Stochastic Quantum Hamiltonian Descent (SQHD)¶

Implements Stochastic Quantum Hamiltonian Descent (SQHD), a stochastic-gradient variant of Quantum Hamiltonian Descent for finite-sum objectives.

Quantum Hamiltonian Descent evolves a wavefunction \(|\psi\rangle\) under a time-dependent Hamiltonian whose kinetic term drives exploration and whose potential term encodes the objective \(f\). SQHD targets finite-sum problems \(f(x) = \tfrac{1}{m}\sum_{j=1}^{m} f_j(x)\) by replacing the full potential with a single randomly sampled component each step, the quantum analogue of stochastic gradient descent. Each iteration is realized as a symmetric Trotter (Strang) splitting that alternates a half-step of free kinetic evolution, a full-step of evolution under the sampled potential, and another half-step of kinetic evolution.

The continuous Hamiltonian is \(H(t) = e^{\psi(t)}\left(-\tfrac{1}{2}\Delta\right) + e^{\chi(t)}\hat{f}\), where \(\hat{f} = \int f(x)\,|x\rangle\langle x|\,\mathrm{d}x\). Discretizing with step \(\eta\) over \(N\) epochs gives the per-step update:

\[ \begin{aligned} a_j &= \exp\!\big(\psi((j+\tfrac{1}{2})\eta)\big), \qquad b_j = \exp\!\big(\chi((j+\tfrac{1}{2})\eta)\big), \\ \xi_j &\sim \mathrm{Uniform}\{1,\dots,m\}, \\ |\psi_{j+1}\rangle &= \exp\!\Big(-i\tfrac{\eta}{2}\,a_j\big(-\tfrac{1}{2}\Delta\big)\Big)\, \exp\!\big(-i\,\eta\,b_j\,\hat{f}_{\xi_j}\big)\, \exp\!\Big(-i\tfrac{\eta}{2}\,a_j\big(-\tfrac{1}{2}\Delta\big)\Big)\,|\psi_j\rangle . \end{aligned} \]

where \(|\psi_j\rangle\) is the state after epoch \(j\), \(\Delta\) is the Laplacian (kinetic term), \(\hat{f}_{\xi_j} = \int f_{\xi_j}(x)\,|x\rangle\langle x|\,\mathrm{d}x\) is the operator for the sampled component potential, \(\eta\) is the discretization step, \(\psi(t)\) and \(\chi(t)\) are the QHD scheduling functions setting the relative kinetic and potential weights \(a_j, b_j\), and the measured position of the final state \(|\psi_N\rangle\) yields the candidate minimizer.

Reference: Sirui Peng, Shengminjie Chen, Xiaoming Sun, Hongyi Zhou, "Stochastic Quantum Hamiltonian Descent", arXiv 2025. https://arxiv.org/abs/2507.15424

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