All-photonic one-way quantum repeaters Daoheng Niu12Yuxuan Zhang12Alireza Shabani3and Hassan Shapourian1 1Cisco Quantum Lab San Jose CA 95134 USA

2025-04-30 0 0 2.1MB 11 页 10玖币

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All-photonic one-way quantum repeaters

Daoheng Niu,1, 2 Yuxuan Zhang,1, 2 Alireza Shabani,3and Hassan Shapourian1

1Cisco Quantum Lab, San Jose, CA 95134, USA

2Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA

3Cisco Quantum Lab, Los Angeles, CA 90049, USA

Quantum repeater is the key technology enabler for long-distance quantum communication. To

date, most of the existing quantum repeater protocols are designed based on speciﬁc quantum

codes or graph states. In this paper, we propose a general framework for all-photonic one-way

quantum repeaters based on the measurement-based error correction, which can be adapted to any

Calderbank-Shor-Steane code including the recently discovered quantum low density parity check

(QLDPC) codes. We present a novel decoding scheme, where the error correction process is carried

out at the destination based on the accumulated data from the measurements made across the net-

work. This procedure not only outperforms the conventional protocols with independent repeaters

but also simpliﬁes the local quantum operations at repeaters. As an example, we numerically show

that the [[48,6,8]] generalized bicycle code (as a small but eﬃcient QLDPC code) has an equally

good performance while reducing the resources by at least an order of magnitude.

I. INTRODUCTION

Quantum network is one of the key quantum technolo-

gies and plays a central role in enabling unconditionally

secure communication, distributed quantum computing,

and quantum sensing [1,2]. Being an active area of

research, the exact requirements and applications of a

large-scale quantum network remain to be better under-

stood. At the fundamental level nevertheless, a putative

quantum network needs to provide a way for quantum

communication, i.e., transfer of quantum information,

among diﬀerent network nodes where photons constitute

the medium of choice. Realizing a large-scale quantum

network requires transmitting quantum information over

long distances, that is challenging due to the photon loss

which grows exponentially with distance. To circumvent

this issue, quantum repeaters have been proposed [3],

and there have been tremendous eﬀorts over the past

decade [4–18]. The basic idea is to place a number of re-

peater stations at intermediate distances and use quan-

tum correlations in multi-qubit entangled states to eﬀec-

tively enhance the transmission rate between two distant

nodes.

Quantum repeater protocols are generally divided into

two categories: The ﬁrst category [3,4] is based on the

heralded quantum entanglement distribution, where a

pairwise entanglement between adjacent repeater nodes

is established so that a long-range entanglement between

the end nodes can be achieved via the entanglement

swapping, i.e., performing Bell state measurement at

each intermediate node. Quantum information is then

transferred via the quantum teleportation. The success

of a teleportation attempt relies on successfully estab-

lishing entanglement links between neighboring nodes

and performing Bell measurements. Hence, a two-way

classical channel is required to communicate the success

of both processes to the adjacent nodes for every iter-

ation. Two-way communication limits the performance

of these protocols and may necessitate long-lived quan-

tum memories at repeater stations, although the latter

requirement in principle can be relaxed in all-photonic

schemes [10,11]. The second category of repeater proto-

cols [12–17] involves sending encoded quantum informa-

tion in the form of multi-qubit loss tolerant states which

are received and (typically) error corrected at intermedi-

ate repeater stations. Such protocols only involve one-

way communication and hence their performance is not

impacted by the two-way communication requirement in

the ﬁrst category. Furthermore, the one-way protocols

are far more eﬃcient than the two-way protocols when

it comes to network traﬃc in a large scale quantum net-

work.

In this paper, we introduce an all-photonic architecture

for one-way quantum repeaters based on stabilizer codes

realized by graph states of photons, where the photon loss

is treated as a qubit erasure error and corrected through

a measurement-based error correction scheme. Our pro-

posed architecture provides a general formalism that can

be adapted to any Calderbank-Shor-Steane (CSS) stabi-

lizer code. In particular, one can leverage the remarkable

properties (including large code distance) of the recently

developed quantum low-density parity check (QLDPC)

codes [19,20] in this formalism. We should contrast

our repeater protocol with previous code-speciﬁc pro-

tocols such as those based on the quantum parity code

(QPC) [11–16], where a teleportation-based error correc-

tion is performed to deal with erasure and possible op-

erational errors, or other protocols based on tree graph

states [17,18,21], which can be viewed as teleportation

path multiplexers. Our repeater architecture in short in-

volves encoding logical qubits in a graph state of photons

corresponding to a CSS code and performing logical Bell

state measurements at each repeater. The classical infor-

mation obtained from measurement outcomes (which also

contains loss events) is not processed until received by the

recipient party who performs the error correction [22,23]

