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arXiv:1501.01013 · International Journal of Quantum Information

Making a circulant 2-qubit entangling gate

Claire Levaillant

Published Mar 1, 2016• 2 citations
arXivHTMLPDF

Abstract

We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2-qubit entangling gates which we generate in the present work, we produce in particular the circulant gate [Formula: see text] where J denotes the permutation matrix associated with the cycle [Formula: see text] and I denotes the identity matrix.

DOI
10.1142/s0219749916500064

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We present a way to physically realize a circulant 2-qubit entangling gate in the Kauffman–Jones version of SU(2) Chern–Simons theory at level 4. Our approach uses qubit and qutrit ancillas, braids, fusions and interferometric measurements. Our qubit is formed by four anyons of topological charges 1221. Among other 2-qubit entangling gates which we generate in the present work, we produce in particular the circulant gate [Formula: see text] where J denotes the permutation matrix associated with the cycle [Formula: see text] and I denotes the identity matrix.
Abstract As part of a protocol, we braid in a certain way six anyons of topological charges 222211 in the Kauffman-Jones version of SU(2) Chern-Simons theory at level 4. The gate we obtain is a braid for the usual qutrit 2222 but with respect to a different basis than the usual basis. With respect to that basis, the Freedman group of J. Phys. A: Math. Theor. 47, 285203 (2014) is identical to the D-group D(18, 1, 1; 2, 1, 1) from the Blichfeldt classification. We provide a physical realization of each Blichfeldt generator of the SU(3) finite subgroup D(18, 1, 1; 2, 1, 1). The link of this group to elementary particle physics was already well-known by the physicists. This paper uncovers its realization in quantum physics. Inspired by these new techniques for the qutrit, we are able to make new ancillas, namely 12(∣1> +∣3>)\frac{1}{\sqrt{2}}(|1>\,+|3>)2​1​(∣1>+∣3>) 1 2 ( | 1 > + | 3 > ) and 12(∣1> −∣3>)\frac{1}{\sqrt{2}}(|1>\,-|3>)2​1​(∣1>−∣3>) 1 2 ( | 1 > - | 3 > ) , for the qubit 1221.
The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.
We examine a class of operations for topological quantum computation based on fusing and measuring topological charges for systems with SU(2)4\mathrm{SU}{(2)}_{4}SU(2)4​ or k=4k=4k=4 Jones-Kauffman anyons. We show that such operations augment the braiding operations, which, by themselves, are not computationally universal. This augmentation results in a computationally universal gate set through the generation of an exact, topologically protected irrational phase gate and an approximate, topologically protected controlled-ZZZ gate.
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.
Topological quantum computation may provide a robust approach for encoding and manipulating information utilizing the topological properties of anyonic quasiparticle excitations. We develop an efficient means to map between dense and sparse representations of quantum information (qubits) and a simple construction of multiqubit gates, for all anyon models from Chern-Simons-Witten SU(2)k{}_{k}k​ theory that support universal quantum computation by braiding ($k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$). In the process, we show how the constructions of topological quantum memory and gates for k=2,4k=2,4k=2,4 connect naturally to those for $k\ensuremath{\geqslant}3,\phantom{\rule{4pt}{0ex}}k\ensuremath{\ne}4$, unifying these concepts in a simple framework. Furthermore, we illustrate potential extensions of these ideas to other anyon models outside of Chern-Simons-Witten field theory.
Fibonacci anyons are attractive for use in topological quantum computation because any unitary transformation of their state space can be approximated arbitrarily accurately by braiding. However there is no known braid that entangles two qubits without leaving the space spanned by the two qubits. In other words, there is no known "leakage-free" entangling gate made by braiding. In this paper, we provide a remedy to this problem by supplementing braiding with measurement operations in order to produce an exact controlled rotation gate on two qubits.