where the unitary operationsUj are indifferent of the values of the variablesxi andOx is the oracle as defined above. Therefore, the final state of the algorithm is
|ψf inali=UtOxUt−1. . . U1OxU0|ψstarti
and the output is decided by some measurement of the state|ψf inali. A quantum algorithm is said to exactly compute f if for all (x1, x2, . . . , xn) it outputs the value of the function correctly with probability 1. The minimum number of queries needed by a Quantum Algorithm to achieve this is called the Exact Quantum Query Complexity QE(f) of the function. We also define the term separable as we often use it.
Definition 2. A Boolean functionf is called separable if QE(f)< D(f) and non-separable other- wise.
Now, it is easy to see that in both the models the maximum query complexity of a function can be n, in any model. One can also show that the least exact quantum query complexity of a non-degenerate function onnvariables is Ω √
logn
. Let us now move to query friendly functions.
nature. In this regard we design a new algorithmic technique that gives us separation for a class of functions with Ω
2
√n
functions built using direct sum constructions.
Finally In Chapter 4 we study the D(f) and QE(f) of the Maiorana-Mcfarland (MM) type bent functions. This is a class of size super-exponential in n, with more than 22
n
4 functions for any n.
The motivation behind this study is the uniformity in the algebraic normal form of these functions.
We obtain that D(f) = nfor functions in this class. Then we design a parity decision tree based technique with query complexity ofdn4e. We further reduce it tod5n8 eusing the techniques we have developed in Chapter 3. We finish the chapter with open questions in this area that we are yet to solve.
Chapter 2
Query Friendly Functions
2.1 Introduction
In this chapter we concentrate on the deterministic and exact quantum query complexity of different Boolean function classes. There are other computational models such as the classical randomized model and the bounded error quantum model [1] and there exists rich literature on work on these models as well. However, those are not in the scope of this work.
In this regard one may note that the work by Barnum et.al [5] can be used to find the ex- act quantum query complexity of any function onn variables by repetitively solving semi definite programs (SDP). Montanaro et.al [10] have used this method to find exact quantum query com- plexity of all Boolean functions upto four variables as well as describe a procedure of formulating the quantum algorithm to achieve the said exact quantum query complexity. This method is not yet found to be suitable for finding the exact quantum query complexity of a general classes of Boolean functions. Additionally, the SDP are resource intensive in nature and solving the SDP for large values of nis computationally challenging. But for the cases where the number of variables is low, this does offer an exhaustive view of the exact quantum query complexities of all Boolean functions.
As an example, in a very recent paper Chen et.al [8] have shown thatf(x) =xiorf(x) =xi1⊕xi2 are the only Boolean functions withQE(f) = 1. However the work of Montanaro et.al [10, Section 6.1] show that the Boolean functions f with 2 or lesser variables andQE(f) = 1 are
• The single variable function xi.
• The two variable functionsxi1 ⊕xi2.
Then it is shown in [10, Section 6.2] that the minimum quantum exact quantum query complexity of any Boolean function with 3 or more influencing variables is 2. This essentially implies that the work of [8] is in fact a direct corollary of [10].
We now lay out of the structure of the rest of the chapter.
2.1.1 Organization & Contribution
In Section 2.2, we start by describing the fact that the maximum number of influencing variables that a function withkdeterministic query complexity can have is (2k−1). We first construct such a function using the decision tree model. The decision tree representation of such a function is a
k-depth fully-complete binary tree in which every internal node queries a unique variable. We first
prove in Theorem 4 that any function with 2k−1 influencing variables and kdeterministic query complexity must have the same exact quantum query complexity (k).
Next, we define a special class of Boolean functions in Section 2.2.1, called the “Query Friendly”
functions. A function f withn influencing variables is called query friendly if there does not exist any other function with n influencing variables with lesser deterministic query complexity than f. If n lies between 2k−1 and 2k−1 (both inclusive) then all functions with deterministic query complexityk are called query friendly functions. The proof in Theorem 4 directly implies that all query friendly functions withn= 2k−1 influencing variables are non-separable.
Then in Section 2.2.2 we identify a class of non-separable query friendly functions for all values ofn. We conclude this section by showing that all query friendly functions withn= 2k−2 (k >2) influencing variables are non-separable as well.
In Section 2.3, we describe the parity decision tree model. We first discuss the simple result that a k-depth parity decision tree can describe functions with upto 2k+1 −2 influencing variables.
In Section 2.3.1 we define another set of query friendly functions on n influencing variables that exhibit minimum separation (i.e., one) between deterministic and exact quantum query complexity for certain generalized values ofn. We prove by construction that if 2k−1 ≤n <2k−1+ 2k−2 then there exists a class of query friendly functions such that for any function f in that class we have QE(f) =D(f)−1. We conclude the section by showing that for other values of nthere does not exist separable query friendly functions that can be completely described by the parity decision tree model.
We conclude the study in Section 2.4 outlining the future direction of our work. We further state open problems that we have encountered in this work. Solution to these problems will help us better understand the limitations of the parity decision tree model.