Total Boolean Function Research Articles

Exact Quantum 1-Query Algorithms and Complexity

Deutsch–Jozsa problem (D–J) has exact quantum 1-query complexity (“exact” means no error), but requires super-exponential queries for the optimal classical deterministic decision trees. D–J problem is equivalent to a symmetric partial Boolean function, and in fact, all symmetric partial Boolean functions having exact quantum 1-query complexity have been found out and these functions can be computed by D–J algorithm. A special case is that all symmetric Boolean functions with exact quantum 1-query complexity follow directly and these functions are also all total Boolean functions with exact quantum 1-query complexity obviously. Then there are pending problems concerning partial Boolean functions having exact quantum 1-query complexity and new results have been found, but some problems are still open. In this paper, we review these results regarding exact quantum 1-query complexity and in particular, we also obtain a new result that a partial Boolean function with exact quantum 1-query complexity is constructed and it cannot be computed by D–J algorithm. Further problems are pointed out for future study.

Conflict complexity is lower bounded by block sensitivity

It is shown that conflict complexity of every total Boolean function, recently introduced in [1] to prove a composition theorem of randomized decision tree complexity, is at least a half of its block sensitivity. The paper proposes to compare conflict complexity with certificate complexity and explains why it could be interesting.

Characterization of exact one-query quantum algorithms

The quantum query models is one of the most important models in quantum computing. Several well-known quantum algorithms are captured by this model, including the Deutsch-Jozsa algorithm, the Simon algorithm, the Grover algorithm and others. In this paper, we characterize the computational power of exact one-query quantum algorithms. It is proved that a total Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ can be exactly computed by a one-query quantum algorithm if and only if $f(x)=x_{i_1}$ or ${x_{i_1} \oplus x_{i_2} }$ (up to isomorphism). Note that unlike most work in the literature based on the polynomial method, our proof does not resort to any knowledge about the polynomial degree of $f$.

On the modulo degree complexity of Boolean functions

For each integer m≥2, every Boolean function f can be expressed as a unique multilinear polynomial modulo m, and the degree of this multilinear polynomial is called its modulo m degree. In this paper we investigate the modulo degree complexity of total Boolean functions initiated by Parikshit Gopalan et al. [9], in which they asked the following question: whether the degree complexity of a Boolean function is polynomially related with its modulo m degree. For m be a power of primes, it is already known that the module m degree can be arbitrarily smaller compare to the degree complexity (see Section 2 for details). When m has at least two distinct prime factors, the question remains open. Towards this question, our results include: (1) we obtain some nontrivial equivalent forms of this question; (2) we affirm this question for some special classes of functions; (3) we prove a no-go theorem, explaining why this problem is difficult to attack from the computational complexity point of view; (4) we show a super-linear separation between the degree complexity and the modulo m degree.

Totally optimal decision rules

Optimality of decision rules (patterns) can be measured in many ways. One of these is referred to as length. Length signifies the number of terms in a decision rule and is optimally minimized. Another, coverage represents the width of a rule’s applicability and generality. As such, it is desirable to maximize coverage. A totally optimal decision rule is a decision rule that has the minimum possible length and the maximum possible coverage. This paper presents a method for determining the presence of totally optimal decision rules for “complete” decision tables (representations of total functions in which different variables can have domains of differing values). Depending on the cardinalities of the domains, we can either guarantee for each tuple of values of the function that totally optimal rules exist for each row of the table (as in the case of total Boolean functions where the cardinalities are equal to 2) or, for each row, we can find a tuple of values of the function for which totally optimal rules do not exist for this row.

Separations in Query Complexity Based on Pointer Functions

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total Boolean function is given by the function f on n = 2 k bits defined by a complete binary tree of NAND gates of depth k , which achieves R 0 ( f ) = O ( D ( f ) 0.7537… ). We show that this is false by giving an example of a total Boolean function f on n bits whose deterministic query complexity is Ω( n ) while its zero-error randomized query complexity is Õ(√ n ). We further show that the quantum query complexity of the same function is Õ( n 1/4 ), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total Boolean function g on n variables that has zero-error randomized query complexity Ω( n / log ( n )) and bounded-error randomized query complexity R ( g ) = Õ(√ n ). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is Q E ( g ) = Õ(√ n ). These functions show that the relations D ( f ) = O ( R 1 ( f ) 2 ) and R 0 ( f ) = Õ( R ( f ) 2 ) are optimal, up to polylogarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R 0 , a 3/2-power separation between Q E and R , and a 4th-power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Göös, Pitassi, and Watson, which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

Communication is Bounded by Root of Rank

We prove that any total boolean function of rank r can be computed by a deterministic communication protocol of complexity O (√ ċ log( r )). Equivalently, any graph whose adjacency matrix has rank r has chromatic number at most 2 O (√ r ċlog( r )) . This gives a nearly quadratic improvement in the dependence on the rank over previous results.

Superlinear Advantage for Exact Quantum Algorithms

A quantum algorithm is exact if, on any input data, it outputs the correct answer with certainty (probability 1). A key question is, how big is the advantage of exact quantum algorithms over their classical counterparts: deterministic algorithms? We present the first example of a total Boolean function $f(x_1,\ldots, x_N)$ for which exact quantum algorithms have superlinear advantage over deterministic algorithms. Any deterministic algorithm that computes our function must use $N$ queries but an exact quantum algorithm can compute it with $O(N^{0.8675\ldots})$ queries. A modification of our function gives a similar result for communication complexity: there is a function $f$ which can be computed by an exact quantum protocol that communicates $O(N^{0.8675\ldots}\log N)$ quantum bits but requires $\Omega(N)$ bits of communication for classical protocols.

How Low can Approximate Degree and Quantum Query Complexity be for Total Boolean Functions?

