Demonstration of teleportation across a quantum network code

2025-04-22 0 0 4.21MB 14 页 10玖币

侵权投诉

Hjalmar Rall∗and Mark Tame

Department of Physics, Stellenbosch University, Matieland 7602, South Africa

(Dated: August 12, 2024)

In quantum networks an important goal is to reduce resource requirements for the transport and

communication of quantum information. Quantum network coding presents a way of doing this by

distributing entangled states over a network that would ordinarily exhibit contention. In this work,

we study measurement-based quantum network coding (MQNC), which is a protocol particularly

suitable for noisy intermediate-scale quantum devices. In particular, we develop techniques to adapt

MQNC to state-of-the-art superconducting processors and subsequently demonstrate successful tele-

portation of quantum information, giving new insight into MQNC in this context after a previous

study was not able to produce a useful degree of entanglement. The teleportation in our demonstra-

tion is shown to occur with ﬁdelity higher than could be achieved via classical means, made possible

by considering qubits from a polar cap of the Bloch Sphere. We also present a generalization of

MQNC with a simple mapping onto the heavy-hex processor layout and a direct mapping onto a

proposed logical error-corrected layout. Our work provides some useful techniques for testing and

successfully carrying out quantum network coding.

I. INTRODUCTION

Quantum communication enables interactions between

physically distant quantum systems and opens the path

to applications such as distributed quantum computing,

quantum key distribution (QKD), and communication

within quantum processors [1–3]. Signiﬁcant progress

towards practical quantum communication has been

made in recent years: high ﬁdelity quantum commu-

nication between a single source and destination has

been realized in a number of experiments, including

free-space quantum communication [4], a large scale

QKD network with satellite link [5], a 3-node quantum

network utilising solid-state qubits [6], and research

into optical communication between superconducting

quantum computers is ongoing [7, 8]. These systems are

too small at present to be of practical use, but larger

and more complex networks are becoming feasible,

necessitating the study of quantum networks in a prac-

tical context. Of equally great importance are internal

communication networks inside quantum processors, as

the superconducting processors of Google, IBM, and

Rigetti are rapidly growing to sizes where the standard

entanglement and qubit swapping approaches become

impractical [9]. In addition, methods for external

communication networks that enable the linking up of

small quantum processors to make an eﬀective larger

processor have started to gain attention recently [10, 11],

which is relevant in light of computational techniques for

distributed processors such as entanglement forging [12].

Quantum networks have been studied at length in the

literature [13–15], but many practical issues remain, es-

pecially in the current noisy intermediate-scale quantum

(NISQ) era [16], where entanglement is imperfect and

∗hjalmar.rall1@gmail.com

many rounds of puriﬁcation may be required to achieve

a suﬃcient degree of entanglement. Thus the pre-shared

entanglement required for a teleportation-based network

is diﬃcult to establish and the bandwidth of the network

may be severely limited. Quantum networking is further

restricted by the need for quantum routing which takes

up valuable resources in terms of the number of qubits

required and also introduces additional noise into the

system. It is therefore necessary to ﬁnd eﬃcient schemes

for quantum networking with limited qubit number and

bandwidth. A solution to this is provided by quantum

network coding (QNC) [17, 18]. In classical networks

with limited bandwidth, network coding [19–21] solves

the problem of contention by encoding messages which

must pass through bottlenecks and using uncontended

channels to send decoding instructions. In certain

communication scenarios classical network coding can

utilise all available bandwidth for useful communication

despite the presence of bottlenecks. QNC mimics the

classical case in that it makes use of local operations to

achieve simultaneous transmission of messages through

a bottleneck in a quantum network. In contrast to

the classical case, this is achieved by redistributing the

available channels (entanglement) so as to eliminate the

bottleneck entirely.

QNC has experienced considerable interest since its

introduction in 2007 [22], and has been studied both as

a theoretical tool and as a practical protocol in quantum

networks and processors [18, 23]. It has also recently

been demonstrated experimentally in an optical setup

[24]. Measurement-based quantum network coding

(MQNC) [25] is a very recent development which is

well-suited to the NISQ regime by virtue of requiring

shallower circuits than existing QNC protocols. As

a result of shorter circuit depth, the eﬀect of qubit

loss, gate errors, and qubit decoherence is reduced.

