Communication Complexity of Statistical Distance

We prove nearly matching upper and lower bounds on the randomized communication complexity of the following problem: Alice and Bob are each given a probability distribution over $n$ elements, and they wish to estimate within +-epsilon the statistical (total variation) distance between their distributions. For some range of parameters, there is up to a log(n) factor gap between the upper and lower bounds, and we identify a barrier to using information complexity techniques to improve the lower bound in this case. We also prove a side result that we discovered along the way: the randomized communication complexity of n-bit Majority composed with n-bit Greater-Than is Theta(n log n)

Communication Complexity of Set-Disjointness for All Probabilities

We study set-disjointness in a generalized model of randomized two-party communication where the probability of acceptance must be at least alpha(n) on yes-inputs and at most beta(n) on no-inputs, for some functions alpha(n)>beta(n). Our main result is a complete characterization of the private-coin communication complexity of set-disjointness for all functions alpha and beta, and a near-complete characterization for public-coin protocols. In particular, we obtain a simple proof of a theorem of Braverman and Moitra (STOC 2013), who studied the case where alpha=1/2+epsilon(n) and beta=1/2-epsilon(n). The following contributions play a crucial role in our characterization and are interesting in their own right. (1) We introduce two communication analogues of the classical complexity class that captures small bounded-error computations: we define a "restricted" class SBP (which lies between MA and AM) and an "unrestricted" class USBP. The distinction between them is analogous to the distinction between the well-known communication classes PP and UPP. (2) We show that the SBP communication complexity is precisely captured by the classical corruption lower bound method. This sharpens a theorem of Klauck (CCC 2003). (3) We use information complexity arguments to prove a linear lower bound on the USBP complexity of set-disjointness

StoqMA Meets Distribution Testing

$\mathsf{StoqMA}$ captures the computational hardness of approximating the ground energy of local Hamiltonians that do not suffer the so-called sign problem. We provide a novel connection between $\mathsf{StoqMA}$ and distribution testing via reversible circuits. First, we prove that easy-witness $\mathsf{StoqMA}$ (viz. $\mathsf{eStoqMA}$, a sub-class of $\mathsf{StoqMA}$) is contained in $\mathsf{MA}$. Easy witness is a generalization of a subset state such that the associated set's membership can be efficiently verifiable, and all non-zero coordinates are not necessarily uniform. This sub-class $\mathsf{eStoqMA}$ contains $\mathsf{StoqMA}$ with perfect completeness ($\mathsf{StoqMA}_1$), which further signifies a simplified proof for $\mathsf{StoqMA}_1 \subseteq \mathsf{MA}$ [BBT06, BT10]. Second, by showing distinguishing reversible circuits with ancillary random bits is $\mathsf{StoqMA}$-complete (as a comparison, distinguishing quantum circuits is $\mathsf{QMA}$-complete [JWB05]), we construct soundness error reduction of $\mathsf{StoqMA}$. Additionally, we show that both variants of $\mathsf{StoqMA}$ that without any ancillary random bit and with perfect soundness are contained in $\mathsf{NP}$. Our results make a step towards collapsing the hierarchy $\mathsf{MA} \subseteq \mathsf{StoqMA} \subseteq \mathsf{SBP}$ [BBT06], in which all classes are contained in $\mathsf{AM}$ and collapse to $\mathsf{NP}$ under derandomization assumptions.Comment: 24 pages. v2: mostly adds corrections and clarifications. v3: add a connection between eStoqMA and Guided Stoquastic Hamiltonian Proble

Perfect Zero Knowledge: New Upperbounds and Relativized Separations

We investigate the complexity of problems that admit perfect zero-knowledge interactive protocols and establish new unconditional upper bounds and oracle separation results. We establish our results by investigating certain distribution testing problems: computational problems over high-dimensional distributions represented by succinct Boolean circuits. A relatively less-investigated complexity class SBP emerged as significant in this study. The main results we establish are: 1. A unconditional inclusion that NIPZK is in CoSBP. 2. Construction of a relativized world in which there is a distribution testing problem that lies in NIPZK but not in SBP, thus giving a relativized separation of NIPZK (and hence PZK) from SBP. 3. Construction of a relativized world in which there is a distribution testing problem that lies in PZK but not in CoSBP, thus giving a relativized separation of PZK from CoSBP. These results refine the landscape of perfect zero-knowledge classes in relation to traditional complexity classes

The complexity of estimating min-entropy

Goldreich et al. (CRYPTO 1999) proved that the promise problem for estimating the Shannon entropy of a distribution sampled by a given circuit is NISZK-complete. We consider the analogous problem for estimating the min-entropy and prove that it is SBP-complete, where SBP is the class of promise problems that correspond to approximate counting of NP witnesses. The result holds even when the sampling circuits are restricted to be 3-local. For logarithmic-space samplers, we observe that this problem is NP-complete by a result of Lyngsø and Pedersen on hidden Markov models (JCSS 65(3):545–569, 2002)