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Squeezing anyons for braiding on small lattices

N. S. Srivatsa, Xikun Li, Anne E. B. Nielsen

Published Sep 1, 2021• 9 citations
OpenAlex

Abstract

Adiabatically exchanging anyons gives rise to topologically protected operations on the quantum state of the system, but the desired result is only achieved if the anyons are well separated, which requires a sufficiently large area. Being able to reduce the area needed for the exchange, however, would have several advantages, such as enabling a larger number of operations per area and allowing anyon exchange to be studied in smaller systems that are easier to handle. Here, we use optimization techniques to squeeze the charge distribution of Abelian anyons in lattice fractional quantum Hall models, and we show that the squeezed anyons can be exchanged within a smaller area with a close to ideal outcome. We first use a toy model consisting of a modified Laughlin trial state to show that one can shape the anyons without altering the exchange statistics under certain conditions. We then squeeze and braid anyons in the Kapit-Mueller model and an interacting Hofstadter model by adding suitable potentials. We consider a fixed system size, for which the charge distributions of the normal anyons overlap, and we find that the outcome of the exchange process is closer to the ideal value for the squeezed anyons. The time needed for the exchange is also important, and for a particular example we find that the duration needed for the process to be close to the adiabatic limit is about five times longer for the squeezed anyons when the path length is the same. Finally we show that the exchange outcome is robust with respect to small modifications of the potential away from the optimized value.

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Optimising Qubit Designs forTopological QuantumComputation

Robert Ainsworth

The goal of this thesis is to examine some of the ways in which we might optimise the design of topological qubits. The topological operations which are imposed on qubits, in order to perform logic gates for topological quantum computations, are governed by the exchange group of the constituent particles. We examine representations of these exchange groups and investigate what restrictions their structure places on the effciency, reliability and universality of qubits (and multi-qubit systems) as a function of the number of particles composing them. Specific results are given for the limits placed on d-dimensional qudits where logic gates are imposed by braiding anyons in 2+1 dimensions. We also study qudits designed from ring-shaped, anyon-like excitations in 3+1 dimensions, where logic gates are implemented by elements of the loop braid group. We introduce the concept of local representations, where the generators of the loop braid group are defined to act non-trivially only on the local vector spaces associated with the rings which undergo the motion. We present a method for obtaining local representations of qudits and show how any such representation can be decomposed into representations which come from the quantum doubles of groups. Due to the dimension of the local representation being related to the number of generators, any non-Abelian properties of the representation are not compromised with an addition of extra operations, we conclude that universal representations may be easier to find than in previously discussed cases (though not for topological operations alone). We model a ring of Ising anyons in a fractional quantum Hall uid to study how interactions in a real environment may impact any qubits we have created. Fractional quantum Hall liquids are currently one of the most promising possibilities for the physical realisation of TQC and so present a natural choice of system in which to study these effects. We show how interactions between the anyons compromise the practicality of qubits defined by the fusion channels of anyon pairs and explore the use of the fermion number parity sectors as qubit states. Interactions between the anyon ring and the edge of the liquid are modelled to study the effect they will have on the state of the qubit. We perform numerical simulations, for a small system, to give some indication of how the edge interaction will in uence the reliability of the qubit.

