Vector spaces are the mathematical structures in which the state vectors of quantum computation live. When endowed with an inner product, a vector space becomes an inner product space. Of particular relevance are complex inner product spaces. An objective of this chapter is to provide a formal definition of a vector space and give several examples. You will see that, in this context, a ‘vector’ is a much more general object than vectors such as those in https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> ℝ 3 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math6_2.tif "/> . However, the already familiar operations of vector addition and scalar multiplication are essential ingredients of all vector spaces. We demonstrate how vectors in a vector space can be constructed using a set of vectors called a basis and see how vectors can be written in terms of different bases. Of particular importance are orthonormal bases in which the basis vectors are mutually orthogonal and normalised. Finally, we introduce cosets and explain how a quotient space is constructed.
From here, we take up the task of reformulating quantum mechanics in the language of inner product space. We start by introducing vector space through the familiar example of a set of finite dimensional, complex, column matrices under addition and multiplication by scalars. The inner product is then introduced as a dot product generalized to complex vectors leading to the idea of an inner product space . This is adequate to show how the concept of a quantum state can be described by a normalized vector in an inner product space so that the state space can be identified with the inner product space. Amplitudes are associated with inner products , and observables with orthonormal bases in this space. Hence the laws of quantum mechanics are reformulated. We then point out that the now reformulated laws are independent of the space of description and the reader is urged to contemplate an universality which would reflect the independence with respect to the representation space . This creates the stage for the introduction of the abstract inner product space and the formulation of quantum mechanics in this setting. The chapter concludes with this description. The transformed laws are now called postulatesof quantum mechanics .
In Chapter 15 , quantum emulations of some familiar invertible digital gates were discussed; specifically the not , cnot , Toffoli and Peres gates, each of which is significant in quantum computation, were considered. The emulations were defined on basis elements only – i.e., on https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_3.tif "/> or https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_4.tif "/> or https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_5.tif "/> etc. On these restricted inputs the quantum emulations were seen to produce outputs identical to their digital equivalents. Unlike the digital gates the quantum emulations may be extended beyond the basis sets to the entire tensor product spaces so defined – i.e., to https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 , ℂ 2 ⊗ ℂ 2 , ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_6.tif "/> etc. Such generalised inputs to the quantum gates lead to more general quantum outputs to which the simple deterministic quantum measurement processes of Chapter 15 are not applicable. The quantum gates defined on the extended domains are shown to be unitary, and are denoted in the same way as their restrictions to the basis vectors. We first consider a sufficient condition for an operator to be unitary; it has applications in our discussions below and in Chapter 17 .
In this chapter, the simplest quantum measurement situation, i.e., where the state is an eigenvector of the observable to be measured, is considered. The states are therefore non-superimposed and measurement Axiom 3.2 of Section 13.7 in the previous chapter is applicable. We consider states in https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 , ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_1.tif "/> and https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_2.tif "/> . The generalisation to https://www.w3.org/1998/Math/MathML" display="inline"> ⊗ n ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math14_3.tif "/> is then clear.
Vectors are reviewed since they are extensively used in electric and magnetic field calculations and in quantum mechanics. A general definition of vectors is given followed by the more familiar Euclidean definition. The algebra of vectors is discussed including their addition, subtraction, multiplication and group characteristics. The matrix and Dirac notation for vectors is introduced and used to present vector transformations via operator matrices. A short description of Hilbert space and its fundamental definition is given. The merging of vectors spaces is discussed in preparation for the development of multiple-spin density matrices.
Black box representations of functions, by definition, do not provide information on how a function, so represented, is computed. By a ‘circuit’ for the computation of a function we mean a sequence of ‘basic’ gates that implement the function. It can be shown that, given a classical circuit for the Boolean function https://www.w3.org/1998/Math/MathML" display="inline"> f ^ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math18_1.tif "/> , there is a quantum circuit of equal ‘efficiency’ for the implementation of the unitary operator https://www.w3.org/1998/Math/MathML" display="inline"> U f ^ https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math18_2.tif "/> and hence the quantum computation of f . In this chapter we consider examples of invertible digital functions and their evaluation using basic invertible Boolean gates; specifically the Toffoli and Feynman gates T D and F D . We recall that the Toffoli gate is complete for invertible digital computation, however, the inclusion of the Feynman gate, F D , simplifies the circuits. We proceed by example to demonstrate the process and consequential side-effects of such implementations and their quantum emulations. It is seen that the cost of using only Toffoli gates is that additional input bits/qubits may be required, and the computation of intermediate results (usually referred to as garbage or junk) may be necessary. We also consider a general circuit for re-setting registers containing ‘junk’, this is helpful to avoid problems with downstream computation.
Vectors are reviewed since they are extensively used in electric and magnetic field calculations and in quantum mechanics. A general definition of vectors is given followed by the more familiar Euclidean definition. The algebra of vectors is discussed including their addition, subtraction, multiplication and group characteristics. The matrix and Dirac notations for vectors are introduced and used to present vector transformations via operator matrices. A short description of Hilbert space and its fundamental definition is given. The merging of vector spaces is discussed in preparation for the development of multiple-spin density matrices.