Efficient Estimation of Resonant Coupling between Quantum Systems

Abstract

We present an efficient method for the characterization of two coupled discrete quantum systems, one of which can be controlled and measured. For two systems with transition frequencies ωq, ωr, and coupling strength g we show how to obtain estimates of g and ωr whose error decreases exponentially in the number of measurement shots rather than as a power law expected in simple approaches. Our algorithm can thereby identify g and ωr simultaneously with high precision in a few hundred measurement shots. This is achieved by adapting measurement settings upon data as it is collected. We also introduce a method to eliminate erroneous estimates with small overhead. Our algorithm is robust against the presence of relaxation and typical noise. Our results are applicable to many candidate technologies for quantum computation, in particular, for the characterization of spurious two-level systems in superconducting qubits or stripline resonators. Parameter estimation in many microscopic and some macroscopic systems inevitably involves quantum measurements. This implies that parameters cannot be identified with a single measurement shot since the outcome of such a measurement is generally random. Instead, the standard approach is to determine ensemble averages for many experiments and fit the parameters of certain quantitative models to those averages. The most common example for this is spectroscopy: it involves direct measurement of the energy splittings between quantum states in the form of resonances to incoming radiation. Typically, a large ensemble average is produced by gathering data from a large number of independent trials, either simultaneously on an ensemble of molecules (in nuclear magnetic resonance [1]) or from many repetitions of a specific experiment (in optical spectroscopy of single molecules, quantum dots, or superconducting qubits [2]).While being reliable in many contexts, this approach is often too resource intensive. Specifically, the error in the estimate of a single expectation value at a fixed measurement setting decreases in proportion to M − 1 2 r after M r measurement shots. Moreover, many choices of measurement settings are usually required for complex measurement tasks. Such slowness of parameter estimation can also turn into imprecision in the estimate if the parameters of interest drift as a function of time, broadening spectroscopic signatures.Imprecision in system characterization is particularly problematic for quantum information processing applications. These require extremely precise logic operations, usually implemented as pulses. The pulse parameters such as length, amplitude, and carrier frequency, depend on the system parameters. In manufactured solid state qubits, this is rather central as they are subject to fabrication uncertainty.In this Letter, we demonstrate that advanced spectroscopy can be performed far more efficiently. Our re-(b) (a) Figure 1: (color online) (a) Theoretically obtained swap spectrum in frequency-waiting time plane.Here, δ = (ωq − ωr,0)/(2g0), with ωq the q...

Structured filtering

“…Current efficient methods for parameter estimation in quantum systems are implicitly biased via structural assumptions about the posterior distribution [6,9,14,39]. Here we show that even the subtle biases introduced through filtering with Liu-West resampling can catastrophically fail to estimate parameters for problems where the experiments chosen are incapable of distinguishing between two equivalent hypotheses.…”

“…In metrology, for instance, experimental data is used to infer parameters of interest such as magnetic fields, temperature, or other physical quantities [1]. The process by which these parameters are learned from data is thus of critical importance to tasks as diverse as metrology and quantum information processing [2][3][4][5][6].…”

Characterization of qubit chains by Feynman probes

“…This happens for several quantities of interest in quantum technology and in all these cases, quantum estimation theory [15][16][17] provides tools to evaluate the ultimate precision attainable by any estimation procedure and to design optimal measurement schemes. Examples include the estimation of the phase [18][19][20][21], quantum correlations [22][23][24], temperature [25,26], characterization of classical processes or environmental parameters [27][28][29][30], and, indeed, the coupling constants of different kinds of interactions [31][32][33][34][35].…”

Adaptive identification of coherent states

“…Bayesian inference techniques have been developed, e.g, for phase [14,17,23,31,42,43], state [44,45], and Hamiltonian [46][47][48][49][50][51] estimation. For certain one-parameter estimation problems it is possible to perform local Bayesian optimization of the measurement settings analytically [14,23,31,48,49].…”

“…For a larger number of unknown parameters, however, finding optimal measurement settings adaptively becomes generally analytically intractable. To perform the Bayesian updates numerically, a sequential Monte-Carlo approach [44,[49][50][51][52][53][54][55] has recently undergone a strong development.…”