[Paper Review] Riemannian quantum circuit optimization for Hamiltonian simulation

Ayse Kotil, Rahul Banerjee+2 more|arXiv (Cornell University)|Dec 15, 2022

Quantum Computing Algorithms and Architecture35 references4 citations

TL;DR

This paper proposes a Riemannian optimization framework to improve Trotter-Suzuki-based quantum circuits for Hamiltonian simulation by treating quantum gates as general unitary matrices on a manifold. Using the Riemannian trust-region algorithm with tensor network techniques, the method achieves orders-of-magnitude accuracy improvements over standard fourth-order splitting methods for one-dimensional Ising and Heisenberg models.

ABSTRACT

Hamiltonian simulation, i.e., simulating the real time evolution of a target quantum system, is a natural application of quantum computing. Trotter-Suzuki splitting methods can generate corresponding quantum circuits; however, a faithful approximation can lead to relatively deep circuits. Here we start from the insight that for translation invariant systems, the gates in such circuit topologies can be further optimized on classical computers to decrease the circuit depth and/or increase the accuracy. We employ tensor network techniques and devise a method based on the Riemannian trust-region algorithm on the unitary matrix manifold for this purpose. For the Ising and Heisenberg models on a one-dimensional lattice, we achieve orders of magnitude accuracy improvements compared to fourth-order splitting methods. The optimized circuits could also be of practical use for the time-evolving block decimation (TEBD) algorithm.

Motivation & Objective

• Address the challenge of deep quantum circuits in Hamiltonian simulation due to high-order Trotter-Suzuki approximations.
• Overcome the limitations of parametrized variational circuits by optimizing full unitary gates instead of parametrized gates.
• Leverage translation invariance and tensor network methods to enable efficient optimization of quantum circuits for large systems.
• Develop a scalable classical optimization framework for quantum circuits that improves fidelity and reduces circuit depth.
• Enable practical use of optimized circuits in time-evolving block decimation (TEBD) and future two-dimensional simulations.

Proposed method

• Formulate the quantum circuit optimization problem as minimizing the distance between the target time-evolution operator and the circuit's unitary transformation on the unitary matrix manifold.
• Apply the Riemannian trust-region algorithm to optimize gate matrices $ G_{ heta} $, treating them as points on the manifold $ \mathcal{U}(d^2) $, using Riemannian geometry for efficient descent.
• Use tensor network techniques to compute gradients and update directions efficiently, avoiding full state-vector reconstruction.
• Employ the Wirtinger formalism for complex differentiation to compute Riemannian gradients of the cost function with respect to unitary matrices.
• Utilize a brick-wall circuit layout with periodic boundary conditions to suppress finite-size effects and enable scaling to larger systems.
• Optimize gates for a fixed evolution time $ t \sim 1 $, enabling time evolution to integer multiples via circuit repetition.

Experimental results

Research questions

• RQ1 Can Riemannian optimization of full unitary gates significantly improve the accuracy of Trotter-Suzuki-based quantum circuits for Hamiltonian simulation?
• RQ2 How does optimizing general unitary matrices compare to parametrized gate optimization in terms of fidelity and circuit depth?
• RQ3 To what extent can optimized circuits for small systems be reliably extrapolated to larger, translation-invariant systems?
• RQ4 Can the proposed method enhance the performance of classical simulation algorithms like TEBD?
• RQ5 What is the impact of periodic boundary conditions on the optimization and scalability of the method?

Key findings

• The proposed method achieves orders-of-magnitude improvements in fidelity over standard fourth-order Trotter-Suzuki splitting methods for the one-dimensional Ising and Heisenberg models.
• Optimized circuits maintain high accuracy even when applied to larger systems due to the light-cone structure and translation invariance, enabling faithful simulation beyond the training size.
• The Riemannian trust-region algorithm successfully converges to high-fidelity solutions, demonstrating the effectiveness of manifold-based optimization for quantum circuit design.
• The method is applicable to systems with longer-range interactions as long as the light-cone constraints are satisfied.
• Optimized gates can be reused across multiple time steps by concatenating the circuit, enabling scalable time evolution with reduced depth.
• The framework is extendable to two-dimensional lattices and non-translation-invariant systems, though computational cost increases with system size.