QUIVER¶

Implements QUIVER, an adaptive forward-gradient optimizer for parameterized quantum circuits.

QUIVER avoids the parameter-shift rule by estimating the gradient from a tunable number \(V\) of random directional derivatives, each measured with a finite-difference pair of \(M\)-shot circuit evaluations. Averaging \(V\) such directional probes gives an unbiased forward-gradient estimator, and the descent step uses this estimator in place of the exact gradient.

The distinctive part is the measurement budget: rather than fixing the number of directions and shots, QUIVER derives a closed-form minimum-cost allocation for the shots \(M_\ell\) assigned to each direction, balancing measurement variance against the contribution that direction makes to the descent gain. Rademacher directions (\(\kappa = 1\)) minimize the estimator's second moment.

\[ \begin{aligned} \tilde\nabla_{v}^{\,\epsilon} f &= \frac{f^{M}(\theta + \epsilon v) - f^{M}(\theta - \epsilon v)}{2\epsilon}, \\ \tilde\nabla^{F} f(\theta_t) &= \frac{1}{V}\sum_{\ell=1}^{V} \big(\tilde\nabla_{v^{\ell}}^{\,\epsilon} f\big)\, v^{\ell}, \\ M_\ell^{*} &= \frac{2\,\mathrm{Var}_m\!\big[\tilde\nabla_{v^{\ell}} \mathcal{L}_m\big]}{C\,\lVert \nabla\mathcal{L}\rVert^{2} - (\nabla_{v^{\ell}}\mathcal{L})^{2}}, \qquad C = \frac{2V}{L\,\eta\,(N + V + \kappa - 2)}, \\ \theta_{t+1} &= \theta_t - \eta\, \tilde\nabla^{F} f(\theta_t). \end{aligned} \]

where \(\theta\) are the circuit parameters, \(\eta\) the learning rate, \(f^{M}(\cdot)\) the \(M\)-shot sample-mean of the loss, \(\epsilon\) the finite-difference step, \(v^{\ell}\) random unit directions, \(V\) the number of directions per step, \(M_\ell^{*}\) the optimal shots for direction \(\ell\), \(N\) the number of parameters, \(L\) the loss smoothness constant, \(\kappa\) the fourth-moment constant of the direction distribution (\(\kappa=1\) for Rademacher, \(\kappa=3\) for Gaussian), \(\mathcal{L}_m\) the single-shot loss, and \(\mathrm{Var}_m\) the per-shot measurement variance.

Reference: Brian Coyle, Snehal Raj, Virag Umathe, El Amine Cherrat, Elham Kashefi, "Adaptive directional gradients for parameterised quantum circuits", arXiv 2026. https://arxiv.org/abs/2606.09734

Back to the Canon