[Paper Review] Simulating spin systems with Majorana networks

Alex Thomson, Falko Pientka|arXiv (Cornell University)|Jul 24, 2018

Advanced Condensed Matter Physics158 references4 citations

TL;DR

This paper proposes a topological quantum simulator using networks of Majorana-based superconducting islands to emulate a wide range of quantum spin systems, including frustrated and spin-1/2 models. By leveraging topological protection and mesoscopic island physics, the system enables long coherence times and robust simulation of ground and excited states, with measurable dynamic correlations via coupling to resonators and external baths.

ABSTRACT

With the discovery of Majorana quasiparticles in semiconductor-superconductor hybrid structures, topologically protected qubits have emerged as a promising contender for quantum information processing. While the construction of a universal quantum computer with topological protection likely requires significant advances in materials science, intermediate-scale devices are nearly within the reach of current technology. As a near-term milestone for topological qubits, we propose a network of topological superconductors as a simulator of a large variety of quantum spin systems, including those with frustration. Our proposal is founded on existing technology, combining advantages of semiconducting and superconducting qubits. We identify local measurement protocols that give access to information about ground and excited states as well as dynamic correlations. The topological protection of the qubits results in longer coherence times, and relaxation to the groundstate can be controlled by coupling the network an external bath. We conclude by pointing out specific applications of the quantum simulator, e.g., spin liquids, quantum criticality, and thermalization.

Motivation & Objective

• To develop a scalable, topologically protected quantum simulator for complex spin Hamiltonians using existing semiconductor-superconductor hybrid technology.
• To overcome decoherence and error susceptibility in conventional quantum simulators by utilizing Majorana zero modes with nonlocal encoding.
• To enable measurement of ground and excited states, as well as dynamic correlation functions, via coupling to external baths and resonators.
• To demonstrate feasibility of simulating spin-1/2 and higher-spin models on various lattices using tunable Josephson and charging energies.
• To establish a near-term milestone toward topological quantum computing by realizing a functional, controllable quantum simulator with minimal error overhead.

Proposed method

• Each spin-1/2 degree of freedom is encoded in a mesoscopic superconducting island hosting four Majorana zero modes, forming a non-Abelian anyon-like qubit.
• The system is modeled using a Hamiltonian combining Josephson tunneling (H_tun) and charging energy (H_C), with mutual capacitance included for inter-island coupling.
• The effective spin model emerges from the low-energy subspace of the Majorana fermions, with spin interactions generated via tunable tunneling terms between adjacent islands.
• Ground state degeneracy and topological protection are preserved through nonlocal encoding of qubit states across spatially separated Majorana modes.
• Dynamic correlations and excitation spectra are accessed by coupling the network to transmission line resonators and measuring response functions.
• Numerical diagonalization in a truncated Fock space (H_N_max) confirms the recovery of four-fold ground state degeneracy at large N_max, validating the effective spin model.

Experimental results

Research questions

• RQ1 Can a network of topological superconducting islands with Majorana zero modes simulate a broad class of quantum spin Hamiltonians, including frustrated and spin-1/2 models?
• RQ2 How does topological protection in the Majorana-based qubits enhance coherence times and suppress decoherence in the simulated spin system?
• RQ3 What local measurement protocols enable access to ground and excited state properties, as well as dynamic correlation functions, in the simulated spin system?
• RQ4 To what extent do charging energy and finite island capacitance affect the emergence of the effective spin model and its low-energy spectrum?
• RQ5 Can the system be tuned to simulate spin-1/2 and higher-spin models on various lattices using only existing semiconductor-superconductor platform technology?

Key findings

• The system exhibits a four-fold degenerate ground state in the exact Majorana basis, confirming the topological nature of the encoded qubit.
• Numerical simulations show that the ground state degeneracy is recovered at N_max = 4, with energy levels converging to E_exact = -8.4853t for increasing N_max.
• The effective spin model emerges robustly from the low-energy subspace, with spin interactions generated via tunable Josephson coupling between islands.
• The system supports long coherence times due to topological protection, and relaxation to the ground state can be controlled via coupling to an external bath.
• Dynamic correlation functions and excitation spectra can be probed via coupling to transmission line resonators, enabling experimental access to key physical observables.
• The proposed simulator is realizable with current semiconductor-superconductor hybrid technology, offering a near-term platform for studying spin liquids, quantum criticality, and thermalization.