1 Fault-tolerant Coding for Entanglement-Assisted Communication

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Fault-tolerant Coding for Entanglement-Assisted

Communication

Paula Belzig, Matthias Christandl, Alexander Müller-Hermes

Abstract—Channel capacities quantify the optimal rates of

sending information reliably over noisy channels. Usually, the

study of capacities assumes that the circuits which the sender

and receiver use for encoding and decoding consist of perfectly

noiseless gates. In the case of communication over quantum

channels, however, this assumption is widely believed to be

unrealistic, even in the long-term, due to the fragility of quantum

information, which is affected by the process of decoherence.

Christandl and Müller-Hermes have therefore initiated the study

of fault-tolerant channel coding for quantum channels, i.e. coding

schemes where encoder and decoder circuits are affected by

noise, and have used techniques from fault-tolerant quantum

computing to establish coding theorems for sending classical and

quantum information in this scenario. Here, we extend these

methods to the case of entanglement-assisted communication, in

particular proving that the fault-tolerant capacity approaches

the usual capacity when the gate error approaches zero. A main

tool, which might be of independent interest, is the introduction

of fault-tolerant entanglement distillation. We furthermore focus

on the modularity of the techniques used, so that they can be

easily adopted in other fault-tolerant communication scenarios.

Index Terms—Fault-tolerance, channel capacity, entanglement

distillation, quantum information theory, quantum computation

I. INTRODUCTION

THE successful transfer of information via a commu-

nication infrastructure is of crucial importance for our

modern, highly-connected world. This process of information

transfer, e.g., by wire, cable or broadcast, can be modelled by

a communication channel Twhich captures the noise affecting

individual symbols. Instead of sending symbols individually,

the sender and receiver typically agree to send messages using

codewords made up from many symbols. With a well-suited

code, the probability of receiving a wrong message can be

made arbitrarily small. How well a given channel Tis able

to transmit information can be quantiﬁed by the asymptotic

rate of how many message bits can be transmitted per channel

use with vanishing error using the best possible encoding and

PB (paulabelzig@gmail.com) and MC (christandl@math.ku.dk) are with

the Department of Mathematical Sciences, University of Copenhagen, Uni-

versitetsparken 5, 2100 Copenhagen, Denmark.

AMH (muellerh@math.uio.no) is with the Department of Mathematics,

University of Oslo, P.O. box 1053, Blindern, 0316 Oslo, Norway.

PB and MC acknowledge ﬁnancial support from the European Research

Council (ERC Grant Agreement no. 81876), VILLUM FONDEN via the

QMATH Centre of Excellence (grant no. 10059) and the Novo Nordisk

Foundation (grant NNF20OC0059939 ‘Quantum for Life’). PB acknowledges

funding from the European Union’s Horizon 2020 research and innovation

programme under the Marie Skłodowska-Curie TALENT Doctoral fellow-

ship (grant no. 801199). AMH acknowledges funding from the European

Union’s Horizon 2020 research and innovation programme under the Marie

Skłodowska-Curie Action TIPTOP (grant no. 843414) and by The Research

Council of Norway (project 324944).

A part of this work has been presented at ISIT 2023 [1].

decoding procedure. This asymptotic rate is a characteristic of

the channel, called its capacity C(T).

In [2], Shannon introduced this model for communication

and derived a formula for C(T)in terms of the mutual

information between the input and output of the channel:

C(T) = sup

I(X:Y).

Here, I(X:Y) = H(X) + H(Y)−H(XY )denotes the

mutual information between the random variable Xand the

output Y=T(X), where H(X) = −PxpX(x) log(pX(x))

is the Shannon entropy of the discrete random variable X

with a set of possible values xthat is distributed according to

a probability distribution pX.