across the quantum network based on the accumulated

data (See Fig. 1). This feature is fundamentally diﬀerent

from conventional methods, where the error correction is

performed at every repeater node, and oﬀers several ad-

arXiv:2210.10071v1 [quant-ph] 18 Oct 2022

vantages. First, the overall performance is improved over

doing error correction at each repeater [24]. Second, since

there is no decoding at each repeater, there is no need

for matter qubits and adaptive measurements. Third, for

the same reason, the quantum gates and measurements

within each repeater is independent of the choice of the

stabilizer code. The latter two properties are in stark

contrast with the previous studies where both the quan-

tum and classical hardware as well as the error correc-

tion software were designed for speciﬁc encoding schemes

such as the quantum parity code [12,13] or tree graph

state [17]. This ﬂexibility of our protocol would lead

to a long-term advantage as the hardware technology is

improved and new generation of quantum codes will be

available.

As we explain, the error correction in our architec-

ture is eﬀectively carried out on a one-dimensional clus-

ter state concatenated by the CSS code as depicted in

Fig. 1(c). We derive the condition for a successful trans-

mission of the logical states across the cluster state and

provide a decoding algorithm. We illustrate details of

our framework using the [[7,1,3]] Steane code [25] and

[[48,6,8]] generalized bicycle code [19], and numerically

show that their performance is equal or better than ex-

isting protocols while requiring less resources.

II. RESULTS

A. Quantum repeater protocol

In this section, we introduce our quantum repeater ar-

chitecture. As we explain, we use a measurement-based

quantum error correction protocol so that the photon loss

is treated as unheralded, and there is no need for long-

lived matter-based quantum memory. As shown in Fig. 1,

quantum information is encoded in a graph state realiza-

tion of a quantum code, and repeaters are placed along

the channel to correct errors occurring during the trans-

mission through a lossy channel. Compared to existing

quantum repeater proposals, our protocol does not re-

quire any extra quantum hardware overhead in addition

to a resource-state generator (RSG) and single-photon

detectors per each repeater. Furthermore, no classical

data processing is required within the repeaters, and

measurement outcomes are transmitted via a classical

channel to the receiver.

We consider using single photons in the discrete-

variable formalism such as time-bin encoding which is

generally not sensitive to dephasing error and suitable for

long-distance quantum communication [26]. The main

source of error in our case is then photon loss which

is detected during the measurement process and can be

viewed as a quantum erasure channel where the error lo-

cation is known but the error type is not. According to

our protocol, the sender encodes the quantum informa-

tion (logical qubits) in multi-photon graph states. There

are two kinds of qubits in these graph states: data and

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Alice

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Bob

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Classical channel

Lost qubits

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E(| iL)

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RSG

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(c)

Alice

Bob

FIG. 1. All-photonic quantum repeater architecture

(a) Repeater chain, where R1,R2,· · · are the repeater nodes

placed between Alice (sender) and Bob (receiver). Curved

lines represent the quantum channel (optical ﬁber). (b) Inside

a repeater station, where measurement-based error correction

occurs. Two graph states associated with the X(yellow) and

Z(red) stabilizers are generated at RSG with a transversal

controlled-phase gate applied to them. The incoming logical

qubit undergoes another controlled-phase gate with the graph

state corresponding to the Xstabilizer. Measurement out-

comes are relayed to the classical channel. Here, the [[7,1,3]]

Steane code is used for illustration purposes, where missing

qubits are shown in light colors encircled with dashed lines

(See Fig. 2for further information on graph state representa-

tion). (c) Syndrome graph for the error correction is realized

at Bob’s location and eﬀectively forms a linear cluster state

concatenated by the CSS stabilizer code (a.k.a. foliated quan-

tum code [23]).

ancilla qubits. Data qubits collectively encode the logi-

cal information, while ancilla qubits are used to measure

the quantum code stabilizers. The size and shape of the

graph are determined by the deployed CSS quantum code

as will be explained in Sec. II B.

At each repeater station (Fig. 1(b)), upon receiving

the incoming graph state (from the sender or previous

repeater station), two graph states (associated with the

Xand Zstabilizer generators of the CSS code) are pre-

pared by an RSG and form a (logical) Bell pair by apply-

ing a transversal controlled-phase gate. We note that this

transversal gate can be incorporated into the state gen-

eration in the RSG. Next, a transversal controlled-phase

gate is applied between the received logical qubit and the

local logical qubit corresponding to Xstabilizer; then,

both qubits are sent to single-photon detectors where

the physical qubits are all measured in X-basis. This

step eﬀectively teleports the input state to the remaining

(unmeasured) local logical qubit which is transmitted to

the next repeater. The teleportation process may seem to

be reminiscent of the teleportation-based error correction

schemes [27,28]; however, the usage of CSS error cor-

recting codes across the network results in a signiﬁcantly

greater loss tolerance. The key idea is that our decoder

uses all the classical information obtained by measuring

qubits across all repeaters as opposed to breaking it down

to two qubit measurements per repeater. In other words,

our measurement-based protocol leads to a loss tolerant

channel by eﬀectively realizing a linear cluster state of

logical qubits between the sender and receiver as shown

in Fig. 1(c).

