Experimental differentiation and extremization with analog quantum circuits

TL;DR

The paper demonstrates, for the first time, experimental differentiation and extremization of differentiable quantum circuits (DQC and QEL) on a commercial analog quantum computer using neutral-atom qubits. By encoding a solvable first-order ODE with a feature map and trainable ansatz, and employing agpsr for derivative evaluation in an analog setting, the authors train a surrogate solution $f(x)$ and locate its extremum $x_{ ext{opt}}$ with close agreement to the analytical solution. The work showcases a practical pathway to use variational quantum algorithms for scientific computing on near-term hardware, including a closed-loop experimental protocol and multiplexing to reduce resource use. It also outlines necessary hardware considerations and future improvements toward more expressive digital-analog circuits and gradient-based optimization on analog platforms.

Abstract

Solving and optimizing differential equations (DEs) is ubiquitous in both engineering and fundamental science. The promise of quantum architectures to accelerate scientific computing thus naturally involved interest towards how efficiently quantum algorithms can solve DEs. Differentiable quantum circuits (DQC) offer a viable route to compute DE solutions using a variational approach amenable to existing quantum computers, by producing a machine-learnable surrogate of the solution. Quantum extremal learning (QEL) complements such approach by finding extreme points in the output of learnable models of unknown (implicit) functions, offering a powerful tool to bypass a full DE solution, in cases where the crux consists in retrieving solution extrema. In this work, we provide the results from the first experimental demonstration of both DQC and QEL, displaying their performance on a synthetic usecase. Whilst both DQC and QEL are expected to require digital quantum hardware, we successfully challenge this assumption by running a closed-loop instance on a commercial analog quantum computer, based upon neutral atom technology.

Figures (4)

• Figure 1: The neutral-atom register. A pictorial representation of the setup used for the experiment, along with an indication of the main components. a A zoom on the optical clamping of the Rb atoms, attained via the tweezers inside the vacuum chamber. b Perspective representation of the regular hexagonal 61-atoms array generated for each circuit execution, with the two multiplexed qubit sets targeted for the experiment highlighted in green. cFeature Map and trainable ansatz via laser pulse sequence. The area of the first pulse $\Omega$ maps the value of the feature variable $x$, whereas the phase $\phi$ of the last pulse corresponds to the ansatz parameter $\theta$. The modulation of the ideal square pulses (as solid lines) is visible as overlapped shaded areas. This inset was generated using pulserpulser and shows the longest sequence, with the highest phase value that was used in this work.
• Figure 2: Derivatives calculated after applying the (orange dots) to the output of the the parameterised quantum circuit of Fig. \ref{['fig:register']}c (blue dots), representing $f_\theta(x)$. For comparison, we report alongside the function and its derivative, as calculated from the interpolated smoothed data (grey lines). We present the results for each of the 9 values of the ansatz parameter $\theta$, as elicited in the legend of each plot. The x-axis represents the value of the input variable $x$.
• Figure 3: Experimental results for the execution of DQC and QEL protocols on a .a (On the y-axis) the loss as calculated from the $8$ chosen collocation points $\{x_i\}$, (on the x-axis) for various values of the ansatz parameter $\theta$. As blue dots we plot the root of $L_d(\theta)$ in Eq. \ref{['eq:loss']}, whereas as orange dots we plot the boundary loss $L_b(\theta)$. b As blue dots, the total magnetization as obtained from experimental data, representing the target function $f(x)$ as estimated via the parameterised quantum circuit, with the pink cross marking its minimum. For comparison, the green line shows the analytical solution of the differential equation, Eq. \ref{['eq:de_rounded']}. As orange dots, the derivative $df/dx$ calculated by . Error bars for the function (derivative) estimates represent the (propagated) shot noise from the device. On the x-axis we report the value of the input variable $x$. The gray dashed lines mark the analytical minimum location, and the zero of the y-axis.
• Figure 4: Data and simulation. Data obtained from the (blue dots) for 9 values of the ansatz parameter $\theta$ (legended for each plot), compared against pulser noiseless simulations. Error bars represent the Poissonian shot noise expected from the attained number of shots at each point. The orange line is the result of the simulation using a default hardware configuration, whereas the green line is obtained by including the contribution from a detuning offset $\delta_{\text{offset}}$, as explained in the text. The x-axis represents the value of the input variable $x$, whilst the y-axis represents the total magnetization of the output, as defined in Eq. \ref{['eq:circ_output']}.