Abstract Alice and Bob explain the difference between classical (traditional) computing and quantum computing, deploying a gentle introduction to classical binary digits (bits) beginning with a brief history of the development of quantum mechanics and computer architecture. The abstract backbone of classical computing is logic gates, which represent changes to input bits under specific rules. These rules, governed by Boolean, logic can be summarized in truth tables, giving the output values for specific input values. The most common classical gates—NOT, AND, NAND, and XOR gates—are introduced. The vocabulary of classical computing provides the basis for understanding quantum computing.
Abstract Alice and Bob explain the difference between classical (traditional) computing and quantum computing, deploying a gentle introduction to classical binary digits (bits) beginning with a brief history of the development of quantum mechanics and computer architecture. The abstract backbone of classical computing is logic gates, which represent changes to input bits under specific rules. These rules, governed by Boolean, logic can be summarized in truth tables, giving the output values for specific input values. The most common classical gates—NOT, AND, NAND, and XOR gates—are introduced. The vocabulary of classical computing provides the basis for understanding quantum computing.
Abstract As discussed in Chap. 10.1007/978-3-030-61601-4_2 , information in classical computers is represented by bits. However, if the bits did not change, then the computer would remain the same forever and would not be very useful! Therefore, it is necessary to change the values of bits depending on what you want the computer to do. For example, if you want a computer to multiply the number 2 and the number 3 together to produce the number 6, then you need to put each of the numbers 2 and 3 into an 8-bit binary representation, and then have a computational operation to multiply the two 8-bit values together to produce 6. The operation of changing bits in a classical computer is performed by classical logic gates.
Quantum computing is an emerging form of computing that uses the rules of quantum physics to process information in ways that are far different from how computers do it now. Quantum computing uses quantum bits, or qubits, which can exist in superpositions of states and get entangled with each other, which is different from classical computing, which uses a bit that represents binary values (0 or 1). These characteristics allow quantum devices to execute parallel computations, enabling significant speedups for specific categories of problems considered intractable for classical machines. This chapter gives a thorough introduction to quantum computing, starting with a historical overview of its foundations in quantum physics and significant theoretical advancements. It analyzes fundamental ideas including superposition, entanglement, quantum interference, and the no-cloning theorem, explaining how these elements distinguish quantum computation from traditional approaches. The advancement of quantum hardware is examined through several architectures, trapped ions, superconducting circuits, photonics, and topological qubits, together with software frameworks like Qiskit, Cirq, and PennyLane, which support algorithm development and simulation. The chapter examines fundamental quantum algorithms, such as Grover’s algorithm for database search, Shor’s algorithm for integer factorization, and hybrid variational methods designed for noisy intermediate-scale quantum devices. The practical applications in domains including as encryption, optimization, machine learning, chemistry, and finance emphasize the real-world significance of quantum technology. Challenges like decoherence, noise, scalability, and algorithmic maturity are rigorously analyzed, followed by an anticipated discourse on future trajectories including fault-tolerant architectures, quantum networking, and the development of quantum cloud services. This chapter presents a fundamental introduction for students, researchers, and practitioners seeking to comprehend the essential principles, functionalities, and possibilities of quantum computing in the post–Moore’s law period.
This lecture continues Lectures One and Two, where we introduced the basic concepts of quantum computing and reviewed the essential linear algebra background for undergraduate students. In this lecture, we focus on the basic computational units of quantum computation, known as quantum gates. Quantum gates play a central role in quantum algorithms, as each quantum algorithm can be implemented as a sequence of quantum gate operations. This idea has a close analogy with classical computing, where computations can be represented by Boolean circuits constructed from elementary logic gates. In principle, classical algorithms can be implemented through combinations of such logic gates. To establish this connection, we first review several fundamental logic gates in classical computing, including the AND, OR, and NOT gates. These elementary Boolean gates provide a useful foundation for understanding quantum gates and how classical computational concepts are extended into the quantum domain. In general, a quantum processing unit (QPU), which executes quantum gate operations, can be viewed as the quantum counterpart of a classical central processing unit (CPU), where computations are carried out through logical operations and circuits. This lecture establishes the foundation for understanding the quantum circuit model and for designing and analyzing quantum algorithms in subsequent lectures. To provide a smooth transition to the topics that follow, we also introduce several tiny quantum algorithms and examples in this lecture. These examples illustrate how quantum gates can be combined to perform meaningful computational tasks and prepare students for the more advanced quantum algorithms discussed later.
Abstract Alice and Bob introduce and explain in detail some of the algorithms that convinced physicists and computer scientists that there might be something to this crazy idea of quantum computing. Each algorithm provides a case study of how the properties of quantum states can be used to provide an advantage for using a quantum computer compared to a classical computer. Alice and Bob describe the Deutsch algorithm, designed to answer a simple question about the properties of functions of binary digits (0 and 1), and the Simon algorithm, which finds a “secret code.” The Simon algorithm, like many other quantum algorithms, is a mix of quantum state algorithms and classical computational methods. Alice and Bob also introduce Hadamard gates that act on multi-qubit systems. Those gates play an important role in many quantum algorithms.
In this work, we develop quantum reversible counterparts of the logical AND, NAND, OR, NOR, and XOR gates within a single architectural template: a quantum Fourier transform (QFT)–based adder that emulates a digital half‐adder by computing the sum and carry of two input bits. Each member of the gate family is obtained from the same adder by inserting only X and SWAP gates as pre‐ and postprocessing steps, with the required Boolean identities supplied by De Morgan’s laws. We prove that every proposed circuit is unitary prior to readout, so the input data remain recoverable from the output state. The construction generalizes to N ‐input gates through an in‐place population‐count (Hamming‐weight) register, and we derive explicitly that the resulting qubit requirement is N + log 2 N , compared with the 2 N − 1 qubits of the standard Toffoli‐ladder synthesis; the derivation is validated by exhaustive simulation for N ≤ 8. We provide a comprehensive resource analysis‐qubit count, gate count, circuit depth, CX count, T count, transpiled depth, and success probability‐against Toffoli‐, CNOT‐, and X‐gate‐based reversible circuits as well as three multicontrolled‐NOT decompositions with different ancilla‐depth trade‐offs. The analysis shows that the qubit saving is obtained at the price of a larger two‐qubit‐gate count, and identifies the regimes in which the QFT‐based design is advantageous. All two‐input gates and a four‐input NAND instance are executed for every input combination on IBM quantum hardware and on a calibrated noise‐model simulator, and the measured outputs consistently reproduce the corresponding truth tables, with hardware success probabilities of 0.97–0.99 for the two‐input gates and 0.88–0.93 for the four‐input NAND.