•Jan 1, 2014•1 citation
Adiabatically exchanging anyons gives rise to topologically protected operations on the quantum state of the system, but the desired result is only achieved if the anyons are well separated, which requires a sufficiently large area. Being able to reduce the area needed for the exchange, however, would have several advantages, such as enabling a larger number of operations per area and allowing anyon exchange to be studied in smaller systems that are easier to handle. Here, we use optimization techniques to squeeze the charge distribution of Abelian anyons in lattice fractional quantum Hall models, and we show that the squeezed anyons can be exchanged within a smaller area with a close to ideal outcome. We first use a toy model consisting of a modified Laughlin trial state to show that one can shape the anyons without altering the exchange statistics under certain conditions. We then squeeze and braid anyons in the Kapit-Mueller model and an interacting Hofstadter model by adding suitable potentials. We consider a fixed system size, for which the charge distributions of the normal anyons overlap, and we find that the outcome of the exchange process is closer to the ideal value for the squeezed anyons. The time needed for the exchange is also important, and for a particular example we find that the duration needed for the process to be close to the adiabatic limit is about five times longer for the squeezed anyons when the path length is the same. Finally we show that the exchange outcome is robust with respect to small modifications of the potential away from the optimized value.
There is a growing interest in searching for topology in fractal dimensions with the aim of finding different properties and advantages compared to the integer dimensional case. Here we construct a local Hamiltonian on a fractal lattice whose ground state exhibits topological braiding properties. The fractal lattice is obtained from a second-generation Sierpinski carpet with Hausdorff dimension 1.89. We use local potentials to trap and exchange anyons in the model, and the numerically obtained results for the exchange statistics of the anyons are close to the ideal statistics for quasiholes in a bosonic Laughlin state at half filling. For the considered system size, the energy gap is about three times larger for the fractal lattice than for a two-dimensional square lattice, and we find that the braiding results obtained on the fractal lattice are more robust against disorder. We propose a scheme to implement both fractal lattices and our proposed local Hamiltonian with ultracold atoms in optical lattices.
The common approach to topological quantum computation is to implement quantum gates by adiabatically moving non-Abelian anyons around each other. Here we present an alternative perspective based on the possibility of realizing the exchange (braiding) operators of anyons by adiabatically varying pairwise interactions between them rather than their positions. We analyze a system composed by four anyons whose couplings define a T junction, and we show that the braiding operator of two of them can be obtained through a particular adiabatic cycle in the space of the coupling parameters. We also discuss how to couple this scheme with anyonic chains in order to recover the topological protection.
Anyons obeying fractional exchange statistics arise naturally in two dimensions: Hard-core two-body constraints make the configuration space of particles not simply-connected. The braid group describes how topologically-inequivalent exchange paths can be associated to non-trivial geometric phases for abelian anyons. Braid-anyon exchange statistics can also be found in one dimension (1D), but this requires broken Galilean invariance to distinguish different ways for two anyons to exchange. However, recently it was shown that an alternative form of exchange statistics can occur in 1D because hard-core three-body constraints also make the configuration space not simply-connected. Instead of the braid group, the topology of exchange paths and their associated non-trivial geometric phases are described by the traid group. In this article we propose a first concrete model realizing this alternative form of anyonic exchange statistics. Starting from a bosonic lattice model that implements the desired geometric phases with number-dependent Peierls phases, we then define anyonic operators so that the kinetic energy term in the Hamiltonian becomes local and quadratic with respect to them. The ground-state of this traid-anyon-Hubbard model exhibits several indications of exchange statistics intermediate between bosons and fermions, as well as signs of emergent approximate Haldane exclusion statistics. The continuum limit results in a Galilean invariant Hamiltonian with eigenstates that correspond to previously constructed continuum wave functions for traid anyons. This provides not only an a-posteriori justification of our lattice model, but also shows that our construction serves as an intuitive approach to traid anyons, i.e. anyons intrinsic to 1D.
Coherent control and braiding of anyons remain central challenges in realizing topologically protected quantum operations. We propose a Ramsey interferometry protocol to directly access the geometric phases associated with anyons in fractional Chern insulators. Our approach employs impurities with individually addressable internal states that bind to the anyons, allowing their adiabatic motion and exchange under full spatial control. By combining Ramsey and spin-echo sequences using one and two impurities, the protocol gives independent access to the Aharonov-Bohm and exchange contributions to the total geometric phase, thereby providing an unambiguous probe of anyonic statistics. Our scheme can potentially be implemented in cold-atom quantum simulators as well as in van der Waals heterostructures. Complementary finite-size simulations in non-interacting Chern insulators quantify the system sizes required to faithfully extract geometric phases, highlighting the role of edge effects. Our results establish impurity-based interferometry as a feasible route toward direct anyon braiding experiments in quantum simulators and lay the groundwork for future explorations of non-Abelian braiding and topological quantum control.
Observation of Time-Domain Braiding of Non-Abelian Anyons at ν=5/2ν= 5/2ν=5/2 State
Unlike elementary particles, which obey either bosonic or fermionic exchange statistics, certain quasiparticles, known as anyons, are predicted to exhibit Abelian or non-Abelian braiding statistics. While braiding Abelian anyons modifies the wavefunction by a 'statistical phase', braiding non-Abelian anyons implements a unitary transformation of the state within a degenerate subspace of states. Experimental evidence of non-Abelian braiding has thus far remained elusive. Here, we report a 'time-domain braiding' signature of non-Abelian anyons in the ν=5/2ν= 5/2ν=5/2 fractional quantum Hall state, by extending our previously demonstrated approach with Abelian anyons at ν=1/3ν= 1/3ν=1/3. Our approach is based on measurements of the current fluctuations arising from weak partitioning of a highly dilute one-dimensional edge mode. We independently probe the partition noise of the downstream charged mode and also that of the upstream neutral mode. These independent measurements agree with our theoretical predictions for 'time-domain braiding' of the downstream Abelian and the upstream non-Abelian anyons, respectively, in the 'particle-hole Pfaffian' topological order. Together, these results provide evidence for the presence of non-Abelian anyons.
The goal of this thesis is to examine some of the ways in which we might optimise the design of topological qubits. The topological operations which are imposed on qubits, in order to perform logic gates for topological quantum computations, are governed by the exchange group of the constituent particles. We examine representations of these exchange groups and investigate what restrictions their structure places on the effciency, reliability and universality of qubits (and multi-qubit systems) as a function of the number of particles composing them. Specific results are given for the limits placed on d-dimensional qudits where logic gates are imposed by braiding anyons in 2+1 dimensions. We also study qudits designed from ring-shaped, anyon-like excitations in 3+1 dimensions, where logic gates are implemented by elements of the loop braid group. We introduce the concept of local representations, where the generators of the loop braid group are defined to act non-trivially only on the local vector spaces associated with the rings which undergo the motion. We present a method for obtaining local representations of qudits and show how any such representation can be decomposed into representations which come from the quantum doubles of groups. Due to the dimension of the local representation being related to the number of generators, any non-Abelian properties of the representation are not compromised with an addition of extra operations, we conclude that universal representations may be easier to find than in previously discussed cases (though not for topological operations alone). We model a ring of Ising anyons in a fractional quantum Hall uid to study how interactions in a real environment may impact any qubits we have created. Fractional quantum Hall liquids are currently one of the most promising possibilities for the physical realisation of TQC and so present a natural choice of system in which to study these effects. We show how interactions between the anyons compromise the practicality of qubits defined by the fusion channels of anyon pairs and explore the use of the fermion number parity sectors as qubit states. Interactions between the anyon ring and the edge of the liquid are modelled to study the effect they will have on the state of the qubit. We perform numerical simulations, for a small system, to give some indication of how the edge interaction will in uence the reliability of the qubit.