In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave a strongly-explicit construction of infinite families of ($p+1$)regular Ramanujan Cayley graphs, for infinitely many primes p. In this paper we extend the work of LPS and its successors to bigraphs (biregular bipartite graphs): we investigate the combinatorial properties of various generalizations of the notion of Ramanujan graphs, define a notion of Cayley bigraphs, and give strongly-explicit constructions of infinite families of ($p^{3}+1, p+1$)-regular Ramanujan Cayley bigraphs, for infinitely many primes p. In addition, we present a pseudorandomness characterization of Ramanujan bigraphs, and a more general notion of biexpanders. We also show that the graphs we construct exhibit the cutoff phenomenon with bounded window size for the mixing time of non-backtracking random walks, and present some other applications, such as optimal unitary gates in quantum computation.

Consider the following problem: Given an$n$×$n$multiplication table, decide whether it is a Cayley multiplication table of a group. Among deterministic algorithms for this problem, the best known algorithm is implied by F. W. Light's associativity test (1949) and has running time of${O}(n^{2}\log n)$. Allowing randomization. the best known algorithm has running time of$O(n^{2}\log(1/\delta))$, where$\delta > 0$is the error probability of the algorithm (Rajagopalan and Schulman, FOCS 1996, SICOMP 2000). In this work, we improve upon both of the above known algorithms. Specifically, we present a deterministic algorithm for the above problem whose running time is$O(n^{2})$. This performance is optimal up to constants. A central tool we develop is an efficient algorithm for finding a subset$A$of a group$G$satisfying$A^{2}=G$while$\vert A\vert=O(\sqrt{\vert G\vert })$.

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich–Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

A locally testable code (LTC) is an error correcting code that has a property-tester. The tester reads q bits that are randomly chosen, and rejects words with probability proportional to their distance from the code. The parameter q is called the locality of the tester.

Constructing quantum LDPC codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large X-distance but a much smaller Z-distance. However, together with a classical expander LDPC code and a tensoring method that generalises a construction of Hastings and also the Tillich-Zemor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 3-dimensional Ramanujan complex, we show that its 2-systole behaves like a square of the log of the complex size, which results in an overall quantum code of minimum distance n1/2logn, and sets a new record for quantum LDPC codes. When we use a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance n1/2log1/2n. We then exploit the expansion properties of the complex to devise the first polynomial time algorithm that decodes above the square root barrier for quantum LDPC codes.

In recent years a high dimensional theory of expanders has emerged. The notion of combinatorial expansion of graphs (i.e. the Cheeger constant of a graph) has seen two generalizations to high dimensional simplicial complexes. One generalization, known as coboundary expansion, is due to Linial and Meshulem; the other, which we term here cosystolic expansion, is due to Gromov, who showed that cosystolic expanders have the topological overlapping property. No construction (either random or explicit) of bounded degree combinational expanders (according to either definition) were known until a recent work of Kaufman, Kazhdan and Lubotzky, which provided the first bounded degree cosystolic expanders of dimension two. No bounded degree combinatorial expanders are known in higher dimensions. In this work we present explicit bounded degree cosystolic expanders of every dimension. This solves affirmatively an open question raised by Gromov, who asked whether there exist bounded degree complexes with the topological overlapping property in every dimension. Moreover, we provide a local to global criterion on a complex that implies cosystolic expansion: Namely, for a d-dimensional complex, X, if its underlying graph is a good expander, and all its links are both coboundary expanders and good expander graphs, then the (d-1)-dimensional skeleton of the complex is a cosystolic expander.