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Journal of Physics A Mathematical and Theoretical

Hamiltonian identification in presence of large control field perturbations

Ying Fu, HERSCHEL A. RABITZ, Gabriel Turinici

Published Nov 16, 2016• 3 citations
Publisher

Abstract

Abstract Quantum system inversion concerns learning the characteristics of the underlying Hamiltonian by measuring suitable observables from the responses of the system’s interaction with members of a set of applied fields. Various aspects of inversion have been confirmed in theoretical, numerical and experimental works. Nevertheless, the presence of noise arising from the applied fields may contaminate the quality of the results. In this circumstance, the observables satisfy probability distributions, but often the noise statistics are unknown. Based on a proposed theoretical framework, we present a procedure to recover both the unknown parts of the Hamiltonian and the unknown noise distribution. The procedure is implemented numerically and seen to perform well for illustrative Gaussian, exponential and bi-modal noise distributions.

DOI
10.1088/1751-8113/49/49/495301

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Abstract Quantum system inversion concerns learning the characteristics of the underlying Hamiltonian by measuring suitable observables from the responses of the system’s interaction with members of a set of applied fields. Various aspects of inversion have been confirmed in theoretical, numerical and experimental works. Nevertheless, the presence of noise arising from the applied fields may contaminate the quality of the results. In this circumstance, the observables satisfy probability distributions, but often the noise statistics are unknown. Based on a proposed theoretical framework, we present a procedure to recover both the unknown parts of the Hamiltonian and the unknown noise distribution. The procedure is implemented numerically and seen to perform well for illustrative Gaussian, exponential and bi-modal noise distributions.
The problem of recovering the Hamiltonian and dipole moment is considered in a bilinear quantum control framework. The process uses as inputs some measurable quantities (observables) for each admissible control. If the implementation of the control is noisy the data available is only in the form of probability laws of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to phase factors). Both additive and multiplicative noises are considered. Numerical illustrations support the theoretical results.
The identification of quantum system Hamiltonians through the use of experimental data remains an important research goal. Seeking a Hamiltonian that is consistent with experimental measurements constitutes an excursion over a Hamiltonian inversion landscape, which is the quality of reproducing the data as a function of the Hamiltonian parameters. Recent theoretical work showed that with sufficient experimental data there should be local convexity about the true Hamiltonian on the landscape. The present paper builds on this result and performs simulations to test whether such convexity is observed. A gradient-based Hamiltonian search algorithm is incorporated into an inversion routine as a means to explore the local inversion landscape. The simulations consider idealized noise-free as well as noise-ridden experimental data. The results suggest that a sizable convex domain exists about the true Hamiltonian, even with a modest amount of experimental data and in the presence of a reasonable level of noise.
An approximate quantum Hamiltonian identification algorithm is presented with the assumption that the system initial state and observation matrix can be set appropriately. We sample the system with a fixed period and using the sampled data we estimate the Hamiltonian based on a Taylor expansion of the matrix exponential function. We prove the estimation error is linear in the variance of the additive Gaussian noise. We also propose a heuristic formula to find the order of magnitude of the optimal sampling period. Two numerical examples are presented to validate the theoretical results.
A method for inverting observations on quantum mechanical systems to obtain estimates of unknown parameters residing in the Hamiltonian is presented. The quantal system is represented in matrix form with respect to a chosen basis, and it is assumed that the associated expansion coefficients are truncated to a finite dimension. The uncontrollable laboratory noise will be modeled by means of an inhomogeneous white noise process so that the experimental observations are represented as stochastic variables satisfying a stochastic differential equation. It will be assumed that measurements obtained from an experiment are now equivalent to a realization of these stochastic variables. It is known from filtering theory that the minimum variance estimate of the unknown parameters in the quantal model is now given by the expectation of the unknowns conditional on this realization. This estimator can be calculated analytically from the associated a posteriori probability density if the original quantal system does not contain any random elements. This probability density for the unknown matrix elements is calculated, and it is demonstrated that for a full Hamiltonian matrix the asymptotic variance of the parameter estimator decreases as a third power in time and a fourth power in the initial conditions. Some differences with the minimum least-square method are mentioned, and a few issues of numerical implementation are discussed.
Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.
We introduce optimal identification (OI), a collaborative laboratory/computational algorithm for extracting quantum Hamiltonians from experimental data specifically sought to minimize the inversion error. OI incorporates the components of quantum control and inversion by combining ultrafast pulse shaping technology and high throughput experiments with global inversion techniques to actively identify quantum Hamiltonians from tailored observations. The OI concept rests on the general notion that optimal data can be measured under the influence of suitable controls to minimize uncertainty in the extracted Hamiltonian despite data limitations such as finite resolution and noise. As an illustration of the operating principles of OI, the transition dipole moments of a multilevel quantum Hamiltonian are extracted from simulated population transfer experiments. The OI algorithm revealed a simple optimal experiment that determined the Hamiltonian matrix elements to an accuracy two orders of magnitude better than obtained from inverting 500 random data sets. The optimal and nonlinear nature of the algorithm were shown to be capable of reliably identifying the Hamiltonian even when there were more variables than observations. Furthermore, the optimal experiment acted as a tailored filter to prevent the laboratory noise from significantly propagating into the extracted Hamiltonian.