Classiﬁcation of Hybrid Quantum-Classical Computing Frank Phillipson

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Classiﬁcation of Hybrid Quantum-Classical

Computing

Frank Phillipson

TNO

Dpt. Applied Cryptography & Quantum Algorithms

The Hague, The Netherlands

Maastricht University

School of Business and Economics

Maastricht, The Netherlands

Niels Neumann and Robert Wezeman

TNO

Dpt. Applied Cryptography & Quantum Algorithms

The Hague, The Netherlands

Abstract—As quantum computers mature, the applicability

in practice becomes more important. Many uses of quantum

computers will be hybrid, with classical computers still playing

an important role in operating and using the quantum computer.

The term hybrid is however diffuse and multi-interpretable. In

this work we deﬁne two classes of hybrid quantum-classical

computing: vertical and horizontal. The ﬁrst is application-

agnostic and concerns using quantum computers. The second

is application-speciﬁc and concerns running an algorithm. For

both, we give a further subdivision in different types of hybrid

quantum-classical computing and we coin terms for them.

Index Terms—hybrid quantum computing, classiﬁcation, hy-

brid quantum algorithm, workﬂow

I. INTRODUCTION

Quantum computing is the technique of using quantum

mechanical phenomena such as superposition, entanglement

and interference for doing computational operations. The type

of devices which are capable of doing such quantum opera-

tions are still being actively developed and named quantum

computers. We distinguish between two paradigms of quan-

tum computing devices: gate-based quantum computers and

quantum annealers. A gate-based quantum computing system

uses basic quantum circuit operations on qubits, similar to the

classical operations on regular bits, that can be put together in

any sequence to form algorithms. A quantum annealer brings

a collection of qubits into an equal superposition and then

applies a problem speciﬁc magnetic ﬁeld. The qubits will

interact under this magnetic ﬁeld and move towards the state

with the lowest energy, which encodes the solution of an

optimisation problem.

Theory predicts that quantum computers will solve spe-

ciﬁc problems much faster than classical computers. Where

classical computers have been under development for decades

and are therefore quite mature, quantum computers are still in

the early stages of development. They are not yet capable of

solving real world problems, due to the low number of qubits

and their unstable nature causing noise, errors and loss of

information. This current state of quantum computers is called

the noisy intermediate-scale quantum (NISQ) era introduced

by Preskill [1], where quantum computers have 50-200 qubits

and their noise places serious limitations on their capabilities.

Only recently IBM passed the 100-qubit barrier for the ﬁrst

time with its 127-qubit Eagle processor and they plan a 1,000-

qubit chip, the Condor, in 2023 [2].

Other aspects than number of qubits affect the capabilities

of quantum computers in practice, such as parallelism of

operations and the topology layout of the qubits. Researchers

are investigating innovative ways to solve valuable problems

using already available NISQ systems and to achieve quantum

advantage by demonstrating a signiﬁcant performance advan-

tage over today’s classical computers. The latter was only quite

recently claimed for the ﬁrst time for an artiﬁcially created

problem [3]. As quantum computers mature in the coming

years, their computational power will increase and they can be

applied in more settings and actually provide help in speciﬁc

practical areas.

For the ﬁrst practical applications in the near future, quan-

tum computers will only execute a small part of a larger

total workﬂow where a classical computer executes the other

steps. See for example the workﬂow of retrieving data, training

a classiﬁcation algorithm and evaluating the obtained classi-

ﬁer [4], as shown in Fig. 1. In this example only a small part of

the workﬂow is performed on a quantum computer. Already

in 2005 such a combination was described in literature and

was named hybrid quantum computing that “combine both

classical and quantum computing architectures in order to

leverage the beneﬁts of both” [5].

Murray Thom, the Vice President of Software and Cloud

Services at D-Wave, compared this with jets and ”normal”

vehicles: ”Consider that while jet airplanes transformed the

way we travel long distances, we still need vehicles that take

us to our front door.” Thus, ”quantum applications will always

and only be hybrid” [6].

This term, hybrid quantum-classical computing, was and is

used in various contexts, each time used for (slightly) different

settings. This is confusing for many researchers and practi-

tioners in quantum computing, both on the application side

as on the hardware side. The interest in quantum computing

arXiv:2210.15314v1 [quant-ph] 27 Oct 2022

causes an enormous amount of research output1and, resulting,

survey papers. One example of this kind is [7], where a

survey of NISQ era hybrid quantum-classical machine learning

research on hybrid quantum-classical systems is given, without

explaining what is meant by hybrid quantum-classical exactly.

This is totally left to the reader. Our goal is to provide a

classiﬁcation framework where authors can refer to, to make

the scope of their contribution more clear to the reader. For

this, we connect quantum research with classical computer

science and workﬂow terminology. So, many of the proposed

terminology in this paper exists already in computer sci-

ence, think of compilation between computer languages, and

workﬂow research, think of decomposition and activities. Our

contribution is bringing them together in a clear framework

for hybrid quantum-classical computing.