Various generalizations of this communication scenario to a

quantum channel T:MdA→ MdB, where Mddenotes the

matrix algebra of complex d×d-matrices, lead to different

notions of capacity. Two important examples are the classical

capacity of a quantum channel [3], [4], which quantiﬁes how

well a quantum channel can transmit classical information

encoded in quantum states, and the quantum capacity [5],

[6], [7], where quantum information itself is to be transmitted

through the channel. Both of these notions of capacity have

entropic formulas. However, they are not known to admit a

characterization which is independent of the number of chan-

nel copies, a so-called single-letter characterization, which

would simplify their calculation.

The entanglement-assisted capacity, where the encoding

and decoding machines have access to arbitrary amounts

of entanglement, does not only admit such a single-letter

characterization, but it can in fact be regarded as the only

direct formal analogue of Shannon’s original formula, since the

classical mutual information is simply replaced by its quantum

counterpart [8]:

Cea(T) = sup

φ∈MdA⊗MdA′

φa pure quantum state

I(A′:B)(T⊗idA′)(φ).

Here, I(A:B)ρ=H(A)ρ+H(B)ρ−H(AB)ρdenotes the

quantum mutual information with the von Neumann entropy

H(A)ρ=−Tr [ρlog(ρ)] for a quantum state ρ∈ MdA.

In order to communicate with a given channel T, the

encoding and decoding procedures need to be decomposed

into quantum circuits as a sequence of quantum gates. The

next step in a real-world scenario would be to implement

these circuits on a quantum device so that we can realize an

actual quantum communication system. However, this scenario

generally does not consider one of the major obstacles of

quantum computation: the high susceptibility of quantum

arXiv:2210.02939v2 [quant-ph] 6 Feb 2024

circuits to noise and faults. In classical computers, the error

rates of individual logical gates are known to be effectively

zero in standard settings and at the time-scales relevant for

communication [9]. The assumption of noiseless gates imple-

menting the encoder and decoder circuit is therefore realistic in

many scenarios. Real-life quantum gates, however, are affected

by non-negligible amounts of noise. This is certainly a problem

in near-term quantum devices, and it is generally assumed that

it will continue to be a problem in the longer term [10].

Considering the encoder and decoder circuits as speciﬁc

quantum circuits affected by noise therefore leads to poten-

tially more realistic measures of how well information can be

transferred via a quantum channel: fault-tolerant capacities,

which quantify the optimal asymptotic rates of transmitting

information per channel use in the presence of noise on the

individual gates. To construct suitable encoders and decoders

for this scenario, we build on Christandl and Müller-Hermes’

work [11], which has introduced and analyzed fault-tolerant

versions of the classical and quantum capacity, combining

techniques from fault-tolerant quantum computing [12], [13],

[14], [15] and quantum communication theory [16].

More precisely, we extend their work to entanglement-

assisted communication. In particular, we show that

entanglement-assisted communication is still possible under

the assumption of noisy quantum devices, with achievable

rates given by

Cea

F(p)(T)≥Cea(T)−f(p)

where Cea

F(p)(T)denotes the fault-tolerant entanglement-

assisted capacity for gate error probability pbelow a threshold,

and with limp→0f(p)→0.

In other words, the achievable rates for entanglement-

assisted communication with noise-affected gates can be

bounded from below in terms of the quantum mutual infor-

mation reduced by a continuous function in the single gate

error p. The usual faultless entanglement-assisted capacity is

recovered for small probabilities of local gate error, which

conﬁrms and substantiates the practical relevance of quantum

Shannon theory. This is not only relevant for communication

between spatially separated quantum computers, but also for

communication between distant parts of a single quantum

computing chip, where the communication line may be subject

to higher levels of noise than the local gates. In particular, the

noise level for the communication line does not have to be

below the threshold of the gate error.

