6 research outputs found
Systematic Design and Optimization of Quantum Circuits for Stabilizer Codes
Quantum computing is an emerging technology that has the potential to achieve
exponential speedups over their classical counterparts. To achieve quantum
advantage, quantum principles are being applied to fields such as
communications, information processing, and artificial intelligence. However,
quantum computers face a fundamental issue since quantum bits are extremely
noisy and prone to decoherence. Keeping qubits error free is one of the most
important steps towards reliable quantum computing. Different stabilizer codes
for quantum error correction have been proposed in past decades and several
methods have been proposed to import classical error correcting codes to the
quantum domain. However, formal approaches towards the design and optimization
of circuits for these quantum encoders and decoders have so far not been
proposed. In this paper, we propose a formal algorithm for systematic
construction of encoding circuits for general stabilizer codes. This algorithm
is used to design encoding and decoding circuits for an eight-qubit code. Next,
we propose a systematic method for the optimization of the encoder circuit thus
designed. Using the proposed method, we optimize the encoding circuit in terms
of the number of 2-qubit gates used. The proposed optimized eight-qubit encoder
uses 18 CNOT gates and 4 Hadamard gates, as compared to 14 single qubit gates,
33 2-qubit gates, and 6 CCNOT gates in a prior work. The encoder and decoder
circuits are verified using IBM Qiskit. We also present optimized encoder
circuits for Steane code and a 13-qubit code in terms of the number of gates
used.Comment: arXiv admin note: substantial text overlap with arXiv:2309.1179
Structure of CSS and CSS-T Quantum Codes
We investigate CSS and CSS-T quantum error-correcting codes from the point of
view of their existence, rarity, and performance. We give a lower bound on the
number of pairs of linear codes that give rise to a CSS code with good
correction capability, showing that such pairs are easy to produce with a
randomized construction. We then prove that CSS-T codes exhibit the opposite
behaviour, showing also that, under very natural assumptions, their rate and
relative distance cannot be simultaneously large. This partially answers an
open question on the feasible parameters of CSS-T codes. We conclude with a
simple construction of CSS-T codes from Hermitian curves. The paper also offers
a concise introduction to CSS and CSS-T codes from the point of view of
classical coding theory
Coding for Parallel Channels: Gallager Bounds for Binary Linear Codes with Applications to Repeat-Accumulate Codes and Variations
This paper is focused on the performance analysis of binary linear block
codes (or ensembles) whose transmission takes place over independent and
memoryless parallel channels. New upper bounds on the maximum-likelihood (ML)
decoding error probability are derived. These bounds are applied to various
ensembles of turbo-like codes, focusing especially on repeat-accumulate codes
and their recent variations which possess low encoding and decoding complexity
and exhibit remarkable performance under iterative decoding. The framework of
the second version of the Duman and Salehi (DS2) bounds is generalized to the
case of parallel channels, along with the derivation of their optimized tilting
measures. The connection between the generalized DS2 and the 1961 Gallager
bounds, addressed by Divsalar and by Sason and Shamai for a single channel, is
explored in the case of an arbitrary number of independent parallel channels.
The generalization of the DS2 bound for parallel channels enables to re-derive
specific bounds which were originally derived by Liu et al. as special cases of
the Gallager bound. In the asymptotic case where we let the block length tend
to infinity, the new bounds are used to obtain improved inner bounds on the
attainable channel regions under ML decoding. The tightness of the new bounds
for independent parallel channels is exemplified for structured ensembles of
turbo-like codes. The improved bounds with their optimized tilting measures
show, irrespectively of the block length of the codes, an improvement over the
union bound and other previously reported bounds for independent parallel
channels; this improvement is especially pronounced for moderate to large block
lengths.Comment: Submitted to IEEE Trans. on Information Theory, June 2006 (57 pages,
9 figures
Statistical cryptanalysis of block ciphers
Since the development of cryptology in the industrial and academic worlds in the seventies, public knowledge and expertise have grown in a tremendous way, notably because of the increasing, nowadays almost ubiquitous, presence of electronic communication means in our lives. Block ciphers are inevitable building blocks of the security of various electronic systems. Recently, many advances have been published in the field of public-key cryptography, being in the understanding of involved security models or in the mathematical security proofs applied to precise cryptosystems. Unfortunately, this is still not the case in the world of symmetric-key cryptography and the current state of knowledge is far from reaching such a goal. However, block and stream ciphers tend to counterbalance this lack of "provable security" by other advantages, like high data throughput and ease of implementation. In the first part of this thesis, we would like to add a (small) stone to the wall of provable security of block ciphers with the (theoretical and experimental) statistical analysis of the mechanisms behind Matsui's linear cryptanalysis as well as more abstract models of attacks. For this purpose, we consider the underlying problem as a statistical hypothesis testing problem and we make a heavy use of the Neyman-Pearson paradigm. Then, we generalize the concept of linear distinguisher and we discuss the power of such a generalization. Furthermore, we introduce the concept of sequential distinguisher, based on sequential sampling, and of aggregate distinguishers, which allows to build sub-optimal but efficient distinguishers. Finally, we propose new attacks against reduced-round version of the block cipher IDEA. In the second part, we propose the design of a new family of block ciphers named FOX. First, we study the efficiency of optimal diffusive components when implemented on low-cost architectures, and we present several new constructions of MDS matrices; then, we precisely describe FOX and we discuss its security regarding linear and differential cryptanalysis, integral attacks, and algebraic attacks. Finally, various implementation issues are considered