It has long been known that any Boolean function that depends on n input variables has both degree and exact quantum query complexity of Ω(log n), and that this bound is achieved for some functions. In this paper, we study the case of approximate degree and bounded-error quantum query complexity. We show that for these measures, the correct lower bound is Ω(log n/ log log n), and we exhibit quantum algorithms for two functions where this bound is achieved.

On Exact Quantum Query Complexity

We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs characterising quantum query complexity for small problem sizes. We include numerical results giving the optimal success probabilities achievable by quantum algorithms computing all boolean functions on up to 4 bits, and all symmetric boolean functions on up to 6 bits.

Unbounded-error quantum query complexity

This work studies the quantum query complexity of Boolean functions in an unbounded-error scenario where it is only required that the query algorithm succeeds with a probability strictly greater than 1/2. We show that, just as in the communication complexity model, the unbounded-error quantum query complexity is exactly half of its classical counterpart for any (partial or total) Boolean function. Moreover, connecting the query and communication complexity results, we show that the “black-box” approach to convert quantum query algorithms into communication protocols by Buhrman–Cleve—Wigderson [STOC’98] is optimal even in the unbounded-error setting. We also study a related setting, called the weakly unbounded-error setting, where the cost of a query algorithm is given by q + log ( 1 / 2 ( p − 1 / 2 ) ) , where q is the number of queries made and p > 1 / 2 is the success probability of the algorithm. In contrast to the case of communication complexity, we show a tight multiplicative Θ ( log n ) separation between quantum and classical query complexity in this setting for a partial Boolean function. The asymptotic equivalence between them is also shown for some well-studied total Boolean functions.

Nonadaptive quantum query complexity

We study the power of nonadaptive quantum query algorithms, which are algorithms whose queries to the input do not depend on the result of previous queries. First, we show that any bounded-error nonadaptive quantum query algorithm that computes a total boolean function depending on n variables must make Ω ( n ) queries to the input in total. Second, we show that, if there exists a quantum algorithm that uses k nonadaptive oracle queries to learn which one of a set of m boolean functions it has been given, there exists a nonadaptive classical algorithm using O ( k log m ) queries to solve the same problem. Thus, in the nonadaptive setting, quantum algorithms for these tasks can achieve at most a very limited speed-up over classical query algorithms.

Quantum communication complexity of block-composed functions

A major open problem in communication complexity is whether or not quantum protocolscan be exponentially more efficient than classical ones for computing a total Booleanfunction in the two-party in...

To the problem of expressibility in the algebra of partial Boolean functions

The paper considers the problem of expressibility of total Boolean functions by superpositions over a system of partial Boolean functions. The problem is solved in terms of precomplete Boolean classes, i.e., extensions of Post classes in the algebra of partial Boolean functions.

Average-case quantum query complexity

We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity. A preliminary version of this paper appeared in the Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS'2000) (Lecture Notes in Computer Science vol 1770) (Berlin: Springer).

Lower bounds of quantum black-box complexity and degree of approximating polynomials by influence of Boolean variables

We prove that, to compute a total Boolean function f on N variables with bounded error probability, any quantum black-box algorithm has to query at least Ω(ρ fN)=Ω( S f) times, where ρ f is the average influence of variables in f , and S f is the average sensitivity. It is interesting to contrast this result with the known lower bound of Ω( S f ) , where S f is the sensitivity. This lower bound is tight for some functions. We also show for any polynomial f ̃ that approximates f , deg( f ̃ )=Ω(ρ fN)=Ω( S f) . For some functions, this bound is better than the previously known lower bound of Ω( BS f ) , where BS f is the block sensitivity. Our technique may be of interest in itself: by applying Fourier analysis to functions mapping {0,1} N to unit vectors in a Hilbert space, we can write the state of the quantum computer at step t as ∑ s∈{0,1} N, |s|≤t φ ̂ s(−1) s·x , which is handy for lower bound analysis.

Bidual Horn functions and extensions

Partially defined Boolean functions (pdBf) ( T, F), where T, F⊆{0,1} n are disjoint sets of true and false vectors, generalize total Boolean functions by allowing that the function values on some input vectors are unknown. The main issue with pdBfs is the extension problem, which is deciding, given a pdBf, whether it is interpolated by a function f from a given class of total Boolean functions, and computing a formula for f. In this paper, we consider extensions of bidual Horn functions, which are the Boolean functions f such that both f and its dual function f d are Horn. They are intuitively appealing for considering extensions because they give a symmetric role to positive and negative information (i.e., true and false vectors) of a pdBf, which is not possible with arbitrary Horn functions. Bidual Horn functions turn out to constitute an intermediate class between positive and Horn functions which retains several benign properties of positive functions. Besides the extension problem, we study recognition of bidual Horn functions from Boolean formulas and properties of normal form expressions. We show that finding a bidual Horn extension and checking biduality of a Horn DNF is feasible in polynomial time, and that the latter is intractable from arbitrary formulas. We also give characterizations of shortest DNF expressions of a bidual Horn function f and show how to compute such an expression from a Horn DNF for f in polynomial time; for arbitrary Horn functions, this is NP-hard. Furthermore, we show that a polynomial total algorithm for dualizing a bidual Horn function exists if and only if there is such an algorithm for dualizing a positive function.

On Partial Classes Containig All Monotone and Zero‐Preserving Total Boolean Functions

AbstractWe describe sets of partial Boolean functions being closed under the operations of superposition. For any class A of total functions we define the set 𝔐(A) consisting of all partial classes which contain precisely the functions of A as total functions. The cardinalities of such sets 𝔐(A) can be finite or infinite. We state some general results on 𝔐(A). In particular, we describe all 30 closed sets of partial Boolean functions which contain all monotone and zero‐preserving total Boolean functions.