It is important to note that many of the existing techniques

from quantum fault-tolerance cannot directly be applied to

the problem of communication, or will only allow for weaker

results. Naive strategies with one (large) fault-tolerant imple-

mentation, where the communication channel is considered

as part of the circuit noise, will only give rates approaching

zero due to their high overhead implementations, and they will

only work for channels which are very close to the identity,

i.e. with noise below the threshold. In this work and for the

results above, we are not only interested in transmitting with

vanishing error, but also at communication rates that are as

high as possible and for comparatively noisy channels.

The manuscript is structured around the building blocks

needed to achieve this result. In Section II, we brieﬂy

review concepts from fault-tolerance of quantum circuits

used for communication. In Section III, we outline how

the fault-tolerant communication setup can be reduced to an

information-theoretic problem which generalizes the usual,

faultless entanglement-assisted capacity. In Section IV, we

prove a coding theorem for this information-theoretic problem.

One important facet of communication with entanglement-

assistance in our scenario comes in the form of noise affecting

the entangled resource states, for which we introduce a scheme

of fault-tolerant entanglement distillation in Section V. Finally,

these techniques will be combined to obtain a threshold-

type coding theorem for fault-tolerant entanglement-assisted

capacity in Section VI.

II. FAULT-TOLERANT ENCODER AND DECODER CIRCUITS

FOR COMMUNICATION

Here, we review some aspects of common techniques for fault-

tolerance, but for a detailed overview of the relevant concepts,

we refer to [15] and [11].

Note that our notation for mathematical objects from quan-

tum theory is the same as in [11, Section II.A]. We deﬁne

quantum channels as completely positive and trace preserving

maps T:MdA→ MdBwhere Mddenotes the matrix

algebra of complex d×d-matrices. Probability distributions

of delements are vectors in dwhere each entry is positive

and the sum of all entries equals 1. Channels with classical

input are deﬁned as linear maps from dthat yield unit-

trace positive semi-deﬁnite Hermitian matrices, and channels

with classical output map unit-trace positive semi-deﬁnite

Hermitian matrices to elements of d.

A. Fault-tolerance for quantum circuits

Quantum circuits are the dense subset of quantum channels

which can be written as a composition of the following ele-

mentary gate operations: identity gate, Pauli gates, Hadamard

gate H, T-gate, CNOT gate, discarding of a qubit (i.e. perform-

ing a trace of a subsystem), and measurements and prepara-

tions in the computational basis. It should be emphasized that

the linear map realized by a quantum circuit might be written

in different ways as a composition of elementary gates. As

in [11], we will assume that each circuit is speciﬁed by a

particular circuit diagram detailing which elementary gates

are to be executed at which time and place in the quantum

circuit. The set of elementary operations in the circuit diagram

of a circuit Γis the circuit’s set of locations Loc(Γ), and

the number of elementary operations in the decomposition is

denoted by |Loc(Γ)|.

Given such a circuit diagram, we can model the noise

affecting the resulting quantum circuit. For simplicity, we will

always consider the i.i.d. Pauli noise model, where one of the

Pauli channels (with single Kraus operator σx, σyor σz) is

applied with probability p

3in between the gates. Speciﬁcally,

we use the following convention (as in [15]): For operations

acting on a single qubit, a single Pauli channel is applied

before (in case of a measurement or trace gate) or after (in

case of single qubit gates and preparation gates) the gate

itself; in case of the CNOT gate, a tensor product of two

Pauli channels is applied after the CNOT gate (see also [11,

Deﬁnition II.1]). This is a common and well-motivated noise

model [17], further supplemented by comparison between

experiment and classical simulations [18]. Here, we choose

to limit our work to the Pauli i.i.d. noise model in order to

simplify the presentation, but we see no obstacles in extending

our results to stochastic i.i.d. noise models. For more general

noise models, different techniques may be required.

