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arXiv:1612.03767 · Physical Review Letters

Estimating the Error of an Analog Quantum Simulator by Additional Measurements

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

Published Dec 13, 2017• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.119.240502

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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.
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 nnn-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.
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.
Simulation of interacting electron systems is one of the great challenges of modern quantum chemistry and solid state physics. Controllable quantum systems offer the opportunity to create artificial structures which mimic the system of interest. An interesting quantity to extract from these quantum simulations is the spectral function. We map a noisy quantum simulator onto a fermionic system and investigate the influence of decoherence on the simulation of the spectral density using a diagrammatic approach on the Keldysh contour. We show that features stronger than the single-qubit decoherence rate can be resolved, while weaker features wash out. For small systems, we compare our Keldysh approach to master-equation calculations.
Different approaches in quantifying environmentally-induced decoherence are considered. We identify a measure of decoherence, derived from the density matrix of the system of interest, that quantifies the environmentally induced error, i.e., deviation from the ideal isolated-system dynamics. This measure can be shown to have several useful features. Its behavior as a function of time has no dependence on the initial conditions, and is expected to be insensitive to the internal dynamical time scales of the system, thus only probing the decoherence-related time dependence. For a spin-boson model - a prototype of a qubit interacting with environment - we also demonstrate the property of additivity: in the regime of the onset of decoherence, the sum of the individual qubit error measures provides an estimate of the error for a several-qubit system, even if the qubits are entangled, as expected in quantum-computing applications. This makes it possible to estimate decoherence for several-qubits quantum computer gate designs for which explicit calculations are exceedingly difficult.
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.
We use Deutsch's algorithm as a stand in for more complex quantum algorithms in order to determine how quantum properties of an environment manifest themselves in results that can be obtained on quantum computers. We model pure dephasing in two different ways; one keeps the full density matrix of the qubits and environments (quantum) while the other uses Kraus operators (classical). We find that a single run of the algorithm yields the same effect in both cases, but running the algorithm twice leads to stark differences. Taking correlations and interplay between different decoherence processes into account leads to a slowing of decoherence effects for balanced functions. For constant functions, the effect is much more pronounced, and there is a qualitative change in the dependence of measurement outcomes on decoherence. We present results obtained on one of the IBM Quantum processors, which fully reproduce the predicted effect regardless of the assumptions made in the derivation. We further illustrate the findings on NV center spin qubits, which show more complex behavior due to a small size of the environment.