Hessian-vector products for tensor networks via recursive tangent-state propagation

TL;DR

This paper introduces an efficient Hessian-vector product kernel for tensor networks, enabling scalable second-order optimization that significantly improves quantum circuit compression performance.

Contribution

It presents a recursive tangent-state propagation method to compute Hessian-vector products efficiently for tensor networks, facilitating scalable second-order optimization.

Findings

Achieves up to four orders of magnitude fidelity improvement in quantum circuit compression.

Provides smoother and faster convergence compared to first-order methods like Riemannian ADAM.

Demonstrates scalability through bounded virtual bond dimension in recursive tangent-state propagation.

Abstract

Optimizing tensor networks with standard first-order methods often leads to slow convergence and entrapment in local minima. Although second-order optimization offers enhanced robustness, explicitly constructing the full Hessian matrix is computationally prohibitive for large-scale systems. In this work, we bypass this bottleneck by introducing an analytical Hessian-vector product kernel designed for arbitrary compositions of linear maps. This two-pass algorithm leverages recursive tangent-state propagation with a bounded virtual bond dimension to guarantee scalability. We demonstrate the practical utility of this kernel by integrating it into a Riemannian trust-region framework for quantum circuit compression. Evaluated on time-evolution circuits for various spin chains, our second-order approach achieves up to a four-order-of-magnitude improvement in fidelity over naive Trotterization, while delivering significantly smoother, faster convergence than conventional first-order methods such as Riemannian ADAM.