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Continuous Variable Quantum Algorithms: an Introduction
Samantha Buck ∗1, Robin Coleman †1, and Hayk Sargsyan ‡2
1University of Guelph, 50 Stone Rd E, Guelph, ON. Canada
2Department of Physics, Yerevan State University, 1 Alex Manoogian, 0025 Yerevan, Armenia
Quantumcomputingisusuallyassociatedwithdiscretequantumstatesandphysicalquantitiespossessing
discrete eigenvalue spectrum. However, quantum computing in general is any computation accomplished
by the exploitation of quantum properties of physical quantities, discrete or otherwise. It has been shown
that physical quantities with continuous eigenvalue spectrum can be used for quantum computing as well.
Currently, continuous variable quantum computing is a rapidly developing field both theoretically and
1202 experimentally. Inthispedagogicalintroductionwepresentthebasictheoreticalconceptsbehinditandthe
toolsforalgorithmdevelopment. Thepapertargetsreaderswithdiscretequantumcomputingbackground,
who are new to continuous variable quantum computing.
luJ
Contents
the beauty of discreteness all-around quantum me-
chanics and forget about continuous quantum phe-
5 Introduction 1 nomena. Thereisaclearmoderntrendintransform-
ing the curricula of quantum mechanics into discrete
]hp-tnauq[
1 Quantum Continuous Variables 2 quantumcomputingbiasedones,wherediscretephe-
1.1 General Concepts . . . . . . . . . . . . 2 nomenaandcalculationsinfinitedimensionalHilbert
1.2 Eigenvalues and eigenfunctions . . . . 3 spaces are emphasized. The ”older” way of teaching
1.3 Bra-ket notation . . . . . . . . . . . . 4 based on coordinate representation wave functions,
de Broglie waves, Fourier transforms, etc. seems to
2 Hilbert Space 5 be fading away, however, this kind of knowledge is
crucial for continuous variable quantum computing
3 Quantum Harmonic Oscillator 7 (CVQC). In this pedagogical introduction we aim to
1v15120.7012:viXra
bridgethisgap. Herewepresentthebasicmathemat-
4 Continuous Variable Quantum Algo- ical concepts and tools used when dealing with con-
rithms 9 tinuousquantitiesinquantummechanicsandexplain
4.1 Grover’s algorithm. . . . . . . . . . . . 9 how to use them for CVQC algorithm development.
4.2 The Deutsch–Jozsa algorithm . . . . . 10 Manyphysicalquantities,suchaspositionandmo-
mentum or the quadratures of electromagnetic field,
5 Summary 11 can accept values from a continuous spectrum in
quantum mechanics. Due to the nature of quantum
References 12 mechanicsandHeisenberg’suncertaintyrelationsthe
precise manipulation of continuous quantum quanti-
Appendices 14
ties is fundamentally impossible. Moreover, the ex-
istence of noise in quantum systems further worsens
the situation and it seems that there is no perspec-
Introduction
tiveinusingcontinuousquantumquantitiesforcom-
putation. However, the developments in relevant ex-
Nowadays, one usually encounters quantum mechan-
perimentalrealizationsandquantumerrorcorrection
ics either because they are a physics (or related) stu-
codes have motivated the investigation of CVQC as
dent, or they got interested in quantum computing.
an independent computational paradigm. The ques-
In any case it’s extremely easy to be carried away by
tionofuniversalityofCVQCisasubtleone,however
ithasbeenaddressedintherestrictedcaseofpolyno-
∗Email: sbuck@uoguelph.ca
†Email: rcolem01@uoguelph.ca mial hamiltonians [1]. Since then, a number of algo-
‡Email: sargsyan.hayk@ysu.am rithms have been developed for CVQC. Besides the
1
algorithmsoriginallyspecifictocontinuousvariables, contrast to discrete variables which can accept only
most of the well-known quantum algorithms in dis- distinct values. An example of a continuous variable
crete quantum computing (Deutsch–Jozsa, Grover, is the x coordinate of a particle. In quantum me-
Shor,etc.) havebeenadaptedtothecontinuousvari- chanics physical quantities are represented in terms
able setting. of operators2 and their continuity properties are en-
Quantum computing, in general, is any computa- coded in the eigenvalue spectrum of these operators.
tion which is done by the exploitation of quantum If the eigenvalue spectrum of some operator is con-
properties of physical quantities, be they discrete, tinuousthenthecorrespondingphysicalquantitycan
continuousorthecombinationofboth1. Moreover,it serve as a continuous variable for computation. In
is also possible to combine systems with continuous this paper we use the term continuous variable as a
variables with traditional discrete quantum comput- collectivereferencetothephysicalquantityitself,the
ing and implement the so called hybrid computing corresponding operator and the eigenvalue.
[2, 3]. Our focus in this paper is the CVQC as an