QHDOPT

Implements QHDOPT, a quantum solver that minimizes nonlinear functions by evolving Quantum Hamiltonian Descent (QHD) dynamics on quantum hardware.

QHD is the quantum analogue of gradient descent: instead of a single point moving downhill, a wavefunction \(\Psi(t,x)\) evolves under a Schrodinger equation whose Hamiltonian combines a kinetic term (the Laplacian, which spreads and tunnels the state through barriers) with the objective \(f\) acting as a potential. Measuring the state after evolution yields a low-energy configuration, i.e. an approximate minimizer. The two time-dependent coefficients act as a damping schedule that gradually shifts weight from the kinetic term toward the potential, annealing the dynamics from broad exploration to localization near a minimum.

\[ \begin{aligned} i\,\frac{\partial}{\partial t}\Psi(t,x) &= \left[\, e^{\varphi_t}\left(-\tfrac{1}{2}\Delta\right) + e^{\chi_t} f(x) \,\right]\Psi(t,x), \\ \varphi_t &= -\log\!\left(1+\gamma t^2\right), \qquad \chi_t = \log\!\left(1+\gamma t^2\right). \end{aligned} \]

where \(\Psi(t,x)\) is the quantum state over the search domain \(\Omega\) (with \(\Psi=0\) on \(\partial\Omega\)), \(\Delta\) is the Laplacian giving the kinetic energy, \(f(x)\) is the objective acting as the potential, and \(\gamma>0\) controls the damping schedule that decays the kinetic coefficient \(e^{\varphi_t}\) while growing the potential coefficient \(e^{\chi_t}\).

Reference: Samuel Kushnir, Jiaqi Leng, Yuxiang Peng, Lei Fan, Xiaodi Wu, "QHDOPT: A Software for Nonlinear Optimization with Quantum Hamiltonian Descent", INFORMS Journal on Computing 2024. https://pubsonline.informs.org/doi/10.1287/ijoc.2024.0587

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