QN-SPSA¶

Implements QN-SPSA, a quantum natural gradient method that approximates the Fubini-Study metric with simultaneous perturbation stochastic approximation.

Quantum natural gradient preconditions the gradient with the Fubini-Study metric tensor \(g\) (one quarter of the quantum Fisher information), but evaluating \(g\) exactly costs \(O(d^2)\) circuit evaluations. QN-SPSA estimates both the loss gradient and the metric with simultaneous perturbation: a first-order SPSA stochastic gradient, and a second-order SPSA point estimate of \(g\) built from fidelity differences along two random directions \(\Delta_1,\Delta_2 \in \{-1,+1\}^d\). The raw metric estimate is averaged over iterations and regularized to stay positive definite before being inverted.

\[ \begin{aligned} \hat g_t &= -\tfrac{1}{2}\,\frac{\delta F}{2\epsilon^2}\,\frac{\Delta_1\Delta_2^\top + \Delta_2\Delta_1^\top}{2}, \\ \delta F &= F\!\left(\theta_t,\theta_t+\epsilon\Delta_1+\epsilon\Delta_2\right) - F\!\left(\theta_t,\theta_t+\epsilon\Delta_1\right) - F\!\left(\theta_t,\theta_t-\epsilon\Delta_1+\epsilon\Delta_2\right) + F\!\left(\theta_t,\theta_t-\epsilon\Delta_1\right), \\ \bar g_t &= \frac{t}{t+1}\,\bar g_{t-1} + \frac{1}{t+1}\,\hat g_t, \\ \tilde g_t &= \sqrt{\bar g_t \bar g_t} + \beta I, \\ \hat \nabla f_t &= \frac{f(\theta_t+\epsilon\Delta) - f(\theta_t-\epsilon\Delta)}{2\epsilon}\,\Delta, \\ \theta_{t+1} &= \theta_t - \eta\, \tilde g_t^{-1}\, \hat \nabla f_t, \end{aligned} \]

where \(f\) is the loss, \(F(\theta,\theta') = |\langle\psi(\theta)|\psi(\theta')\rangle|^2\) is the state fidelity, \(\Delta,\Delta_1,\Delta_2\) are random \(\pm 1\) perturbation vectors, \(\epsilon\) is the perturbation size, \(\eta\) is the learning rate, \(\bar g_t\) is the running-averaged metric, and \(\beta>0\) regularizes the matrix square root to keep \(\tilde g_t\) positive definite and invertible.

Reference: Julien Gacon, Christa Zoufal, Giuseppe Carleo, Stefan Woerner, "Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information", Quantum 2021. https://quantum-journal.org/papers/q-2021-10-20-567/

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