ExcitationSolve¶

Implements ExcitationSolve, a gradient-free, hyperparameter-free optimizer for excitation parameters in variational quantum eigensolvers.

Excitation generators \(G\) used in unitary coupled-cluster ansätze satisfy \(G^3 = G\), so their eigenvalues lie in \(\{-1, 0, +1\}\). As a consequence, the energy expectation along any single parameter \(\theta_j\) (holding all others fixed) is exactly a second-order trigonometric polynomial. ExcitationSolve reconstructs this one-dimensional landscape from a handful of energy evaluations, then jumps directly to its global minimizer along that coordinate, sweeping over parameters one at a time. The same quantum resources a gradient-based step would consume suffice to locate the exact per-parameter optimum.

For a single coordinate \(\theta_j\), the energy takes the closed form

\[ \begin{aligned} E(\theta_j) &= c + a_1 \cos\theta_j + b_1 \sin\theta_j + a_2 \cos 2\theta_j + b_2 \sin 2\theta_j, \\ \theta_j &\leftarrow \arg\min_{\theta \in [0, 2\pi)} E(\theta), \end{aligned} \]

where the five coefficients \(c, a_1, b_1, a_2, b_2\) are fixed by sampling \(E\) at five values \(\theta_j^{(0)} + \tfrac{2\pi \ell}{5}\) for \(\ell = 0,\dots,4\) (one reused from the previous step) and solving the resulting linear system; the global minimizer \(\arg\min E\) is obtained analytically via the roots of \(E'(\theta) = 0\) (a companion-matrix eigenvalue problem), and \(\theta\) is the parameter, \(E\) the measured energy expectation.

Reference: Jonas Jäger, Thierry Nicolas Kaldenbach, Max Haas, Erik Schultheis, "Fast gradient-free optimization of excitations in variational quantum eigensolvers", Communications Physics 2025. https://www.nature.com/articles/s42005-025-02375-9

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