Riemannian gradient descent-based quantum algorithms for ground state preparation with guarantees

TL;DR

The paper develops randomized Riemannian gradient descent (RGD) methods for preparing ground states of a target Hamiltonian on quantum devices, showing that convergence depends on spectral structure via the gap $\Delta$, initial overlap $|C_0|$, and norm $\|H\|_{\infty}$. It introduces projection of the Riemannian gradient into polynomial-sized Pauli subspaces, enabling efficient steps at the cost of potential loss in guarantees, and explores trade-offs between subspace size and convergence speed. Two quantum-device implementations are analyzed: a Trotterization-based approach and a qDRIFT-inspired randomized protocol, with numerical and IBM-device results indicating reduced circuit depth for the randomized method and highlighting hardware noise as a limiting factor for scaling. The work provides a framework for practical, convergence-guaranteed ground-state preparation on near-term devices and motivates further advances in randomized manifold optimization and hardware-aware compilation.

Abstract

We investigate Riemannian gradient flows for preparing ground states of a desired Hamiltonian on a quantum device. We show that the number of steps of the corresponding Riemannian gradient descent (RGD) algorithm that prepares a ground state to a given precision depends on the structure of the Hamiltonian. Specifically, we develop an upper bound for the number of RGD steps that depends on the spectral gap of the Hamiltonian, the overlap between ground and initial state, and the target precision. In numerical experiments we study examples where we observe for a 1D Ising chain with nearest-neighbor interactions that the RGD steps needed to prepare a ground state scales linearly with the number of spins. For all-to-all couplings a quadratic scaling is obtained. To achieve efficient implementations while keeping convergence guarantees, we develop RGD approximations by randomly projecting the Riemannian gradient into polynomial-sized subspaces. We find that the speed of convergence of the randomly projected RGD critically depends on the size of the subspace the gradient is projected into. Finally, we develop efficient quantum device implementations based on Trotterization and a quantum stochastic drift-inspired protocol. We implement the resulting quantum algorithms on IBM's quantum devices and provide data for small-scale problems.

Figures (7)

• Figure 1: Schematic representation of the Riemannian gradient $\mathrm{grad}\,J[U_{k}]$ at a point $U_{k}$ in the tangent space $T_{U_{k}}\mathrm{SU}(2^{n})$ (illustrated as two-dimensional) for an $n$-qubit system. The Riemannian gradient is approximated by $\widetilde{\mathrm{grad}\,J[U_{k}]}$ that lies in a subspace (illustrated as one-dimensional) of dimension polynomial in $n$, enabling efficient implementation of the retraction $e^{\gamma\widetilde{\mathrm{grad}\,J[U_{k}]}}$ (red arrow) as an adaptive step in \ref{['eq:discretizedsolution']} for ground-state preparation on a quantum device.
• Figure 2: Number of adaptive steps required to prepare the ground state of the Ising Hamiltonian \ref{['eq:IsingHam']} using Riemannian gradient descent defined by \ref{['eq:discretizedsolution']} for a 1D chain with nearest-neighbor interactions (blue) and a complete graph (red) versus the number of spins $n$. Solid lines indicate an approximately linear fit (chain) and a quadratic fit (complete graph), respectively.
• Figure 3: Number of a adaptive steps needed to prepare the ground state of the 1D Ising chain with nearest neighbor interactions \ref{['eq:IsingHam']} for different approximations of the Riemannian gradient given by \ref{['eq:approxGrad']} as a function of the number of spins $n$ on a semi-logarithmic scale. The red curve corresponds to randomly picking $n$ Pauli operators in each adaptive step and projecting the Riemannian gradient into the corresponding subspace. The yellow and green curve was obtained by projecting into $n^{2}$ and $n^{3}$ dimensional subspaces. For comparison we show again the linear scaling behavior of the full RGD algorithm (blue curve in Fig. \ref{['fig:ScalingGradFlow']}) as a black curve. Each data point corresponds to an average taken over 100 samples.
• Figure 4: Randomized-Subspace Riemannian Gradient Descent
• Figure 5: qDRIFT-inspired: Randomized implementation of the approximated Riemannian Gradient Descent