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arXiv:1701.02683 · Physical Review A

Reconstructing the ideal results of a perturbed analog quantum simulator

Iris Schwenk, Jan-Michael Reiner, Sebastian Zanker, Lin Tian, Juha Leppäkangas, Michael Marthaler

Published Apr 6, 2018• 7 citations
arXivHTMLPDF

Abstract

Well-controlled quantum systems can potentially be used as quantum simulators. However, a quantum simulator is inevitably perturbed by coupling to additional degrees of freedom. This constitutes a major roadblock to useful quantum simulations. So far there are only limited means to understand the effect of perturbation on the results of quantum simulation. Here we present a method which, in certain circumstances, allows for the reconstruction of the ideal result from measurements on a perturbed quantum simulator. We consider extracting the value of the correlator $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{j}(0)\ensuremath{\rangle}$ from the simulated system, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}$ are the operators which couple the system to its environment. The ideal correlator can be straightforwardly reconstructed by using statistical knowledge of the environment, if any $n$-time correlator of operators ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}$ of the ideal system can be written as products of two-time correlators. We give an approach to verify the validity of this assumption experimentally by additional measurements on the perturbed quantum simulator. The proposed method can allow for reliable quantum simulations with systems subjected to environmental noise without adding an overhead to the quantum system.

DOI
10.1103/physreva.97.042310

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Well-controlled quantum systems can potentially be used as quantum simulators. However, a quantum simulator is inevitably perturbed by coupling to additional degrees of freedom. This constitutes a major roadblock to useful quantum simulations. So far there are only limited means to understand the effect of perturbation on the results of quantum simulation. Here we present a method which, in certain circumstances, allows for the reconstruction of the ideal result from measurements on a perturbed quantum simulator. We consider extracting the value of the correlator \ensuremath⟨O\ifmmode^\elseˆ\fii(t)O\ifmmode^\elseˆ\fij(0)\ensuremath⟩\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}(t){\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{j}(0)\ensuremath{\rangle}\ensuremath⟨O\ifmmode^\elseˆ\fi​i(t)O\ifmmode^\elseˆ\fi​j(0)\ensuremath⟩ from the simulated system, where O\ifmmode^\elseˆ\fii{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}O\ifmmode^\elseˆ\fi​i are the operators which couple the system to its environment. The ideal correlator can be straightforwardly reconstructed by using statistical knowledge of the environment, if any nnn-time correlator of operators O\ifmmode^\elseˆ\fii{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{O}}^{i}O\ifmmode^\elseˆ\fi​i of the ideal system can be written as products of two-time correlators. We give an approach to verify the validity of this assumption experimentally by additional measurements on the perturbed quantum simulator. The proposed method can allow for reliable quantum simulations with systems subjected to environmental noise without adding an overhead to the quantum system.
We study an analog quantum simulator coupled to a reservoir with a known spectral density. The reservoir perturbs the quantum simulation by causing decoherence. The simulator is used to measure an operator average, which cannot be calculated using any classical means. Since we cannot predict the result, it is difficult to estimate the effect of the environment. Especially, it is difficult to resolve whether the perturbation is small or if the actual result of the simulation is in fact very different from the ideal system we intend to study. Here, we show that in specific systems a measurement of additional correlators can be used to verify the reliability of the quantum simulation. The procedure only requires additional measurements on the quantum simulator itself. We demonstrate the method theoretically in the case of a single spin connected to a bosonic environment.
Tracing out the environmental degrees of freedom is a necessary procedure when simulating open quantum systems. While being an essential step in deriving a tractable master equation it represents a loss of information. In situations where there is strong interplay between the system and environmental degrees of freedom this loss makes understanding the dynamics challenging. These dynamics, when viewed in isolation, have no time-local description: they are non-Markovian and memory effects induce complex features that are difficult to interpret. To address this problem, we here show how to use system correlations, calculated by any method, to infer any correlation function of a Gaussian environment, so long as the coupling between system and environment is linear. This not only allows reconstruction of the full dynamics of both system and environment, but also opens avenues into studying the effect of a system on its environment. In order to obtain accurate bath dynamics, we exploit a numerically exact approach to simulating the system dynamics, which is based on the construction and contraction of a tensor network that represents the process tensor of this open quantum system. Using this we are able to find any system correlation function exactly. To demonstrate the applicability of our method we show how heat moves between different modes of a bosonic bath when coupled to a two-level system that is subject to an off-resonant drive.
An important step in quantum simulation is to measure the many-body correlations of the simulated model. For a practical quantum simulator composed of finite number of qubits and cavities, in contrast to ideal many-body systems in the thermodynamic limit, a measurement device can generate strong backaction on the simulator, which could prevent the accurate readout of the correlation functions. Here we calculate the readout of a detector coupled to an analog quantum simulator. We show that reliable characterization of the many-body correlations in the simulator can be achieved when the coupling operators obey the Wick's theorem. Our results are illustrated with two examples: a simulator for an harmonic oscillator and a simulator for the free electron gas.
Analog quantum simulators (AQS) will likely be the first nontrivial application of quantum technology for predictive simulation. However, there remain questions regarding the degree of confidence that can be placed in the results of AQS since they do not naturally incorporate error correction. Specifically, how do we know whether an analog simulation of a quantum model will produce predictions that agree with the ideal model in the presence of inevitable imperfections? At the same time there is a widely held expectation that certain quantum simulation questions will be robust to errors and perturbations in the underlying hardware. Resolving these two points of view is a critical step in making the most of this promising technology. In this work we formalize the notion of AQS reliability by determining sensitivity of AQS outputs to underlying parameters, and formulate conditions for robust simulation. Our approach naturally reveals the importance of model symmetries in dictating the robust properties. To demonstrate the approach, we characterize the robust features of a variety of quantum many-body models.
Reconstructing the effective equation of motion for the time evolution of a subset of degrees of freedom of a larger system remains a problem of interest in quantum physics. Many methods have been developed, but they either rely on an ad hoc ansatz, demand data that is not experimentally accessible, or lack physical interpretability. The authors employ machine-learning methods to infer the effective dynamical generator from a noisy, finite set of local measurements. Their method yields interpretable results that may be used to infer noise models on quantum simulators, or to study thermalization dynamics in quantum many-body systems.
Various fundamental phenomena of strongly correlated quantum systems such as high-T(c) superconductivity, the fractional quantum-Hall effect and quark confinement are still awaiting a universally accepted explanation. The main obstacle is the computational complexity of solving even the most simplified theoretical models which are designed to capture the relevant quantum correlations of the many-body system of interest. In his seminal 1982 paper (Feynman 1982 Int. J. Theor. Phys. 21 467), Richard Feynman suggested that such models might be solved by 'simulation' with a new type of computer whose constituent parts are effectively governed by a desired quantum many-body dynamics. Measurements on this engineered machine, now known as a 'quantum simulator,' would reveal some unknown or difficult to compute properties of a model of interest. We argue that a useful quantum simulator must satisfy four conditions: relevance, controllability, reliability and efficiency. We review the current state of the art of digital and analog quantum simulators. Whereas so far the majority of the focus, both theoretically and experimentally, has been on controllability of relevant models, we emphasize here the need for a careful analysis of reliability and efficiency in the presence of imperfections. We discuss how disorder and noise can impact these conditions, and illustrate our concerns with novel numerical simulations of a paradigmatic example: a disordered quantum spin chain governed by the Ising model in a transverse magnetic field. We find that disorder can decrease the reliability of an analog quantum simulator of this model, although large errors in local observables are introduced only for strong levels of disorder. We conclude that the answer to the question 'Can we trust quantum simulators?' is … to some extent.