Universal AQC neural-network training¶

Implements universal adiabatic quantum neural-network training, learning weights by adiabatically evolving a qubit register into the ground state of a Hamiltonian that encodes the loss.

Each weight is discretized over \(N\) qubits and promoted to a Pauli-spin operator, so the training loss becomes a problem Hamiltonian \(\hat H = V(\hat w)\). Starting from the ground state of a trivial mixer \(\hat H_0\), the system is evolved under a time-dependent interpolating Hamiltonian on a linear schedule \(s(t)=t/t_{\mathrm{final}}\). The adiabatic theorem guarantees that, if the evolution is slow enough, the register stays in the instantaneous ground state and ends in the minimizer of the loss; measuring the qubits then reads off the trained weights. There is no iterative gradient step — optimization is the continuous-time Hamiltonian flow itself.

\[ \begin{aligned} \hat H_A(t) &= \bigl[1 - s(t)\bigr]\,\hat H_0 + s(t)\,\hat H, \qquad s(t) = \frac{t}{t_{\mathrm{final}}}, \\ \hat H_0 &= -\sum_{\ell} X_\ell, \\ \hat w &= \sum_{\ell=1}^{N} 2^{-\ell}\,\frac{1 - Z_\ell}{2}, \\ \hat H &= V(\hat w), \qquad \mathcal{L} = \sum_a \bigl(Y(x_a) - y_a\bigr)^2. \end{aligned} \]

where \(\hat H_A(t)\) is the interpolating Hamiltonian with schedule \(s(t)\) running from \(s(0)=0\) to \(s(t_{\mathrm{final}})=1\), \(\hat H_0\) is the transverse-field mixer whose ground state is the uniform superposition, \(X_\ell\) and \(Z_\ell\) are Pauli operators on qubit \(\ell\), \(\hat w\) is the \(N\)-qubit discretization of a weight into bins on \([0,1]\), \(V(\cdot)\) encodes the loss as a polynomial in the weight operators, and \(\mathcal{L}\) is the mean-squared error over data \((x_a, y_a)\) with network output \(Y\). The trained weights are the eigenvalues recovered by measuring all \(Z_\ell\) in the final ground state.

Reference: Steve Abel, Juan Carlos Criado, Michael Spannowsky, "Training neural networks with universal adiabatic quantum computing", Frontiers in Artificial Intelligence 2024. https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2024.1368569/full

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