Quantum optimization with exact geodesic transport

TL;DR

The paper tackles the practical bottleneck of training variational quantum algorithms by exploiting the Riemannian geometry of quantum state space. It introduces Exact Geodesic Transport (EGT) with a diagonal metric in hyperspherical amplitude encoding, enabling analytic geodesics and exact parameter updates, plus a conjugate-gradient extension (EGT-CG) with convergence guarantees. Through thorough numerical benchmarks on molecular ground-state problems and 1D spin chains, EGT-CG achieves large reductions in optimization steps (often >10x) and maintains competitive quantum-resource scaling, including successful demonstrations on IonQ Forte hardware with error mitigation. This work bridges quantum machine learning, differential geometry, and optimal control to deliver more efficient, globally convergent VQAs with practical implications for quantum simulation.

Abstract

We introduce an architecture for variational quantum algorithms that can be efficiently trained via parameter updates along exact geodesics on the Riemannian state manifold. This features a parameter-optimal circuit ansatz which supersedes known quantum natural gradient methods by removing expensive estimations of the metric tensor and provably reducing gradient estimation costs by $62.5\%$. Moreover, the framework also naturally incorporates conjugate gradients as a built-in feature, giving an accelerated descent method with convergence guarantees that we dub exact geodesic transport with conjugate gradients. Numerical benchmarks against state-of-the-art variational methods for ground-state preparation of molecular Hamiltonians or $1$-dimensional spin chains (both with and without particle-number conservation) up to $n=16$ qubits show reductions of over one order of magnitude in the number of optimization steps, with global convergence even for degenerate cases and competitive quantum-resource scalings. In addition, we perform proof-of-principle demonstrations on IonQ's Forte quantum processor, showcasing deployment of pre-trained circuits for the $H_{3}^{+}$ molecule and experimental training for $H_{2}$. Our work enables quantum machine learning applications with shorter training runtime, with implications at the interface of quantum simulation, differential geometry, and optimal control theory.

Figures (8)

