Except for the NOT gate, all the other familiar classical gates are irreversible in the sense that we cannot uniquely reconstruct the input states from the output states. For example, OR, AND, NAND, NOR, etc. are irreversible. They all map a two-bit input state into a single bit output state. Thus one bit is erased during operation of each of these gates, and according to Landauer’s principle that requires dissipation of a minimum amount of energy. Now a simple question arises in our curious mind: Is it possible to avoid this loss of energy? The answer is yes. If we don’t erase any bit then we can circumvent this energy loss. Thus we need to map n bit input states to n bit output states. In addition, if a one-toone correspondence exists between the input states and the output states then only we will be able to uniquely reconstruct the input states from the output states. In such a case the gate is called reversible. For example, if we have f(00) = 00, f(01) = 10, f(10) = 01, f(11) = 11 then f represents a reversible gate. Quantum evolution operators are unitary so for every operator U we have an inverse operator U−1 = U†. Therefore, quantum evolution operators are the natural choice for the construction of energy efficient reversible gates. However, it is not the only choice. We can have classical reversible gates, too. Usually by reversible gates we refer to classical reversible gates, and quantum gates are specifically referred to as quantum gates. Reversible gates and quantum gates are similar but there exists a fundamental difference that reversible gates cannot accept superposition states (e.g. α|0〉 + β|1〉) as input states, whereas quantum gates can. Thus all quantum gates are essentially reversible but the converse is not true.
Except for the NOT gate, all the other familiar classical gates are irreversible in the sense that we cannot uniquely reconstruct the input states from the output states. For example, OR, AND, NAND, NOR, etc. are irreversible. They all map a two-bit input state into a single bit output state. Thus one bit is erased during operation of each of these gates, and according to Landauer’s principle that requires dissipation of a minimum amount of energy. Now a simple question arises in our curious mind: Is it possible to avoid this loss of energy? The answer is yes. If we don’t erase any bit then we can circumvent this energy loss. Thus we need to map n bit input states to n bit output states. In addition, if a one-toone correspondence exists between the input states and the output states then only we will be able to uniquely reconstruct the input states from the output states. In such a case the gate is called reversible. For example, if we have f(00) = 00, f(01) = 10, f(10) = 01, f(11) = 11 then f represents a reversible gate. Quantum evolution operators are unitary so for every operator U we have an inverse operator U−1 = U†. Therefore, quantum evolution operators are the natural choice for the construction of energy efficient reversible gates. However, it is not the only choice. We can have classical reversible gates, too. Usually by reversible gates we refer to classical reversible gates, and quantum gates are specifically referred to as quantum gates. Reversible gates and quantum gates are similar but there exists a fundamental difference that reversible gates cannot accept superposition states (e.g. α|0〉 + β|1〉) as input states, whereas quantum gates can. Thus all quantum gates are essentially reversible but the converse is not true.
This work presents the logical reversibility. The inputs and outputs of reversible logic gates can be uniquely retrievable from each other. The reversible logic operations can't erase information and dissipate zero heat. The circuit actually operates in a backward operation, allows reproducing the inputs from the outputs and consumes zero power. As the basic elements of any logic circuit, logic gates are used to realize Boolean functions. By combining reversible logic gates, reversible circuits can perform complex logical and arithmetic operations. A one-to-one mapping between inputs and outputs is realized. The logical operations run backwards by cascading a reversible logic gate with its dual (inverse). Reversible circuits are also called lossless circuits, as there is neither energy loss nor information loss. These circuits are very attractive for applications where extremely low power consumption, or heat dissipation, is desirable in areas ranging from communications, low power VLSI (very large-scale integration) technology, optical computing to nanotechnology. Reversible logic found to be very useful in quantum computing where the quantum evolution is inherently reversible.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
Quantum is an emerging technology in future computers. Reversibility is the main advantage of quantum computers. In conventional computers, the computation is irreversible i.e. the input bits are lost once the logic block generates the output and input bits cannot be restored but it can be done in reversible computation because in reversible computation the inputs and outputs have a one-to-one correspondence. Therefore, a reversible gate input could even be uniquely determined from their output which leads to less power consumption. Hence the complexity of the digital circuits can be reduced by using reversible computing. In quantum computer to perform reversible operations, we need to implement the reversible gates using quantum gates. In this paper, we discussed various reversible logic gates like Feynman, Toffoli, R, Peres and TR gates using basic quantum gates like CNOT, Pauli, Swap gates and their implementation using IBM quantum experience.
In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50
Quantum computing represents a paradigm shift in computation by leveraging the principles of quantum mechanics such as superposition, entanglement, and interference to perform operations beyond classical limitations. Unlike classical computing systems that rely on irreversible logic operations and suffer from energy dissipation due to information loss, quantum systems utilize reversible unitary transformations that preserve information and improve computational efficiency. This paper presents a comprehensive framework for the design, simulation, and validation of quantum gates and reversible arithmetic circuits, including half adders and half subtractors. The proposed approach systematically maps classical arithmetic logic into quantum circuits using fundamental quantum gates such as Pauli-X, Hadamard, Controlled-NOT (CNOT), and Toffoli gates. The circuits are implemented and simulated using the Qiskit Aer framework, enabling detailed analysis of quantum state evolution and measurement outputs. The results obtained from simulation include program-level outputs, state vector representations, measurement counts, and probability distributions, all of which demonstrate strong agreement with classical truth tables. The analysis confirms the correctness, reversibility, and scalability of the proposed designs. Furthermore, a comparative performance evaluation highlights the advantages of quantum computing over classical approaches in terms of parallelism and energy efficiency. The proposed framework establishes a robust foundation for future implementation of complex Boolean logic and arithmetic operations in quantum systems.
This chapter makes us walk a few preliminary, but decisive, steps towards quantum information theory (QIT), which will be the focus of the rest of this book. Here, we shall remain in the classical world, yet getting a hint that it is possible to think of a different world where computations may be reversible, namely, without any loss of information. One key realization through this paradigm shift is that “information is physical.” As we shall see, such a nonintuitive and striking conclusion actually results from the age-long paradox of Maxwell's demon in thermodynamics, which eventually found an elegant conclusion in Landauer's principle. This principle states that the erasure of a single bit of information requires one to provide an energy that is proportional to log 2, which, as we know from Shannon's theory, is the measure of information and also the entropy of a two-level system with a uniformly distributed source. This consideration brings up the issue of irreversible computation. Logic gates, used at the heart of the CPU in modern computers, are based on such computation irreversibility. I shall then describe the computers' von Newman's architecture, the intimate workings of the ALU processing network, and the elementary logic gates on which the ALU is based. This will also provide some basics of Boolean logic, expanding on Chapter 1, which is the key to the following logic-gate concepts.