Classical feature map surrogates and metrics for quantum control landscapes

TL;DR

The paper addresses the challenge of understanding and learning quantum cost landscapes by introducing three feature-map representations of parametrized quantum dynamics: a Lie-Fourier partial-sum expansion, a Taylor polynomial expansion, and a low-dimensional sinc kernel regression. It derives fundamental bounds on landscape derivatives, proves Lipschitz continuity, and reveals how symmetries shape the spectral content, including selection rules and resonances in the Ising model. It then develops a kernel-based surrogate (sinc kernel) and analyzes its learning performance, showing trade-offs between data size and bandwidth, with a local Taylor representation providing efficient approximations when time-energy budgets are limited. The work connects landscape structure to optimizer design, offering quantitative guidance on step sizes, stopping criteria, and global-search strategies, and highlights how spectral properties constrain the hardness of optimization, including barren plateaux and quantum speed limits. Overall, these results provide principled, physics-informed surrogates and metrics to guide learning and optimization in quantum control and variational quantum algorithms.

Abstract

We derive and analyze three feature map representations of parametrized quantum dynamics, which generalize variational quantum circuits. These are (i) a Lie-Fourier partial sum, (ii) a Taylor expansion, and (iii) a finite-dimensional sinc kernel regression representation. The Lie-Fourier representation is shown to have a dense spectrum with discrete peaks, that reflects control Hamiltonian properties, but that is also compressible in typically found symmetric systems. We prove boundedness in the spectrum and the cost function derivatives, and discrete symmetries of the coefficients, with implications for learning and simulation. We further show the landscape is Lipschitz continuous, linking global variation bounds to local Taylor approximation error - key for step size selection, convergence estimates, and stopping criteria in optimization. This also provides a new form of polynomial barren plateaux originating from the Lie-Fourier structure of the quantum dynamics. These results may find application in local and general surrogate model learning, which we benchmark numerically, in characterizations of hardness in the problem instances, and for meta-parameter heuristics in quantum optimizers.

Figures (6)

• Figure 1: The time-evolved expectation value $\langle \hat{O}(\boldsymbol{u})\rangle$ of an observable (e.g. state fidelity $\hat{O} = \ket{\chi} \bra{\chi}$) for a system controlled through stepwise-constant pulses $\boldsymbol{u}$ is a Parametrized Quantum Circuit (Panel $(a)$). As such, it can be represented as a shallow computational network (Panel $(b)$), which consists in a linear combination of non-linear functions $\phi_i(\boldsymbol{u})$ called "features", with weights $w_i$ (Panel $(c)$).
• Figure 2: Lie-Fourier representation coefficients of the Ising model dynamical landscape for selected state transfer problems. The results are computed for $n=200$ using the DFT algorithm described in App. \ref{['appendix_e']}. (Upper panel) The $c_{\omega}$ coefficients for a single timestep landscape $N=1$ are plotted for several values of drift strength $\alpha_d \pi^{-1} = 0.0, 0.89, 1.78, 2.67, 4.0$ (color scale). The line styles correspond to the real part of the even (solid) and odd (dashed) frequency index branches and to the imaginary part of the even branch (dot-dashed). The imaginary part of the odd branch is numerically zero in all cases. (Lower panel) The real parts of the $c_{\omega}$ coefficients for a double timestep landscape $N=2$ are plotted as a function of the vector frequencies $\boldsymbol{\omega}$ for $\alpha_d \pi^{-1} = 5.59$. In all cases but for random states $\ket{r} \to \ket{r'}$ the coefficients can be reorganized in two branches which exhibit continuous behaviour inside intervals defined by the resonant frequencies.
• Figure 3: Given an error threshold $\epsilon$, we plot the solution for $P$ of the equation $\epsilon(P)=\epsilon$. This quantity represents the minimum order of Taylor expansion $P$ to represent $J$ up to an error $\epsilon$. The different lines show the results for several values for the error threshold $\epsilon=\{10^{-1}, 10^{-3}, 10^{-5}, 10^{-7}, 10^{-9}, 10^{-11}, 10^{-13}, 10^{-15}\}.$ There is a crossover from ${u_{\mathrm{max}}}L < 1$, where $P\lesssim 10$ and it depends weakly on ${u_{\mathrm{max}}}L$, enabling an efficient local representation of the landscape, to the region ${u_{\mathrm{max}}}L \gg 1$ where the dependence becomes linear $P\sim e{u_{\mathrm{max}}}L$.
• Figure 4: Prediction performance $\epsilon_{\mathrm{rms}}$ of surrogate models as a function of training dataset size $N_{\mathrm{train}}$ for the system given by Eq. \ref{['eq:ising']}. The results shown in the plot are obtained for $Q=5$ qubits, time $T=1.0$, where the colors correspond to the choice of feature map. Each of the six plots corresponds to a value of the parameters $(N, u_{\mathrm{max}})$$\in \{2,4\} \times \{1.0,2.0,4.0\}$. For Taylor and Fourier features we have $\lambda_{R}=10^{-6}$, and only the result for the optimal value of $N_{\mathrm{weights}} \leq N_{\mathrm{train}}$ is shown, while for the sinc kernel we have $\lambda_{R}=10^{-12}$. The training datasets are sampled from a pool of $12672$ controls, while the test datasets with $N_{\mathrm{test}}=128$ are sampled from another pool of $3200$ and each training/test is repeated $32$ times. The solid lines correspond to the median of the prediction errors $\epsilon_{\mathrm{rms}}$ over the samples, while the shaded area corresponds to the $25-75$ percentile range. The dotted line shows the square root of the variance of the sampled landscape. Compared to the other feature maps, the sinc kernel model typically shows lower values of $\epsilon_{\mathrm{rms}}$ for large values of $N_{\mathrm{train}}$, while the opposite is true for the Taylor representation.
• Figure 5: Sinc kernel regression with reduced kernel bandwidth $\omega_{\mathrm{ker}} \leq \omega_{\mathrm{max}}$ on the system given by Eq. \ref{['eq:ising']}. The results shown in the plot are obtained for $Q=5, T=1.0, N=4, u_{\mathrm{max}}=1.0$ and the different colors (from blue to yellow) corresponding to $\omega_{\mathrm{ker}} / \omega_{\mathrm{max}} = 0.1,0.3,0.5,0.8,0.9,1.0$. The training datasets are sampled from a pool of $12672$ points and each training is repeated $32$ times. The median (solid lines) and interquartile range (shaded area) of the prediction errors $\epsilon_{\mathrm{rms}}$ over the samples are shown. The dotted line shows the square root of the variance of the sampled landscape. Reducing $\omega_{\mathrm{ker}}$ considerably reduces prediction error for small training datasets, but also increases it for large datasets.

Theorems & Definitions (40)

• Lemma 1: Bandwidth limitation
• proof
• Lemma 2: $L_1$ Boundedness of the coefficients
• proof
• Lemma 3: $L_2$ Boundedness of the coefficients
• proof
• Lemma 4: Symmetries and selection rules
• proof
• Lemma 5: Boundedness of the derivatives
• proof