Quantum Network Utility A Framework for Benchmarking Quantum Networks Yuan Lee1Wenhan Dai2 3Don Towsley3and Dirk Englund1 4 1Department of Electrical Engineering and Computer Science

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Quantum Network Utility: A Framework for Benchmarking Quantum Networks

Yuan Lee,1, ∗Wenhan Dai,2, 3, †Don Towsley,3and Dirk Englund1, 4, ‡

1Department of Electrical Engineering and Computer Science,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Quantum Photonics Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

3College of Information and Computer Sciences,

University of Massachusetts, Amherst, MA 01003, USA

4Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Dated: October 14, 2022)

The absence of a common framework for benchmarking quantum networks is an obstacle to

comparing the capabilities of diﬀerent quantum networks. We propose a general framework for

quantifying the performance of a quantum network, which is based on the value created by con-

necting users through quantum channels. In this framework, we deﬁne the quantum network utility

metric UQN to capture the social and economic value of quantum networks. While the quantum

network utility captures a variety of applications from secure communications to distributed sensing,

we study the example of distributed quantum computing in detail. We hope that the adoption of

the utility-based framework will serve as a foundation for guiding and assessing the development of

new quantum network technologies and designs.

I. INTRODUCTION

Quantum networks transmit quantum information be-

tween quantum systems separated by large distances, en-

abling applications like quantum cryptography and quan-

tum sensing that are not possible with classical communi-

cation networks alone [1,2]. Eﬀorts are underway across

the globe to develop the cornerstones of such quantum

networks, with the goal of distributing quantum infor-

mation between quantum memories. These eﬀorts are

diverse, employing a range of protocols and hardware

to support long-distance, unconditionally secure com-

munication, precision sensing/navigation and distributed

quantum computing. On-demand quantum entangle-

ment is now possible between separated quantum memo-

ries [3] and quantum memories have been shown to oﬀer

a clear advantage in the quantum secure information ca-

pacity compared to direct transmission over equivalent

loss channels [4]. Recent theory established the secret

key capacity for arbitrary quantum communication net-

works [5,6], and various quantum network routing proto-

cols are being developed [7–10] to try and approach these

capacities.

But given this multitude of applications, is there a way

to quantify the usefulness of a quantum network? Even

though Ref. [6] established the maximum point-to-point

quantum communication capacity, it considers channel

losses as the only limit on quantum communication rates.

In practice, local operation errors and sub-optimal link

layer network protocols can also limit the performance of

quantum networks. Furthermore, we would like to bench-

mark quantum networks with respect to other network

tasks, such as computing and sensing. In the absence

∗leeyuan@mit.edu; Equal contribution

†whdai@mit.edu; Equal contribution

‡englund@mit.edu

of a commonly agreed-upon framework that can accom-

modate various quantum network-enabled applications,

comparing the capabilities of diﬀerent quantum networks

at various global network tasks remains diﬃcult.

Fundamentally, the value of a network derives from the

applications it enables by connecting people and things.

A telephone network’s worth derives from the utility seen

by the callers it connects. The value of a datacenter net-

work derives from the added utility of networked, rather

than isolated, computers. A wireless sensor network en-

abled by 5G technology adds value by connecting sensors

that, taken in isolation, would be less valuable: i.e. the

utility of the whole is greater than the sum of its parts.

When it comes to quantifying the worth of a quan-

tum network, we are guided by the same principle: we

consider how the ability to pass quantum information be-

tween devices or people creates new value. Speciﬁcally,

we introduce utility-based metrics to quantify the perfor-

mance of a quantum network in servicing the diversity of

envisioned quantum network applications. These met-

rics form a general framework for comparing the utility

of diﬀerent quantum networks, such as those illustrated

in Fig. 1. These metrics can also be used to guide the

design of quantum networks, so they can best serve the

quantum information needs of their users.

Analyzing these metrics reveals new insights in the

form of scaling laws for the performance of quantum net-

works. These laws serve the same purpose for quantum

networks as Metcalfe’s law does for classical networks:

they provide network designers and users with a prac-

tical measure of network utility, which in turn informs

the expansion of existing networks or the construction of

new ones. In addition, these scaling laws can also serve

the same purpose for quantum networks as Moore’s law

does for classical computing: they provide a measure of

progress for quantum networks in their applications of

communication, computing or sensing.

After explaining the general framework for quantify-

arXiv:2210.10752v1 [quant-ph] 19 Oct 2022

Coalition 2

Coalition 1

Coalition 3

Coalition 4

Quantum network nodes

Coalitions for quantum communication

/ quantum computing

/ quantum sensing tasks

Quantum network utility UQN

= sum of utility from all coalitions

=u1(p1) + u2(p2) + u3(p3)+ ...

