A reminder of Dan Browne's talk at 4pm in the Time room at PI.
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