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arXiv:2310.13634

Optimising the exchange of Majorana zero modes in a quantum nanowire network

Tomasz Maciążek, Aaron Conlon

Published Oct 20, 2023• 0 citations
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Abstract

Determination of optimal control protocols for Majorana zero modes during their exchange is a crucial step towards the realisation of the topological quantum computer. In this paper, we study the finite-time exchange process of Majorana zero modes on a network formed by coupled p-wave superconducting one-dimensional nanowires. We provide scalable computational tools for optimising such an exchange process relying on deep learning techniques. To accomplish the scalability, we derive and implement an analytic formula for the gradient of the quantum infidelity which measures the error in the topological quantum gate generation in the Majorana zero modes exchange. Our optimisation strategy relies on learning the optimised transport protocol via a neural net which is followed by direct gradient descent fine tuning. The optimised exchange protocols in the super-adiabatic regime discover the fact that the Majorana zero modes must necessarily stop before crossing a junction point in the network. We explain that this is caused by fast changes in the energy gap of the system whenever one of the Majorana zero modes approaches a junction point. In particular, the energy gap exhibits oscillations followed by a sharp jump. We explain this phenomenon analytically in the regime where the Majorana zero modes are completely localised. Finally, we study how the disorder in the quantum nanowire affects the exchange protocols. This shows that understanding the disorder pattern would allow one to improve quantum gate fidelity by one to two orders of magnitude.

DOI
10.48550/arxiv.2310.13634

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Determination of optimal control protocols for Majorana zero modes during their exchange is a crucial step towards the realisation of the topological quantum computer. In this paper, we study the finite-time exchange process of Majorana zero modes on a network formed by coupled p-wave superconducting one-dimensional nanowires. We provide scalable computational tools for optimising such an exchange process relying on deep learning techniques. To accomplish the scalability, we derive and implement an analytic formula for the gradient of the quantum infidelity which measures the error in the topological quantum gate generation in the Majorana zero modes exchange. Our optimisation strategy relies on learning the optimised transport protocol via a neural net which is followed by direct gradient descent fine tuning. The optimised exchange protocols in the super-adiabatic regime discover the fact that the Majorana zero modes must necessarily stop before crossing a junction point in the network. We explain that this is caused by fast changes in the energy gap of the system whenever one of the Majorana zero modes approaches a junction point. In particular, the energy gap exhibits oscillations followed by a sharp jump. We explain this phenomenon analytically in the regime where the Majorana zero modes are completely localised. Finally, we study how the disorder in the quantum nanowire affects the exchange protocols. This shows that understanding the disorder pattern would allow one to improve quantum gate fidelity by one to two orders of magnitude.
The prospects for realizing a topological quantum computer have brightened since the apparent detection of Majorana fermions at the ends of semiconducting nanowires. These Majorana zero-modes persist in the presence of the strong disorder that may be present in such systems, protected by the mobility gap of localized systems. Recent proposals to manipulate quantum information in a Majorana chain involve adiabatically adjusting gate voltages in one-dimensional nanowire networks. However, as we show, in the adiabatic limit, a disordered system with multiple zero-mode Majorana fermions will undergo a non-local response to a local perturbation via a multi-level Landau-Zener transition. This will disperse the quantum information stored in the target zero-modes amongst other zero-modes that are present in the system, complicating the realization of controlled gates. There is however a solution that may work in practice: we show that there is an optimum window of intermediate time-scales during which the quantum information is approximately preserved under manipulation.
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.
In topological quantum computing, unitary operations on qubits are performed by adiabatic braiding of non-Abelian quasiparticles, such as Majorana zero modes, and are protected from local environmental perturbations. In the adiabatic regime, with timescales set by the inverse gap of the system, the errors can be made arbitrarily small by performing the process more slowly. To enhance the performance of quantum information processing with Majorana zero modes, we apply the theory of optimal control to the diabatic dynamics of Majorana-based qubits. While we sacrifice complete topological protection, we impose constraints on the optimal protocol to take advantage of the nonlocal nature of topological information and increase the robustness of our gates. By using the Pontryagin's maximum principle, we show that robust equivalent gates to perfect adiabatic braiding can be implemented in finite times through optimal pulses. In our implementation, modifications to the device Hamiltonian are avoided. Focusing on thermally isolated systems, we study the effects of calibration errors and external white and 1/f1/f1/f (pink) noise on Majorana-based gates. While a noise-induced antiadiabatic behavior, where a slower process creates more diabatic excitations, prohibits indefinite enhancement of the robustness of the adiabatic scheme, our fast optimal protocols exhibit remarkable stability to noise and have the potential to significantly enhance the practical performance of Majorana-based information processing.
Abstract Braiding Majorana zero-modes around each other is a promising route toward topological quantum computing. Yet, two competing maxims emerge when implementing Majorana braiding in real systems: on the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that decoherence effects introduced by the environment are negligible, which are generally unavoidable in finite-size systems. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g. environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.
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.
Braiding Majorana zero-modes around each other is a promising route towards topological quantum computing. Yet, two competing maximes emerge when implementing Majorana braiding in real systems: On the one hand, perfect braiding should be conducted adiabatically slowly to avoid non-topological errors. On the other hand, braiding must be conducted fast such that unavoidable decoherence effects introduced by the environment are negligible. This competition results in an intermediate time scale for Majorana braiding that is optimal, but generally not error-free. Here, we calculate this intermediate time scale for a T-junction of short one-dimensional topological superconductors coupled to a bosonic bath that generates fluctuations in the local electric potential, which stem from, e.g., environmental photons or phonons of the substrate. We thereby obtain boundaries for the speed of Majorana braiding with a predetermined gate fidelity. Our results emphasize the general susceptibility of Majorana-based information storage in finite-size systems and can serve as a guide for determining the optimal braiding times in future experiments.