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The problem of computing distances of error-correcting codes is fundamental in both the classical and quantum settings. While hardness for the classical version of these problems has been known for some time (in both the exact and approximate settings), it was only recently that Kapshikar and Kundu showed these problems are also hard in the quantum setting. As our first main result, we reprove this using arguably simpler arguments based on hypergraph product codes. In particular, we get a direct reduction to CSS codes, the most commonly used type of quantum code, from the minimum distance problem for classical linear codes. Our second set of results considers the distance of a graph state, which is a key parameter for quantum codes obtained via the codeword stabilized formalism. We show that it is NP-hard to compute/approximate the distance of a graph state when the adjacency matrix of the graph is the input. In fact, we show this is true even if we only consider X-type errors of a graph state. Our techniques moreover imply an interesting classical consequence: the hardness of computing or approximating the distance of classical codes with rate equal to 1/2. One of the main motivations of the present work is a question raised by Kapshikar and Kundu concerning the NP-hardness of approximation when there is an additive error proportional to a quantum code’s length. We show that no such hardness can hold for hypergraph product codes. These observations suggest the possibility of a new kind of square root barrier. 

Differential privacy and sublinear algorithms are both rapidly emerging algorithmic themes in times of big data analysis. Although recent works have shown the existence of differentially private sublinear algorithms for many problems including graph parameter estimation and clustering, little is known regarding hardness results on these algorithms. In this paper, we initiate the study of lower bounds for problems that aim for both differentially-private and sublinear-time algorithms. Our main result is the incompatibility of both the desiderata in the general case. In particular, we prove that a simple problem based on one-way marginals yields both a differentially-private algorithm, as well as a sublinear-time algorithm, but does not admit a "strictly" sublinear-time algorithm that is also differentially private. 

The goal of the trace reconstruction problem is to recover a string x E {0, 1} given many independent traces of x, where a trace is a subsequence obtained from deleting bits of x independently with some given probability. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the state-of-the-art result of Chase (STOC 2021), which is based on statistics of arbitrary length-k subsequences, can also be obtained by considering the “k-mer statistics”, i.e., statistics regarding occurrences of contiguous k-bit strings (a.k.a, k-mers) in the initial string x, for k = Mazooji and Shomorony (ISIT 2023) show that such statistics (called k-mer density map) can be estimated within accuracy from poly(n, 2k, l/e) traces. We call an algorithm to be k-mer-based if it reconstructs x given estimates of the k-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that any k-mer-based algorithm for trace reconstruction must use exp n)) traces, under the assumption that the estimator requires poly(2k, 1 e) traces, thus establishing the optimality of this number of traces. Our analysis also shows that the analysis technique used by Chase is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, i.e., up to a factor of n in the number of samples needed for an optimal algorithm, and show that this factor of n loss may be necessary under general “model estimation” settings. 

The goal of the trace reconstruction problem is to recover a string x E {0, 1} given many independent traces of x, where a trace is a subsequence obtained from deleting bits of x independently with some given probability. In this paper we consider two kinds of algorithms for the trace reconstruction problem. We first observe that the state-of-the-art result of Chase (STOC 2021), which is based on statistics of arbitrary length-k subsequences, can also be obtained by considering the “k-mer statistics”, i.e., statistics regarding occurrences of contiguous k-bit strings (a.k.a, k-mers) in the initial string x, for k = Mazooji and Shomorony (ISIT 2023) show that such statistics (called k-mer density map) can be estimated within accuracy from poly(n, 2k, l/e) traces. We call an algorithm to be k-mer-based if it reconstructs x given estimates of the k-mer density map. Such algorithms essentially capture all the analyses in the worst-case and smoothed-complexity models of the trace reconstruction problem we know of so far. Our first, and technically more involved, result shows that any k-mer-based algorithm for trace reconstruction must use exp n)) traces, under the assumption that the estimator requires poly(2k, 1 e) traces, thus establishing the optimality of this number of traces. Our analysis also shows that the analysis technique used by Chase is essentially tight, and hence new techniques are needed in order to improve the worst-case upper bound. Our second, simple, result considers the performance of the Maximum Likelihood Estimator (MLE), which specifically picks the source string that has the maximum likelihood to generate the samples (traces). We show that the MLE algorithm uses a nearly optimal number of traces, i.e., up to a factor of n in the number of samples needed for an optimal algorithm, and show that this factor of n loss may be necessary under general “model estimation” settings.