FIG. 1: Illustration of a global quantum network. Nodes are

connected by quantum channels. Diﬀerent coalitions (i.e.

subsets) of nodes in a quantum network perform diﬀerent

tasks repeatedly. Each task is associated with a utility

function. The performance of a quantum network is

measured by the aggregate utility provided by the network.

ing the performance of a quantum network, we propose

one example of a quantum network metric that extends

IBM’s quantum volume [11] for benchmarking quantum

computers to quantum networks.1This framework can

also be used to construct metrics for other common quan-

tum network tasks. By incorporating information on the

network’s hardware, connectivity and link layer proto-

cols, these metrics provide realistic and physically rele-

vant measures of a quantum network’s performance.

Our utility-based model for analyzing quan-

tum networks is similar in spirit to the approach

Refs. [12] and [13] take to analyze classical networks.

These papers focus on an eﬃcient framework for rate

control, whereas we focus on valuing and comparing

networks. Nevertheless, our analysis can potentially be

extended to the development of a rate control algorithm.

1We provide some code to compute the example metric in an

interactive notebook, which can be found at https://tinyurl.

com/quantumNetworkUtility2022.

II. BENCHMARKING QUANTUM NETWORKS

Quantum networks establish quantum communication

channels between distant nodes, who can use these chan-

nels to perform quantum computation, distribute secure

keys or send quantum states. For concreteness, we as-

sume that the quantum network establishes such chan-

nels by distributing entanglement between end users, but

our framework also applies to other quantum communi-

cation protocols.

A fundamental description of a quantum network’s ca-

pabilities is given by its rate region, which describes the

communication rates it can simultaneously enable among

all groups of users. The rate region captures the scarcity

of entanglement resources, as an entangled state that is

used to connect one group of users cannot also be used

to connect another group of users. The rate region incor-

porates information on the link-layer and network-layer

protocols [8,9] of the quantum network, such as the rout-

ing algorithms between nodes and the error-correction

procedures at individual nodes.

The quantum communication enabled by the network

can be applied to a series of tasks from which users of

the network derive value. We attribute a utility function

to each of the tasks the quantum network performs. It

will be convenient to think of this utility as the users’

willingness to pay for the output of these tasks, but more

generally, this abstract “utility” can reﬂect a monetary

value, an equivalent cost of classical cryptography, or an

abstract social good. Then the performance of a quantum

network is summarized by the maximum aggregate utility

the network can provide.

The rate region alone is poorly suited for comparing

the performance of quantum networks, because the rate

region may not reﬂect the reality that some communica-

tion channels are more valuable than others. Moreover,

diﬀerent quantum networks may connect diﬀerent groups

of users, so their rate regions are incomparable. Perhaps

most importantly, quantum networks may be used for

diﬀerent applications, and the performance of a quantum

network must be evaluated in the context of the appli-

cation for which it is used. The aggregate utility metric

incorporates information about how value is derived from

the quantum network by assigning utilities to tasks that

can be completed using quantum communication. Not

only can these utilities guide the design of quantum net-

work protocols, but they can also be used to enforce a

fair entanglement sharing policy between groups of users.

We introduce some notation for concreteness. Let

the quantum network be described by the graph (V,Ep),

where Vis the set of nodes and Epis the set of physical

links connecting pairs of nodes. The edge set Epalso de-

scribes the set of multiparty communication channels the

quantum network can enable without performing entan-

glement swaps. This entanglement can be used to per-

form tasks, each of which involves a subset of nodes in

V. (Note that tasks and channels are distinct concepts.)

Suppose there are Dtasks the quantum network per-

forms repeatedly. The quantum network completes tasks

as frequently as the output entanglement rates allow. Let

the feasible task region Wbe the set of task completion

rate vectors P= (p1, p2, . . . , pD) that can be sustained

by the quantum network. Here, piis the rate at which

the ith task is completed. A vector of task completion

rates can be sustained if the rate at which the quantum

network must consume entanglement lies within the rate

region of the network.

Users derive value from the quantum network when

tasks are completed. In our proposed framework, we al-

low the utility derived from a task to depend nonlinearly

on the rate at which this task is completed, but the util-

ity users enjoy from one task is independent of the rates

at which other tasks are completed.

Let ui:R+→Rbe the utility function associated

with the ith task. If the quantum network performs task

iat rate pi, users derive utility ui(pi) from this task.

Thus, if the network achieves the task completion rate

P= (pi)D

i=1, users derive an aggregate utility PD

i=1 ui(pi)

over all tasks.

We choose the task completion rate vector that maxi-

mizes the aggregate utility users derive from the network.

