Q2Graph a modelling tool for measurement-based quantum computing Greg Bowenand Simon Devitty Centre for Quantum Software and Information University of Technology Sydney Sydney NSW 2007 Australia

2025-05-02 0 0 780.83KB 9 页 10玖币

侵权投诉

Q2Graph: a modelling tool for measurement-based quantum computing

Greg Bowen∗and Simon Devitt†

Centre for Quantum Software and Information, University of Technology Sydney, Sydney, NSW 2007, Australia

(Dated: October 4, 2022)

The quantum circuit model is the default for encoding an algorithm intended for a NISQ computer

or a quantum computing simulator. A simple graph and through it, a graph state - quantum state

physically manifesting an abstract graph structure - is syntactically expressive and tractable. A

graph representation is well-suited for algorithms intended for a quantum computing facility founded

on measurement-based quantum computing (MBQC) principles. Indeed, the process of creating an

algorithm-speciﬁc graph can be eﬃciently realised through classical computing hardware. A graph

state is a stabiliser state, which means a graph is a (quantum) intermediate representation at all

points of the algorithm-speciﬁc graph process. We submit Q2Graph, a software package for designing

and testing of simple graphs as algorithms for quantum computing facilities based on MQBC design

principles. Q2Graph is a suitable modelling tool for NISQ computing facilities: the user is free to

reason about structure or characteristics of its graph-as-algorithm without also having to account

for (quantum) errors and their impact upon state.

I. INTRODUCTION

Quantum computing is an evolutionary step, to the

extent that it exhibits solutions to some well-structured

computational problems that remain prohibitive, even in

theory, to classical facilities. It is the diﬀerence in fun-

damentals of the two computing models that accounts

for much of the distance between classical and quantum

computing capabilities. Unlike the voltages that deter-

mine a bit in classical computing, quantum computing

derives from properties of an elementary particle and its

mathematical abstraction, the qubit (Cf. [1]). As such,

a qubit presents as a two-state system with an orthonor-

mal basis, within a complex 2n-dimensional vector space.

The qubit as a vector can be manipulated in accordance

with standard linear algebra and as implied by its two-

state system, a qubit of the ‘computational basis’ of |0i

and |1icorresponds to the binary bit-state capabilities

(i.e. {0,1}) of classical computing. The reader is referred

to [2] for a comprehensive overview of both the notations

and the linear algebra to appear in this article.

The state vector of a qubit’s orthonormal basis marks

the point of departure of quantum- from classical com-

puting. Expressing an arbitrary two-state system, |ψiin

the computational basis gives,

|ψi=α|0i+β|1i,(1)

by which the amplitudes α,βare complex numbers and

by virtue of orthonormality,

|α|2+|β|2= 1.(2)

This third, ‘superposition’ state of a qubit as represented

by (1) has no equivalent in classical computing insofar as

it describes a linear combination of |0iand |1i.

∗gregory.bowen@student.uts.edu.au

†simon.devitt@uts.edu.au

As with superposition, the property of entanglement

is a feature of quantum computing not found in the bi-

nary state of classical computing. An entangled pair of

particles is an inseparable state even after subsequent

transformation of the state or physical relocation of ei-

ther particle. Of more direct relevance, entanglement of

qubits is a precondition to the property of quantum tele-

portation in computational processes, which is laid out

in Section II, below.

The future of quantum computing appears promising

although the reality of those quantum computing facili-

ties presently available is more nuanced. Quantum com-

puting facilities as at December 2021 are classed as noisy

intermediate-scale quantum (NISQ) computers and are

characterised both by low counts of physical qubits (n

≤130) [3–5] and sensitivity to errors arising from en-

vironmental decoherence [6, 7]. Furthermore, real-time

access to a NISQ computing facility is limited to services

provided by a few cloud-based vendors. The average user

must accept the risk of realising an error(s) when pro-

cessing an algorithm on a NISQ computing facility while

accruing a ﬁnancial cost.

A number of oﬀerings are available to simulate the

workings of an idealised quantum computer on a clas-

sical computer, commonly in the form of a software de-

velopment kit (SDK) or a platform as a service (PaaS)

oﬀering[8]. Due to their ready availability, these ‘simula-

tors’ are in the forefront of experimentation in quantum

computing: the user of a given simulator can expect to

prepare the state of (simulated) qubits and transform

those qubits by means of standard operators to then ob-

serve an output free of errors. It is noteworthy that

the format of passing instructions to a NISQ computer

or its simulators is far closer to the punch-cards asso-

ciated with classical mainframes of the 1940s through

1960s than to the familiar, human-friendly programming

languages dominant since the 1970s. Indeed, a circuit

format has become the default for encoding an algorithm

intended for a NISQ computer or a quantum computing

simulator. To be sure, some simulator oﬀerings have co-

arXiv:2210.00657v1 [quant-ph] 3 Oct 2022

opted such programming languages as Python so that the

user can organise and approximate its (quantum) com-

putation yet such language extensions are an aid to the

(classical computing) user transitioning to the quantum

computing paradigm. The computing model founded on

the (quantum) circuit, as proposed by [9] dominates those

simulator resources presently available.

