Quantum lowdensity paritycheck (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform faulttolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain faulttolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D dimensional hyperbolic space.
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Locally testable codes via highdimensional expanders
Locally testable codes (LTC) are errorcorrecting codes that have a local tester which can distinguish valid codewords from words that are far from all codewords, by probing a given word only at a very small (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. A major open problem is whether there exist LTCs with positive rate, constant relative distance and testable with a constant number of queries.
In this paper, we present a new approach towards constructing such LTCs using the machinery of highdimensional expanders.
To this end, we consider the Tanner representation of a code, which is specified by a graph and a base code. Informally, our result states that if this graph is part of an {\em agreement expander} then the local testability of the code follows from the local testability of the base code. Agreement expanders allow one to stitch together many mostlyconsistent local functions into a single global function. Highdimensional expanders are known to yield agreement expanders with constant degree.
This work unifies and generalizes the known results on testability of the Hadamard, ReedMuller and lifted codes, all of which are proved via a single round of local selfcorrection: the corrected value at a vertex v depends on the values of all vertices that share a constraint with v. In the above codes this set includes all of the vertices. In contrast, in our setting the degree of a vertex might be a constant, so we cannot hope for oneround selfcorrection. We overcome this technical hurdle by performing iterative self correction with logarithmically many rounds and tightly controlling the error in each iteration using properties of the agreement expander.
Given this result, the missing ingredient towards constructing a constantquery LTC with positive rate and constant relative distance is an instantiation of a base code and a constantdegree agreement expander that interact well with each other.
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 Award ID(s):
 1900460
 NSFPAR ID:
 10169301
 Date Published:
 Journal Name:
 Electronic colloquium on computational complexity
 Volume:
 27
 Issue:
 72
 ISSN:
 14338092
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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