MQNC has previously been studied on an IBM Q

superconducting processor by Pathumsoot et al. [26],

arXiv:2210.02878v2 [quant-ph] 9 Aug 2024

but the study was severely limited by the high degree

of noise in the processor and did not realize quantum

communication over the network code. In the time

since, IBM has made new processors available, with

greatly reduced noise, and a standardised layout which

it is said will remain ﬁxed for the forseeable future. It

is of interest then, to see how MQNC performs on this

new hardware, and what further insight can be gained

into practical implementaiton of MQNC beyond general

predictions made in the previous work. We overcome the

challenges of translating MQNC to the new processor

layout, and show that - even with the extra overhead

incurred - genuine quantum information transfer using

teleportation over an MQNC network is possible on

these processors, provided that the input states are

restricted to a polar cap of the Bloch sphere, as in a

recent theory proposal by Roy et al. [27].

With a view to the future, we also present a general-

isation of butterﬂy MQNC to a non-blocking network

switch with an arbitrary number of nodes. Interestingly,

the switch may be created directly on square grid

topologies which are already in use or planned for use

in superconducting quantum processors - examples

include the Google Sycamore processor [28] and the

planned error-corrected logical topology of the IBM

processors [29]. While this is important for transferring

quantum information within processors, it also has

implications for networking within a quantum inter-

net, where switching within quantum routers [30] is

essential if entanglement between arbitrary pairs of end

nodes is to be established using a shared physical link

layer instead of private direct connections between nodes.

The paper is structured as follows: In section II we give

a brief overview of measurement-based quantum com-

puting, QNC, and MQNC, and introduce the generalised

switch. In section III we present the particulars of the

protocol and the method used to adapt the previous work

to a newer processor. Section IV forms the main body,

where the results of teleportation using MQNC are pre-

sented. Section V presents a general mapping of MQNC

onto IBM processors. We end with a discussion and con-

cluding remarks in Section VI.

II. BACKGROUND

A. Measurement-Based Quantum Computing

A graph state is an entangled state |G⟩with qubits

and entanglement between qubits corresponding to the

vertices and edges of an undirected graph G= (V, E). An

N-qubit graph state with edge set Eis deﬁned according

|G⟩=Y

{i,j}∈E

CZi,j |+⟩⊗N,(1)

where CZij is the controlled phase operation on qubits i

and j, and |+⟩is the Pauli-Xeigenstate with eigenvalue

+1. A number of quantum operations transform between

graph states, and can thus be viewed as operations on

the underlying graph. The following are the most

common transformation rules that will be used in this

work [31]:

•T1: A Z-basis measurement on a qubit aremoves

the corresponding vertex and incident edges from

the graph.

•T2: A Y-basis measurement on a qubit aremoves

the corresponding vertex and incident edges, and

complements the subgraph induced by the neigh-

bourhood Na. i.e. G(V, E)→G(V /{a}, E∆K)

with Kthe edge set for the complete subgraph in-

duced by Na∪{a}and ∆ the symmetric diﬀerence.

In other words, neighbours of aare connected un-

less a connection already exists, in which case it is

broken.

•T3: X-basis measurements on two adjacent qubits

aand bremoves them and complements the bi-

partite subgraph induced by Naand Nb. i.e.

G(V, E)→G(V/{a, b}, E∆K) with Kthe com-

plete bipartite subgraph induced by Na∪ {a}and

Nb∪{b}. In other words, all the neighbours of aare

connected to all the neighbours of bunless a con-

nection already exists, in which case it is broken.

•T4: Local complementation on a qubit agiven by

√XaQb∈Na√Zb|G⟩complements the subgraph in-

duced by Naand leaves aand its incident edges un-

changed. In other words, neighbours of aare con-

nected unless a connection already exists, in which

case it is broken. This is a non-destructive ver-

sion of the Y-measurement transformation rule T2

where the qubit ais not removed from the graph.

It should be noted that Pauli byproducts are intro-

duced for certain measurement outcomes for the above

operations. These must be tracked by a classical com-

puter and either adaptively corrected or commuted

through to the end of the circuit before being corrected

(possibly via post-selection). For an in-depth discus-

sion of graph states we refer the reader to Ref. [31].

