We establish a lower bound of $\Omega{(\sqrt{n})}$ on the bounded-error quantum query complexity of read-once Boolean functions, providing evidence for the conjecture that $\Omega(\sqrt{D(f)})$ is a lower bound for all Boolean functions.Our technique extends a result of Ambainis, based on the idea that successful computation of a function requires ``decoherence'' ... more >>>

In Direct Sum problems [KRW], one tries to show that for a given computational model, the complexity of computing a collection $F = \{f_1(x_1), \ldots f_l(x_l)\}$ of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ... more >>>

Let $\mathcal{M}$ be a bridgeless matroid on ground set $\{1,\ldots, n\}$ and $f_{\mathcal{M}}: \{0,1\}^n \to \{0, 1\}$ be the indicator function of its independent sets. A folklore fact is that $f_\mathcal{M}$ is ``evasive," i.e., $D(f_\mathcal{M}) = n$ where $D(f)$ denotes the deterministic decision tree complexity of $f.$ Here we prove ... more >>>

We prove that the randomized decision tree complexity of the recursive majority-of-three is $\Omega(2.6^d)$, where $d$ is the depth of the recursion. The proof is by a bottom up induction, which is same in spirit as the one in the proof of Saks and Wigderson in their FOCS 1986 paper ... more >>>

For a Boolean function $f,$ let $D(f)$ denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine $f.$ In a classic paper,
Rivest and Vuillemin \cite{rv} show that any non-constant monotone property $\mathcal{P} : \{0, 1\}^{n \choose 2} \to ... more >>>

We construct a total Boolean function $f$ satisfying
$R(f)=\tilde{\Omega}(Q(f)^{5/2})$, refuting the long-standing
conjecture that $R(f)=O(Q(f)^2)$ for all total Boolean functions.
Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions,
we improve this to $R(f)=\tilde{\Omega}(Q(f)^3)$.
Our construction is motivated by the Göös-Pitassi-Watson function
but does not ... more >>>

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity ... more >>>

Given a problem which is intractable for both quantum and classical algorithms, can we find a sub-problem for which quantum algorithms provide an exponential advantage? We refer to this problem as the "sculpting problem." In this work, we give a full characterization of sculptable functions in the query complexity setting. ... more >>>

We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.1 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a ... more >>>

We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f ... more >>>

Adaptivity is known to play a crucial role in property testing. In particular, there exist properties for which there is an exponential gap between the power of \emph{adaptive} testing algorithms, wherein each query may be determined by the answers received to prior queries, and their \emph{non-adaptive} counterparts, in which all ... more >>>

Let $X$ be a random variable distributed over $n$-bit strings with $H(X) \ge n - k$, where $k \ll n$. Using subadditivity we know that a random coordinate looks random. Meir and Wigderson [TR17-149] showed a random coordinate looks random to an adversary who is allowed to query around $n/k$ ... more >>>

We show a deterministic simulation (or lifting) theorem for composed problems $f \circ EQ_n$ where the inner function (the gadget) is Equality on $n$ bits. When $f$ is a total function on $p$ bits, it is easy to show via a rank argument that the communication complexity of $f\circ EQ_n$ ... more >>>

Instance complexity is a measure of goodness of an algorithm in which the performance of one algorithm is compared to others per input. This is in sharp contrast to worst-case and average-case complexity measures, where the performance is compared either on the worst input or on an average one, ... more >>>

The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of $O(1)$ vs. ... more >>>

Let R_eps denote randomized query complexity for error probability eps, and R:=R_{1/3}. In this work we investigate whether a perfect composition theorem R(f o g^n)=Omega(R(f).R(g)) holds for a relation f in {0,1}^n * S and a total inner function g:{0,1}^m \to {0, 1}.

Let D^(prod) denote the maximum distributional query ... more >>>