Topological quantum compiling

Hormozi, Layla; Zikos, Georgios; Bonesteel, N. E.; Simon, Steven H.

doi:10.1103/physrevb.75.165310

Cited by 85 publications

(142 citation statements)

References 44 publications

Supporting

Mentioning

134

Contrasting

Order By: Relevance

“…One of the simplest models of non-Abelian statistics is known as the Fibonacci anyon model, or "Golden theory" (Bonesteel et al, 2005;Freedman et al, 2002a;Hormozi et al, 2007;Preskill, 2004). In this model, there are only two fields, the identity (1) as well as single nontrivial field usually called τ which represents the non-Abelian quasiparticle.…”

Section: B Fibonacci Anyons: a Simple Example Which Is Universal Formentioning

confidence: 99%

Non-Abelian anyons and topological quantum computation

Simon²,

et al. 2008

View full text Add to dashboard Cite

Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a fault-tolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as Non-Abelian anyons, meaning that they obey non-Abelian braiding statistics. Quantum information is stored in states with multiple quasiparticles, which have a topological degeneracy. The unitary gate operations which are necessary for quantum computation are carried out by braiding quasiparticles, and then measuring the multi-quasiparticle states. The fault-tolerance of a topological quantum computer arises from the non-local encoding of the states of the quasiparticles, which makes them immune to errors caused by local perturbations. To date, the only such topological states thought to have been found in nature are fractional quantum Hall states, most prominently the ν = 5/2 state, although several other prospective candidates have been proposed in systems as disparate as ultra-cold atoms in optical lattices and thin film superconductors. In this review article, we describe current research in this field, focusing on the general theoretical concepts of non-Abelian statistics as it relates to topological quantum computation, on understanding non-Abelian quantum Hall states, on proposed experiments to detect non-Abelian anyons, and on proposed architectures for a topological quantum computer. We address both the mathematical underpinnings of topological quantum computation and the physics of the subject using the ν = 5/2 fractional quantum Hall state as the archetype of a non-Abelian topological state enabling fault-tolerant quantum computation.

show abstract

Section: B Fibonacci Anyons: a Simple Example Which Is Universal Formentioning

confidence: 99%

Non-Abelian anyons and topological quantum computation

Simon²,

et al. 2008

View full text Add to dashboard Cite

show abstract

“…For example, by use of braiding matrices (4.31) of the 4-quasi-particle states in the Pfaffian state, we can construct the NOT gate and the Hadamard gate by [20] [16] that the gate of large Fourier transformation cannot be realized exactly. In fact several unitary gates were approximated using these braiding matrices [4,27,55]. It may be interesting to construct fundamental gates such as NOT, Hadamard, and CNOT, using the braiding matrices of SU(2) K theory presented here.…”

Section: Concluding Remarks and Discussionmentioning

confidence: 99%

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

Hikami

2008

Annals of Physics

View full text Add to dashboard Cite

ABSTRACT. We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read-Rezayi state whose effective theory is the SU (2) K Chern-Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.

show abstract

“…13, when using the Fibonacci code, Fibonacci anyons can be associated with holes in the lattice subject to certain boundary conditions and proper initialization. These Fibonacci anyons can then be used to encode logical qubits and universal quantum computation can be carried out purely by braiding them, [17][18][19] without the need for magic state distillation.…”

Section: -512mentioning

confidence: 99%

Quantum circuits for measuring Levin-Wen operators

Bonesteel

DiVincenzo

2012

Phys. Rev. B

Self Cite

View full text Add to dashboard Cite

We construct quantum circuits for measuring the commuting set of vertex and plaquette operators that appear in the Levin-Wen model for doubled Fibonacci anyons. Such measurements can be viewed as syndrome measurements for the quantum error-correcting code defined by the ground states of this model (the Fibonacci code). We quantify the complexity of these circuits with gate counts using different universal gate sets and find these measurements become significantly easier to perform if n-qubit Toffoli gates with n = 3, 4 and 5 can be carried out directly. In addition to measurement circuits, we construct simplified quantum circuits requiring only a few qubits that can be used to verify that certain self-consistency conditions, including the pentagon equation, are satisfied by the Fibonacci code.

show abstract

Topological quantum compiling

Cited by 85 publications

References 44 publications

Non-Abelian anyons and topological quantum computation

Non-Abelian anyons and topological quantum computation

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

Quantum circuits for measuring Levin-Wen operators

Contact Info

Product

Resources

About