Quantum Information Processing, Vol. 4, No. 3, August 2005 (© 2005)
Quantum Universality from Magic States Distillation
Applied to CSS Codes
Ben W. Reichardt
Received February 15, 2005; accepted May 23, 2005
Given Clifford group operations and the ability to repeatedly prepare a single-
qubit mixed state ρ, can one do universal quantum computation? We show a
sharp threshold in the Hadamard “magic” direction of the Bloch sphere between
those ρ allowing universal quantum computation, and those for which any cal-
culation can be efﬁciently classically simulated. As a corollary, the ability to
repeatedly prepare any pure state which is not a stabilizer state (e.g., any single-
qubit pure state which is not a Pauli eigenstate), together with Clifford group
operations, gives quantum universality. As motivation for this question, “magic
state” distillation procedures can reduce the general fault-tolerance problem to
that of performing fault-tolerant Clifford group circuits.
KEY WORDS: Universal quantam computing; protocols; stabilizer states.
PACS: 03.67.Lx, 03.67.Pp.
In “magic states distillation,” introduced by Bravyi and Kitaev in Ref.
1, we try to achieve universal quantum computation using only Clifford
group unitaries, preparation and measurement in the computational basis
|0, |1, and the ability to prepare a given single-qubit mixed state ρ.
If ρ, considered as a point in the Bloch sphere of Fig. 1, lies within
O the octahedral closed convex hull of the six eigenvectors of the Pauli
operators X, Y and Z, then the calculation is classically simulable by the
Gottesman–Knill theorem. Bravyi and Kitaev show universality if ρ is one
of certain pure states: either states symmetrical, under the symmetries of
Research Supported in Part by NSF ITR Grant CCR-0121555, and ARO Grant DAAD
EECS Department, Computer Science Division, University of California, Berkeley, CA
94720, USA. E-mail: email@example.com
1570-0755/05/0800-0251/0 © 2005 Springer Science+Business Media, Inc.