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Additional Topic: Quantum Computing

Quantum computing is a rapidly evolving field that leverages the principles of quantum mechanics to perform computations that are infeasible for classical computers. It is, however, often misunderstood and misrepresented in popular media, which tends to oversimplify or exaggerate its capabilities. In this section, we will provide a comprehensive introduction to quantum computing, covering its fundamental concepts, principles, and potential applications.

In computer science, we divide concepts into different levels of abstraction.

  • In a classical computer, the lowest level of abstraction is the physical layer, which consists of the hardware components that perform computations. This involves the atomic composition of the computer, such as transistors and wires.
  • The next level of abstraction is the logical layer, which abstracts away the physical details and instead represents them as simple s and s. We perform operations on these bits using logical gates, which manipulate the bits to perform computations. These can be classified based on the number of bits they operate on. For example, a NOT gate operates on one bit, while an AND gate operates on two bits. Combined together, they form a powerful computational model known as the Turing machine.
  • The third level of abstraction is the algorithmic layer, which consists of algorithms that perform computations on the logical layer. These algorithms are designed to solve specific problems and can be implemented using various programming languages.
  • The fourth level of abstraction is the programming language layer, which provides a high-level interface for programmers to write algorithms. Programming languages have different levels of abstraction, with some being closer to the hardware and others being more abstract. For example, assembly language is much closer to the hardware than C, which is still closer to the hardware than Python.

On the other hand, quantum computing is a new paradigm that introduces a different set of abstractions.

  • The lowest level of abstraction in quantum computing is the physical layer, which consists of quantum systems. These systems can be atoms, ions, photons, or superconducting circuits, among others. The physical layer is governed by the principles of quantum mechanics, which describe the behavior of these systems at the atomic and subatomic levels. We have already thoroughly studied quantum mechanics in the previous sections.

  • The next level of abstraction is the logical layer, which consists of qubits. Many quantum systems can be represented by the same Hilbert space, thus allowing us to use the same mathematics to describe them. One such category of systems is two-level systems, which can be represented by a two-dimensional Hilbert space. For example, spin-1/2 electrons, polarized photons, and superconducting qubits can all be represented by a two-dimensional Hilbert space. We collectively refer to these systems as qubits in this layer of abstraction. By doing so, we ignore the physical details of the systems and focus solely on their mathematical representation. To manipulate qubits, we use quantum gates, which are the quantum analogs of classical logical gates. Mathematically, they are just matrices that act on the qubits.

    Combined together, multiple qubits exponentially increase the dimensionality of the Hilbert space. The total Hilbert space is the tensor product of the individual Hilbert spaces of the qubits. For qubits, the total Hilbert space is -dimensional.

  • The third level of abstraction is the algorithmic layer, which consists of quantum algorithms. These algorithms are designed to solve specific problems using quantum gates and qubits. They can be implemented using various programming languages, such as Qiskit, Cirq, and Q#.

Because we have already studied quantum mechanics, we will not go into detail about the physical layer of quantum computing. Instead, we will focus on the logical layer and the algorithmic layer. We will also briefly discuss the programming language layer, but we will not go into detail about specific programming languages.