SGLBO¶

Implements SGLBO (Stochastic Gradient Line Bayesian Optimization), a shot-efficient optimizer for parameterized quantum circuits.

SGLBO decouples the choice of update direction from the choice of step size. At each iteration it estimates the gradient of the cost \(f\) with respect to the circuit parameters via the parameter-shift rule, fixing the descent direction \(-g_t\). It then restricts the cost to the one-dimensional line through \(\theta_t\) along that direction and uses Bayesian optimization (a Gaussian-process surrogate with Thompson sampling) to locate the step size \(\eta\) that minimizes the cost on the line, which is far cheaper in measurement shots than a high-dimensional search. The optimizer pairs this with \(\alpha\)-suffix averaging of the final iterates to suppress statistical and hardware noise.

\[ \begin{aligned} g_t &= \nabla_\theta f(\theta_t), \qquad \frac{\partial f(\theta)}{\partial \theta_i} = \tfrac{1}{2}\left[\, f\!\left(\theta + \tfrac{\pi}{2}e_i\right) - f\!\left(\theta - \tfrac{\pi}{2}e_i\right) \right], \\ \eta_t^{*} &= \arg\min_{\eta \in [-\eta_{\max},\,\eta_{\max}]} f(\theta_t - \eta\, g_t), \\ \theta_{t+1} &= \theta_t - \eta_t^{*}\, g_t, \\ \bar{\theta}_{\alpha,T} &= \frac{1}{\alpha T} \sum_{t=(1-\alpha)T-1}^{T-1} \theta_t . \end{aligned} \]

where \(\theta\) are the circuit parameters, \(g_t\) is the parameter-shift gradient estimate, \(e_i\) is the \(i\)-th unit vector, \(\eta_t^{*}\) is the step size chosen by Bayesian optimization over the line \(\{\theta_t - \eta\, g_t : \eta \in [-\eta_{\max}, \eta_{\max}]\}\), \(\eta_{\max}\) bounds the line search, \(T\) is the total number of iterations, and \(\alpha \in (0,1]\) sets the fraction of final iterates averaged in the suffix-averaged output \(\bar{\theta}_{\alpha,T}\).

Reference: Shiro Tamiya, Hayata Yamasaki, "Stochastic Gradient Line Bayesian Optimization for Efficient Noise-Robust Optimization of Parameterized Quantum Circuits", npj Quantum Information 2022. https://www.nature.com/articles/s41534-022-00592-6

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