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.