Adaptive Observation Cost Control for Variational Quantum Eigensolvers

TL;DR

This work tackles the high cost of measurements in variational quantum eigensolvers by introducing SubsCoRe, an adaptive observation strategy that allocates measurement shots based on Gaussian-process confidence within the subspace updated during 1D SMO. By proving that equidistant, 1+2V_d observations yield uniform posterior uncertainty, the method enables a min-max optimal shot distribution without relying on acquisition functions. The authors present two practical variants, SubsCoRe-Bound and SubsCoRe-Center, with SubsCoRe-Center demonstrating superior efficiency and accuracy over state-of-the-art baselines (NFT, SGLBO, EMICoRe) on Ising/Heisenberg benchmarks. The approach promises substantial reductions in quantum hardware budget while maintaining predictive accuracy, and it establishes a meaningful link between GP regression with the VQE kernel and Fourier analysis, suggesting avenues for further algorithmic advances and real-device validation.

Abstract

The objective to be minimized in the variational quantum eigensolver (VQE) has a restricted form, which allows a specialized sequential minimal optimization (SMO) that requires only a few observations in each iteration. However, the SMO iteration is still costly due to the observation noise -- one observation at a point typically requires averaging over hundreds to thousands of repeated quantum measurement shots for achieving a reasonable noise level. In this paper, we propose an adaptive cost control method, named subspace in confident region (SubsCoRe), for SMO. SubsCoRe uses the Gaussian process (GP) surrogate, and requires it to have low uncertainty over the subspace being updated, so that optimization in each iteration is performed with guaranteed accuracy. The adaptive cost control is performed by first setting the required accuracy according to the progress of the optimization, and then choosing the minimum number of measurement shots and their distribution such that the required accuracy is satisfied. We demonstrate that SubsCoRe significantly improves the efficiency of SMO, and outperforms the state-of-the-art methods.

Figures (6)

• Figure 1: Illustration of SubsCoRe. (a) After iteration $t-1$, the function value at the best point $\widehat{\boldsymbol{x}}^{t-1}$, which is in the CoRe (yellow rectangle) due to the previous observations, is predicted by the GP with sufficiently low uncertainty (red curves). Two equidistant shifts (blue points) at $\widehat{\boldsymbol{x}}^{t-1} \pm \frac{2\pi}{3} \boldsymbol{e}_{d_t}$ are chosen, yet not measured, along the line parallel to the $d_t$ axis at the current step. (b) SubsCoRe finds the minimum number of measurement shots for measuring the points $\{\widehat{\boldsymbol{x}}^{t-1},\boldsymbol{x}_\pm\}$ such that the entire line will be included in the CoRe, i.e., the posterior uncertainty at any point on the line is smaller than the required threshold. The three points are measured (green circles), with the middle point $\widehat{\boldsymbol{x}}^{t-1}$ typically requiring fewer shots because of its lower prior uncertainty. (c) Using the trigonometric polynomial regression, the minimum $\widehat{\boldsymbol{x}}^{t}$ (orange point) along the line (where the GP mean function is fully identified by predicting any three points) is computed along with the corresponding energy, thus becoming the starting point for the next iteration.
• Figure 2: Uncertainty of GP with the VQE kernel trained on equidistant (darker green, round markers) and non-equidistant (olive, triangle markers) observation points. The left and right plots show the $V_d=1$ and $V_d=3$ cases, respectively. The green and olive solid lines refer to the GP posterior uncertainty obtained after observing $1 + 2 V_d$ points with an equidistant and non-equidistant spacing. For instance, in the $V_d=1$ case, the observations along a one-dimensional subspace (parallel to the $d$-axis) are performed at $x_d=\pi - \alpha, \pi, \pi + \alpha$ with $\alpha$ being $\frac{\pi}{2}$ (olive) and $\frac{2\pi}{3}$ (green) in the non-equidistant and equidistant cases, respectively. Following our theory, observing $1 + 2V_d$ equidistant points leads to uniform posterior uncertainty.
• Figure 3: Energy (left) and fidelity (right), as the difference from the ground-truth (see Eqs. \ref{['eq:DeltaEnergy']} and \ref{['eq:DeltaFidelity']}), achieved by our SubsCoRe-Center and the baselines, NFT, SGLBO, and EMICoRe for the Ising model with $(Q, L) = (5,3)$. Both plots are on a logarithmic scale, and the horizontal axis indicates the cumulative number of shots (per operator group) as the total quantum computation cost. The fidelity by SGLBO is not shown since the original code does not store the optimal parameters $\widehat{\boldsymbol{x}}$ to reproduce the quantum state $\vert\psi_{\widehat{\boldsymbol{x}}}\rangle$. However, given its slower convergence in terms of energy (left), we expect that the achieved fidelity is also worse than our SubsCoRe-Center.
• Figure 4: Prior (top) and posterior (middle) GP with SubsCoRe-Center (left) and SubsCoRe-Bound (right) in an SMO step. The red solid lines are the true function $f^*(\boldsymbol{x})$, and the blue solid lines and shadows are the GP mean and uncertainty, respectively. The vertical lines represent the previous best point $\widehat{x}_0$ and the equidistant shifts ($x_{\pm}$) as visualized in the cartoon in \ref{['fig:cartoon_SubsCoRe']}. The bottom row reports the number of shots required by SubsCoRe-Center and SubsCoRe-Bound at each observed point. Before the new observations (i.e., prior), the previous best point $\widehat{x}_0$ is already in the CoRe, i.e., the uncertainty of GP is within the CoRe requirement (shown as black dashed lines). After the new observations (i.e., posterior), the entire subspace is in the CoRe. We see the gap (highlighted in orange) between the CoRe requirement and the posterior uncertainty with SubsCoRe-Bound. This is because SubsCoRe-Bound relies on a looser bound and assigns many shots to the previous best point $\widehat{x}_0$, on which the prior GP was already trained accurately (see bottom-right plot).
• Figure 5: Energy (left) and fidelity (right), as the difference from the ground-truth (see Eqs. \ref{['eq:DeltaEnergy']} and \ref{['eq:DeltaFidelity']}), achieved by SubsCoRe-Center and SubsCoRe-Bound. The inferior efficiency of SubsCoRe-Bound is due to the fact that it relies on a looser bound.

Theorems & Definitions (3)

• Theorem 3.1
• Corollary 3.2
• Corollary 3.1