Dynamics of Majorana Fermions on a Quantum Computer

TL;DR

The paper tackles real-time dynamics of Majorana fermions in a transverse-field Ising model (TFIM) on NISQ devices, addressing circuit-depth and noise challenges with a constant-depth circuit (CDC) built from matchgates. By mapping each spin to a pair of Majorana modes and using a 7-site open chain, the authors identify two dynamical regimes set by the relative magnitudes of the inter-site coupling $J$ and the transverse field $h$, and they demonstrate that impurities can probe and control Majorana signatures. CDC-based simulations on IBM hardware reproduce ground-truth dynamics with high fidelity, enabling observation of edge Majorana modes in the $J > h$ regime and uniform, field-dominated behavior in the $h > J$ regime, with impurities acting as tunable barriers and confinement knobs. The work provides a first demonstration of quasiparticle dynamics, including Majorana features, on a quantum computer and highlights the practical potential of CDC for studying topological quasiparticles on current hardware.

Abstract

The study of quasiparticle dynamics is central to understanding non-equilibrium phenomena in quantum many-body systems. Direct simulation of such dynamics on quantum hardware has been limited by circuit depth and noise constraints. In this work, we use a recently developed constant-depth circuit algorithm to examine the real-time evolution of site-resolved magnetization in a transverse-field Ising chain on noisy intermediate-scale quantum devices. By representing each spin as a pair of Majorana fermions, we identify two distinct dynamical regimes governed by the relative strength of spin interaction. Furthermore, we show how local impurities can serve as probes of Majorana modes, acting as dynamical barriers in the weak coupling regime. These results demonstrate that constant-depth quantum circuits provide a viable route for studying quasiparticle propagation and for probing Majorana signatures on currently available quantum processors.

Figures (12)

• Figure 1: Illustration of the (a) transverse field Ising model on an open ended 7 qubit chain and (b) its representation using Majorana fermions with no impurities.
• Figure 2: The constant depth quantum circuit composed of matchgates used for the TFIM.
• Figure 3: Total scaled magnetization (magnetization divided by the initial magnetization), averaged over the spins, as a function of time for the CDC on ibm_kyiv (red), the Trotter circuit on ibm_kyiv (blue) and the Trotter circuit on a noisy simulator (green), for two different parameter regimes in Eq. \ref{['eq:tfim_hamiltonian']}. (a) Field-dominated regime with ($J = 0.5$ and $h = 10.0$). (b) Coupling-dominated regime with ($J = 10.0$ and $h = 0.5$). The duration of each time step is 0.05 fs, the total evolution time is 2 fs.
• Figure 4: Time evolution of qubit magnetizations and their corresponding Fourier transforms (in power spectral density). The system is initialized in fully polarized states along the x- and z-directions respectively. (a) $x$-component of the site-resolved magnetizations for $J = 0.5$, $h = 10.0$, and $\lambda = 1.0$ (local magnetic field at the central site is changed to $h \to (1-\lambda) h$). Each sub-figure from top to bottom represents individual qubits, from qubit one to qubit seven; (b) Fourier transform of the data in (a); (c) $z$-component of the site-resolved z-magnetizations for $J = 10.0$, $h = 0.5$, and $\lambda = 1.0$; (d) Fourier transform of the data in (c). The time step is $\Delta t=0.05 ~\text{fs}$, the total evolution time is $10.0~\text{fs}$.
• Figure 5: (a) Heat map showing the difference in peak frequency between one of the edge sites (site 1) and a bulk site (site 2), extracted from the Fourier transform of the time-evolved site-resolved magnetization for a wide range of model parameters. This way, a phase diagram can be constructed experimentally by monitoring the system dynamics. (b) Peak frequency difference between the impurity site ($i=4$) and a non-impurity site ($i=1$).