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arXiv:1112.1516 · Physical Review A

Nonlocality as a benchmark for universal quantum computation in Ising anyon topological quantum computers

Mark Howard, Jiří Vala

Published Feb 3, 2012• 16 citations
arXivHTMLPDF

Abstract

An obstacle affecting any proposal for a topological quantum computer based on Ising anyons is that quasiparticle braiding can only implement a finite (nonuniversal) set of quantum operations. The computational power of this restricted set of operations (often called stabilizer operations) has been studied in quantum information theory, and it is known that no quantum-computational advantage can be obtained without the help of an additional nonstabilizer operation. Similarly, a bipartite two-qubit system based on Ising anyons can not exhibit nonlocality (in the sense of violating a Bell inequality) when only topologically protected stabilizer operations are performed. To produce correlations that can not be described by a local hidden variable model again requires the use of a nonstabilizer operation. Using geometric techniques, we relate the sets of operations that enable universal quantum computing (UQC) with those that enable violation of a Bell inequality. Motivated by the fact that nonstabilizer operations are expected to be highly imperfect, our aim is to provide a benchmark for identifying UQC-enabling operations that is both experimentally practical and conceptually simple. We show that any (noisy) single-qubit nonstabilizer operation that, together with perfect stabilizer operations, enables violation of the simplest two-qubit Bell inequality, can also be used to enable UQC. This benchmarking requires finding the expectation values of two distinct Pauli measurements on each qubit of a bipartite system.

DOI
10.1103/physreva.85.022304

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An obstacle affecting any proposal for a topological quantum computer based on Ising anyons is that quasiparticle braiding can only implement a finite (nonuniversal) set of quantum operations. The computational power of this restricted set of operations (often called stabilizer operations) has been studied in quantum information theory, and it is known that no quantum-computational advantage can be obtained without the help of an additional nonstabilizer operation. Similarly, a bipartite two-qubit system based on Ising anyons can not exhibit nonlocality (in the sense of violating a Bell inequality) when only topologically protected stabilizer operations are performed. To produce correlations that can not be described by a local hidden variable model again requires the use of a nonstabilizer operation. Using geometric techniques, we relate the sets of operations that enable universal quantum computing (UQC) with those that enable violation of a Bell inequality. Motivated by the fact that nonstabilizer operations are expected to be highly imperfect, our aim is to provide a benchmark for identifying UQC-enabling operations that is both experimentally practical and conceptually simple. We show that any (noisy) single-qubit nonstabilizer operation that, together with perfect stabilizer operations, enables violation of the simplest two-qubit Bell inequality, can also be used to enable UQC. This benchmarking requires finding the expectation values of two distinct Pauli measurements on each qubit of a bipartite system.
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.
We have studied SU(2)k{\rm SU}(2)_kSU(2)k​ anyon models, assessing their prospects for topological quantum computation. In particular, we have compared the Ising (k=2k=2k=2) anyon and Fibonacci (k=3k=3k=3) anyon models, motivated by their potential for future realizations based on Majorana fermion quasiparticles or exotic fractional quantum-Hall states, respectively. The quantum computational performance of the different anyon models is quantified at single qubit level by the difference between a target unitary operator and its approximation realised by anyon braiding. To facilitate efficient comparisons, we have developed a Monte Carlo enhanced Solovay-Kitaev quantum compiler algorithm that finds optimal braid words in polynomial time from the exponentially large search tree. Since universal quantum computation cannot be achieved within the Ising anyon model by braiding alone, we have introduced an additional elementary phase gate to model a non-topological measurement process, which restores universality of the anyon model at the cost of breaking the full topological protection. We model conventional kinds of decoherence processes algorithmically by introducing a controllable noise term to all non-topological gate operations. We find that for reasonable levels of decoherence, even the hybrid Ising anyon model retains a significant topological advantage over a conventional, non-topological, quantum computer. Furthermore, we find that only surprisingly short anyon braids are ever required to be compiled due to the gate noise exceeding the intrinsic error of the braid words already for word lengths of the order of 100100100 elementary braids. We conclude that the future for hybrid topological quantum computation remains promising.
Universal Quantum Computation with the S3S_3S3​ Quantum Double: A Pedagogical Exposition
Non-Abelian topological order (TO) enables topologically protected quantum computation with its anyonic quasiparticles. Recently, TO with S3S_3S3​ gauge symmetry was identified as a sweet spot -- simple enough to emerge from finite-depth adaptive circuits yet powerful enough to support a universal topological gate-set. In these notes, we review how anyon braiding and measurement in S3S_3S3​ TO are primitives for topological quantum computation and we explicitly demonstrate universality. These topological operations are made concrete in the S3S_3S3​ quantum double lattice model, aided by the introduction of a generalized ribbon operator. This provides a roadmap for near-term quantum platforms.
Topological quantum computation encodes quantum information in the internal fusion space of non-Abelian anyonic quasiparticles, whose braiding implements logical gates. This goes beyond Abelian topological order (TO) such as the toric code, as its anyons lack internal structure. However, the simplest non-Abelian generalizations of the toric code do not support universality via braiding alone. Here we demonstrate that such minimally non-Abelian TOs can be made universal by treating anyon fusion as a computational primitive. We prepare a 54-qubit TO wavefunction associated with the smallest non-Abelian group, S3S_3S3​, on Quantinuum's H2 quantum processor. This phase of matter exhibits cyclic anyon fusion rules, known to underpin universality, which we evidence by trapping a single non-Abelian anyon on the torus. We encode logical qutrits in the nonlocal fusion space of non-Abelian fluxes and, by combining an entangling braiding operation with anyon charge measurements, realize a universal topological gate set and read-out, which we further demonstrate by topologically preparing a magic state. This work establishes S3S_3S3​ TO as simple enough to be prepared efficiently, yet rich enough to enable universal topological quantum computation.
Abstract Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.