Spin glass reflection of the decoding transition for quantum error correcting codes

We study the decoding transition for quantum error correcting codes with the help of a mapping to random-bond Wegner spin models. Families of quantum low density parity-check (LDPC) codes with a finite decoding threshold lead to both known models (e.g., random bond Ising and random plaquette $\Z2$ gauge models) as well as unexplored earlier generally non-local disordered spin models with non-trivial phase diagrams. The decoding transition corresponds to a transition from the ordered phase by proliferation of extended defects which generalize the notion of domain walls to non-local spin models. In recently discovered quantum LDPC code families with finite rates the number of distinct classes of such extended defects is exponentially large, corresponding to extensive ground state entropy of these codes. Here, the transition can be driven by the entropy of the extended defects, a mechanism distinct from that in the local spin models where the number of defect types (domain walls) is always finite.Comment: 15 pages, 2 figure

Density Evolution for Asymmetric Memoryless Channels

Density evolution is one of the most powerful analytical tools for low-density parity-check (LDPC) codes and graph codes with message passing decoding algorithms. With channel symmetry as one of its fundamental assumptions, density evolution (DE) has been widely and successfully applied to different channels, including binary erasure channels, binary symmetric channels, binary additive white Gaussian noise channels, etc. This paper generalizes density evolution for non-symmetric memoryless channels, which in turn broadens the applications to general memoryless channels, e.g. z-channels, composite white Gaussian noise channels, etc. The central theorem underpinning this generalization is the convergence to perfect projection for any fixed size supporting tree. A new iterative formula of the same complexity is then presented and the necessary theorems for the performance concentration theorems are developed. Several properties of the new density evolution method are explored, including stability results for general asymmetric memoryless channels. Simulations, code optimizations, and possible new applications suggested by this new density evolution method are also provided. This result is also used to prove the typicality of linear LDPC codes among the coset code ensemble when the minimum check node degree is sufficiently large. It is shown that the convergence to perfect projection is essential to the belief propagation algorithm even when only symmetric channels are considered. Hence the proof of the convergence to perfect projection serves also as a completion of the theory of classical density evolution for symmetric memoryless channels.Comment: To appear in the IEEE Transactions on Information Theor

Homological Product Codes

Quantum codes with low-weight stabilizers known as LDPC codes have been actively studied recently due to their simple syndrome readout circuits and potential applications in fault-tolerant quantum computing. However, all families of quantum LDPC codes known to this date suffer from a poor distance scaling limited by the square-root of the code length. This is in a sharp contrast with the classical case where good families of LDPC codes are known that combine constant encoding rate and linear distance. Here we propose the first family of good quantum codes with low-weight stabilizers. The new codes have a constant encoding rate, linear distance, and stabilizers acting on at most $\sqrt{n}$ qubits, where $n$ is the code length. For comparison, all previously known families of good quantum codes have stabilizers of linear weight. Our proof combines two techniques: randomized constructions of good quantum codes and the homological product operation from algebraic topology. We conjecture that similar methods can produce good stabilizer codes with stabilizer weight $n^a$ for any $a>0$. Finally, we apply the homological product to construct new small codes with low-weight stabilizers.Comment: 49 page

Complexity and second moment of the mathematical theory of communication

The performance of an error correcting code is evaluated by its block error probability, code rate, and encoding and decoding complexity. The performance of a series of codes is evaluated by, as the block lengths approach infinity, whether their block error probabilities decay to zero, whether their code rates converge to channel capacity, and whether their growth in complexities stays under control. Over any discrete memoryless channel, I build codes such that: for one, their block error probabilities and code rates scale like random codes’; and for two, their encoding and decoding complexities scale like polar codes’. Quantitatively, for any constants π, ρ > 0 such that π+2ρ < 1, I construct a series of error correcting codes with block length N approaching infinity, block error probability exp(−Nπ), code rate N−ρ less than the channel capacity, and encoding and decoding complexity O(N logN) per code block. Over any discrete memoryless channel, I also build codes such that: for one, they achieve channel capacity rapidly; and for two, their encoding and decoding complexities outperform all known codes over non-BEC channels. Quantitatively, for any constants τ, ρ > 0 such that 2ρ < 1, I construct a series of error correcting codes with block length N approaching infinity, block error probability exp(−(logN)τ ), code rate N−ρ less than the channel capacity, and encoding and decoding complexity O(N log(logN)) per code block. The two aforementioned results are built upon two pillars—a versatile framework that generates codes on the basis of channel polarization, and a calculus–probability machinery that evaluates the performances of codes. The framework that generates codes and the machinery that evaluates codes can be extended to many other scenarios in network information theory. To name a few: lossless compression with side information, lossy compression, Slepian–Wolf problem, Wyner–Ziv Problem, multiple access channel, wiretap channel of type I, and broadcast channel. In each scenario, the adapted notions of block error probability and code rate approach their limits at the same paces as specified above

Group, Lattice and Polar Codes for Multi-terminal Communications.

We study the performance of algebraic codes for multi-terminal communications. This thesis consists of three parts: In the rst part, we analyze the performance of group codes for communications systems. We observe that although group codes are not optimal for point-to-point scenarios, they can improve the achievable rate region for several multi-terminal communications settings such as the Distributed Source Coding and Interference Channels. The gains in the rates are particularly signicant when the structure of the source/channel is matched to the structure of the underlying group. In the second part, we study the continuous alphabet version of group/linear codes, namely lattice codes. We show that similarly to group codes, lattice codes can improve the achievable rate region for multi-terminal problems. In the third part of the thesis, we present coding schemes based on polar codes to practically achieve the performance limits derived in the two earlier parts. We also present polar coding schemes to achieve the known achievable rate regions for multi-terminal communications problems such as the Distributed Source Coding, the Multiple Description Coding, Broadcast Channels, Interference Channels and Multiple Access Channels.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108876/1/ariaghs_1.pd