Hi all,
next week, we have a talk by Dan Browne on Wednesday at 4pm in the
Time Room.
Best wishes,
Markus
Title: Quantum Reed-Muller codes and Magic State distillation in
all prime dimensions
Speaker: Dan Browne (University College London)
Abstract:
Joint work with Earl Campbell (FU-Berlin) and Hussain Anwar
(UCL)
Magic state distillation is a key component of some
high-threshold schemes for fault-tolerant quantum computation [1],
[2]. Proposed by Bravyi and Kitaev [3] (and implicitly by Knill
[4]), and improved by Reichardt [4], Magic State Distillation is a
method to broaden the vocabulary of a fault-tolerant computational
model, from a limited set of gates (e.g. the Clifford group or a
sub-group[2]) to full universality, via the preparation of mixed
ancilla qubits which may be prepared without fault tolerant
protection.
Magic state distillation schemes have a close relation with
quantum error correcting codes, since a key step in such protocols
[5] is the projection onto a code subspace. Bravyi and Kitaev
proposed two protocols; one based upon the 5-qubit code, the
second derived from a punctured Reed-Muller code. Reed Muller
codes are a very important family of classical linear code. They
gained much interest [6] in the early years of quantum error
correction theory, since their properties make them well-suited to
the formation of quantum codes via the CSS-construction [7].
Punctured Reed-Muller codes (loosely speaking, Reed-Muller
codewords with a bit removed) in particular lead to quantum codes
with an unusual property, the ability to implement non-Clifford
gates transversally [8].
Most work in fault-tolerant quantum computation focuses on
qubits, but fault tolerant constructions can be generalised to
higher dimensions [9] - particularly readily for prime
dimensions. Recently, we presented the first magic state
distillation protocols [10] for non-binary systems, providing
explicit protocols for the qutrit case (complementing a recent
no-go theorem demonstrating bound states for magic state
distillation in higher dimensions [11]). In this talk, I will
report on more recent work [12], where the properties of punctured
Reed-Muller codes are employed to demonstrate Magic State
distillation protocols for all prime dimensions. In my talk, I
will give a technical account of this result and present numerical
investigations of the performance of such a protocol in the qutrit
case. Finally, I will discuss the potential for application of
these results to fault-tolerant quantum computation.
This will be a technical talk, and though some concepts of
linear codes and quantum codes will be briefly revised, I will
assume that listeners are familiar with quantum error correction
theory (e.g. the stabilizer formalism and the CSS construction)
for qubits.
[1] E. Knill. Fault-tolerant postselected quantum computation:
schemes, quant-ph/0402171
[2] R. Raussendorf, J. Harrington and K. Goyal, Topological
fault-tolerance in cluster state quantum computation,
arXiv:quant-ph/0703143v1
[3] S. Bravyi and A. Kitaev. Universal quantum computation
based on a magic states distillation, quant- ph/0403025
[4] B. W. Reichardt, Improved magic states distillation for
quantum universality, arXiv:quant-ph/0411036v1
[5] E.T. Campbell and D.E. Browne, On the Structure of Protocols for
Magic State Distillation, arXiv:0908.0838
[6] A. Steane, Quantum Reed Muller Codes, arXiv:quant-ph/9608026[7]
Nielsen and Chuang, Quantum Information and Computation, chapter 10
[8] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for
quantum computation, quant-ph/9610011
[9] D. Gottesman, Fault-Tolerant Quantum Computation with
Higher-Dimensional Systems, quant-ph/9802007
[10] H. Anwar, E.T Campbell and D.E. Browne, Qutrit Magic State
Distillation, arXiv:1202.2326
[11] V. Veitch, C. Ferrie, J. Emerson, Negative
Quasi-Probability Representation is a Necessary Resource for
Magic State Distillation, arXiv:1201.1256v3
[12] H. Anwar, E.T Campbell and D.E. Browne, in preparation
Date: May 02, 2012 - 4:00 pm
Series: Perimeter Institute Quantum Discussions
Location: Time Rm