96,948 research outputs found
Deterministic Rateless Codes for BSC
A rateless code encodes a finite length information word into an infinitely
long codeword such that longer prefixes of the codeword can tolerate a larger
fraction of errors. A rateless code achieves capacity for a family of channels
if, for every channel in the family, reliable communication is obtained by a
prefix of the code whose rate is arbitrarily close to the channel's capacity.
As a result, a universal encoder can communicate over all channels in the
family while simultaneously achieving optimal communication overhead. In this
paper, we construct the first \emph{deterministic} rateless code for the binary
symmetric channel. Our code can be encoded and decoded in time per
bit and in almost logarithmic parallel time of , where
is any (arbitrarily slow) super-constant function. Furthermore, the error
probability of our code is almost exponentially small .
Previous rateless codes are probabilistic (i.e., based on code ensembles),
require polynomial time per bit for decoding, and have inferior asymptotic
error probabilities. Our main technical contribution is a constructive proof
for the existence of an infinite generating matrix that each of its prefixes
induce a weight distribution that approximates the expected weight distribution
of a random linear code
Perfect zero knowledge for quantum multiprover interactive proofs
In this work we consider the interplay between multiprover interactive
proofs, quantum entanglement, and zero knowledge proofs - notions that are
central pillars of complexity theory, quantum information and cryptography. In
particular, we study the relationship between the complexity class MIP, the
set of languages decidable by multiprover interactive proofs with quantumly
entangled provers, and the class PZKMIP, which is the set of languages
decidable by MIP protocols that furthermore possess the perfect zero
knowledge property.
Our main result is that the two classes are equal, i.e., MIP
PZKMIP. This result provides a quantum analogue of the celebrated result of
Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP
PZKMIP (in other words, all classical multiprover interactive protocols can be
made zero knowledge). We prove our result by showing that every MIP
protocol can be efficiently transformed into an equivalent zero knowledge
MIP protocol in a manner that preserves the completeness-soundness gap.
Combining our transformation with previous results by Slofstra (Forum of
Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we
obtain the corollary that all co-recursively enumerable languages (which
include undecidable problems as well as all decidable problems) have zero
knowledge MIP protocols with vanishing promise gap
Constructions and Noise Threshold of Hyperbolic Surface Codes
We show how to obtain concrete constructions of homological quantum codes
based on tilings of 2D surfaces with constant negative curvature (hyperbolic
surfaces). This construction results in two-dimensional quantum codes whose
tradeoff of encoding rate versus protection is more favorable than for the
surface code. These surface codes would require variable length connections
between qubits, as determined by the hyperbolic geometry. We provide numerical
estimates of the value of the noise threshold and logical error probability of
these codes against independent X or Z noise, assuming noise-free error
correction
The Stability of Quantum Concatenated Code Hamiltonians
Protecting quantum information from the detrimental effects of decoherence
and lack of precise quantum control is a central challenge that must be
overcome if a large robust quantum computer is to be constructed. The
traditional approach to achieving this is via active quantum error correction
using fault-tolerant techniques. An alternative to this approach is to engineer
strongly interacting many-body quantum systems that enact the quantum error
correction via the natural dynamics of these systems. Here we present a method
for achieving this based on the concept of concatenated quantum error
correcting codes. We define a class of Hamiltonians whose ground states are
concatenated quantum codes and whose energy landscape naturally causes quantum
error correction. We analyze these Hamiltonians for robustness and suggest
methods for implementing these highly unnatural Hamiltonians.Comment: 18 pages, small corrections and clarification
Lowering qubit requirements for quantum simulations of fermionic systems
The mapping of fermionic states onto qubit states, as well as the mapping of
fermionic Hamiltonian into quantum gates enables us to simulate electronic
systems with a quantum computer. Benefiting the understanding of many-body
systems in chemistry and physics, quantum simulation is one of the great
promises of the coming age of quantum computers. One challenge in realizing
simulations on near-term quantum devices is the large number of qubits required
by such mappings. In this work, we develop methods that allow us to trade-off
qubit requirements against the complexity of the resulting quantum circuit. We
first show that any classical code used to map the state of a fermionic Fock
space to qubits gives rise to a mapping of fermionic models to quantum gates.
As an illustrative example, we present a mapping based on a non-linear
classical error correcting code, which leads to significant qubit savings
albeit at the expense of additional quantum gates. We proceed to use this
framework to present a number of simpler mappings that lead to qubit savings
with only a very modest increase in gate difficulty. We discuss the role of
symmetries such as particle conservation, and savings that could be obtained if
an experimental platform could easily realize multi-controlled gates.Comment: 11+13 pages, 5 figures, 2 tables, see ArXiv files for Mathematica
code (text file) and documentation (pdf); fixed typos in this new versio
Self-adaptive node-based PCA encodings
In this paper we propose an algorithm, Simple Hebbian PCA, and prove that it
is able to calculate the principal component analysis (PCA) in a distributed
fashion across nodes. It simplifies existing network structures by removing
intralayer weights, essentially cutting the number of weights that need to be
trained in half
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