1,045 research outputs found
Unfolding the color code
The topological color code and the toric code are two leading candidates for
realizing fault-tolerant quantum computation. Here we show that the color code
on a -dimensional closed manifold is equivalent to multiple decoupled copies
of the -dimensional toric code up to local unitary transformations and
adding or removing ancilla qubits. Our result not only generalizes the proven
equivalence for , but also provides an explicit recipe of how to decouple
independent components of the color code, highlighting the importance of
colorability in the construction of the code. Moreover, for the -dimensional
color code with boundaries of distinct colors, we find that the
code is equivalent to multiple copies of the -dimensional toric code which
are attached along a -dimensional boundary. In particular, for , we
show that the (triangular) color code with boundaries is equivalent to the
(folded) toric code with boundaries. We also find that the -dimensional
toric code admits logical non-Pauli gates from the -th level of the Clifford
hierarchy, and thus saturates the bound by Bravyi and K\"{o}nig. In particular,
we show that the -qubit control- logical gate can be fault-tolerantly
implemented on the stack of copies of the toric code by a local unitary
transformation.Comment: 46 pages, 15 figure
Clifford algebras, Spin groups and qubit trees
Representations of Spin groups and Clifford algebras derived from structure
of qubit trees are introduced in this work. For ternary trees the construction
is more general and reduction to binary trees is formally defined by deleting
of superfluous branches. Usual Jordan-Wigner construction also may be formally
obtained in such approach by bringing the process up to trivial qubit chain
("trunk"). The methods can be also used for effective simulations of some
quantum circuits corresponding to the binary tree structure. Modeling of more
general qubit trees and relation with mapping used in Bravyi-Kitaev
transformation are also briefly outlined.Comment: LaTeX 12pt, 36 pages, 9 figures; v5: updated, with two new
appendices. Comments are welcom
Examples of minimal-memory, non-catastrophic quantum convolutional encoders
One of the most important open questions in the theory of quantum
convolutional coding is to determine a minimal-memory, non-catastrophic,
polynomial-depth convolutional encoder for an arbitrary quantum convolutional
code. Here, we present a technique that finds quantum convolutional encoders
with such desirable properties for several example quantum convolutional codes
(an exposition of our technique in full generality will appear elsewhere). We
first show how to encode the well-studied Forney-Grassl-Guha (FGG) code with an
encoder that exploits just one memory qubit (the former Grassl-Roetteler
encoder requires 15 memory qubits). We then show how our technique can find an
online decoder corresponding to this encoder, and we also detail the operation
of our technique on a different example of a quantum convolutional code.
Finally, the reduction in memory for the FGG encoder makes it feasible to
simulate the performance of a quantum turbo code employing it, and we present
the results of such simulations.Comment: 5 pages, 2 figures, Accepted for the International Symposium on
Information Theory 2011 (ISIT 2011), St. Petersburg, Russia; v2 has minor
change
Matchgates and classical simulation of quantum circuits
Let G(A,B) denote the 2-qubit gate which acts as the 1-qubit SU(2) gates A
and B in the even and odd parity subspaces respectively, of two qubits. Using a
Clifford algebra formalism we show that arbitrary uniform families of circuits
of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines,
can be classically efficiently simulated. This reproduces a result originally
proved by Valiant using his matchgate formalism, and subsequently related by
others to free fermionic physics. We further show that if the n.n. condition is
slightly relaxed, to allowing the same gates to act only on n.n. and next-n.n.
qubit lines, then the resulting circuits can efficiently perform universal
quantum computation. From this point of view, the gap between efficient
classical and quantum computational power is bridged by a very modest use of a
seemingly innocuous resource (qubit swapping). We also extend the simulation
result above in various ways. In particular, by exploiting properties of
Clifford operations in conjunction with the Jordan-Wigner representation of a
Clifford algebra, we show how one may generalise the simulation result above to
provide further classes of classically efficiently simulatable quantum
circuits, which we call Gaussian quantum circuits.Comment: 18 pages, 2 figure
Entanglement Generation of Clifford Quantum Cellular Automata
Clifford quantum cellular automata (CQCAs) are a special kind of quantum
cellular automata (QCAs) that incorporate Clifford group operations for the
time evolution. Despite being classically simulable, they can be used as basic
building blocks for universal quantum computation. This is due to the
connection to translation-invariant stabilizer states and their entanglement
properties. We will give a self-contained introduction to CQCAs and investigate
the generation of entanglement under CQCA action. Furthermore, we will discuss
finite configurations and applications of CQCAs.Comment: to appear in the "DPG spring meeting 2009" special issue of Applied
Physics
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