Abstract After a quick review of Chapter 2’s bits and gates, Bob and Alice introduce Cardy to several concepts and symbolic representations that will be of crucial importance in understanding quantum computing: state vectors, pictorial representations of state vectors, and simple mathematical forms that help us manipulate and reason about state vectors. State vectors may be represented by arrows in a state space diagram. States may also be represented more abstractly using Dirac notation symbols |⟩. Logic gates, critically important for carrying out computational work, can be represented by mathematical objects called matrices. States are represented by column vectors. Everything you need to know about matrices is presented over just a few pages.
Abstract After a quick review of Chapter 2’s bits and gates, Bob and Alice introduce Cardy to several concepts and symbolic representations that will be of crucial importance in understanding quantum computing: state vectors, pictorial representations of state vectors, and simple mathematical forms that help us manipulate and reason about state vectors. State vectors may be represented by arrows in a state space diagram. States may also be represented more abstractly using Dirac notation symbols |⟩. Logic gates, critically important for carrying out computational work, can be represented by mathematical objects called matrices. States are represented by column vectors. Everything you need to know about matrices is presented over just a few pages.
Abstract Moving completely into the quantum realm, the key concept of quantum state vector is described in three ways: (1) arrows in state space, (2) Dirac notation symbols that are important for the formal mathematical manipulation of quantum states, and (3) collections of 1s and 0s grouped in columns or rows. A qubit (two-level system) has two basis states. These basis state vectors can be added to give superposition states and multiplied using scalar products to change the state coefficients. These abstract concepts are given a physical meaning in a system of polarized light, a familiar phenomenon because of its use in polarized sunglasses. Light polarization has been widely used to demonstrate many fundamental aspects of quantum computing and quantum information science.
Abstract Quantum gates, equivalent to logic gates, modify the state vectors. Linear algebra calls these gates linear transformations. In quantum language, they are “operators.” These gates include the NOT gate, the Pauli gates, and the Hadamard gate. The operations of the gates are represented both as abstract symbols and by matrices multiplying state vectors (expressed as column vectors). Quantum computers use a series of gates called a quantum circuit. The chapter explains how to read a quantum circuit diagram and translate it into linear algebra mathematics. The mathematical requirements on the matrices allowed to be used is delineated. Gates are reversible by applying the same gate twice, but measurements are not reversible.
Abstract With this chapter, we begin to transit from digital information processing to quantum information processing. However, to be able to master this transition, we first need to familiarize ourselves with quantum mechanical principles and the mathematics used to describe them. Since we already know about abstract state spaces, tensor products, and Hamiltonian operators and their eigenvectors and eigenvalues, this should not be too difficult or too alien. And yet, there remain a few challenges: Mathematical descriptions of quantum states usually involve the Dirac notation, require complex linear algebra in high dimensional spaces, and encode unintuitive probabilistic interpretations. Rising to these challenges, we briefly study the notion of Hilbert spaces, introduce the Dirac notation, and discuss the axioms of quantum mechanics and some of their consequences. All this will admittedly be dense and abstract but prepare us for everything that is still to follow. Our main goal is to emphasize that quantum mechanics and therefore quantum computing is just complex linear algebra in very high dimensional spaces so that quantum computing needs a different kind of thinking than we are used to in digital computing.
If mathematics is the queen of science, then physics might well be the king. It successes are obvious. However, as a science, physics may have failed in one regard, and that is to explain what its basic concepts – such as state vectors, wavefunctions, and transformation matrices – actually represent. When studying quantum mechanics, it is, effectively, hard to keep up the initial enthusiasm, and those who branch out to other fields – which is most of us – quickly end up going through the motions only: we regurgitate models and equations and know how to solve the standard problems, so as to pass the exam, but then forget about them as soon as possible. This paper explores a very intuitive sentiment about the issue: the wavefunction is a rather ‘flat’ mathematical object – it is two-dimensional, basically – so it can’t do the trick, perhaps. In contrast, Maxwell’s equations have real vectors in them, which is why a deeper or more intuitive understanding of electromagnetism comes relatively easily. Indeed, when everything is said and done, we are just human beings living in three-dimensional space, and that is why vector equations (or systems of vector equations), as a mathematical tool, make sense to us. This paper further explores this sentiment. It also offers a way out by, predictably, presenting yet another possible physical interpretation of the wavefunction. More importantly (for the reviewer of this paper, at least), this paper offers a sensible response to the mainstream view that three-dimensional physical interpretations of the wavefunction cannot make any sense because of the weird 720° symmetry of the wavefunction when describing spin-1/2 particles (fermions or – for all practical purposes – electrons). The author does so by analyzing (1) Dirac’s belt trick more in detail – and what it implies in terms of the interaction between the observer and the object – as well as (2) Feynman’s derivation of the transformation matrices for spin-1/2 two-state systems.
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.