A function is a mathematical concept that establishes a special relationship between the elements of two sets. The first set consists of the elements that form the inputs of the function, while the second set consists of the elements that form the outputs of the function. From this perspective, it can be stated that functions possess data processing capabilities. Consequently, we can infer that for each input, a function assigns only one output, which is the defining characteristic of functions.
2.1.1. States and Functions of Digital Computers
The state of digital computers we use in everyday life at time t corresponds to the digital data stored in the computer’s memory units at that moment. Computer programs advance this digital state through a series of operations, transforming it from one state to another. This process is carried out by structures known as Boolean Gates, which perform the logical operations of the computer.
In the subsequent section of the study, titled Quantum Gate-Based Computation, we examine these gates from the perspective of computer engineering. Here, however, we will focus on analyzing these gates from a mathematical standpoint, while adhering to the mathematical concept of the work. From a mathematical perspective, Boolean gates can be treated as functions.
These gates:
- Affect inputs (sequences of bits consisting of 0 and 1 values).
- Process these inputs to produce outputs (consisting of 0 and 1 values).
As can be understood, the logical evolution of a computer is modeled by the effect of Boolean gates on inputs, and this process represents the transition of the system from one state to another.
2.1.2. The Importance of Functions Defined on the {0, 1} Set
The definition of functions on the {0, 1} set is of critical importance, particularly in digital systems and the field of quantum computing. This is because both classical and quantum computers fundamentally operate based on binary logic. Inputs and outputs typically consist of bits, which is why such functions play a central role at both the theoretical and practical levels.
2.1.3. Bijective Functions and Inverse Functions
Throughout this study, particular focus will be given to two types of functions: (1) Bijective Functions and (2) Inverse Functions. Let us now discuss these functions in detail.
- Bijective Functions: A bijective function is both injective and surjective. This means that each input has a unique output, and each output corresponds to only one input. Bijective functions possess the property of invertibility.
- Inverse Functions: Invertible functions are those that allow the reversal of the process, mapping outputs back to inputs. In other words, starting from the output of a function, you can retrieve the corresponding input. Mathematically, the inverse of a function f is denoted as 𝒇−𝟏, and it satisfies the following property:
𝒇−𝟏𝒇 (𝒙) = 𝒙
These types of functions form the foundation for reversible digital gates and quantum gates. In quantum computing, the reversibility of gates is a necessity dictated by quantum mechanical principles. This means that quantum gates can only be modeled using bijective functions. We have discussed this topic several times under the heading Quantum Gate-Based Computation.
2.1.4. Reversible Digital Gates and Quantum Gates
Reversible digital gates operate in such a way that their inputs can be uniquely obtained from their outputs. This property is critical for quantum computing because, according to quantum mechanics, the evolution of the system must be reversible. Reversible digital gates also contribute to energy efficiency, as they do not result in information loss.
As mentioned in previous studies, Boolean Gates or Classical Computer Gates are not always reversible. For example, AND and OR gates are not reversible. However, quantum gates are completely reversible, meaning they can always be modeled as such. Therefore, the concepts of bijective functions and reversibility hold a fundamental role in the mathematical definition of both digital and quantum gates.