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Chapman & Hall/CRC eBooks

Quantum information processing 2: the quantum emulation of arbitrary Boolean functions

Helmut Bez, Tony Croft

Published Jan 6, 2023• 0 citations
Publisher

Abstract

Chapters 15 and 16 relate to the quantum emulation, and unitary extension, of a number of special invertible Boolean functions. In this chapter, we turn our attention to the quantum emulation of arbitrary Boolean functions. This extends the work of Chapters 15 and 16 in two respects: to arbitrary invertible Boolean functions, and to arbitrary non-invertible Boolean functions.

DOI
10.1201/9781003264569-17

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Chapman & Hall/CRC eBooks•Nov 28, 2023•0 citations
Chapters 15 and 16 relate to the quantum emulation, and unitary extension, of a number of special invertible Boolean functions. In this chapter, we turn our attention to the quantum emulation of arbitrary Boolean functions. This extends the work of Chapters 15 and 16 in two respects: to arbitrary invertible Boolean functions, and to arbitrary non-invertible Boolean functions.
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.
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.
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 .
This chapter introduces Boolean quantum gates that allow us to map any Boolean circuit in such a way that the circuit becomes reversible. It describes quantum gates for one qubit, like for example Clifford gates that are the elements of the Clifford group and can be efficiently simulated with a classical computer. The chapter introduces the controlled-U gates and the unitary decomposition and formulate the process of transpilation. A Boolean quantum circuit represents a permutation in Hilbert space and can be represented by a unitary permutation matrix composed of the unitary matrices representing the quantum gates. Such a mapping does not alter the distribution of the amplitudes; the distribution remains unchanged during the execution of the quantum Boolean gates.