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arXiv:2305.18190 · Physical review. B./Physical review. B

Ancilla-free scheme of deterministic topological quantum gates for Majorana qubits

Su-Qi Zhang, Jian-Song Hong, Yuan Xue, Xun-Jiang Luo, Li-Wei Yu, Xiong-Jun Liu, Xin Liu

Published Apr 5, 2024• 2 citations
arXivHTMLPDF

Abstract

The realization of quantum gates in topological quantum computation still confronts significant challenges in both fundamental and practical aspects. Here, we propose an ancilla-free, deterministic, and topologically protected measurement-based scheme to realize the implementation of Clifford quantum gates on the Majorana qubits. Our scheme is based on a rigorous proof that the single-qubit gate can be performed by leveraging the neighboring Majorana qubit but not disturbing its carried quantum information, enabling an ancilla-free scheme for the topological quantum computing with Majorana qubits. Benefiting from the ancilla-free construction, we show the minimum measurement sequences with four steps to achieve two-qubit Clifford gates by constructing their geometric visualization. The uncertainty of the current strategy can be avoided by manipulating the Majorana modes in their parameter space, as shown in a concrete Majorana platform, correcting the undesired measurement outcomes while maintaining topological protection. Our scheme identifies the minimal operations of measurement-based topological and deterministic Clifford gates and offers an ancilla-free design of topological quantum computation.

