Experimental graybox quantum system identification and control

TL;DR

The paper tackles the challenge of characterizing and controlling quantum devices when the control-to-Hamiltonian dependence is unknown. It introduces a graybox architecture that combines a trainable black-box mapping controls to Hamiltonian elements with a white-box Hamiltonian-evolution pipeline, yielding both the unitary $U=e^{-i H T}$ and measurement statistics via the Born rule. Experiments on a 3-mode voltage-controlled photonic chip demonstrate high-fidelity synthesis of target unitaries and output distributions, with the graybox approach outperforming traditional whitebox and pure blackbox models while exposing the effective Hamiltonian's dependence on controls. The method provides physically interpretable, controllable models for quantum devices and is extendable to time-dependent and open quantum systems, with potential impact on quantum noise spectroscopy and cancellation.

Abstract

Understanding and controlling engineered quantum systems is key to developing practical quantum technology. However, given the current technological limitations, such as fabrication imperfections and environmental noise, this is not always possible. To address these issues, a great deal of theoretical and numerical methods for quantum system identification and control have been developed. These methods range from traditional curve fittings, which are limited by the accuracy of the model that describes the system, to machine learning methods, which provide efficient control solutions but no control beyond the output of the model, nor insights into the underlying physical process. Here we experimentally demonstrate a "graybox" approach to construct a physical model of a quantum system and use it to design optimal control. We report superior performance over model fitting, while generating unitaries and Hamiltonians, which are quantities not available from the structure of standard supervised machine learning models. Our approach combines physics principles with high-accuracy machine learning and is effective with any problem where the required controlled quantities cannot be directly measured in experiments. This method naturally extends to time-dependent and open quantum systems, with applications in quantum noise spectroscopy and cancellation.

Figures (5)

• Figure 1: Physical and machine learning models of the class of quantum devices considered in this paper. These are described by a time-independent Hamiltonian in the absence of interaction with the environment. a) Representation of the class of quantum devices considered in this work. b) A schematic of an integrated photonic voltage-controlled reconfigurable waveguide array chip, implementing a noiseless time-independent Hamiltonian. Photons enter from the input port (on the left), undergo a voltage-controlled propagation along the chip, and are then measured at the output port of the chip (on the right). c) The structure of the proposed graybox model. The input to the model is the set of $M$ controls, while the outputs are the quantum measurements for the set of computational basis as initial states. $P_{a \to b}$ indicates the transition probability from input port $a$ to output port $b$. The graybox is a combination of black and white boxes. The blackbox estimates the real and imaginary components of each matrix element of the Hamiltonian. The whitebox layers construct the Hamiltonian matrix and perform the quantum evolution and measurements. d) A fully whitebox architecture where a physical model is utilized. The first layer generates predefined Hamiltonian parameters that follow a known analytical dependence. The remaining layers perform the quantum evolution and measurements. e) A fully blackbox model where only a generic neural network is utilized with no physical model.
• Figure 2: Protocol schematic. The first step is to construct an experimental dataset by applying controls to the system and measuring the corresponding outputs. The dataset is then used to train the machine learning models (1). Next, the trained models are tested (2) for generalization by comparing their output predictions against a different experimental testing dataset. After that, the trained models can be used to optimize controls (3) to achieve a certain target, which could be a Hamiltonian, a unitary gate, or an output probability distribution. Finally, the obtained controls are tested (4) experimentally and the controlled system output is compared against the desired target.
• Figure 3: Experimental performance of the machine learning models. The whitebox model consists of fan-in, reconfigurable, and fan-out sections, each modelled as a real-valued tri-diagonal Hamiltonian in addition to a linear dependence on voltage for the reconfigurable section. (a) Results of training the different models on the experimental dataset. The MSE is plotted versus iteration number (b) The results of evaluating the different models on the testing set.
• Figure 4: Experimental quantum control performance. The distribution of the fidelity between the experimentally measured output power distribution and the desired target distribution for the three models. The whitebox model utilizes a real-valued tri-diagonal Hamiltonian with linear dependence on voltages, in addition to fan-in and fan-out sections. The results are for (a) the output controller, and (b) the unitary controller. The reported values are the average over the three distributions corresponding to each possible initial state. (c) Violin plot showing the statistics of the MSE obtained for the training, testing, and control datasets. The horizontal lines represent from bottom to top, the minimum, median, and maximum respectively. The plot also shows an estimated kernel density for the data.
• Figure 5: Dependence of the Hamiltonian elements to a subset of input voltages, as predicted by the graybox model. (a) Real and (b) imaginary parts of the Hamiltonian matrix elements as a function of voltage when all electrodes are grounded except the first electrode. It should be noted that the imaginary parts of $H_{11}$, $H_{22}$, and $H_{33}$ are by definition equal to zero. The non-linear dependence, the second off-diagonal elements, and the imaginary components indicate an effective Hamiltonian being estimated for a time-dependent system, attributed to non-homogeneity of the chip along the propagation direction.