Introduction to Topological Quantum Computation Xin Wan Zhejiang University

Quantum Information

  EPR paradox (1935): Can quantum-mechanical description of physical reality be considered complete?

  Bell’s inequality (1964): No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.

  Quantum no-cloning theorem: It is impossible to create an identical copy of an arbitrary unknown quantum state [Wootters and Zurek (1982), Dieks (1982)].

  Quantum entanglement, quantum teleportation, quantum cryptography….

Quantum Computation

  [David DiVincenzo] It is the prospect of building a quantum computer, rather than the fascinating properties of quantum physics or of entanglement, that is responsible for much of today’s interest in quantum information.

Classical vs Quantum

  Information encoded in bits.

  Possible bit states: 0 or 1

  Information encoded in qubits.

  Possible qubit states: any superposition described by wave function

0 H

Ψ = a 0 + b 1

0

120 + 1( )

1200 + 11( )

entanglement X

Shor’s Algorithm (1994)

DiVincenzo’s Five Criteria

  Well defined extendible qubit array – stable memory

Preparable in the “000…” state

  Long decoherence time (>104 operation time)

  Universal set of gate operations

  Single-quantum measurements

D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” Fort. der Physik 48, 771 (2000), quant-ph/0002077.

Systems Considered for QC

  Liquid-state NMR

  NMR spin lattices

  Linear ion-trap spectroscopy

  Neutral-atom optical lattices

  Cavity QED + atoms

  Linear optics with single photons

  Nitrogen vacancies in diamond

  Electrons on liquid He

  Josephson junctions (charge, flux, phase, transmon)

  Spin spectroscopies, impurities in semiconductors & fullerines

  Coupled quantum dots

  Topological systems (FQHE, quantum wires, …)

Superconducting Qubits

H. Wang Group

No Evidence of Quantum Speedup

http://science.sciencemag.org/content/early/2014/06/18/science.1252319.abstract

Steane’s 7-qubut Code Error correction: Circuit does non-demolition measurement of operators

Disadvantages:

•  Lots of qubits •  Long-distance couplings

(regularity is not geometric)

Encoding qubits Ancilla qubits

Surface Codes

Topological 2D Surface Codes

A. Kitaev, in Quantum Communication, Computing, and Measurement , O. Hirota et al., Eds. (Plenum, New York, 1997); R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007).

Topological Quantum Computation

  Topological systems   Degenerate ground states protected by a spectral gap

  Braiding of anyonic excitations = unitary evolution

  Robust against noises (local perturbations)

  Perform error correction on the physical level

  Topological quantum computation

Matrices form a non-Abelian representation of the braid group.

Fractional Quantum Hall States

  Laughlin state (ν = 1/3)

  Moore-Read state (ν = 5/2)

Conceptual Design

Kitaev’s Toy Model

1D Quantum Wire

Experimental Progress

  See, most recently, S. M. Albrecht et al., Exponential protection of zero modes in Majorana islands, Nature 531, 206 (2016).

Following theoretical proposals, several experiments have identified signatures of Majorana modes in nanowires with proximity-induced superconductivity and atomic chains, with small amounts of mode splitting potentially explained by hybridization of Majorana modes.

Initialiazion, Fusion and Braiding

Summary

  The first quantum evolution occurred at the beginning of the 20th century, arising out theoretical attempts to explain experiments on blackbody radiation. The achievements led to the computer-chip industry and the information age.

  We are in the midst of the second quantum revolution: the engineering of quantum matter with arbitrary precision. To build a fault-tolerant quantum computer is at the research forefront. We can remain optimistic but must recognize the great challenges lying ahead.