Essential Mathematics for Quantum Computing Quotes

“The other major inequality is the triangle inequality. It comes from our old friend Euclid in his book The Elements. Succinctly stated, it says that the length of two sides of a triangle must always be more than the length of one side. They will only be equal in the corner case when the triangle has zero area.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“Cauchy-Schwarz and triangle inequalities The Cauchy-Schwarz inequality is one of the most important inequalities in mathematics. Succinctly stated, it says that the absolute value of the inner product of two vectors is less than or equal to the norm of those two vectors multiplied together. In fact, they are only equal if the two vectors are linearly dependent: There are several proofs of this inequality, which I encourage you to seek out if you are interested. But, in the totality of things, knowing this inequality is all that is really required for quantum computing.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“and Invertibility If the determinant of a matrix equals zero, it is not invertible. Otherwise, the matrix is invertible.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“Bloch sphere This is the big payoff of the chapter, understanding the Bloch sphere! The Bloch sphere, named after Felix Bloch, is a way to visualize a single qubit. From Chapter 1, Superposition with Euclid, we know that a qubit can be represented in the following way:”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“It is hard to overstate the beauty of this equation. It combines in one equation what are arguably the most important symbols and operations in mathematics. Along with the operations of addition, multiplication, and exponentiation, you have 0, the additive identity, 1, the multiplicative identity, i, the imaginary unit, and two of the most important mathematical constants, e and π. This equation is also integral to quantum computing. You will see eiθ all over the place in quantum computing, so you better get used to it!”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“There are three main ways of representing a complex number: Cartesian form (aka the general form) Polar form Exponential form”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“Even the great physicist Erwin Schrödinger was perplexed by the occurrence of complex numbers in quantum mechanics. Yet, complex numbers have been found to be inherent to quantum mechanics and, hence, quantum computing.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“A linear functional is a special case of a linear transformation that takes in a vector and spits out a scalar:”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“In quantum computing, you will rarely see the term linear transformation, but it is a common term used in mathematics. It will almost always be linear operator in quantum computing, and now you know that it is just a special type of linear transformation.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“rotations, and projections are all linear operators. In quantum, we put a "hat" or caret on the top of the letter of the linear operator when we want to distinguish it from its representation as a matrix.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“In quantum mechanics, observables such as momentum are represented by linear transformations. All of this leads to the famous uncertainty principle that states that two observables that do not commute cannot be measured simultaneously.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“What is a linear transformation? To be precise, a linear transformation is a function T from a vector space U to a vector space V. A capital letter "T" is traditionally used by mathematicians to denote a generic transformation, and we use the same syntax that we introduced for functions in Anchor 3, Foundations: Similarly, the vector space U is the domain, and the vector space V is the codomain. Each vector that is transformed is called the image of the original vector in the domain. The set of all images is the range.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“mechanics is a linear theory. That's what makes linear algebra crucial to understanding quantum computing.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“In quantum computing, we call the standard basis the computational basis, and that is the term we will use in this book.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“A set of vectors is linearly independent if they are not linearly dependent.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“If you have a set of vectors and you can create a linear combination of one of the vectors from a subset of the other vectors, then all of those vectors are linearly dependent.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“A vector space is defined as having the following mathematical objects: An Abelian group ⟨V,+⟩ with an identity element e. We call members of the set V vectors. We define the identity element to be the zero vector, and we denote this by 0. The operation + is called vector addition. A field {F, +, ⋅}. We say that V is a vector space over the field F, and we call the members of F scalars.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“A field is a set (denoted by S) and two operations (+ and ⋅) that we will notate as {S, +, ⋅}, which follows these rules: ⟨S, +⟩ is an Abelian group with the identity element e = 0. If you exclude the number 0 from the set S to produce a new set S', then ⟨S', ⋅ ⟩ is an Abelian group with the identity element e = 1. For the rule of distributivity, let a, b, c ∈ S. Then, a ⋅ (b + c) = a ⋅ b + a ⋅ c.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“A group builds upon the concept of a set by adding a binary operation to it. We denote a group by putting the set and the operation in angle brackets (⟨⟩). For example, ⟨A, ֎⟩ for set A and operation ֎. The operation has to follow certain rules to be considered a group, namely, the rules of identity, associativity, invertibility, and closure. If the operation ֎ also has the property of commutativity, then it is called an Abelian group (also known as a commutative group).”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

“Invertible functions are key in quantum computing because the laws of quantum mechanics only allow these types of functions in certain situations.”
― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities

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― Leonard S Woody III, Essential Mathematics for Quantum Computing: A beginner's guide to just the math you need without needless complexities