Note that the feasible task region implicitly depends on

the rate region of the network. The quantum network’s

performance is measured by the following metric, known

as the quantum network utility:

UQN = maximum aggregate utility that can be

provided by the network

= max

P∈W

i=1

ui(pi).(1)

Diﬀerent applications of quantum networks call for dif-

ferent tasks and utility functions, thereby giving diﬀer-

ent quantum network metrics. If a quantum network has

multiple applications, the set of tasks should include rele-

vant tasks across all applications. Regardless, the metric

is deﬁned by an optimization problem over the rate re-

gion of the network.

The feasible task region Wcan also account for a rate-

ﬁdelity tradeoﬀ through the rate region. Supplementary

Note 1 provides more details on the connection between

the rate region and the feasible task region.

Moreover, when the quantum network utility is treated

as a dimensioned quantity, it is a universal measure of the

value provided by any quantum network. More details

can be found in Supplementary Note 2.

III. QUANTUM NETWORK UTILITY FOR

DISTRIBUTED QUANTUM COMPUTATION

In this section, we present a quantum network metric

that applies the quantum volume proposed in [11] to the

aggregate utility framework described above. This utility

metric Ucomp quantiﬁes the value derived from perform-

ing distributed quantum computing tasks, and can be

viewed as a “quantum volume throughput”.

A. Quantum volume

The quantum volume is deﬁned with respect to a com-

puting task known as Heavy Output Generation (HOG).

The HOG task comprises some number of layers d. If

mmemories are involved in the HOG task, each layer

performs bm/2cSU(4) operations over pairs of memo-

ries, chosen uniformly at random without replacement. A

quantum device is said to perform the m-memory HOG

task up to depth dif the circuit can be implemented with

a given minimum overall accuracy. The value associated

with performing an m-memory, depth-dHOG task is

v=βmin(m,d),

where β= 2 in Ref. [11].

The quantum volume is then the maximum value over

all HOG tasks that can be performed by the device. Some

memories that are otherwise available for computation

may be excluded from the HOG task in order to reduce

the error per layer, thereby allowing a deeper and thus

higher-value HOG task to be performed.

Evaluating the quantum volume of a quantum device

is complicated in general, but Ref. [11] approximates the

quantum volume by stating that an m-memory, depth-d

HOG task can be performed only if

md ≤1

eﬀ

,(2)

where eﬀ is the average error probability per two-qubit

gate.

The quantum volume can be interpreted as the cost

of classical computing needed to simulate the equivalent

quantum circuit.

B. Extension to networks

Now we extend the quantum volume to a network set-

ting. This quantum network utility metric diﬀers from

the quantum volume in two main ways.

1. The quantum network utility explicitly considers

the rate at which multi-qubit operations can be per-

formed, because transmitting qubits between nodes

incurs a signiﬁcant time cost.

2. The quantum network utility accounts for the util-

ity derived simultaneously from tasks performed on

diﬀerent parts of the network, not just the task that

generates the highest value.

Without loss of generality, we assume that each node

in the network has one single-qubit memory allocated

for computation. If a physical node in the network has

multiple computation memories, we can split this node

into multiple virtual nodes connected by appropriate lo-

cal links, such that each virtual node has one computa-

tion memory. As before, let Vbe the set of all (possibly

virtual) nodes. We also assume that the network only

produces bipartite entanglement. This simplifying as-

sumption happens to be realistic for near-term quantum

networks.

In the above aggregate utility framework, a task is

deﬁned by a coalition of at least two nodes Mi⊆ V

(with |Mi| ≥ 2) performing a HOG task of depth di,

i= 1,2, . . . , D.

The utility ui(pi) derived from completing a HOG task

(Mi, di)∈ D at a given rate piis taken to be proportional

to the rate of task completion and the quantum volume

of the task:

ui(pi) = βmin(|Mi|,di)pi.

The network utility is then the maximum aggregate

utility that can be achieved over all task completion rates

in the feasible task region W:

Ucomp = max

P∈W

i=1

ui(pi).

Network utility can be interpreted as the quantum vol-

ume throughput enabled by the quantum network. Fol-

lowing the interpretation of quantum volume as the

equivalent cost of classical computation [11], we can also

think of the above metric as the cost savings enabled by

the quantum network.

C. Modelling the feasible task region

We now provide a simple model for the feasible task

region, so that we can calculate the network utility of

distributed quantum computation. This model maps the

rate region to the feasible task region, then constructs

the rate region itself.

To construct the rate region, we follow Ref. [14] and

assume that entanglement swapping occurs with a given

eﬃciency qcin node c∈ V. We also assume that, in

the absence of any entanglement swaps, the network pro-

duces entanglement between aand b∈ V at rate fab.

