Lattice quantum codes and exotic topological phases of matter

This thesis addresses whether it is possible to build a robust memory device for quantum information. Many schemes for fault-tolerant quantum information processing have been developed so far, one of which, called topological quantum computation, makes use of degrees of freedom that are inherently insensitive to local errors. However, this scheme is not so reliable against thermal errors. Other fault-tolerant schemes achieve better reliability through active error correction, but incur a substantial overhead cost. Thus, it is of practical importance and theoretical interest to design and assess fault-tolerant schemes that work well at finite temperature without active error correction.

In this thesis, a three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square.

This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.

Information-theoretic Studies and Capacity Bounds: Group Network Codes and Energy Harvesting Communication Systems

Network information theory and channels with memory are two important but difficult frontiers of information theory. In this two-parted dissertation, we study these two areas, each comprising one part. For the first area we study the so-called entropy vectors via finite group theory, and the network codes constructed from finite groups. In particular, we identify the smallest finite group that violates the Ingleton inequality, an inequality respected by all linear network codes, but not satisfied by all entropy vectors. Based on the analysis of this group we generalize it to several families of Ingleton-violating groups, which may be used to design good network codes. Regarding that aspect, we study the network codes constructed with finite groups, and especially show that linear network codes are embedded in the group network codes constructed with these Ingleton-violating families. Furthermore, such codes are strictly more powerful than linear network codes, as they are able to violate the Ingleton inequality while linear network codes cannot. For the second area, we study the impact of memory to the channel capacity through a novel communication system: the energy harvesting channel. Different from traditional communication systems, the transmitter of an energy harvesting channel is powered by an exogenous energy harvesting device and a finite-sized battery. As a consequence, each time the system can only transmit a symbol whose energy consumption is no more than the energy currently available. This new type of power supply introduces an unprecedented input constraint for the channel, which is random, instantaneous, and has memory. Furthermore, naturally, the energy harvesting process is observed causally at the transmitter, but no such information is provided to the receiver. Both of these features pose great challenges for the analysis of the channel capacity. In this work we use techniques from channels with side information, and finite state channels, to obtain lower and upper bounds of the energy harvesting channel. In particular, we study the stationarity and ergodicity conditions of a surrogate channel to compute and optimize the achievable rates for the original channel. In addition, for practical code design of the system we study the pairwise error probabilities of the input sequences.

Combinatorial properties of finite geometric lattices

Let L be a finite geometric lattice of dimension n, and let w(k) denote the number of elements in L of rank k. Two theorems about the numbers w(k) are proved: first, w(k) ≥ w(1) for k = 2, 3, ..., n-1. Second, w(k) = w(1) if and only if k = n-1 and L is modular. Several corollaries concerning the "matching" of points and dual points are derived from these theorems.

Both theorems can be regarded as a generalization of a theorem of de Bruijn and Erdös concerning ʎ= 1 designs. The second can also be considered as the converse to a special case of Dilworth's theorem on finite modular lattices.

These results are related to two conjectures due to G. -C. Rota. The "unimodality" conjecture states that the w(k)'s form a unimodal sequence. The "Sperner" conjecture states that a set of non-comparable elements in L has cardinality at most max/k {w(k)}. In this thesis, a counterexample to the Sperner conjecture is exhibited.

Varieties generated by modular lattices of width four

A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let M^∞_n denote the lattice variety generated by all modular lattices of width not exceeding n. M^∞₁ and M^∞₂ are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that M^∞₃ is also finitely based. On the other hand, K. Baker has shown that M^∞_n is not finitely based for 5 ≤ n ˂ ω. This thesis settles the finite basis problem for M^∞₄. M^∞₄ is shown to be finitely based by proving the stronger result that there exist ten varieties which properly contain M^∞₄ and such that any variety which properly contains M^∞₄ contains one of these ten varieties.

The methods developed also yield a characterization of sub-directly irreducible width four modular lattices. From this characterization further results are derived. It is shown that the free M^∞₄ lattice with n generators is finite. A variety with exactly k covers is exhibited for all k ≥ 15. It is further shown that there are 2^Ӄo sub- varieties of M^∞₄.