3,146 research outputs found
Fault-tolerant logical gates in quantum error-correcting codes
Recently, Bravyi and K\"onig have shown that there is a tradeoff between
fault-tolerantly implementable logical gates and geometric locality of
stabilizer codes. They consider locality-preserving operations which are
implemented by a constant depth geometrically local circuit and are thus
fault-tolerant by construction. In particular, they shown that, for local
stabilizer codes in D spatial dimensions, locality preserving gates are
restricted to a set of unitary gates known as the D-th level of the Clifford
hierarchy. In this paper, we elaborate this idea and provide several extensions
and applications of their characterization in various directions. First, we
present a new no-go theorem for self-correcting quantum memory. Namely, we
prove that a three-dimensional stabilizer Hamiltonian with a
locality-preserving implementation of a non-Clifford gate cannot have a
macroscopic energy barrier. Second, we prove that the code distance of a
D-dimensional local stabilizer code with non-trivial locality-preserving m-th
level Clifford logical gate is upper bounded by . For codes with
non-Clifford gates (m>2), this improves the previous best bound by Bravyi and
Terhal. Third we prove that a qubit loss threshold of codes with non-trivial
transversal m-th level Clifford logical gate is upper bounded by 1/m. As such,
no family of fault-tolerant codes with transversal gates in increasing level of
the Clifford hierarchy may exist. This result applies to arbitrary stabilizer
and subsystem codes, and is not restricted to geometrically-local codes. Fourth
we extend the result of Bravyi and K\"onig to subsystem codes. A technical
difficulty is that, unlike stabilizer codes, the so-called union lemma does not
apply to subsystem codes. This problem is avoided by assuming the presence of
error threshold in a subsystem code, and the same conclusion as Bravyi-K\"onig
is recovered.Comment: 13 pages, 4 figure
Clifford Gates by Code Deformation
Topological subsystem color codes add to the advantages of topological codes
an important feature: error tracking only involves measuring 2-local operators
in a two dimensional setting. Unfortunately, known methods to compute with them
were highly unpractical. We give a mechanism to implement all Clifford gates by
code deformation in a planar setting. In particular, we use twist braiding and
express its effects in terms of certain colored Majorana operators.Comment: Extended version with more detail
On Subsystem Codes Beating the Hamming or Singleton Bound
Subsystem codes are a generalization of noiseless subsystems, decoherence
free subspaces, and quantum error-correcting codes. We prove a Singleton bound
for GF(q)-linear subsystem codes. It follows that no subsystem code over a
prime field can beat the Singleton bound. On the other hand, we show the
remarkable fact that there exist impure subsystem codes beating the Hamming
bound. A number of open problems concern the comparison in performance of
stabilizer and subsystem codes. One of the open problems suggested by Poulin's
work asks whether a subsystem code can use fewer syndrome measurements than an
optimal MDS stabilizer code while encoding the same number of qudits and having
the same distance. We prove that linear subsystem codes cannot offer such an
improvement under complete decoding.Comment: 18 pages more densely packed than classically possibl
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