Note that if nodes aand bare not connected by a phys-

ical link, then any communication between these nodes

must be generated using entanglement swaps, so fab = 0.

Then, the feasible task region can be inserted into the

optimization problem for the quantum network utility:

Ucomp = max

(pi),(rab ),(wac

ab )

i=1

βmin(|Mi|,di)pi(3)

subject to pi≥0∀i, rab ≥0∀a, b ∈ V, wac

ab ≥0∀a, b, c ∈ V;

rab =

i=1

2pidi((|Mi| − 1)−1if |Mi|is even

|Mi|−1if |Mi|is odd )a∈Mib∈Mi∀a, b ∈ V;

|Mi|di≤1

eﬀ ∀i= 1, . . . , D such that pi>0;

rab ≤fab +X

c∈V\{a,b}

qcwac

ab +wbc

2−X

c∈V\{a,b}wab

ac +wab

bc ∀a, b ∈ V;

wac

ab =wbc

ab ∀a, b, c ∈ V.

This optimization problem is a linear program, and the

number of variables is polynomial in |V| and D.

Supplementary Note 3 discusses this optimization

problem in greater detail.

D. Case study: repeater chains

There is a signiﬁcant remaining issue: the number of

possible coalitions, and thus D, grows exponentially with

the number of nodes |V| in the network. To make the

problem tractable, it is desirable to reduce the set of can-

didate coalitions that needs to be searched to obtain the

utility-maximizing solution set of candidate coalitions.

We use repeater chains (Fig. 2) as one example of how

we can reduce the set of candidate coalitions when cal-

culating the quantum network utility. In such networks,

the connectivity of physical links corresponds to a chain,

so that fab = 0 for all non-adjacent a, b ∈ V.

We say that a coalition is connected if there is a path

in the coalition between any two nodes of the coalition.

For repeater chains, the following proposition holds.

0 1 2 M− 1M− 2

∙∙∙∙

f01 f12 fM−2, M−1

q1q2qM−2

FIG. 2: Illustration of an M-node repeater chain. Physical

links are shown as edges between nodes. In this ﬁgure,

V={0,1,2,...,M −1}and fi,j = 0.6|i−j|=1. Relevant

coalitions of nodes are Mi⊆ V such that |Mi| ≥ 2.

Proposition 1. In the problem (3), there exists an op-

timal solution such that any coalition Miwith pi>0is

connected.

For a repeater chain with M=|V| nodes, the number

of connected coalitions is M(M−1)/2. This is substan-

tially smaller than the number of coalitions in V, which

grows exponentially with M.

Proposition 1provides a lower bound on the size of the

largest coalition. We now consider homogeneous repeater

chains, which are repeater chains with fab =ffor all

adjacent nodes a, b ∈ V and qc=qfor all nodes c∈ V,

where f, q > 0 are constants.

Proposition 2. In a homogeneous repeater chain with

perfect quantum memories and no gate errors (i.e. eﬀ =

0), the size of the largest coalition with nonzero task rate

in an optimal solution is bounded from below by

M+ logβ

Mlog q

(1 + q)M3(M−1)2/4.

Proposition 2states that the size of the largest coali-

tion increases as M−O(log M). Therefore, for large

networks with perfect memories and gates, almost all

the nodes in the network should be involved in the same

coalition, performing a computation task.

We next consider the case where the quantum network

produces entanglement of imperfect ﬁdelity. In this case,

the size of the largest coalition depends on the ﬁdelity

and does not increase as M−O(log M).

Proposition 3. In a quantum network with errors (i.e.

eﬀ >0), the size of the largest coalition with nonzero

task rate in an optimal solution is bounded from above by

b1/√eﬀc.

Proposition 3states that the size of the largest coali-

tion is dependent on the error rate eﬀ and not on the size

of the network M, assuming the network is suﬃciently

large. Intuitively, coalitions with size |Mi|>1/√eﬀ can

only perform HOG tasks up to depth di≤1/√eﬀ|Mi|<

|Mi|. Completing one such task provides the same utility

as completing another task with coalition size and depth

|Mj|=dj=di<1/√eﬀ, but consumes more entan-

glement. Therefore, it is never optimal to perform HOG

tasks over coalitions with more than 1/√eﬀ nodes.

Proposition 4. In a homogeneous repeater chain with

errors (i.e. eﬀ >0), the size of the largest coalition with

FIG. 3: Example of a dumbbell network. The long link

connecting the two star-shaped subnetworks is the bar; the

Quantum Network Utility A Framework for Benchmarking Quantum Networks Yuan Lee1Wenhan Dai2 3Don Towsley3and Dirk Englund1 4 1Department of Electrical Engineering and Computer Science

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