20,401 research outputs found
Entanglement of purification: from spin chains to holography
Purification is a powerful technique in quantum physics whereby a mixed
quantum state is extended to a pure state on a larger system. This process is
not unique, and in systems composed of many degrees of freedom, one natural
purification is the one with minimal entanglement. Here we study the entropy of
the minimally entangled purification, called the entanglement of purification,
in three model systems: an Ising spin chain, conformal field theories
holographically dual to Einstein gravity, and random stabilizer tensor
networks. We conjecture values for the entanglement of purification in all
these models, and we support our conjectures with a variety of numerical and
analytical results. We find that such minimally entangled purifications have a
number of applications, from enhancing entanglement-based tensor network
methods for describing mixed states to elucidating novel aspects of the
emergence of geometry from entanglement in the AdS/CFT correspondence.Comment: 40 pages, multiple figures. v2: references added, typos correcte
Cross-level Validation of Topological Quantum Circuits
Quantum computing promises a new approach to solving difficult computational
problems, and the quest of building a quantum computer has started. While the
first attempts on construction were succesful, scalability has never been
achieved, due to the inherent fragile nature of the quantum bits (qubits). From
the multitude of approaches to achieve scalability topological quantum
computing (TQC) is the most promising one, by being based on an flexible
approach to error-correction and making use of the straightforward
measurement-based computing technique. TQC circuits are defined within a large,
uniform, 3-dimensional lattice of physical qubits produced by the hardware and
the physical volume of this lattice directly relates to the resources required
for computation. Circuit optimization may result in non-intuitive mismatches
between circuit specification and implementation. In this paper we introduce
the first method for cross-level validation of TQC circuits. The specification
of the circuit is expressed based on the stabilizer formalism, and the
stabilizer table is checked by mapping the topology on the physical qubit
level, followed by quantum circuit simulation. Simulation results show that
cross-level validation of error-corrected circuits is feasible.Comment: 12 Pages, 5 Figures. Comments Welcome. RC2014, Springer Lecture Notes
on Computer Science (LNCS) 8507, pp. 189-200. Springer International
Publishing, Switzerland (2014), Y. Shigeru and M.Shin-ichi (Eds.
Classification of topologically protected gates for local stabilizer codes
Given a quantum error correcting code, an important task is to find encoded
operations that can be implemented efficiently and fault-tolerantly. In this
Letter we focus on topological stabilizer codes and encoded unitary gates that
can be implemented by a constant-depth quantum circuit. Such gates have a
certain degree of protection since propagation of errors in a constant-depth
circuit is limited by a constant size light cone. For the 2D geometry we show
that constant-depth circuits can only implement a finite group of encoded gates
known as the Clifford group. This implies that topological protection must be
"turned off" for at least some steps in the computation in order to achieve
universality. For the 3D geometry we show that an encoded gate U is
implementable by a constant-depth circuit only if the image of any Pauli
operator under conjugation by U belongs to the Clifford group. This class of
gates includes some non-Clifford gates such as the \pi/8 rotation. Our
classification applies to any stabilizer code with geometrically local
stabilizers and sufficiently large code distance.Comment: 6 pages, 2 figure
Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence
We propose a family of exactly solvable toy models for the AdS/CFT
correspondence based on a novel construction of quantum error-correcting codes
with a tensor network structure. Our building block is a special type of tensor
with maximal entanglement along any bipartition, which gives rise to an
isometry from the bulk Hilbert space to the boundary Hilbert space. The entire
tensor network is an encoder for a quantum error-correcting code, where the
bulk and boundary degrees of freedom may be identified as logical and physical
degrees of freedom respectively. These models capture key features of
entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi
formula and the negativity of tripartite information are obeyed exactly in many
cases. That bulk logical operators can be represented on multiple boundary
regions mimics the Rindler-wedge reconstruction of boundary operators from bulk
operators, realizing explicitly the quantum error-correcting features of
AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and
bibliographic amendments and minor correction
Algebraic geometric construction of a quantum stabilizer code
The stabilizer code is the most general algebraic construction of quantum
error-correcting codes proposed so far. A stabilizer code can be constructed
from a self-orthogonal subspace of a symplectic space over a finite field. We
propose a construction method of such a self-orthogonal space using an
algebraic curve. By using the proposed method we construct an asymptotically
good sequence of binary stabilizer codes. As a byproduct we improve the
Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this
paper can be understood without knowledge of quantum mechanics.Comment: LaTeX2e, 12 pages, 1 color figure. A decoding method was added and
several typographical errors were corrected in version 2. The description of
the decoding problem was completely wrong in version 1. In version 1 and 2,
there was a critical miscalculation in the estimation of parameters of codes,
and the constructed sequence of codes turned out to be worse than existing
ones. The asymptotically best sequence of quantum codes was added in version
3. Section 3.2 appeared in IEEE Transactions on Information Theory, vol. 48,
no. 7, pp. 2122-2124, July 200
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