NFT

Implements NFT (Nakanishi-Fujii-Todo), sequential minimal optimization for variational quantum-classical hybrid circuits.

When all but one parameter \(\theta_j\) are held fixed, the expectation-value cost of a parameterized quantum circuit is exactly a single sinusoid of period \(2\pi\) in \(\theta_j\). NFT exploits this: at each step it picks one coordinate, reconstructs that sinusoid from three cost evaluations, and jumps directly to its analytic minimizer. The method is gradient-free and hyperparameter-free, sweeping coordinates cyclically (or randomly) and recycling one evaluation per step.

For the active coordinate, write the restricted cost as \(\mathcal{L}_j(\theta_j) = a_1 \cos(\theta_j - a_2) + a_3\). Measuring it at the current point and at the two shifts \(\pm \tfrac{\pi}{2}\) gives \(f_0, f_+, f_-\), which determine the sinusoid and hence its minimizer at \(\theta_j = a_2 + \pi\):

\[ \begin{aligned} f_0 &= \mathcal{L}_j(\theta_j^{(n-1)}), \quad f_\pm = \mathcal{L}_j\!\left(\theta_j^{(n-1)} \pm \tfrac{\pi}{2}\right), \\ \theta_j^{(n)} &= \theta_j^{(n-1)} + \arctan\!\left(\frac{f_+ - f_-}{2 f_0 - f_+ - f_-}\right) + \frac{\pi}{2}\left(1 + \mathrm{sign}\!\left(2 f_0 - f_+ - f_-\right)\right), \\ \theta_k^{(n)} &= \theta_k^{(n-1)} \quad (k \neq j). \end{aligned} \]

where \(\theta_j^{(n)}\) is the updated value of the chosen coordinate after sweep \(n\), \(f_0,f_\pm\) are the three cost evaluations, the \(\arctan\) recovers the phase \(a_2 - \theta_j^{(n-1)}\), and the \(\mathrm{sign}\) term selects the cosine minimum (adding \(\pi\)). The recovered minimum cost \(a_3 - a_1\) is reused as \(f_0\) for the next coordinate.

Reference: Ken M. Nakanishi, Keisuke Fujii, Synge Todo, "Sequential minimal optimization for quantum-classical hybrid algorithms", Phys. Rev. Research 2, 043158 (2020). https://link.aps.org/doi/10.1103/PhysRevResearch.2.043158

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