• Figure 1: Schematic illustration of our method. VQA pipeline with standard gradient descent (GD, red dashed curves) update rule vs. the proposed descent with exact geodesic transport (EGT, blue solid curves). In GD, the search for a minimum of the loss function $\mathcal{L}(\boldsymbol{\theta})$ is done via steps along straight lines on a flat parameter space, whereby the parameters $\boldsymbol{\theta}$ are updated directly. In contrast, EGT implements an amplitude-based update rule, based on paths along geodesics on the curved manifold that contains the state vector $\ket{\psi(\boldsymbol{\theta})}$ output by the variational circuit $U_{\boldsymbol{\theta}}$. For our circuit ansatz, such paths define great-circle arcs on a hypersphere. Both spaces are related via the (exact) coordinate transformation $\boldsymbol{\theta} = \boldsymbol{\theta}(\mathbf{x})$ between circuit parameters $\boldsymbol{\theta}$ and the amplitudes $\mathbf{x}$ of $\ket{\psi(\boldsymbol{\theta})}$ in the computational basis, and its inverse $\mathbf{x}(\boldsymbol{\theta})$. This allows one to perform the amplitudes update via EGT, then recover the updated parameters $\boldsymbol{\theta}_{t+1}$ from the updated amplitudes $\mathbf{x}_{t+1}$, as in Eq. \ref{['eq:update_exact']}.
• Figure 2: Ground-state optimization for molecules. Performance of different optimization schemes using $\operatorname{HWE}_{k}$ as VQA ansatz and the warm start in Eq. \ref{['eq:warm_start']} with $\alpha=0.9$. The different optimizers are: exact geodesic transport with conjugate gradients (EGT-CG, purple), conjugate gradient method with flat-space gradients (CG, blue), quantum natural gradient (QNG) of first (green) and second (pink) orders, and the standard Adam optimizer (orange). See Methods \ref{['ssec:details_numerical']} for the respective learning rate schedulers. (a) Relative ground-state energy error for $\operatorname{H_{2}O}$vs. number of epochs. Using the Jordan-Wigner transformation and the STO-3G basis set, the ground state is represented by $n = 14$ spin-orbitals as the active space with $k = 10$ electrons. EGT-CG and CG display a similar performance, both comfortably beating the other methods ($\eta = 0.005$ for Adam here). (b) Relative ground-state energy error for $\operatorname{H_{5}}$, represented by $n = 10$ spin-orbitals and $k = 5$ electrons. The color code is the same as in (a), and here Adam has $\eta = 0.05$. The infidelity $1 - F_{\text{final}}$ of the state prepared by each scheme at the end of the optimization relative to the true ground state is displayed in the inset. (c) Number of epochs to achieve chemical accuracy for $14$ molecules and different optimizers. For $\operatorname{H_{5}}$ and $\operatorname{CH_{2}}$, some optimization schemes did not converge to chemical accuracy, in which case no bar is shown. $\operatorname{CH_{2}}$ with Adam reached chemical accuracy in 130 epochs ($y$-axis was limited for better presentation). (d) Infidelities relative to the true ground state for the initial state $\ket{\psi_\text{warm}}$ with $\alpha=0.9$ and for the state after 2 epochs of EGT-CG, for the same molecules of (c). Below each molecule label in the $x$-axis, the size of the corresponding parametrized subspace is indicated by $\binom{n}{k}$.
• Figure 3: Ground-state optimization for the XXZ model. Performance of different variational circuit ansatze and optimization schemes as a function of system size $n$. We compare the Hamming-weight encoder ($\operatorname{HWE}_{k}$) with all the considered optimization schemes against the quantum orthogonal layer (QOL), hardware-efficient (HE), and overparametrized HE (HE OP) ansatze optimized with Adam. Each point is averaged over $50$ Haar-random initializations, with error bars indicating the $95\%$ confidence interval. (a) Average number of epochs required to reach chemical accuracy in energy estimation; missing points indicate failure to reach chemical accuracy. (b) Relative energy estimation error at the end of training. Chemical accuracy for each system size is plotted as a dotted black line. (c) Number of CNOT gates per ansatz; for $n>10$, CNOT counts are shown for scaling comparison only. For further details on optimization schemes and quantum resources, see the Methods \ref{['ssec:details_numerical']} and App. \ref{['app:quantum_resources']}.
• Figure 4: Ground state energy of $\operatorname{H_{3}^{+}}$ on IonQ Forte. (main) Energy estimates (in Hartrees) for the $\operatorname{H_{3}^{+}}$ molecule as a function of bond length (in Angstrom). The optimal circuit parameters were trained classically. The dashed red curve indicates the true ground-state energies. The dashed green curve is the result of the deployment on IonQ's Forte quantum processor. The solid purple curve shows the hardware results after debiasing and Hamming-weight-preservation error mitigation (HWEM) for $k = 2$. Error bars indicate the $95\%$ confidence interval for the mean. (inset) Energy estimation error relative to the exact ground state energy as a function of the bond length. The color scheme is the same, with chemical accuracy as a dotted black line.
• Figure 5: EGT-CG optimization for the ground state of the $\operatorname{H_{2}}$ molecule on IonQ Forte. (top) Energy (in Hartree) during ground-state optimization for the $\operatorname{H_{2}}$ molecule, as a function of the number of epochs. In dark purple, we present the quantum hardware results for the EGT-CG optimizer with debiasing and HW $k=2$ error mitigation (HWEM). We used $2 \times 10^{4}$ shots for each measurement basis required to estimate the loss function $\mathcal{L}$ and the gradient $\boldsymbol{\partial}_{\boldsymbol{\theta}} \mathcal{L}$ at the first epoch, and $4 \times 10^{4}$ shots at the last two. In light purple, we show the noiseless simulation of the same process, only with $2 \times 10^4$ shots in all epochs. In orange, we show a noiseless simulation of the iteration using the Adam optimizer with fixed learning rate $\eta = 10^{-2}$, and $2 \times 10^4$ shots in all epochs. The shaded regions represent the $95\%$ confidence interval for the mean. The true ground state energy is plotted as a dotted red line. (bottom) Energy estimation error relative to the true ground state energy as a function of epochs. Chemical accuracy is plotted as a dotted black line.

Theorems & Definitions (2)

• Theorem B.1
• proof