SantaQlaus

Implements SantaQlaus, a shot-adaptive optimizer that tunes the number of quantum measurements so that the intrinsic quantum shot-noise plays the role of the thermal noise in the Santa annealing thermostat.

SantaQlaus builds on the classical Santa optimizer (Stochastic AnNealing Thermostats with Adaptive momentum), which simulates a discretized stochastic differential equation: an adaptive-momentum dynamics with a Nosé–Hoover thermostat \(\boldsymbol{\Xi}\) and injected Gaussian noise of variance \(\propto 1/\beta_t\), where the inverse temperature \(\beta_t\) is annealed from small (exploration) to large (refinement). The central observation is that evaluating a variational-quantum-algorithm gradient with a finite number of shots \(N_t\) already produces an asymptotically Gaussian estimator \(\hat g_t = g_t + \mathcal{N}(0, \Sigma_t / N_t)\). SantaQlaus therefore does not inject artificial thermal noise; instead it chooses the shot count so that the unavoidable quantum shot-noise matches the thermal noise that Santa would have added. Early iterations (small \(\beta_t\), high temperature) need only a few shots to explore, while later iterations (large \(\beta_t\)) demand many shots for precise refinement.

Per iteration, the shot count is set by equating the two noise variances, and the parameters then follow the leading-order Santa step driven by the noisy gradient:

\[ \begin{aligned} \frac{\Sigma_t}{N_t} &\;=\; \frac{2}{\beta_t \, \eta_t} \quad\Longrightarrow\quad N_t \;=\; \left\lceil \frac{\beta_t \, \eta_t \, \Sigma_t}{2} \right\rceil \\ m_t &= \beta_1 m_{t-1} + (1-\beta_1)\, \hat g_t \\ v_t &= \beta_2 v_{t-1} + (1-\beta_2)\, \hat g_t^{\,2}, \qquad G_t = \frac{1}{\sqrt{\,\mathrm{diag}(v_t)\,}+\epsilon} \\ \theta_t &= \theta_{t-1} - \eta_t\, G_t\, m_t + \sqrt{\tfrac{2}{\beta_t}\,\eta_t\, G_t}\;\boldsymbol{\zeta}_t \end{aligned} \]

where \(\theta\) are the variational parameters, \(\eta_t\) the step size, \(\hat g_t\) the shot-estimated gradient, \(\Sigma_t\) its (per-component) shot-noise variance for one shot, \(N_t\) the allocated shot count, \(\beta_t\) the annealed inverse temperature, \(m_t\)/\(v_t\) the first- and second-moment estimates forming the adaptive preconditioner \(G_t\), \(\beta_1,\beta_2\) the decay rates, \(\boldsymbol{\zeta}_t\) standard Gaussian noise, and \(\epsilon\) a stability constant. The defining shot rule \(N_t = \lceil \beta_t \eta_t \Sigma_t / 2 \rceil\) makes the residual quantum shot-noise equal to the Santa thermal-noise term, so the injected-noise contribution can be dropped once \(N_t\) is chosen.

Reference: Kosuke Ito, Keisuke Fujii, "SantaQlaus: A resource-efficient method to leverage quantum shot-noise for optimization of variational quantum algorithms", arXiv 2023. https://arxiv.org/abs/2312.15791

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