A circuit comprises of one or more qubits as input,

which are transformed through the use of single- or

multi-qubit operators collectively referred to as quan-

tum logic gates; the output qubits are speciﬁed or ‘mea-

sured’ (see section 2, below) only once all transforma-

tions are completed. The (quantum) circuit model there-

fore is the method of computation consistent with the

circuit as algorithm: as a method, the quantum circuit

model ‘builds up’ output through the aggregated trans-

formations of input qubits. It is not the case, how-

ever, that the circuit model is the sole option for ex-

ecuting a quantum computation: an alternative model

in measurement-based quantum computing (henceforth,

MQBC) proceeds through single-qubit operations ap-

plied in a certain order and basis to an agglomeration

of qubits formed through quantum entanglement, known

as a ‘cluster state’ [10]. In contrast to the circuit model,

MBQC can be thought of as ‘emerging’ the desired out-

put from a cluster state [11]. The cluster state requires

no additional input qubits to establish a (quantum) state

[12]. Revisiting its purpose as an algorithm, the circuit

is a complicated ﬁt for the MBQC model: a circuit can

be adapted to MBQC but only after deconstructing each

multi-qubit gate to its single-qubit equivalent then se-

curing the output through measurement. In brief, the

circuit as syntax is ineﬃcient for encoding an algorithm

under the MBQC model.

In light of the ineﬃciency of adapting the circuit to the

MBQC model the question becomes, what is an eﬃcient

algorithm representation for the MBQC model? Under

the circuit model, individual qubits are the resource from

which a computation is built up by means of single- or

multi-qubit transformations of individual qubits. In con-

trast, MBQC proceeds through single-qubit operations

that transform/consume a cluster state resource. This

cluster state is therefore pivotal to the question of eﬃ-

ciency in encoding an algorithm for the MBQC model. It

turns out that a subgroup of the quantum state, known

as a graph state has a syntax, in the form of graphs, that

can model a cluster state. A graph, as an expression of

graph state, can eﬃciently implement a prescribed set of

transformations of a cluster state. In eﬀect, the graph

as an algorithm is to MBQC what the circuit is to the

circuit model.

We submit Q2Graph (https://github.com/QSI-

BAQS/Q2Graph), a software package for the design

and testing of a graph as an algorithm for MQBC.

Q2Graph is a standalone executable and draws upon

(simple) graphs as proxies of cluster states to fulﬁl its

requirements as a design tool. Q2Graph diﬀers from

other well-known graph visualisation/design packages

(e.g. [13–18]) by its primary purpose namely, the design

and modelling of a graph as an algorithm for an MBQC

system[19]. More speciﬁcally, the modelling functionality

of Q2Graph includes the ability to reproduce the eﬀect

of single-qubit operations upon the structure of a graph.

The remainder of this article will serve as a detailed

introduction to version 0.1 of Q2Graph. Section II is

a review of MBQC, graphs and graph states, which,

together, form the theoretical underpinnings of the

graph as a unit of instructions for MBQC. With the

previous section for context, Section III is a review of

the design principles that inform Q2Graph, to stimulate

potential use cases for the application. In particular,

the algorithm-speciﬁc graph is introduced as a model to

bridge between classical- and NISQ-computing facilities.

Section IV is a review of Q2Graph functions with an

emphasis upon how a user might use the package to

model a quantum algorithm. The article concludes with

Section V, including some comments on next steps for

development of Q2Graph version 0.2.

II. MEASUREMENT-BASED QUANTUM

COMPUTING, GRAPHS AND GRAPH STATES

Measurement-based quantum computation emerges a

computation from a resource of entangled qubits (‘clus-

ter state’) by transforming single qubits, in a certain or-

der and basis. In preparing the cluster state (denoted

as |ΦiC), each of its qubits, except those designated as

input, are initialised at state |+i. Following on from

initialising the component qubits is the process of creat-

ing quantum entanglement, which is necessary to creating

|ΦiC. As implied by its title, |ΦiCis a composite or mul-

tipartite system of n-interacting qubits. Ordinarily the

state space of a composite physical system (say, |Ψi) is

obtained as the tensor product of its component qubits,

thus,

|Ψi≡|ψ1i⊗|ψ2i ⊗ ... ⊗ |ψni.(3)

In contrast, applying a controlled-Z (‘CZ’) gate between

qubits |iiCand |jiCtransforms to the state space |ijiC

not |iiC|jiC[20, 21] as would be consistent with (3).

This ‘entangled’ state space of |ijiCis inseparable, there

is no observable point of diﬀerence in state between its

component qubits. By deﬁnition then,

|ΦiC6=|iiC⊗ |jiC⊗... ⊗ |niC.(4)

Multipartite states like |ΦiCare eﬃciently described

by the stabiliser formalism (see [1, 22–24]). A stabiliser

state is an eigenvector of an operator, Uwith eigenvalue

+1 (i.e. U|ψi=|ψi). In every instance of a multipar-

tite state, Uwill include operators of the Pauli group,

PN:={±I,±iI,±X,±iX,±Y,±iY,±Z,±iZ}N. More

will be made of both stabiliser states and Pbelow, in

the context of graphs and graph states.

So much for deﬁning the state space of |ΦiC. Com-

putation in MBQC now proceeds through measurement,

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Q2Graph:amodellingtoolformeasurement-basedquantumcomputingGregBowenandSimonDevittyCentreforQuantumSoftwareandInformation,UniversityofTechnologySydney,Sydney,NSW2007,Australia(Dated:October4,2022)ThequantumcircuitmodelisthedefaultforencodinganalgorithmintendedforaNISQcomputeroraquantumcomputingsimul...

展开>> 收起<<

Q2Graph a modelling tool for measurement-based quantum computing Greg Bowenand Simon Devitty Centre for Quantum Software and Information University of Technology Sydney Sydney NSW 2007 Australia.pdf

共9页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Q2Graph a modelling tool for measurement-based quantum computing Greg Bowenand Simon Devitty Centre for Quantum Software and Information University of Technology Sydney Sydney NSW 2007 Australia

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: