Quantum Computing. Hafiz Md. Hasan Babu

14. The construction procedure for a quantum memory unit is also given in this chapter. Three approaches for designing a quantum arithmetic logic unit are described in chapter 15. Finally, several real-life applications of quantum computing technology are given in chapter 16.

      IOP Publishing

      Quantum Computing

      A pathway to quantum logic design

      Hafiz Md Hasan Babu

      Chapter 1

      Quantum logic

      In the age of developing nanotechnology, quantum computing can play an incredibly important role in developing more compact and lower power consumption computers. The main appeal of quantum logic originates in its reflection of the physical law of energy conservation, in which the creation or deletion of energy is impossible, and only transformation from one form to another is possible. Hence, the fundamental law of energy preservation is incorporated into the logic design of circuits and systems by the quantum logic. The motivation to implement circuits and systems using quantum computing is the fact that, theoretically, the internal computations in quantum logic systems consume no power.

1.1 Overview

      Quantum computing is a technology which consumes less power and for which the design is compact. In irreversible logic, energy dissipation is a common phenomenon since every bit loss causes energy loss in the irreversible operation. Quantum logic circuits are necessarily reversible and hence there is no dissipation of energy while processing a bit in quantum computation.

      Quantum technology is one of the most promising nanotechnologies which are useful for designing modern circuits. Logic design with quantum logic is of great interest in recent technologies which allow scaling to atomistic dimensions. In this particular logic design approach, quantum cells are arranged in a particular fashion to define the logic. A classical gate cannot handle the superposition of states represented by a qubit (discussed in section 2.1 of chapter 2). Thus, this forms a special case of quantum device.

      Quantum registers, which are necessary for the implementation of a quantum electronics device, combine n qubits to form larger Hilbert spaces H_n using the tensor product (⊗) operator to form

      ∣Ψ⟩=∣Ψ1⟩⊗∣Ψ2⟩⊗⋯⊗∣Ψn⟩=∑i=0n∣αi∣i⟩;

      where αi∈C,∣Ψi〉 represents a qubit and ∑i=0n∣αi∣2=1. High-speed multiplication has always been a fundamental requirement of high-performance processors and systems. In quantum signal processing (QSP) applications, multiplication is one of the most utilized arithmetic operations. Improving multiplier design directly benefits the high-performance embedded processors and QSP applications used in consumer and industrial electronic products. Moreover, quantum information processing (QIP) is a high-impact research area in quantum information science to construct a quantum computer. The main goal of QIP is to harness the fundamental laws of quantum mechanics to dramatically improve all aspects (e.g. acquisition, transmission, and processing) of information processing as well as enhance the performance of quantum computers.

      There are several tasks for which a quantum computer will be useful. The first, which is mentioned most frequently, is that quantum computers will be able to read secret messages communicated over the Internet using current technologies such as Rivest–Shamir–Adleman (RSA), Diffie–Hellman, and other cryptographic protocols; these protocols are based on the difficulty of number theoretic problems such as factoring and discrete logarithms. In addition, quantum computers are useful for scientists conducting virtual experiments and searching huge amounts of data.

1.2 Motivations towards quantum computing

      Quantum logic is a great achievement for very-large-scale integrated (VLSI) circuit design, and can work faster than classical logic circuits. Quantum circuits are used to build quantum super computers and can solve the complex problems in polynomial time. Quantum algorithms are used to implement quantum circuits.

      Quantum bits have the important property of superposition, which means that the values of quantum bits can stay in more than one position at the same time, which is impossible in the case of classical logic designs. Reversible computation supports binary (0,1) values, whereas quantum logic can hold multiple values (binary, ternary, quaternary, etc). This multi-value support makes quantum circuits more compact and efficient with optimal delay. Unlike the two fixed values 0 or 1 of classical logic, quantum bits can take the values of a linear combination of 0 and 1. Quantum circuits are inherently reversible and there is no dissipation of energy in these circuits. Thus these circuits prevent the loss of information, since energy dissipation causes bit or information loss. The above mentioned properties of quantum logic circuits have motivated researchers to design circuit components using quantum logic.

1.3 The relationship between reversible and quantum logic

      Reversible logic circuits have one-to-one mapping between inputs and outputs. In other words, if the number of outputs in a logic circuit is equal to the number of inputs, and any input pattern may map to a unique output pattern, it is called a reversible logic circuit.

      All the reversible circuits can be represented by quantum logic gates. Quantum circuits maintain the rules of reversible logic. Quantum logic circuits must have a one-to-one relationship between the input and output vectors like reversible logic. Quantum logic gates have unique unitary matrices which are also present in reversible logic. However, there are also differences between these two types of logic. Unlike in quantum logic, the superposition of bits and multiple values is not possible in reversible logic. Thus both similarities and differences exist between reversible and quantum logic.

      Reversible circuits have broad applications in nanocomputing. Nanoelectronics engineering that would enable device scaling down to molecular levels will almost surely imply a cellular architecture with near-neighbor connectivity. The scheme which has been developed to physically realize such a concept is called the quantum-dot cellular automata (QCA). QCA have drawn a lot of attention for their very small feature size and ultra-low power consumption, which make them one candidate for substituting CMOS technology.

1.4 Quantum computers

      In 1959, Richard Feynman noted that ‘as electronic components begin to reach microscopic scales, effects projected by quantum mechanics occur’. He advised that it might be used in the design of more powerful computers. In particular, quantum researchers hope to bind the superposition. In the quantum mechanical world, objects do not necessarily have visibly defined states. A traditional digital computer works with binary digits that can be in one of two states, denoted as 0 and 1; thus, for example, a four-bit computer register can hold any one of 16 (2⁴) possible numbers. However, a quantum bit (qubit) exists in a wavelike superposition of values from 0 to 1; thus, for example, a four-qubit computer register can hold 16 different numbers concurrently. In theory, a quantum computer can therefore control many values in parallel, so that a 30-qubit quantum computer would be equivalent to a digital computer proficient at performing 10 trillion floating-point operations per second.

      During the 1980s and 1990s, the theory of quantum computers advanced considerably beyond Feynman’s early assumptions. In 1985, David Deutsch designated the structure of quantum logic gates for a universal quantum computer. In 1994, Peter Shor established an algorithm to factor numbers with a quantum computer that would involve six qubits (although more qubits would be essential for factoring large numbers in a rational time).

      In 1998, Isaac Chuang, Neil Gershenfeld, and Mark Kubinec produced the first quantum computer (two-qubit) that could be burdened with data and output a result. Although their system was rational for only a few nanoseconds and slight from the perspective of solving meaningful problems, it verified the principles of quantum computation. This type of quantum computer can be prolonged by using molecules with more individually addressable elements. In March 2000, Emanuel Knill, Raymond Laflamme, and Rudy