Quantum Circuit Optimisation Achieves Speedup via Matrix-Free Riemannian Methods.

The pursuit of simulating quantum systems presents a significant computational challenge, demanding increasingly efficient algorithms and high-performance computing techniques. Accurately modelling the time evolution of quantum states requires optimising the parameters within quantum circuits, a process complicated by the exponential growth of the computational space with system size. Researchers at the Technical University of Munich, namely Fabian Putterer, Max M. Zumpe, Isabel Nha Minh Le, Qunsheng Huang, and Christian B. Mendl, address this challenge in their work, “High-Performance Contraction of Quantum Circuits for Riemannian Optimization”. They present a novel matrix-free framework leveraging Riemannian optimisation, a technique employing geometrical properties of mathematical spaces, to efficiently refine quantum circuit parameters without explicitly constructing and storing the large matrices typically required for such calculations. Their approach, optimised for high-performance computing, demonstrates near-linear parallelisation and offers a competitive alternative to established methods like matrix product operators, as benchmarked on the Fermi-Hubbard model with up to 16 sites.

Quantum simulation experiences notable progress through optimised circuits and efficient algorithms, enabling more accurate and scalable computations. Researchers currently refine quantum circuit gates using Riemannian optimisation methods, specifically the trust-region algorithm, without the need to construct and store large unitary matrices. Unitary matrices represent the fundamental mathematical objects describing the evolution of quantum states, and avoiding their explicit construction provides a substantial computational advantage. This innovative approach operates directly on state vectors, utilising memory efficiently and facilitating simulations of increasingly complex quantum systems.

The core of this advancement resides in a matrix-free algorithmic framework, which circumvents the memory limitations inherent in traditional methods. Calculations proceed as sums over state vectors, provided these vectors remain within manageable memory constraints. This technique substantially reduces computational demands and unlocks the potential for simulating larger and more complex quantum systems than previously achievable.

High-Performance Computing (HPC)-optimised kernels form a critical component of this research, facilitating the application of gates to state vectors and enabling the accurate calculation of gradients and Hessians. Gradients and Hessians represent mathematical tools used to optimise the parameters of the quantum circuit, and these kernels, carefully designed to maximise performance on modern architectures, ensure theoretical benefits translate into real-world gains.

Researchers further enhance efficiency by exploiting sparsity structures inherent in Hamiltonian conservation laws, such as parity conservation and lattice translation invariance. The Hamiltonian represents the total energy of the quantum system, and its conservation laws impose constraints that reduce computational demands by recognising and utilising redundancies within the problem. This allows focus only on relevant degrees of freedom, significantly accelerating the simulation process.

Benchmarking on the Fermi-Hubbard model, a widely studied system in condensed matter physics used to understand electron interactions, demonstrates a near-linear speed-up with parallelisation, scaling effectively with up to 112 CPU threads. This scalability confirms the practical viability of the approach and highlights its potential for tackling increasingly complex quantum simulations, opening up new possibilities for scientific discovery.

Comparative analysis against alternative methods, specifically those based on matrix product operators (MPOs), reveals the benefits of this matrix-free framework in terms of both computational efficiency and scalability. MPOs represent a common technique for representing quantum states and operators efficiently, but can exhibit limitations in certain scenarios, particularly when dealing with highly entangled systems. The comparison highlights the strengths of the matrix-free method, demonstrating its potential as a competitive alternative for specific types of quantum simulations.

This research provides a valuable tool for optimising quantum circuits, paving the way for more efficient and accurate simulations of complex physical systems. The emphasis on HPC optimisation and exploitation of Hamiltonian symmetries positions this work as a valuable resource for researchers seeking to push the boundaries of quantum computation, representing a significant step towards realising the full potential of quantum simulation.