Identification of time-varying signals in quantum systems

The identification of time-varying parameters (e.g., directly inaccessible in situ signals in vacuum and low-temperature environments) is prevalent for characterizing the dynamics of quantum processes. Under certain circumstances, they can be identified from time-resolved measurements via Ramsey interferometry experiments, but only with specially designed probe systems can the parameters be explicitly read out, and a rigorous identifiability analysis is lacking, i.e., whether the measurement data are sufficient for unambiguous identification. In this paper we formulate this problem as the invertibility of the input-output mapping associated with the quantum system for which an algebraic identifiability criterion is derived based on the system's relative degree. The invertibility analysis also leads to an inversion-based algorithm for numerically identifying the parameters, which is computationally much more efficient than nonlinear regression methods. The effectiveness of the criterion are demonstrated by numerical examples.

Figure 1

Schematic diagram of the identification of time-varying signals in a superconducting quantum computing system. The in situ signals are delivered from ambient signal generators that experience distortion and crosstalk, but the actual signals are not accessible by room-temperature devices. A quantum probe (e.g., a superconducting qubit) is placed to receive the signal, whose measurement signals are conducted out for identifying the in situ signals.

Figure 2

Ramsey-experiment-based identification. The measured output (a) corresponds to two different time traces of the qubit phase (b) in which one is false. The resulting identified in situ input (c) is thus undecidable.

Figure 3

Identification of the in situ signal by using inversion method and regression method and the denominators in the inversion method. The identified signal by using inversion method by using single measurement and regression method with a good and a bad initial guess on $g\left(t\right)$, respectively. The identification encounters singularity at around $t=130\mathrm{ns}$.

Figure 4

The identification of multiple in situ signals ${\omega }_1\left(t\right),{\omega }_2\left(t\right),$ and $g\left(t\right)$ (all chosen as distorted step functions) in a two-qubit system via measurements on three and five observables. Singularities appear in the three-observable case when a two-point differential is used (red dashed line), but they disappear in the five-observable case (blue solid line) or three-observable case using three-point differentials (yellow dashed line).

Figure 5

(a) The calculated in situ signal when there is low-frequency system noise (${\mathbf{n}}_e$). (b) The calculated in situ signal when there is high-frequency system noise. (c) The calculated in situ signal when there is low-frequency measurement noise (${\mathbf{n}}_m$). (d) The calculated in situ signal when there is high-frequency measurement noise.