We present a scalable protocol for measuring full counting statistics (FCS) in experiments or tensor-network simulations. In this method, an ancilla in the middle of the system acts as a turnstile, with its phase keeping track of the time-integrated particle flux. Unlike quantum gas microscopy, the turnstile protocol faithfully captures FCS starting from number-indefinite initial states or in the presence of noisy dynamics. In addition, by mapping the FCS onto a single-body observable, it allows for stable numerical calculations of FCS using approximate tensor-network methods. We demonstrate the wide-ranging utility of this approach by computing the FCS of the transferred magnetization in a Floquet Heisenberg spin chain, as studied in a recent experiment with superconducting qubits, as well as the FCS of charge transfer in random circuits.
We present a scalable protocol for measuring full counting statistics (FCS) in experiments or tensor-network simulations. In this method, an ancilla in the middle of the system acts as a turnstile, with its phase keeping track of the time-integrated particle flux. Unlike quantum gas microscopy, the turnstile protocol faithfully captures FCS starting from number-indefinite initial states or in the presence of noisy dynamics. In addition, by mapping the FCS onto a single-body observable, it allows for stable numerical calculations of FCS using approximate tensor-network methods. We demonstrate the wide-ranging utility of this approach by computing the FCS of the transferred magnetization in a Floquet Heisenberg spin chain, as studied in a recent experiment with superconducting qubits, as well as the FCS of charge transfer in random circuits.
The dynamical behavior of many-body systems is often richer than what can be anticipated from their static properties. Here we show that in closed quantum systems this becomes evident by considering the statistics of time-integrated observables. In particular, the analytic properties of their generating functions, as estimated by full counting statistics (FCS), allow one to identify FCS phases, i.e., phases with specific fluctuation properties of time-integrated observables, and to locate transitions between these phases. We discuss in detail the case of the quantum Ising chain in a transverse field. We show that this model displays a continuum of full counting statistics transitions, of which the static transition is just an end point. These singularities are not a consequence of particular choices of initial conditions or other external nonequilibrium protocols such as quenches in coupling constants. They can be probed generically through quantum jump statistics of an associated open problem, and for the case of the quantum Ising chain we outline a possible experimental realization of this scheme by digital quantum simulation with cold ions.
Owing to its probabilistic nature, a measurement process in quantum mechanics produces a distribution of possible outcomes. This distribution-or its Fourier transform known as full counting statistics (FCS)-contains much more information than say the mean value of the measured observable, and accessing it is sometimes the only way to obtain relevant information about the system. In fact, the FCS is the limit of an even more general family of observables-the charged moments-that characterize how quantum entanglement is split in different symmetry sectors in the presence of a global symmetry. Here we consider the evolution of the FCS and of the charged moments of a U(1) charge truncated to a finite region after a global quantum quench. For large scales these quantities take a simple large-deviation form, showing two different regimes as functions of time: while for times much larger than the size of the region they approach a stationary value set by the local equilibrium state, for times shorter than region size they show a nontrivial dependence on time. We show that, whenever the initial state is also U(1) symmetric, the leading order in time of FCS and charged moments in the out-of-equilibrium regime can be determined by means of a space-time duality. Namely, it coincides with the stationary value in the system where the roles of time and space are exchanged. We use this observation to find some general properties of FCS and charged moments out of equilibrium, and to derive an exact expression for these quantities in interacting integrable models. We test this expression against exact results in the Rule 54 quantum cellular automaton and exact numerics in the XXZ spin-1/2 chain.
The complete characterization of the charge transport in a mesoscopic device is provided by the full counting statistics (FCS) Pt(m), describing the amount of charge Q=me transmitted during the time t. Although numerous systems have been theoretically characterized by their FCS, the experimental measurement of the distribution function Pt(m) or its moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ is rare and often plagued by strong back-action. Here, we present a strategy for the measurement of the FCS, more specifically its characteristic function $\ensuremath{\chi}(\ensuremath{\lambda})$ and moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$, by a qubit with a set of different couplings ${\ensuremath{\lambda}}_{j}$, $j=1,\ensuremath{\cdots},k,\ensuremath{\cdots}k+p$, $k=\ensuremath{\lceil}n/2\ensuremath{\rceil}$, $p\ensuremath{\ge}0$, to the mesoscopic conductor. The scheme involves multiple readings of Ramsey sequences at the different coupling strengths ${\ensuremath{\lambda}}_{j}$, and we find the optimal distribution for these couplings ${\ensuremath{\lambda}}_{j}$ as well as the optimal distribution Nj of $N=\ensuremath{\sum}{N}_{j}$ measurements among the different couplings ${\ensuremath{\lambda}}_{j}$. We determine the precision scaling for the moments $\ensuremath{\langle}{Q}^{n}\ensuremath{\rangle}$ with the number N of invested resources and show that the standard quantum limit can be approached when many additional couplings $p\ensuremath{\gg}1$ are included in the measurement scheme.
Full counting statistics (FCS) provides a powerful framework to access the statistical information of a system from the characteristic function. However, applications of FCS for generic interacting quantum systems are often hindered by the intrinsic difficulty of classical simulation of quantum many-body problems. Here we propose a quantum algorithm for FCS that can obtain both the particle distribution and cumulants of interacting systems. The algorithm evaluates the characteristic functions by quantum computing and then extracts the distribution and cumulants with classical postprocessing. With digital signal processing theory, we analyze the dependence of accuracy with the number of sampling points for the characteristic functions. We show that the desired number of sampling points for accurate FCS can be reduced by filtering some components of the quantum state that are not of interest. By numeral simulation, we demonstrate the FCS of domain walls for several representative models. Remarkably, we show that the accuracy of our algorithm does not significantly change with the system size for a given number of measurements. The algorithm suggests an avenue for studying full counting statistics on quantum computers.
Full-counting statistics (FCS) provides a systematic framework for characterising current fluctuations in quantum transport, quantum optics, and open quantum systems. We introduce QuantumFCSjl, an open-source Julia package for efficient and flexible numerical FCS calculations. The package defines currents through monitored jump operators and weights, allowing particle, electric, and heat currents to be treated within the same workflow. It implements a recursive cumulant algorithm with dense, sparse, and iterative solver backends, making it suitable for models that are challenging for direct dense methods. We demonstrate the package on photon counting in a driven-dissipative Jaynes-Cummings system near the blockade-breakdown phase transition and on heat-current fluctuations in a non-linear circuit-QED heat engine. These examples show how higher cumulants reveal intermittency between bright and dim emission, antibunching, and provide probes of thermodynamic uncertainty relations. Benchmarks against an existing FCS implementation show substantial speed-ups.
The full counting statistics (FCS) of current has long provided fundamental insights into nonequilibrium systems. Recently, the FCS in quantum many-body systems has attracted growing attention, driven by rapid experimental progress in measuring current fluctuations. Nevertheless, for diffusive quantum many-body dynamics, the FCS of current has yet to be obtained exactly. In this Letter, we present the first exact solution for the FCS of current in a diffusive quantum many-body system, specifically a tight-binding chain with dephasing noise. By leveraging the system's SU(2) symmetry and a mapping to the one-dimensional Hubbard model, we derive an exact Fredholm determinant representation for the moment generating function of the time-integrated current. Our long-time asymptotic analysis shows that the cumulant generating function, and hence the corresponding large-deviation function, exhibit diffusive scaling for any nonzero dephasing. We compare our theoretical predictions with experimentally measured current variance and find consistent diffusive scaling.