Quantum circuit design from a retraction-based Riemannian optimization framework

TL;DR

This work proposes the Riemannian Random Subspace Newton method, a scalable second-order algorithm that constructs a Newton system from measurement data, and establishes a retraction-based Riemannian optimization framework for this setting, ensuring that all algorithmic procedures are implementable on quantum hardware.

Abstract

Designing quantum circuits for ground state preparation is a fundamental task in quantum information science. However, standard Variational Quantum Algorithms (VQAs) are often constrained by limited ansatz expressivity and difficult optimization landscapes. To address these issues, we adopt a geometric perspective, formulating the problem as the minimization of an energy cost function directly over the unitary group. We establish a retraction-based Riemannian optimization framework for this setting, ensuring that all algorithmic procedures are implementable on quantum hardware. Within this framework, we unify existing randomized gradient approaches under a Riemannian Random Subspace Gradient Projection (RRSGP) method. While recent geometric approaches have predominantly focused on such first-order gradient descent techniques, efficient second-order methods remain unexplored. To bridge this gap, we derive explicit expressions for the Riemannian Hessian and show that it can be estimated directly on quantum hardware via parameter-shift rules. Building on this, we propose the Riemannian Random Subspace Newton (RRSN) method, a scalable second-order algorithm that constructs a Newton system from measurement data. Numerical simulations indicate that RRSN achieves quadratic convergence, yielding high-precision ground states in significantly fewer iterations compared to both existing first-order approaches and standard VQA baselines. Ultimately, this work provides a systematic foundation for applying a broader class of efficient Riemannian algorithms to quantum circuit design.

Figures (7)

• Figure 1: The geometric illustration of Riemannian gradient and Riemannian algorithm iterations. (a) For the sake of demonstration, we imagine the curved space $\mathrm{U} (p)$ as a sphere. Given a point $U$ on $\mathrm{U} (p)$, the tangent space $T_U$ is similar to the tangent plane at that point. We plot the contours (dash lines) of the cost function $f$ on $\mathrm{U} (p)$, where the Riemannian gradient $\operatorname{grad} f (U)$ is a special vector in the tangent plane. This vector is orthogonal to the contour and points in the direction of the fastest increase of $f$. (b) The iteration process of a typical Riemannian optimization algorithm $U_{k+1}=\mathrm{R}_{U_k}\left(t_{k} \eta_{k}\right)$, which is an extension of iteration $x_{k+1} = x_k+t_k \eta_k$ in Euclidean space.
• Figure 2: The circuit of the VQA algorithm used in our experiments. It consists of a two-layer hardware-efficient ansatz, where each $R_Y$ and $R_Z$ gate is parameterized by a single free parameter.
• Figure 3: Comparison of typical convergence rate without considering random subspaces. Experiments are conducted on a 4-qubit XXZ, starting from the uniform state. Panels (a) and (b) show the energy error versus iteration number on linear and logarithmic scales, respectively, while (c) and (d) display the evolution of the gradient norm. Although exact line-search accelerates the initial convergence of RRSGP, only RRSN exhibits quadratic convergence, whereas RRSGPs achieve at best linear convergence.
• Figure 4: Impact of VQA warm start on performance for RRSGP and RRSN. Experiments are performed on a 5-qubit XXZ Hamiltonian. Solid curves correspond to cold starts from the uniform state, while dashed curves indicate VQA warm starts. Panels (a) and (c) show that cold start RRSGP is prone to being trapped near a saddle point, whereas the VQA warm start avoids this issue. RRSN, by contrast, bypasses the saddle point even with a cold start. Panels (b) and (d) further demonstrate that VQA warm start enables RRSN to enter the quadratic convergence region earlier.
• Figure 5: Convergence behavior of RSSGP (fixed), RSSGP (exact), and RRSN under practical random subspaces ($d<4^N$). Each curve is averaged over 20 independent runs. Panels (a) and (b) show the energy error of RSSGP (fixed) on linear and logarithmic scales, respectively; panels (c) and (d) show RSSGP (exact); and panels (e) and (f) show RRSN. As $d$ decreases, all methods slow down, while RRSN exhibits a gradual transition from quadratic to superlinear and then linear convergence; for $d=64$ or $128$, it achieves performance close to the full dimensional case. RRSN is more robust than RRSGP to the dimension of random subspaces.

Theorems & Definitions (12)

• Lemma 1: Gradient coefficient estimation
• Proposition 1: Riemannian Hessian
• Lemma 2: Hessian coefficient estimation
• Remark 1
• proof : Proof of \ref{['lem-pqc-1']}
• proof : Proof of \ref{['lem-vqa-2']}
• Proposition 2: Riemannian Hessian
• proof : Proof of \ref{['prop-hessian']}
• Proposition 3: Adjoint-preservation
• proof