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Optimizing and Characterizing Measurement-Only Topological Quantum Computing with Majorana Zero Mode Based Qubits

Alan Tran

Published Nov 17, 2020• 0 citations
OpenAlex

Abstract

Quantum computing is a revolutionizing technology on many fronts. Topological quantum computing is an especially attractive platform for this due to its innate robustness to local sources of noise and scalability. As the advent of quantum computing technologies nears, it becomes increasingly important to both optimize as much as possible the basic operations and to characterize them. In this thesis we study measurement-only topological quantum computing with Majorana zero mode (MZM) based qubits, particularly on their incarnation as the boudnary defects of $1d$ topological superconducting nanowires. The specific qubit device we consider are comprised of collections of such wires connected together on a single island such that 4 or 6 MZMs are hosted (tetron and hexon qubits respectively).This thesis focuses on optimizing and characterizing the basic operations needed for quantum computing with such MZM qubits, concentrating on hexons which are the smallest MZM qubit allowing measurement-only gates on a single island. We begin by optimizing the Clifford gates for a single hexon qubit and for the controlled-Pauli, $\\mathsf{W}$, and $\\mathsf{SWAP}$ gates on two hexon qubits. A brute-force searched is performed with respect to a physically motivated example cost function. Tools and techniques are developed to aid in the compiling and optimization of measurement sequences. Utilizing the developed tools, we apply it towards the optimization of the stabilizer measurements for the surface code, an especially useful quantum error correcting code for the present MZM setting. Finally, we discuss adapting the technique of randomized benchmarking to the hexon setting, addressing simplifications and extensions that a hexon qubit can offer.

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Quantum computing is a revolutionizing technology on many fronts. Topological quantum computing is an especially attractive platform for this due to its innate robustness to local sources of noise and scalability. As the advent of quantum computing technologies nears, it becomes increasingly important to both optimize as much as possible the basic operations and to characterize them. In this thesis we study measurement-only topological quantum computing with Majorana zero mode (MZM) based qubits, particularly on their incarnation as the boudnary defects of 1d1d1d topological superconducting nanowires. The specific qubit device we consider are comprised of collections of such wires connected together on a single island such that 4 or 6 MZMs are hosted (tetron and hexon qubits respectively).This thesis focuses on optimizing and characterizing the basic operations needed for quantum computing with such MZM qubits, concentrating on hexons which are the smallest MZM qubit allowing measurement-only gates on a single island. We begin by optimizing the Clifford gates for a single hexon qubit and for the controlled-Pauli, mathsfW\\mathsf{W}mathsfW, and mathsfSWAP\\mathsf{SWAP}mathsfSWAP gates on two hexon qubits. A brute-force searched is performed with respect to a physically motivated example cost function. Tools and techniques are developed to aid in the compiling and optimization of measurement sequences. Utilizing the developed tools, we apply it towards the optimization of the stabilizer measurements for the surface code, an especially useful quantum error correcting code for the present MZM setting. Finally, we discuss adapting the technique of randomized benchmarking to the hexon setting, addressing simplifications and extensions that a hexon qubit can offer.
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.
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.
Quantum computers are devices that can solve certain problems faster than ordinary, classical computers. The fundamental units of quantum information are qubits, superpositions of two states, a "zero" state and a "one" state. There are various approaches to construct such two-level systems, among others, using superconducting circuits, trapped ions or photons. A common feature of these physical systems is that their coherence times are relatively short compared to the length of useful computations. Superconducting qubits, for instance, are currently the most advanced solid-state qubits, but they decohere after around 100 microseconds, and any information stored in these qubits is lost. On the other hand, useful quantum computations may require quantum information to survive on time scales that are many orders of magnitude longer, as their runtimes can reach several hours or even days. Topological quantum computing is an approach to construct qubits that survive for the entire duration of such a long computation. Topological quantum computing comes in two flavors. The condensed matter approach is to build error-resilient qubits using exotic quasiparticles in topological materials, most prominently Majorana zero modes in topological superconductors. Even though no such qubit has been built to date, the hope is that their coherence times may be significantly longer than the coherence times of currently available solid-state qubits, but are still expected to be too short for large-scale quantum computing. The quantum information approach is to combine many error-prone qubits to build more robust logical qubits using topological error-correcting codes, e.g., surface codes. Even though the first approach is hardware-based and the second approach is software-based, they are deeply related. With Majorana-based qubits, the main logical operations are Majorana fermion parity measurements. By replacing Majorana-based qubits with surface-code patches and parity measurements with lattice-surgery operations, schemes for quantum computation with Majorana-based qubits or with surface codes can be identical. In this thesis, we explore how to construct a large-scale topological fault-tolerant quantum computer that can perform useful quantum computations. Here, topological refers to the nature of the quantum error-correcting code, while the underlying hardware may be based on non-topological qubits, but could also be composed of Majorana-based qubits. We provide a complete picture of such a large-scale device, breaking down large quantum computations into logical qubits and logical operations, describing how these logical operations are performed on the level of physical qubits and physical gates, and finally discussing how these physical qubits can be pieced together in a Majorana-based system using topological superconducting nanowires.
The tetron architecture is a promising candidate for topological quantum computation. Each tetron Majorana island has four Majorana zero modes, and possible measurements are constrained to span zero or two Majoranas per tetron. Such measurements are known to be sufficient for correcting so-called "bosonic errors," which affect an even number of Majoranas per tetron. We demonstrate that such measurements are also sufficient for correcting "fermionic errors," which affect an odd number of Majoranas per tetron. In contrast, previous proposals for "fermionic error correction" on tetrons introduce more experimental challenges. We show that "fermionic codes" can be derived from traditional "bosonic codes" by inclusion of tetrons in the stabilizer group.
Quantum computers hold the promise of solving computational problems that are intractable for classical computers, by harnessing the principles of quantum mechanics. However, quantum computations are highly susceptible to noise and decoherence, which place strict limits on the size and depth of the circuits that can be reliably executed on current hardware. Topological quantum computing (TQC) has emerged as one of the promising approaches to building fault-tolerant quantum computers. The key idea is to store quantum information in the global properties of a physical system, making it naturally resilient to the local noise and errors that affect conventional qubits. This thesis focuses on synthetic topological quantum matter, the physical platform on which TQC is built, and investigates its electronic structure and topological properties. We study the conditions under which these systems enter a topological phase and host Majorana zero modes (MZMs), exotic zero-energy quasiparticles that appear at the boundaries of the system. MZMs are the key building block of TQC, as their quantum mechanical properties allow them to encode and process quantum information in a way that is protected from errors. We demonstrate the existence of MZMs in two canoni- cal topological systems, the Kitaev chain and the semiconducting-superconducting nanowire. To address the experimental challenge of detecting MZMs, we develop an optical spectroscopy technique based on excitonic interactions, which provides clear and measurable signatures of the topological phase. We also explore the variational quantum eigensolver as a quantum algorithmic approach to approximating the ground-state energy of the many-body spectrum of topological systems. Furthermore, we discuss the practical setups for topological quantum matter and simulate them through numerical analysis. Altogether, we present several theoretical and numerical methods that can be used for numerical simulations and practical purposes of detecting MZMs in topological quantum matter.
The idea of topological quantum computation (TQC) is to store and manipulate quantum information in an intrinsically fault-tolerant manner by utilizing the physics of topologically ordered phases of matter. Currently, one of the most promising platforms for a topological qubit is in terms of Majorana fermion zero modes (MZMs) in spin-orbit coupled superconducting nanowires. However, the topologically robust operations that are possible with MZMs can be efficiently simulated on a classical computer and are therefore not sufficient for realizing a universal gate set for TQC. Here, we show that an array of coupled semiconductor-superconductor nanowires with MZM edge states can be used to realize a more sophisticated type of non-Abelian defect: a genon in an Ising ×\times× Ising topological state. This leads to a possible implementation of the missing topologically protected π/8π/8π/8 phase gate and thus universal TQC based on semiconductor-superconductor nanowire technology. We provide detailed numerical estimates of the relevant energy scales, which we show to lie within accessible ranges.