Solving the complex equations governing fermionic systems holds immense promise for advances in fields like materials science and drug discovery, but currently presents a significant computational challenge. Researchers, including D. E. Fisher, S. A. Fldzhyan, and D. V. Minaev, alongside colleagues from Sber Quantum Technology Center and other institutions, now demonstrate a substantial improvement in the efficiency of quantum algorithms designed to tackle these problems. Their work focuses on optimising how fermionic systems are represented on quantum computers, specifically by developing innovative ‘Majorana swap networks’ that dramatically reduce the complexity of the required quantum circuits. This new approach lowers both the depth and the number of operations within these circuits, directly addressing a major limitation of current quantum hardware and paving the way for more reliable simulations of complex fermionic systems.
The application of quantum computing holds promise for simulating complex systems, but mapping nonlocal fermionic operators onto qubits frequently generates circuits that are too deep for implementation on current quantum hardware. This limitation hinders practical simulations, particularly for systems with many interacting particles. To address this challenge, researchers introduce Majorana swap network techniques designed for use with variational quantum eigensolvers, aiming to reduce both circuit depth and the number of two-qubit gates, and thereby limit the accumulation of errors during computation. Specifically, a cyclic compilation algorithm is developed to localize all two-particle interaction terms within a general fermionic Hamiltonian, utilising only auxiliary Majorana-swap gates. This algorithm is particularly suited for quantum processors with all-to-all qubit connectivity, such as those based on trapped-ion technology.
Detailed Mathematical Derivations and Methods
This document provides a detailed supplement to a research paper, offering the mathematical derivations, detailed explanations of methods, and specific parameters used in the research. It serves as a technical appendix, providing supporting information crucial for reproducibility and understanding without cluttering the main paper. It is intended for other researchers in the field who wish to verify the results or adapt the methods. The document covers several interconnected topics related to quantum computation and error mitigation, including an analysis of fidelity and parameters of noise channels, detailing depolarization and transverse relaxation, and a description of linear double excitation rotations and their implementations.
It delves into the mathematical foundations of quantum noise, modelling how imperfections and environmental disturbances affect quantum computations, defining noise channels, and exploring specific models like depolarization and transverse relaxation. The Bloch sphere, a geometrical representation of a qubit, is used to visualize quantum states and the impact of noise, while Kraus operators provide a mathematical tool for describing quantum channels, including noise. Gate fidelity, a measure of a gate’s accuracy, is affected by noise and hardware imperfections. A key challenge lies in the fact that mapping these systems onto qubits often creates extremely complex circuits, hindering practical implementation on current hardware. Researchers developed novel techniques using Majorana swap networks within variational eigensolvers to dramatically reduce circuit complexity and minimize errors. The team’s first breakthrough involves a cyclic compilation algorithm that efficiently localizes interactions between particles in a fermionic Hamiltonian, utilising only auxiliary Majorana-swap gates.
This algorithm is designed for quantum processors with all-to-all qubit connectivity, and it effectively compactifies circuits used in advanced calculations. Furthermore, they designed a specialized Majorana swap network for the UpCCGSD variational ansatz, a method already more compact than its predecessor. This network achieves approximate reductions of 50% in circuit depth and 20% in the total number of two-qubit gates, representing a substantial improvement in computational efficiency. Notably, under more restricted connectivity conditions, these reductions become even more pronounced, reaching approximately 55% for circuit depth and 40% for gate count.
These improvements directly translate to increased robustness against hardware noise, as demonstrated through numerical simulations. The research also addresses the challenge of qubit locality, employing swap networks to dynamically reorder fermionic modes without requiring additional qubits, allowing for constant-locality implementations of physical operators. By utilizing Majorana swap gates, which efficiently reorder modes while preserving the necessary mathematical relationships, the team has created circuits that are significantly shorter and less prone to errors. These networks significantly decrease both the depth of circuits and the number of two-qubit gates required when compiling variational ansätze, achieving reductions of up to 55% in circuit depth and 40% in two-qubit operations compared to existing methods. This improvement stems from a cyclic-permutation strategy that efficiently localizes interactions between particles, and a streamlined swap network incorporated directly into the excitation operators. The resulting circuits demonstrate improved resilience to hardware noise, exhibiting lower energy susceptibility on both superconducting and trapped-ion qubit platforms.
Importantly, the study highlights that circuit robustness depends not only on the total number of gates but also on the specific method used to decompose the excitation operators. These findings suggest that MSNs represent a valuable component for deploying variational quantum algorithms on near-term, noisy intermediate-scale quantum (NISQ) devices. The authors acknowledge that the performance of these networks is influenced by both two-qubit and single-qubit error rates, indicating that improvements in both areas are crucial. Future research directions include combining MSNs with adaptive ansätze, symmetry-tapering techniques, and compiler-level optimization to further minimize error sensitivity and extend the capabilities of quantum simulations on current hardware.