We investigate efficient algorithms for computing Boolean function properties relevant to query complexity. Such properties include, for example, deterministic, randomized, and quantum query complexities; block sensitivity; certificate complexity; and degree as a real polynomial. The algorithms compute the properties, given an n-variable function's truth table (of size N=2 n ) as input. Our main results are the following: O(N log 2 3 log N) algorithms for many common properties, an O(N log 2 5 log N) algorithm for block sensitivity, an O(N) algorithm for testing "quasi symmetry," a notion of a "tree decomposition" of a Boolean function, proof that the decomposition is unique, and an O(N log 2 3 log N) algorithm for finding it, a subexponential-time approximation algorithm for space-bounded quantum query complexity. To develop this algorithm, we give a new way to search systematically through unitary matrices using finite-precision arithmetic. The algorithms discussed have been implemented in a linkable library.
In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially
Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact, with bounded error and even nondeterministic. In this paper, we study the latter class of algorithms. We introduce a new notion in addition to already studied nondeterministic algorithms and introduce dual nondeterministic quantum query algorithms. We examine properties of such algorithms and prove relations with exact and nondeterministic quantum query algorithm complexity. As a result and as an example of the application of discovered properties, we demonstrate a gap of n vs. 2 between classical deterministic and dual nondeterministic quantum query complexity for a specific Boolean function. Finally, we show an approach how to construct examples where quantum nondeterministic complexity of an algorithm is O(1), however classical deterministic algorithm for the same function would require O(n) queries.
Query complexity studies the amount of information about an input that is needed to solve a computational problem. This thesis investigates query complexity in several settings, with applications to cryptography and property testing, as well as connections to communication complexity and other computational models. In property testing, we study tolerant junta testing and establish the first superpolynomial quantum advantage for this problem: within a certain parameter regime, a quantum algorithm uses polynomially many queries, whereas every classical randomised algorithm requires quasipolynomially many queries. In cryptography, we study the average-case complexity of search problems. We show that verifiable delay functions do not exist in the random oracle model, and strengthen known barriers to constructing average-case hard TFNP problems from one-way functions. Finally, we study lower bounds and their behaviour across computational models. We prove new direct sum results for randomised parity decision trees. We also establish a new separation between query complexity measures of total Boolean functions and, through query-to-communication lifting, derive new communication lower bounds and lower bounds for unambiguous finite automata.
Certificate complexity
(C(f)) is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity
(D) and other query models such as bounded-error randomized and quantum complexity
(R,Q), but not as well for quantum zero-error
(Q0) and exact query complexity
(QE), where there is no agreed-upon certificate 'object' (even for
Q). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of
C,RC (randomized certificate complexity) and
QC (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on
QE and
Q0, the quantum analogues of
D and
R0, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for
Q0, but rational degree and
Q are asymptotically smaller.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.