Graph states form a resource for measurement-based

quantum computation (MBQC) [32, 33], many elements

of which are used in MQNC. An arbitrary single-qubit

state |ψ⟩may be attached to a graph state using a

controlled phase gate. This state may subsequently be

transported within the graph by means of quantum tele-

portation along linear sections. An example of this is

shown in Fig. 1(a): First a 2-qubit (linear) graph state

√2(|+0⟩+|−1⟩)12 is created, then the state |ψ⟩0to be

teleported is entangled with this state via a controlled

phase gate, and ﬁnally the measurements Mx,0and Mx,1

in the basis {1

√2(|0⟩+|1⟩),1

√2(|0⟩−|1⟩)}with outcomes

FIG. 1. (a) A state |ψ⟩may be teleported across a lin-

ear section of a graph state. (b) Teleportation can be used

to perform the operation HRZ(−θ). X’s within qubits indi-

cate X-measurements, θindicates a measurement in the basis

√2(|0⟩+eiθ |1⟩),1

√2(|0⟩ − eiθ |1⟩)}.

s0and s1, respectively are performed. This yields the

state Xs1

2Zs0

2|ψ⟩2which may be transformed back to |ψ⟩

if the measurement outcomes are known.

Arbitrary unitary operations may be performed on a

single-qubit state attached to a graph state by way of ap-

propriate projective measurements, which serve to both

teleport and transform the single-qubit state. An exam-

ple is shown in Fig. 1(b): First the state |ψ⟩0is entangled

with the graph state (here the single qubit graph |+⟩1)

via a controlled phase gate, then a measurement in the

basis {1

√2(|0⟩+eiθ |1⟩),1

√2(|0⟩−eiθ |1⟩)}(with θan arbi-

trary angle) is performed on qubit 0. This yields the state

Xs0

1H1RZ(−θ)1|ψ⟩1. Such operations may be composed

to obtain arbitrary unitary operations. Given a suﬃ-

ciently large 2D grid (cluster) graph state, measurement-

based computation is universal. Since all two-qubit oper-

ations are performed during the creation of the resource

state, and they are all commutative, they may be done

simultaneously if the hardware allows. Since all byprod-

ucts are Pauli operators, they can be commuted through

to the end of the circuit either directly or adaptively and

subsequently simpliﬁed, leading to further decreases in

circuit depth.

B. Quantum Network Coding

Quantum network coding is best illustrated through

the example of the butterﬂy network. Given the network

shown in Fig. 2(a) with each channel having capacity

1, the goal is to simultaneously send qubits from S1to

D2and from S2to D1. In Ref. [22] it is shown that

it is not possible to do this perfectly if only quantum

communication is allowed. It was later shown that

perfect quantum network coding is possible if free

classical communication is allowed [18], as shown in

Fig. 2(b). Protocols for perfect QNC have been devel-

FIG. 2. The butterﬂy network (a), and the new network after

performing QNC (b). Blue shading represents network nodes.

Lines represent quantum communication channels. Classical

communication is assumed to be free.

FIG. 3. The procedure for MQNC starting with a 6-qubit

graph state. (a) and (b) show the two diﬀerent conﬁgurations

(cross pairs and straight pairs) of MQNC. Circles and lines

(which are vertices and edges of graphs, respectively) repre-

sent qubits and entanglement between qubits.

oped for the case where transmitters share entanglement

[17] and for the case of the butterﬂy network across

quantum repeaters [23]. The former protocol has been

demonstrated experimentally in an optical setup [24]

with ﬁdelity suﬃcient to enable teleportation of quan-

tum information with ﬁdelity exceeding the classically

achievable bound. These protocols however require

complex circuits and additional steps for resource state

creation. On the other hand, MQNC [25] presents a

measurement-based alternative to the repeater network

protocol with a reduction in circuit depth of 50%

and a corresponding increase in the allowable gate

error to achieve a speciﬁed ﬁdelity. Furthermore, this

protocol contains as an intermediate step a graph state

which also has applications in on-processor teleportation.

The protocol proceeds as follows: Starting with seven

bell pairs, a six-qubit graph state is generated as shown

in Fig. 3(a) on the left hand side. This state may also be

generated directly via controlled phase gates according

to Eq. (1). By measuring the two central qubits in

the X-basis (Fig. 3(a) middle) the entanglement in the

graph state can be redistributed so as to give two cross

pairs (up to Pauli byproducts) using transformation rule

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：

DemonstrationofteleportationacrossaquantumnetworkcodeHjalmarRall∗andMarkTameDepartmentofPhysics,StellenboschUniversity,Matieland7602,SouthAfrica(Dated:August12,2024)Inquantumnetworksanimportantgoalistoreduceresourcerequirementsforthetransportandcommunicationofquantuminformation.Quantumnetworkcodingp...

展开>> 收起<<

Demonstration of teleportation across a quantum network code.pdf

共14页,预览3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！

Demonstration of teleportation across a quantum network code

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: