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arXiv:1303.6365 · Physical Review A

Fault-tolerant quantum random-number generator certified by Majorana fermions

Dong-Ling Deng, L.-M. Duan

Published Jul 23, 2013• 17 citations
arXivHTMLPDF

Abstract

Braiding of Majorana fermions gives accurate topological quantum operations that are intrinsically robust to noise and imperfection, providing a natural method to realize fault-tolerant quantum information processing. Unfortunately, it is known that braiding of Majorana fermions is not sufficient for the implementation of universal quantum computation. Here we show that topological manipulation of Majorana fermions provides the full set of operations required to generate random numbers by way of quantum mechanics and to certify its genuine randomness through violation of a multipartite Bell inequality. The result opens a perspective to apply Majorana fermions for the robust generation of certified random numbers, which has important applications in cryptography and other related areas.

DOI
10.1103/physreva.88.012323

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•Jan 1, 2019•0 citations
Braiding of Majorana fermions gives accurate topological quantum operations that are intrinsically robust to noise and imperfection, providing a natural method to realize fault-tolerant quantum information processing. Unfortunately, it is known that braiding of Majorana fermions is not sufficient for the implementation of universal quantum computation. Here we show that topological manipulation of Majorana fermions provides the full set of operations required to generate random numbers by way of quantum mechanics and to certify its genuine randomness through violation of a multipartite Bell inequality. The result opens a perspective to apply Majorana fermions for the robust generation of certified random numbers, which has important applications in cryptography and other related areas.
Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons. This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions. Unfortunately, these operations are incapable of implementing universal quantum computation. We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations. We also demonstrate that our proposal is optimal in terms of resources, with 10 Majorana fermions shown to be both necessary and sufficient for demonstrating bipartite non-locality. Furthermore, we identify severe restrictions on the possibility of tripartite non-locality. We comment on the potential of such entangled systems to be used in quantum information protocols.
Localized Majorana fermions emerge in many topologically ordered systems and exhibit exchange statistics of Ising anyons. This enables noise-resistant implementation of a limited set of operations by braiding and fusing Majorana fermions. Unfortunately, these operations are incapable of implementing universal quantum computation. We show that, regardless of these limitations, Majorana fermions could be used to demonstrate non-locality (correlations incompatible with a local hidden variable theory) in experiments using only topologically protected operations. We also demonstrate that our proposal is optimal in terms of resources, with 10 Majorana fermions shown to be both necessary and sufficient for demonstrating bipartite non-locality. Furthermore, we identify severe restrictions on the possibility of tripartite non-locality. We comment on the potential of such entangled systems to be used in quantum information protocols.
Among the list of major threats to quantum computation, quantum decoherence poses one of the largest because it generates losses to the environment within a computational system which cannot be recovered via error correction methods. These methods require the assumption that the environmental interaction forces the qubit state into some linear combination of qubit eigenstates. In reality, the environment causes the qubit to enter into a mixed state where the original is no longer recoverable. A promising solution to this problem bases the computational states on the low lying energy excitations within topological materials. The existence of these states is protected by a global parameter within the Hamiltonian which prevents the computational states from coupling locally and decohering. In this paper, the qubit is based on nonlocal, topological Majorana fermions (MF), and the gate operations are generated by swapping or braiding the positions of said MF. The algorithmic calculation for such gate operations is well known, but, the opposite gates-to-braid calculation is currently underdeveloped. Additionally, because one may choose from a number of different possible qubit definitions, the resultant gate operations from calculation to calculation appear different. Here, the calculations for the two- and four-MF cases are recapitulated for the sake of logical flow. This set of gates serves as the foundation for the understanding and construction of the six-MF cases. Using these, a full characterization of the system is made by completely generalizing the list of gates and transformations between possible qubit definitions. A complete description of this system is desirable and will hopefully serve future iterations of topological qubits.
Abstract Majorana fermions and their generalizations to Zn parafermions are considered promising building blocks of fault-tolerant quantum computers for their ability to encode quantum information nonlocally. In such topological quantum computers, highly robust quantum gates are obtained by braiding pairs of these quasi-particles. However, it is well-known that braiding Majorana fermions or parafermions only leads to a Clifford gate, hindering quantum universality. This paper establishes an exact mapping between Majorana fermions to Z4 parafermions in systems under total parity nonconserving and total parity conserving setting. It is revealed that braiding of Majorana fermions may lead to non-Clifford quantum gates in the 4-dimensional qudit representation spanned by Z4 parafermions, whilst braiding of Z4 parafermions may similarly yield non-Clifford quantum gates in the qubit representation spanned by Majorana fermions. This finding suggests that topologically protected universal quantum computing may be possible with Majorana fermions (Z4 parafermions) by supplementing the usual braiding operations with the braiding of Z4 parafermions (Majorana fermions) that could be formed out of Majorana fermions (Z4 parafermions) via the mapping prescribed here. Finally, the paper discusses how braiding of Majorana fermions or Z4 parafermions could be obtained via a series of parity measurements.
Topological quantum computation by way of braiding of Majorana fermions is not universal quantum computation. There are several attempts to make universal quantum computation by introducing some additional quantum gates or quantum states. However, there is a serious problem in embedding an MMM-qubit quantum gate in the NNN-qubit system with N>MN>MN>M. This is an inherent problem to the Majorana system, where quantum gates for logical qubits become nonlocal in terms of physical qubits in general because braiding operations preserve the fermion parity. For instance, the CZ gate could not be embedded in the three-qubit system. We overcome this embedding problem by introducing 2(N+1)2(N+1)2(N+1)-body interactions of Majorana fermions in the NNN-qubit system. A universal set of quantum gates is constructed for NNN logical qubits, leading to topological-nontopological hybrid universal quantum computation. It would be more robust than conventional universal quantum computation because the quantum gates generated by braiding are topologically protected.
Majorana fermions arising in superconducting nanowires are of interest both for fundamental reasons and due to their potential use as building blocks for a topological quantum computer. While recent experiments have reported strong evidence of Majorana fermions in the form of occupation of a zero-energy fermionic mode, there is still disagreement about whether the observed data can be explained by other mechanisms. Furthermore, to be useful for quantum computation, the fermionic parities of zero-energy modes must be measured with high fidelity. In this thesis, we apply a quantum information protocol known as self-testing to a system of Majorana fermions. A self-testing protocol is a statistical test for certifying a quantum state and a set of measurement operators from experimentally determined expectation values, under minimal physical assumptions. Such a test treats the system as a black-box and relies only on the distributions of measurement outcomes given device settings. For the protocol described in this thesis, if the test is passed, then a set of measurements that ideally are of fermionic parities are certified to have the exact anticommutation relations as determined by the anticommutativity of Majorana fermion operators. The protocol is robust in the sense that bounds on the state and operator fidelities can still be established in the presence of small experimental imperfections. Finally, a rigorous statistical analysis is used to derive a confidence interval for the certified fidelities given a finite experimental data set.