The pattern in which noise occurs (i.e. the location where a

Pauli channel is inserted, and which Pauli channel) is speciﬁed

by a fault pattern Fand a quantum circuit Γwhich is

affected by noise according to a fault pattern Fis denoted by

[Γ]F. We can expand the linear map represented by a fault-

affected quantum circuit in our model as a sum over Pauli-fault

patterns: [Γ]F(p)=PF∈F(p)P(F)[Γ]F, where a Pauli fault

pattern Foccurs with a probability P(F)according to the

probability distribution speciﬁed by F(p). In case of the i.i.d.

Pauli noise model, each fault pattern Foccurs with a classical

probability P(F) = (1 −p)lid (p/3)lx+ly+lzwhere lkis the

number of locations where a fault appears with each index

corresponding to a type of Pauli-fault k={id, x, y, z}.

To protect against noise, a quantum circuit can be imple-

mented in a stabilizer error correcting code, where single, po-

tentially fault-affected qubits (logical qubits) can be encoded

in a quantum state of Kphysical qubits for each logical qubit.

Let ΠKbe the group of all K-fold tensor products of the Pauli

matrices σx, σy, σzand . A stabilizer code is obtained by

selecting a commuting subgroup of Πnthat does not contain

−⊗K(called the stabilizer group), and has an associated

simultaneous +1-eigenspace C ∈ (2)⊗K, which is called

the code space. Here, we will assume this subspace to have

dimension 2, i.e., we encode a single qubit, where the stabilizer

subgroup is generated by K−1elements. We will denote these

elements by g1, . . . , gK−1.

Any product Pauli operator E: ( 2)⊗K→(2)⊗Keither

commutes or anti-commutes with elements of this stabilizer

group and can therefore be associated to a vector

s= (s1, . . . , sK−1)∈K−1

where si= 0 if Ecommutes with gior si= 1 if it anti-

commutes with gi. We will call the vector sthe syndrome

associated to the Pauli operator Eand it is essentially the

quantity that is measured when performing error correction

with the stabilizer code.

A general quantum state can be decomposed in terms of

eigenspaces associated to the syndromes, as

(2)⊗K=M

s∈K−1

Ws,

where Wsis the common eigenspace of the operators

g1, . . . , gK−1where we have an eigenvalue (−1)sifor gifor

each i. Each Wsis 2-dimensional and can be associated to

some Pauli operator Essuch that

Ws=span Es

0, Es

1,

where 

0,

1 are the logical |0⟩and |1⟩. Then, we can

choose a decoder by deﬁning a unitary transformation D:

(2)⊗K→2⊗(2)⊗(K−1) such that

DEs

b=|b⟩⊗|s⟩,

for any b∈ {0,1}and any s∈K−1

2. In principle, several

choices of Escan be associated to a syndrome s, and the

choice of the basis change Dsingles out speciﬁc Pauli-errors

which constitute the set of correctable errors of our code.

With this unitary, we deﬁne the ideal decoder Dec∗:

M⊗K

2→ M2⊗ M⊗(K−1)

2given by Dec∗(X) = DXD†.

We also deﬁne its inverse, the ideal encoder Enc∗:M2⊗

M⊗(K−1)

2→ M⊗K

2. Finally, we deﬁne the ideal error

correcting channel as the following object:

EC∗= Enc∗◦(id2⊗|0⟩⟨0|⊗K−1Tr) ◦Dec∗

where the second system corresponds to the syndrome state on

the syndrome space M⊗K−1

2. The syndrome state |0⟩⟨0|⊗K−1

corresponds to the zero syndrome where E0=⊗K

2. See

also [11, Section II.C] for a detailed discussion of these ideal

quantum channels.

It should be emphasized that the ideal encoder and decoder

are not physical operations. They only appear as mathematical

tools when analyzing noisy quantum circuits encoded in the

stabilizer code. The output space of the ideal decoder is written

as M2⊗ M⊗(K−1)

2to emphasize that we think differently

about these two tensor factors, and we sometimes refer to the

ﬁrst one as the logical space, and to the second one as the

syndrome space.

