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arXiv:2308.01255 · Physical Review A

Full counting statistics of particle distribution on a digital quantum computer

Yun-Zhuo Fan, Dan-Bo Zhang

Published Jan 8, 2024• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.109.012412

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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 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.
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.
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.
We demonstrate that the probability distribution of the net number of electrons passing through a quantum system in a junction obeys a steady-state fluctuation theorem (FT) which can be tested experimentally by the full counting statistics (FCS) of electrons crossing the lead-system interface. The FCS is calculated using a many-body quantum master equation combined with a Liouville space generating function formalism. For a model of two coupled quantum dots, we show that the FT becomes valid for long binning times, and provide an estimate for the finite-time deviations. The Mandel (or Fano) parameter associated with the incoming or outgoing electron transfers shows sub-Poissonian (antibunching) statistics.
We study the full counting statistics (FCS) of quantum gases in samples of thousands of interacting bosons, detected atom by atom after a long free-fall expansion.In this far-field configuration, the FCS reveals the many-body coherence from which we characterize iconic states of interacting lattice bosons by deducing their normalized correlations g (n) (0) up to the order n = 6.In Mott insulators, we find a thermal FCS characterized by perfectly contrasted correlations g (n) (0) = n!.In interacting Bose superfluids, we observe small deviations to the Poisson FCS and to the ideal values g (n) (0) = 1 expected for a pure condensate.These deviations become larger as we increase the interaction strength and reveal the role of the quantum depletion of the condensate on many-body coherence.The approach to many-body correlations demonstrated here is readily extendable to characterize a large variety of interacting quantum states and phase transitions.