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Chapman & Hall/CRC eBooks

Unitary extensions of the gates n o t Q , F Q , T Q and P Q : more general quantum inputs

Helmut Bez, Tony Croft

Published Jan 6, 2023• 0 citations
Publisher

Abstract

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 .

DOI
10.1201/9781003264569-16

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Invertible digital circuits and their quantum emulations

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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.

Chapman & Hall/CRC eBooks•Jan 6, 2023•0 citations

More on quantum gates and circuits: those without digital equivalents

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To this point only quantum gates derived from invertible digital counterparts or general invertible Boolean functions have been considered. Here we introduce a number of quantum gates without digital equivalents. In particular, we consider the Pauli gates, the https://www.w3.org/1998/Math/MathML" display="inline"> n o t 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/math21_1.tif "/> gate, the phase-shift gate and the Hadamard gate, some of which occur in quantum algorithms introduced later. These gates impart additional power to quantum computation.

Chapman & Hall/CRC eBooks•Jan 6, 2023•0 citations

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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.

Chapman & Hall/CRC eBooks•Jan 6, 2023•0 citations

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Chapman & Hall/CRC eBooks•Jan 6, 2023•0 citations

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Chapman & Hall/CRC eBooks•Jan 6, 2023•0 citations

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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 .
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.
To this point only quantum gates derived from invertible digital counterparts or general invertible Boolean functions have been considered. Here we introduce a number of quantum gates without digital equivalents. In particular, we consider the Pauli gates, the https://www.w3.org/1998/Math/MathML" display="inline"> n o t 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/math21_1.tif "/> gate, the phase-shift gate and the Hadamard gate, some of which occur in quantum algorithms introduced later. These gates impart additional power to quantum computation.
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.
This chapter aims to define quantum equivalents of a number of familiar digital gates. It argues that the quantum gates, with appropriately restricted state vector inputs, produce exactly the same numerical outputs as their digital counterparts. Hence, although current digital technology is not based on invertible Boolean functions, in order to emulate digital computation quantum mechanically, it is necessary to first construct invertible representations of Boolean functions. The chapter considers only some of the more familiar examples – up to and including the Peres half-adder. In the graphical representation of quantum gates and circuits, the horizontal lines indicate the time-ordered application of the unitary gates to the input qubits.
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.