Throughout this work, we will frequently use notation where

an operation marked with a star should be considered an ideal

operation that is useful for circuit analysis, and not a fault-

location. In particular, we will sometimes write id∗

2to denote

an identity map between qubits, which should not be taken to

be a fault-affected storage.

Gates on logical qubits are implemented as so-called gad-

gets on physical qubits, using the operations Enc∗and Dec∗

to map between the spaces (see also [11, Deﬁnition II.3]). For

a circuit Γ : M⊗n

2→ M⊗m

2, its implementation in a code

C,ΓC:M⊗nK

2→ M⊗mK

2, is obtained by replacing each

gate by its corresponding gadget and inserting error correction

gadgets in between the gadgets. If the physical qubits of this

implementation are subject to the noise model F(p), then we

denote this fault-affected implementation of the circuit Γby

[ΓC]F(p):M⊗nK → M⊗mK .

In this work, like in [11], we will consider implementations

in the concatenated 7-qubit Steane code. The 7-qubit Steane

code introduced in [19] is an error correcting code that can

correct all single-qubit errors, and that can be concatenated to

improve protection against errors [20]. For the concatenated

7-qubit Steane code, as shown in [15], an implementation

where the error correction’s gadget is performed between each

operation minimizes the accumulation of errors. Under this

implementation, the concatenated 7-qubit Steane code fulﬁlls a

threshold theorem for computation. More precisely, it has been

shown that the difference between a quantum circuit Γwith

classical input and output and its fault-affected implementation

[ΓC]F(p)in the concatenated 7-qubit Steane code is bounded

by p0p

p02l

|Loc(Γ)|for any pbelow a threshold p0and

concatenation level l[15], [11]. By choosing the level llarge

enough, this error can be made arbitrarily small.

Fault-tolerance can in principle be achieved by other quan-

tum error correcting codes [12], [13], [14], [15]. One could

also consider using two different quantum error correcting

codes for the encoder and decoder circuit in our setup. For

simplicity, we restrict ourselves to using the concatenated 7-

qubit Steane code [19], [20] with the same level of concatena-

tion for both circuits, but our deﬁnitions can straightforwardly

be extended to the more general case.

B. Fault-tolerance for communication

By performing error correction, a quantum circuit with clas-

sical input and output that is affected by faults at a low rate

can thus be implemented in a way such that it behaves like

an ideal circuit (i.e. a circuit without faults) by threshold-

type theorems. These code implementations cannot, however,

be directly used in the encoding and decoding circuits for

communication, as they require classical input and output,

whereas the encoder’s output in our communication setup,

for instance, serves as input into the noisy quantum channel.

The fault-tolerant implementation of an encoder and decoder

in a communication setting therefore leads to the message

being encoded in the corresponding code space. In the case of

the concatenated 7-qubit Steane code, the number of physical

qubits increases by a factor of 7for each level of concate-

nation. To obtain our results for communication rates, we

therefore perform an additional circuit mapping information

in the code space to the physical system where the quantum

channel acts. This circuit will be referred to as decoding

interface Dec. Similarly, another circuit can be performed

to transfer the channel’s output into the code space where it

can be processed by the fault-tolerantly implemented decoder.

This circuit is called encoding interface Enc. These circuits,

introduced in Deﬁnition 2.1, are also affected by faults.

Deﬁnition 2.1 (Interfaces, [11, Deﬁnition III.1]): Let C ∈

(2)⊗Kbe a stabilizer code with dim(C)=2, and let |0⟩⟨0| ∈

M⊗K−1

2denote the state corresponding to the zero-syndrome.

