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arXiv:2112.06144

Quantum Computing with Two-dimensional Conformal Field Theories

Elias Kokkas, Aaron Bagheri, Zhenghan Wang, George Siopsis

Published Dec 12, 2021• 0 citations
arXivHTMLPDF

Abstract

Conformal field theories have been extremely useful in our quest to understand physical phenomena in many different branches of physics, starting from condensed matter all the way up to high energy. Here we discuss applications of two-dimensional conformal field theories to fault-tolerant quantum computation based on the coset $ SU(2)_1^{\otimes k} / SU(2)_{k}$. We calculate higher-dimensional braiding matrices by considering conformal blocks involving $2N$ anyons, and search for gapped states that can be built from these conformal blocks. We introduce a gapped wavefunction that generalizes the Moore-Read state which is based on the critical Ising model, and show that our generalization leads to universal quantum computing.

DOI
10.48550/arxiv.2112.06144

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Conformal field theories have been extremely useful in our quest to understand physical phenomena in many different branches of physics, starting from condensed matter all the way up to high energy. Here we discuss applications of two-dimensional conformal field theories to fault-tolerant quantum computation based on the coset SU(2)1⊗k/SU(2)k SU(2)_1^{\otimes k} / SU(2)_{k}SU(2)1⊗k​/SU(2)k​. We calculate higher-dimensional braiding matrices by considering conformal blocks involving 2N2N2N anyons, and search for gapped states that can be built from these conformal blocks. We introduce a gapped wavefunction that generalizes the Moore-Read state which is based on the critical Ising model, and show that our generalization leads to universal quantum computing.
We propose a framework for topological quantum computation using newly discovered non-semisimple analogs of topological quantum field theories in 2 + 1 dimensions. These enhanced theories offer more powerful models for quantum computation. The conventional theory of Ising anyons, which is believed to describe excitations in the ν = 5/2 fractional quantum Hall state, is not universal for quantum computation via braiding of quasiparticles. However, we show that the non-semisimple theory introduces new anyon types that extend the Ising framework. By adding just one new anyon type, universal quantum computation can be achieved through braiding alone. This result opens new avenues for realizing fault-tolerant quantum computing in topologically ordered systems. Ising anyons are a promising platform for topological quantum computation, but braiding alone is not universal. Here, authors show that extending to non-semi simple TQFTs introduces new anyon types that enable universal quantum computation with Ising anyons.
Universal Quantum Gate Compilation in SU(2)kSU(2)_kSU(2)k​ Anyon Models via Multiple-Braiding
We investigate the capability of multiple-braiding in SU(2)kSU(2)_kSU(2)k​ anyon models to realize universal quantum computation. The multiple elementary braiding matrices (MEBMs) are derived from the qqq-deformed representation theory of SU(2)SU(2)SU(2). Through a general analysis of the MEBMs, we find that multiple-braiding loses its universality only when the braiding multiplicity mmm causes the MEBMs to collapse to a scalar (up to a global phase). Our numerical analysis of the MEBMs of SU(2)kSU(2)_kSU(2)k​ anyon models with k=3,5,6,7k = 3, 5, 6, 7k=3,5,6,7, and m∈[2,9]m \in [2, 9]m∈[2,9], shows that the values of mmm at which the braid group representation fails to be dense agree exactly with the theoretical prediction. For those mmm that support universality, single-qubit gates are compiled to high precision by a genetic algorithm-enhanced Solovay-Kitaev algorithm (GA-enhanced SKA), and a genetic algorithm (GA) search with progressively increasing braid length yields an approximately the local equivalence class [CNOT][CNOT][CNOT]. Notably, even-order braiding operations offer a physical advantage by reducing the number of non-Abelian anyons required in braiding-based topological quantum computation (TQC). Our analytical and numerical results together provide strong evidence for identifying which braiding multiplicities mmm support universal quantum computation in SU(2)kSU(2)_kSU(2)k​ anyon models.
There exists various suggestions for building a functional and a fault-tolerant large-scale quantum computer. Topological quantum computation is a more exotic suggestion, which makes use of the properties of quasiparticles manifest only in certain two-dimensional systems. These so called anyons exhibit topological degrees of freedom, which, in principle, can be used to execute quantum computation with intrinsic fault-tolerance. This feature is the main incentive to study topological quantum computation. The objective of this thesis is to provide an accessible introduction to the theory. In this thesis one has considered the theory of anyons arising in two-dimensional quantum mechanical systems, which are described by gauge theories based on so called quantum double symmetries. The quasiparticles are shown to exhibit interactions and carry quantum numbers, which are both of topological nature. Particularly, it is found that the addition of the quantum numbers is not unique, but that the fusion of the quasiparticles is described by a non-trivial fusion algebra. It is discussed how this property can be used to encode quantum information in a manner which is intrinsically protected from decoherence and how one could, in principle, perform quantum computation by braiding the quasiparticles. As an example of the presented general discussion, the particle spectrum and the fusion algebra of an anyon model based on the gauge group S_3 are explicitly derived. The fusion algebra is found to branch into multiple proper subalgebras and the simplest one of them is chosen as a model for an illustrative demonstration. The different steps of a topological quantum computation are outlined and the computational power of the model is assessed. It turns out that the chosen model is not universal for quantum computation. However, because the objective was a demonstration of the theory with explicit calculations, none of the other more complicated fusion subalgebras were considered. Studying their applicability for quantum computation could be a topic of further research.
Quaternate generalization of Pfaffian state at ν=5/2ν=5/2ν=5/2
We consider a quaternately generalized Pfaffian QGPf(1J(zi,zj,zk,zl))[J(z1,...,zN)]2(\frac{1}{J(z_i,z_j,z_k,z_l)})[J(z_1,...,z_N)]^2(J(zi​,zj​,zk​,zl​)1​)[J(z1​,...,zN​)]2 in which the square of Vandermonde determinant, [J(z1,...,zN)]2[J(z_1,...,z_N)]^2[J(z1​,...,zN​)]2, implies the upmost Landau level is half filled. This wave function is the unique highest density zero energy state of a special short range interacting Hamiltonian. One can think this quaternate composite fermion liquid as a competing ground state of Moore-Read (MR) Pfaffian state at ν=5/2ν=5/2ν=5/2. The degeneracy of the quasihole excitations above the QGPf is higher than that of Moore-Read even Read-Rezayi quasiholes. The QGPf is related to a unitary conformal field theory with Z2×Z2Z_2\times Z_2Z2​×Z2​ parafermions in coset space SU(3)2/U(1)2SU(3)_2/U(1)^2SU(3)2​/U(1)2 . Because of the level-rank duality between SU(3)2SU(3)_2SU(3)2​ and SU(2)3SU(2)_3SU(2)3​ in conformal field theory, these quasiholes above this QGPf state obeying non-abelian anyonic statistics are expected to support the universal quantum computation at ν=5/2ν=5/2ν=5/2 as Read-Rezayi quasiholes at ν=13/5ν=13/5ν=13/5. The edge states of QGPf are very different from those of the Pfaffian's.