ with subsystem, of size $N_A=4$, chosen from the central sites.
• Figure 2: Magnetization PDF in the quench from tilted ferromagnets: The PDF in a system of $N=12$ and $N_A=4$, for two of the magnetizations, $S_A^z$ (panels (a-c)) and $S_A^x$ (panels (b-d)) are shown. Dashed curves (theory including decoherences during the quench dynamics) match the experimental data (dots and error bars). (a-b) The PDFs are shown as functions of the possible measurement outcomes $q$ for the tilt angles $\theta=0.2\pi$ and $0.5\pi$ at $t=0$. (c-d) We plot the PDFs during the quench dynamics for several values of $q$ for initial state with $\theta=0.5\pi$.
• Figure 3: Magnetization FCS in a quench from tilted ferromagnets: FCS in a system of $N=12$ sites and subsystem of size $N_A=4$ is shown for different initial states. A good match between theory (dashed curves) and experiment (dots and error bars) is observed. (a-b) Real and imaginary parts of the FCS $\chi_z(\alpha)$ for the $S_A^z$ at initial time $t=0$ as functions of the spectral parameter $\alpha$. We see different behavior for different tilt angles $\theta$ (red- $0.2\pi$, green- $0.33\pi$, blue- $0.5\pi$). (c-d) Imaginary part of the FCS $\chi_\mu (t)$ for $\mu=z$ and $x$ under the quench \ref{['eq:IsingH']}. The behavior of FCS with $\alpha$ and $t$ is consistent with the PDFs reported in Fig. \ref{['fig:fig3']} (see main text).
• Figure 4: Histograms of spin projections following applications of random unitaries. On the $x$-axis, the label $m$ is the count of up spin measured at local sites. For a good statistics, we have combined the measurement data on all sites. (Top): uniform distribution of implemented RUs is seen in the experiment beginning with a Néel state. The experiment has 500 RUs with 150 shots per RU. (Down): uniform distribution of implemented RUs is seen in the experiment beginning with tilted ferromagnetic states; blue- data for the state with tilting angle $\theta=0.2\pi$ and orange- data for the state with tilting angle $\theta=0.5\pi$. The experiment has 500 RUs with 30 shots per RU.
• Figure 5: In a quenched Néel state, measurements of expectation of the longitudinal magnetization and its squared (left panel), and that of transverse magnetization and its squared (right panel) are shown. The dotted curve show ideal theory whereas the dashes include initial state preparation and measurement errors. The system comprises of $N=10$ qubits with subsystem $A$ sized $N_A=4$.