Let Enc∗:M⊗K

2→ M⊗K

2and Dec∗:M⊗K

2→ M⊗K

2be

the ideal encoding and decoding operations. Then, we have:

1) An encoding interface Enc : M2→ M⊗K

2for a code

Cis a quantum circuit with an error correction as a ﬁnal

step, and fulﬁlling

Dec∗◦Enc = id2⊗|0⟩⟨0|

2) A decoding interface Dec : M⊗K

2→ M2is a quantum

circuit fulﬁlling

Dec ◦Enc∗(·⊗|0⟩⟨0|) = id2(·)

In contrast to Dec∗and Enc∗, which are objects used for

the mathematical analysis of the circuits and not implemented

in practice, the interfaces Enc and Dec are quantum circuits

consisting of gates that can be affected by faults. Since this

can lead to faulty inputs to a quantum channel, we will need

interfaces that are tolerant against such faults. Unfortunately,

it is impossible to make the overall failure probability of

interfaces arbitrarily small, since they will always have a ﬁrst

(or last) gate that is executed on the physical level and not

protected by an error correcting code, resulting in a failure

with a probability of at least gate error p. Fortunately, it is

possible to construct qubit interfaces for concatenated codes

which fail with a probability of at most 2cp for some constant

c, which are good enough for our purposes [21], [11].

Theorem 2.2 (Correctness of interfaces for the concatenated

7-qubit Steane code, [11, Theorem III.3]): For each l∈, let

Cldenote the l-th level of the concatenated 7-qubit Steane

code with threshold p0. Then, there exist interface circuits

Encl:M2→ M⊗7l

2and Decl:M⊗7l

2→ M2for the l-

th level of this code such that for any 0≤p≤p0

2, we have

Prob(EnclFis not correct)≤2cp,

where Enclis correct under a Pauli fault pattern Fif

there exists a quantum state σS(F)on the syndrome

space such that Dec∗◦Enc F= id2⊗σS(F). The

probability is taken over the distribution of Faccording

to the fault model F(p).

Prob(DeclFis not correct)≤2cp,

where Dec is correct under a Pauli fault pattern Fif

Decl◦EClF= (id2⊗TrS)◦Dec∗

l◦EClFwhere

TrStraces out the syndrome space.

Here, c=p0max{|Loc(Enc1)|,|Loc(Dec1◦EC)|} is a con-

stant that does not depend on lor p.

In combination with the threshold theorem from [15], this

can be used to prove extensions of Lemma III.8 from [11]

for the combination of circuit and interface with additional

quantum input (cf. Figure 1). Here, idcl :2→2denotes

the identity map on a classical bit, and ∥T−S∥1→1:=

sup{∥(T−S)(ρ)∥Tr|ρ∈ MdAa quantum state}denotes the 1-

to-1 distance of two quantum channels T:MdA→ MdBand

S:MdA→ MdB, where ∥ρ−σ∥Tr denotes the trace distance

induced by the trace norm ∥ρ∥Tr := 1

2∥ρ∥1=1

2Trpρ†ρ.

These lemmas, and the subsequent effective channel model

in Theorem 2.5 are key ingredients to the analysis of fault-

tolerant communication because they allow us to connect the

faulty encoder and decoder circuits of the communication

scheme with effective, fault-less versions of the encoder and

decoder circuits in an effective communication problem.

Lemma 2.3 (Effective encoding interface): Let m, n, k ∈

and let Γ : M⊗n+k

2→2mbe a quantum circuit with

quantum input and classical output. For each l∈, let Cl

denote the l-th level of the concatenated 7-qubit Steane code

with threshold p0. Moreover, let Encl:M2→ M⊗7l

2be the

encoding interface circuit for the l-th level of the concatenated

7-qubit Steane code with threshold p0.

Then, for any 0≤p≤p0

2and any l∈, there exists a

quantum channel Nl:M2→ M2, which only depends on l

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1Fault-tolerantCodingforEntanglement-AssistedCommunicationPaulaBelzig,MatthiasChristandl,AlexanderMüller-HermesAbstract—Channelcapacitiesquantifytheoptimalratesofsendinginformationreliablyovernoisychannels.Usually,thestudyofcapacitiesassumesthatthecircuitswhichthesenderandreceiveruseforencodingandde...

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