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arXiv:1905.06923 · Physical review. B./Physical review. B

Error generation and propagation in Majorana-based topological qubits

Aaron Conlon, D. Pellegrino, Johannes K. Slingerland, Shane Dooley, Graham Kells

Published Oct 15, 2019• 23 citations
arXivHTMLPDF

Abstract

We investigate dynamical evolution of a topological memory that consists of two $p$-wave superconducting wires separated by a nontopological junction, focusing on the primary errors (i.e., qubit loss) and secondary errors (bit and phase flip) that arise due to nonadiabaticity. On the question of qubit loss we examine the system's response to both periodic boundary driving and deliberate shuttling of the Majorana bound states. In the former scenario, we show how the frequency-dependent rate of qubit loss is strongly correlated with the local density of states at the edge of wire, a fact that can make systems with a larger gap more susceptible to high-frequency noise. In the second scenario we confirm previous predictions concerning superadiabaticity and critical velocity, but see no evidence that the coordinated movement of edge boundaries reduces qubit loss. Our analysis on secondary bit-flip errors shows that it is necessary that nonadiabaticity occurs in both wires and that interwire tunneling be present for this error channel to be open. We also demonstrate how such processes can be minimized by disordering central regions of both wires. Finally, we identify an error channel for phase-flip errors, which can occur due to mismatches in the energies of states with bulk excitations. In the noninteracting system considered here, this error systematically opposes the expected phase rotation due to finite-size splitting in the qubit subspace.

DOI
10.1103/physrevb.100.134307

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We investigate dynamical evolution of a topological memory that consists of two ppp-wave superconducting wires separated by a nontopological junction, focusing on the primary errors (i.e., qubit loss) and secondary errors (bit and phase flip) that arise due to nonadiabaticity. On the question of qubit loss we examine the system's response to both periodic boundary driving and deliberate shuttling of the Majorana bound states. In the former scenario, we show how the frequency-dependent rate of qubit loss is strongly correlated with the local density of states at the edge of wire, a fact that can make systems with a larger gap more susceptible to high-frequency noise. In the second scenario we confirm previous predictions concerning superadiabaticity and critical velocity, but see no evidence that the coordinated movement of edge boundaries reduces qubit loss. Our analysis on secondary bit-flip errors shows that it is necessary that nonadiabaticity occurs in both wires and that interwire tunneling be present for this error channel to be open. We also demonstrate how such processes can be minimized by disordering central regions of both wires. Finally, we identify an error channel for phase-flip errors, which can occur due to mismatches in the energies of states with bulk excitations. In the noninteracting system considered here, this error systematically opposes the expected phase rotation due to finite-size splitting in the qubit subspace.
Phonon bursts from ionizing radiation cause correlated errors in superconducting quantum processors. We study prescribed nonequilibrium phonon pulses in a Majorana qubit formed by two 2 μm2\,μ\mathrm{m}2μm proximitized nanowires. Combining the lattice Lutchyn--Oreg model, microscopic electron--phonon coupling, and parity-resolved many-body dynamics, we track excitation, relaxation, and logical coherence. Without quasiparticle escape, recombination restores the initial Majorana occupations in about 130 ns, while phonons exchanged during excitation and recovery can retain logical-state information, leaving permanent coherence loss. For mirror-symmetric wires, the fractional coherence loss per wire is 0.75--1.0 times the excitation probability averaged over the two logical states, even at zero Majorana splitting; breaking mirror symmetry can suppress dephasing at small splitting. Factorized Boltzmann kinetics neglects quasiparticle correlations and overestimates residual parity flips. At equal deposited energy density, phonon spectra yield different excitation probabilities and coherence losses. Comparisons with Kitaev chains matched to the wire's gap and Majorana decay length reveal that low electron density suppresses excitation by energetic phonons, whereas hard-wall boundaries in a high-density control increase excitation efficiency by a factor of 5×1045\times10^45×104 relative to the wire for pulses centered at 1 meV. Magnetic-field and chemical-potential tuning yield different phonon sensitivities at the same bulk gap through changes in spin and electron--hole structure. Including hybrid states' semiconductor weights reduces near-threshold excitation rates fourfold when the tunnel coupling equals the parent superconducting gap. Our framework connects phonon spectra to leakage, poisoning, and coherence loss to guide Majorana-device design.
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Majorana zero modes (MZMs) appear as edge states in topological superconductors and are envisioned as promising candidates for application towards fault-tolerant quantum computation. Quantum information may be stored in a degenerate ground state subspace constructed from MZMs while quantum gates may be performed by braiding MZMs and exploiting their non-Abelian properties. While the topological protection of MZMs ensures that they are robust against decoherence, their dynamical manipulation can nonetheless induce undesirable transitions to excited states, manifesting as a “diabatic error.”This thesis studies the diabatic error accumulated as MZMs are transported across a superconducting wire, which is an essential building block for physical braiding schemes. The transport protocol considered relies on appropriately tuning sections (“piano keys”) of a wire between distinct topological phases via electric gates to control local chemical potentials. First, the diabatic error is examined for transport which uses multiple piano keys. The error is calculated through numerical simulation of the transport protocol and is complemented by approximate analytical expressions calculated using an effective, low-energy model. It is found that an optimal number of keys can be used to reduce the error. Furthermore, this thesis explores transport protocols performed on wires which feature disorder on the chemical potential and noise in the piano key tuning. The rise in the diabatic error due to these additional effects with respect to parameters-of-interest including the disorder strength, disorder correlation length, noise amplitude, and noise frequency cut-offs is characterized using numerical methods. For disorder, it is found that the error’s behaviour is principally controlled by the statistics of the minimum bulk energy gap. Spatially correlated disorder is also studied which presents non-monotonic behaviour for the error. Finally, for noise, it is found that optical transitions can cause a substantial increase in the error as the frequency modes within the noise approach the minimum energy gap.
Topological Quantum Computing has largely evolved towards a paradigm of manipulating edge localized Majorana within ppp-wave topological superconducting nanowires. To bridge the gap between physical qubit systems and quantum algorithms, we perform a dynamical analysis to extend what is known in the adiabatic regime, providing time-dependent gate elements for further qubit and algorithm modeling efforts. Our analysis covers dynamical considerations for two methods of shuttling domain edge bound Majoranas in a single nanowire system which both function by applying spatiotemporally dependent onsite and hopping parameters within the system's Hamiltonian. We then complicate this model by converting it into the T-qubit to calculate the finite-time gate representation of the shuttling techniques used in a more practical setting. These contributions provide insight for realistic experimental setups in the next-generation of qubit implementation and will hopefully facilitate fault tolerant scalable systems and universal gate design.
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Recently there have been numerous proposals to create Majorana zero modes in solid state heterojunctions, superconducting wires, and optical lattices. Putatively the information stored in qubits constructed from these modes is protected from various forms of decoherence. Here we present a generic method to study the effect of external perturbations on these modes. We focus on the case where there are no interactions between different Majorana modes either directly or through intermediary fermions. To quantify the rate of loss of the information stored in the Majorana modes we study the two-time correlators for qubits built from them. We analyze a generic gapped fermionic environment (bath) interacting via tunneling with different components of the qubit (different Majorana modes). We present examples with both static and dynamic perturbations (noise), and using our formalism derive a rate of information loss, for Majorana memories, that depends on the spectral density of both the noise and the fermionic bath.