In this contribution we aim to describe clearly the various

contexts hybrid quantum-classical computing can have in

literature and to name these different approaches clearly and

appropriately. In Section II we give an overview of various

forms of hybrid quantum-classical computing that can be

found in literature. Next, in Section III, we distinguish a num-

ber of different types of hybrid quantum-classical computing

from this overview and provide examples for each type. We

end this paper with some conclusions and ideas for further

research.

II. LITERATURE

We consider a global situation where we have a collection

of computational tasks in which both the quantum computer

and the classical computer are used. As such, hybrid forms of

computing that allow for both discrete and continuous vari-

ables [8] and hybrid quantum-classical models of molecules

in chemical and biological studies [9] are out of scope. We

do not try to give an exhaustive overview of all research done

on this topic. Our goal here is to give an overview, based on

some examples, of the various meanings of the term hybrid

quantum-classical computing in literature. This overview will

be the basis of the proposed classiﬁcation later on in this work.

Lanzagorta and Uhlmann presented one of the ﬁrst hybrid

algorithms that used both classical and quantum computers [5].

Later, research appeared on computing schemes and architec-

tures to optimise the interactions between the different type

of computers when executing hybrid quantum algorithms. A

ﬁrst example presents a candidate framework to analyse hybrid

computations by fully integrating the quantum and classical

resources and processes used for measurement-based quantum

computing, where the feed-forward of classical measurement

results is an integral part of the quantum design [10], [11]. A

second example proposes a quantum co-processor to accelerate

a speciﬁc subroutine of a larger task. This is most often seen

as the main reason for hybrid algorithms, for example in [12]–

[15]. The work by Li et al. [14] results in a system-level soft-

ware infrastructure for hybrid quantum–classical computing.

1Google scholar already gives 2090 results for the search on ’hybrid

quantum-classical computing’ for the period January-October 2022.

Endo [16], [17] indicates that for early quantum applications,

a large portion of the computational burden is performed on

a classical computer and hence fully coherent deep quantum

circuits are not required. As the quantum computer takes on

more computational load, noise of the quantum computer will

result in more errors, which will have more impact on the total

calculation. This in turn requires qubits of higher quality and

error mitigation routines to suppress noise.

An important type of hybrid computing appeared with

the introduction of Variational Quantum Algorithms (VQA).

VQAs use a classical optimiser to train a parameterised quan-

tum circuit and provide a framework to tackle a wide array of

tasks, as shown in the extensive overview by Cerezo et al. [18].

Examples include ﬁnding ground states and excited states of

molecules, optimisation, solving linear systems of equations

and machine learning. The ﬁrst VQA, the variational quantum

eigensolver (VQE) algorithm of [19], appeared in 2014. Some

papers see this group of algorithms as “a novel class of hybrid

quantum-classical algorithms”, without explicitly diving in

other groups of algorithms [20]. Also, [21] sees this class of

algorithms as a speciﬁc example of hybrid quantum-classical

computing in a noisy environment. In [22] the main reason

given for hybrid quantum-classical computing is the size of

the problems in combination with the available hardware. They

distinguishes two types of hybrid computing: 1) Pre- or post-

processing of a quantum computation on a classical computer.

Examples are algorithms by Shor [23] and Simon [24] that use

classical post-processing. 2) Algorithms that perform multiple

iterations of quantum and classical computations. Thereby, the

output of the quantum computation is improved in each itera-

tion until the result reaches the required accuracy. An example

they give using this approach is the quantum approximate

optimisation algorithm (QAOA) [25].

In the works [20], [26], the term ‘quassical’ computing is

coined and motivated by “Classical computing and quantum

computing have obvious complementary strengths, so instead

of opposing them it might be better to combine them into

a new type of computing.” They give two reasons for the

combination: First, most quantum computing algorithms “re-

quire some preliminary classical pre-processing to shape the

problem into one the quantum computer can recognise and

then to receive the data returned by the [quantum computer]

and shape it into the answer the engineer needs.” Second,

“all the quantum computers we have heard of are designed

as cyber-physical systems, quantum mechanical systems con-

trolled by digital controllers”, meaning they are quassical in a

trivial sense. In this light, you can also think of the classical

steps needed to transform a quantum circuit to an execution

as performed for example by an openQL framework [27].

They expect that this combination will stay, also when the

quantum computer is in full maturity. The ﬁrst remark is also

mentioned by [28], who indicate that “while hybrid algorithms

and platforms may just be the best ﬁrst step, it is reasonable to

assume that quantum applications will always be hybrid”, for

example, by the need of a pre-processing step which prepares

data for a quantum algorithm or a post-processing step which

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ClassicationofHybridQuantum-ClassicalComputingFrankPhillipsonTNODpt.AppliedCryptography&QuantumAlgorithmsTheHague,TheNetherlandsMaastrichtUniversitySchoolofBusinessandEconomicsMaastricht,TheNetherlandsNielsNeumannandRobertWezemanTNODpt.AppliedCryptography&QuantumAlgorithmsTheHague,TheNetherlandsAbs...

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