Observing and braiding topological Majorana modes on programmable quantum simulators

Majorana fermions, particles that are their own antiparticles, are crucial for topological quantum computing due to their inherent stability against perturbations. However, their experimental demonstration has proven challenging due to issues like disorder and the difficulty in distinguishing them from trivial zero-energy states. This research leverages the capabilities of advanced quantum simulators, specifically superconducting quantum processors, to overcome these obstacles. These processors offer unprecedented control over many-body quantum systems, enabling the simulation of topological phases of matter and the exploration of Majorana modes. Previous studies have hinted at topological modes in photonic experiments and programmable digital quantum processors, but a quantitative study of their properties and, crucially, their manipulation via braiding remained elusive. This paper addresses these challenges by implementing a verifiable method for observing and braiding Majorana modes on a quantum simulator, paving the way for more extensive studies in topological quantum matter and computation.

Existing literature highlights the theoretical potential of Majorana fermions for topological quantum computing, with proposals for their realization in solid-state devices. However, these proposals encountered significant challenges due to disorder, lack of control, and the difficulty of separating Majorana modes from trivial states. While signatures of topological modes have been observed in various systems, including photonic experiments and programmable quantum processors, these often lacked the precision and control needed for a definitive demonstration. The theoretical groundwork for realizing Majorana modes in bosonic multi-qubit devices has been laid for decades, with progress in both theoretical understanding and experimental techniques. This paper builds on previous work by developing novel methods for distinguishing Majorana modes from trivial modes and performing braiding operations on a relatively large scale within the constraints of current noisy quantum hardware.

The researchers employed Floquet engineering, a technique involving time-periodic Hamiltonians, to simulate a one-dimensional topological superconductor known to host Majorana modes at its boundaries. This approach, implemented using a series of local single- and two-qubit gates on a superconducting quantum processor, offers efficient resource utilization compared to Trotterized methods. The Hamiltonian used was a time-periodic function of single-qubit Pauli operators (X and Z) and two-qubit Pauli operators (XX and ZZ), with parameters (J, λ, h) that were periodic in time. The protocol divided a single driving period into three parts, each with a different active term in the Hamiltonian. This dynamics is represented by a Floquet unitary, which was implemented experimentally via a quantum circuit. The Jordan-Wigner transformation mapped qubit Pauli operators into Majorana fermion operators, enabling the identification of Majorana modes. A Fourier transformation of multi-qubit observables was employed to extract the structure of Majorana modes, distinguishing between zero-frequency and π-frequency modes. This process identified Majorana zero modes (MZM) and Majorana π modes (MPM). The researchers developed a method to distinguish topological Majorana modes from trivial zero-energy modes through the analysis of two-point correlation functions. To simulate braiding, a fast approximate swap (FAS) method, a non-adiabatic technique suitable for noisy quantum hardware, was introduced and implemented. This method leverages the approximate conservation of Majorana modes even under the influence of interaction, allowing for identification despite noise and decoherence effects.

The researchers successfully observed and distinguished Majorana modes from trivial modes using their proposed methodology. Fourier transformation of experimental data allowed for the reconstruction of Majorana wavefunctions, both in non-interacting and interacting regimes. The accuracy of the wavefunction reconstruction was validated by comparison to theoretical predictions. Crucially, the two-point correlation function approach enabled distinguishing between trivial and topological zero-energy modes based on their spatial correlations. The experimental implementation of the fast approximate swap (FAS) method successfully demonstrated braiding of Majorana modes, resulting in the expected phase change of the wavefunction. Experiments were performed on multiple IBM quantum devices (ibm_hanoi, ibm_montreal, ibm_mumbai, and ibm_toronto), utilizing different numbers of qubits (5-21) and demonstrating consistent results across devices. The observed decay in signal strength over Floquet cycles was attributed to the natural heating tendency of Floquet systems and compensated through data rescaling.

This work represents a significant advance in the experimental observation and manipulation of Majorana modes. The successful implementation of braiding operations, a fundamental step for topological quantum computing, validates the feasibility of using near-term quantum simulators for this purpose. The methods developed for identifying and distinguishing Majorana modes from trivial states provide valuable tools for future studies of topological phases of matter. The experiments were conducted on currently available noisy quantum hardware, demonstrating the robustness of the proposed techniques. The observed decay of Majorana mode signals underscores the limitations of current quantum hardware in terms of coherence times. While noise affects the fidelity of the results, the methods used successfully account for the noise within the experimental limitations and provide a pathway to improvements in quantum hardware and experimental techniques.

This paper presents a comprehensive framework for observing and braiding Majorana modes on noisy intermediate-scale quantum (NISQ) devices. The successful implementation of this framework opens a new avenue for studying topological phases of matter and advancing the development of topological quantum computing. Future research could explore the application of these techniques to more complex topological systems in higher dimensions, investigate mitigation strategies for the decoherence effects observed in Floquet systems, and explore alternative techniques for braiding to improve robustness and scalability.

The primary limitation of this study is the inherent noise present in current quantum hardware. While the researchers implemented techniques to mitigate the effects of noise, the finite coherence times and errors in quantum gates impact the fidelity of the results. The observed decay in Majorana mode signals due to the heating effect in Floquet systems is another limitation. Further research is needed to fully overcome these challenges and achieve higher fidelity in experimental measurements. The study focused on a one-dimensional system; extending the findings to higher dimensions remains a future research goal.