DOI
10.1103/physrevb.109.165302

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The realization of quantum gates in topological quantum computation still confronts significant challenges in both fundamental and practical aspects. Here, we propose an ancilla-free, deterministic, and topologically protected measurement-based scheme to realize the implementation of Clifford quantum gates on the Majorana qubits. Our scheme is based on a rigorous proof that the single-qubit gate can be performed by leveraging the neighboring Majorana qubit but not disturbing its carried quantum information, enabling an ancilla-free scheme for the topological quantum computing with Majorana qubits. Benefiting from the ancilla-free construction, we show the minimum measurement sequences with four steps to achieve two-qubit Clifford gates by constructing their geometric visualization. The uncertainty of the current strategy can be avoided by manipulating the Majorana modes in their parameter space, as shown in a concrete Majorana platform, correcting the undesired measurement outcomes while maintaining topological protection. Our scheme identifies the minimal operations of measurement-based topological and deterministic Clifford gates and offers an ancilla-free design of topological quantum computation.
One of the main challenges for quantum computation is that while the number of gates required to perform a non-trivial quantum computation may be very large, decoherence and errors in realistic quantum architectures limit the number of physical gate operations that can be performed coherently. Therefore, an optimal mapping of the quantum algorithm into the physically available set of operations is of crucial importance. We examine this problem for a measurement-only topological quantum computer based on Majorana zero modes, where gates are performed through sequences of measurements. Such a scheme has been proposed as a practical, scalable approach to process quantum information in an array of topological qubits built using Majorana zero modes. Building on previous work that has shown that multi-qubit Clifford gates can be enacted in a topologically protected fashion in such qubit networks, we discuss methods to obtain the optimal measurement sequence for a given Clifford gate under the constraints imposed by the physical architecture, such as layout and the relative difficulty of implementing different types of measurements. Our methods also provide tools for comparative analysis of different architectures and strategies, given experimental characterizations of particular aspects of the systems under consideration. As a further non-trivial demonstration, we discuss an implementation of the surface code in Majorana-based topological qubits. We use the techniques developed here to obtain an optimized measurement sequence that implements the stabilizer measurements using only fermionic parity measurements on nearest-neighbor topological qubit islands.
Majorana modes, typically arising at the edges of one-dimensional topological superconductors, are considered to be a promising candidate for encoding nonlocal qubits in fault-tolerant quantum computing. Here we exploit the two-dimensional geometry of Majorana corner modes in second-order topological superconductors to establish measurement-only quantum computation. It is shown that eight Majorana corner modes emerge when such systems are periodically driven, through which two nonlocal logical qubits and one nonlocal ancilla qubit can be constructed. Quantum gate operations can then be implemented by a designed series of parity measurements of topologically protected Majorana corner modes, accomplished via Mach-Zehnder type interference in the conductance between different corners of a second-order topological superconductor. Our theoretical proposal represents a scenario in which topologically protected single- and two-qubit gate operations can be carried out in a minimal setup, thus potentially establishing an efficient and low-cost building block for Majorana-based qubit architectures.
We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit ZZZ rotations. We first design a topologically protected logical ZZZ gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the ZZZ gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.
Topological quantum computing seeks to store and manipulate information in a protected manner using topological phases of matter. Information encoded in the degenerate state space of pairs of non-Abelian anyons or defects is robust to local perturbations, reducing its susceptiblity to environmental errors and potentially providing a scalable approach to quantum computing. However, topological quantum computing faces significant challenges, not least of which is identifying an experimentally accessible platform supporting non-Abelian topological physics. In this thesis, we critically analyze topological quantum computing with Majorana zero modes, non-Abelian defects of a topological superconductor. We identify intrinsic error sources for Majorana-based systems and propose quantum computing architectures that minimize their effects. Additionally, we consider a new approach for realizing and detecting non-Abelian topological defects in fractional Chern insulators. Topological quantum computing is predicated on the idea that braiding non-Abelian anyons adiabatically can implement quantum gates fault tolerantly. However, any braiding experiment will necessarily depart from the strict adiabatic limit. We begin by analyzing the nature of diabatic errors for anyon braiding, paying particular attention to how such errors scale with braiding time. We find that diabatic errors are unfavorably large and worryingly sensitive to details of the time evolution. We present a measurement-based correction protocol for such errors, and illustrate its application in a particular Majorana-based qubit design. We next propose designs for Majorana-based qubits operated entirely by a measurement-based protocol, thereby avoiding the diabatic errors discussed above. Our designs can be scaled into large two dimensional arrays amenable to long-term quantum computing goals, whose core components are testable in near-term devices. These qubits are robust to quasiparticle poisoning, anticipated to be one of the dominant error sources coupling to Majorana zero modes. We demonstrate that our designs support topologically protected Clifford operations and can be augmented to a universal gate set without requiring additional control parameters.While topological protection greatly suppresses errors, residual coupling to noise limits the lifetimes of our proposed Majorana-based qubits. We analyze the dephasing times for our quasiparticle-poisoning-protected qubits by calculating their charge distribution using a particle number-conserving formalism. We find that fluctuations in the electromagnetic environment couple to an exponentially suppressed topological dipole moment. We estimate dephasing times due to 1/f1/f1/f noise, thermal quasiparticle excitations, and phonons for different qubit sizes.The residual errors discussed above will necessarily require error correction for a sufficiently long quantum computation. We develop physically motivated noise models for Majorana-based qubits that can be used to analyze the performance of a quantum error correcting code. We apply this noise model to estimate pseudo-thresholds for a small subsystem code, identifying the relative importance of difference error processes from a fault tolerance perspective. Our results emphasize the necessity of suppressing long-lived quasiparticle excitations that can spread across the code.Finally, we turn our attention to a different platform that could host non-Abelian topological defects: fractional Chern insulators in graphene. We study the edge states of fractional Chern insulators using the field theory of fractional quantum Hall edges supplemented with a symmetry action. We find that lattice symmetries impose a quantized momentum difference for edge electrons in a fractional state of a C=2C=2C=2 Chern band. This momentum difference can be used to selectively contact the different edge states, thereby allowing detection of topological defects in the bulk with a standard four terminal measurement. Our proposal could be implemented in graphene subject to an artificially patterned lattice.
Majorana-based quantum computation seeks to encode information nonlocally in pairs of Majorana zero modes, thereby isolating qubit states from a local noisy environment. In addition to long coherence times, the attractiveness of Majorana-based quantum computing relies on achieving topologically protected Clifford gates from braiding operations. Recent works have conjectured that mean-field BCS calculations may fail to account for nonuniversal corrections to the Majorana braiding operations. Such errors would be detrimental to Majorana-based topological quantum computing schemes. In this work, we develop a particle-number-conserving approach for measurement-based topological quantum computing and investigate the effect of quantum phase fluctuations. We demonstrate that braiding transformations are indeed topologically protected in charge-protected Majorana-based quantum computing schemes.
Topological qubits composed of unpaired Majorana zero modes are under intense experimental and theoretical scrutiny in efforts to realize practical quantum computation schemes. In this work, we show that the minimum four Majorana zero modes required for a topological qubit according to braiding schemes and control of entanglement for gate operations are inherent to multiplicative topological phases, which realize symmetry-protected tensor products—and maximally entangled Bell states—of unpaired Majorana zero modes. We construct and characterize both one-dimensional and two-dimensional multiplicative topological phases with two parent Kitaev chain Hamiltonians. We furthermore characterize topology in the bulk and on the boundary with established methods while also introducing techniques to overcome challenges in characterizing multiplicative topology. In the process, we explore the potential of these multiplicative topological phases for an alternative to braiding-based topological quantum computation schemes, in which gate operations are performed through topological phase transitions. Published by the American Physical Society 2024