Our protocol corrects loss errors in the transmission

via the optical ﬁber as well as during the state gener-

ation process. In terms of their loss probability, there

are two groups of photonic qubits: Those qubits which

travel between the successive repeaters, and others which

are generated and measured within a repeater. For in-

stance, all ancilla qubits belong to the latter group. The

former qubits are subject to the erasure channel with an

overall transmission probability,

η(L) = ηr10−α0

10 L,(1)

where α0is the signal attenuation rate per unit length

in the optical ﬁber (which we set to be 0.2dB/km and

may report as e−L/Latt with the attenuation length of

Latt = 10/(α0ln(10)) ≈22 km), Lis the travel distance,

and ηrdenotes the repeater eﬃciency (or transmittance)

which collectively includes photon-source/detector eﬃ-

ciency, on-chip loss, and in/out coupling losses. Although

the main contributing factor to the repeater eﬃciency de-

pends on the details of generation/detection scheme, it

is usually the case that in/out coupling at the chip-ﬁber

interface is the dominant factor. For this reason, we as-

sume that the transmission eﬃciency of the latter group

of qubits (i.e., internal qubits) is given by √ηr.

B. Measurement-based error correction

As mentioned, we use graph states to implement an

all-photonic quantum code. A graph state (see Ref. [29]

for a detailed review) associated with graph Gof Nver-

tices (i.e., qubits) is deﬁned as a quantum state of N

qubits |ΨGi=Q(i,j)∈Gczi,j |+i⊗N,where subscripts are

qubit labels i, j = 1,··· , N , qubit state |+idenotes the

eigenstate of XPauli operator, and czi,j is a controlled-

phase gate between qubits iand jfor every edge (i, j)

on graph G. An important property of graph states is

that they can be characterized as stabilizer states with

Nstabilizer generators, where the i-th stabilizer gener-

ator associated with the i-th vertex on Gis deﬁned by

Pi=XiN(i,j)∈GZj. In other words, the i-th stabilizer

is a product of the XPauli operator on i-th vertex and

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2k+1

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(a)

(b)

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(c)

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l=2k1

FIG. 2. Graph state construction of CSS codes (a)

Tanner graph of the parity-check matrix HX(or HZ) and (b)

the corresponding graph state for the [[7,1,3]] Steane code.

Circles with integer labels denote the data qubits and squares

with labels S1,2,3denote the ancilla qubits to measure stabi-

lizers. (c) Part of the graph state associated with the three

consecutive sites on the 1D cluster state (Fig. 1(c)). Ancilla

qubits for two diﬀerent sets of stabilizers are shown as red and

yellow for Zand Xstabilizers, respectively. In (b) and (c),

we use graph state representation where solid lines represent

controlled-phase gates. Dotted lines in (c) shows an example

of an inter-site stabilizer operator.

ZPauli operators on the adjacent (in the sense of graph)

vertices. We should note that a graph state is a stabilizer

state (as opposed to a stabilizer code) since there are N

stabilizers which determine a unique state for Nqubits.

A quantum code of distance d, denoted by [[n, k, d]],

encodes klogical qubits into ndata qubits and is sta-

bilized by n−kPauli operators (stabilizer generators

or parity check operators). In the case of CSS codes,

the stabilizer group is divided into two subgroups where

the stabilizer operators are products of either only X

or ZPauli operators. The stabilizer group associated

with Xor Zoperators can conveniently be represented

by a bipartite graph (called Tanner graph) as shown for

example in Fig. 2(a) for the [[7,1,3]] Steane code. A

straightforward implementation of a CSS quantum code

in an all-photonic scheme is as follows: Construct the

Tanner graph associated with Zstabilizers as a graph

state where parity check operators as well as data qubits

are represented as vertices which we call ancilla and data

qubits, respectively. By deﬁnition, measuring the ancilla

qubits in Xbasis then ﬁxes the value of Zparity checks

(See e.g. Fig. 2(b) for the 7-qubit code). Similarly, one

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All-photonicone-wayquantumrepeatersDaohengNiu,1,2YuxuanZhang,1,2AlirezaShabani,3andHassanShapourian11CiscoQuantumLab,SanJose,CA95134,USA2DepartmentofPhysics,TheUniversityofTexasatAustin,Austin,TX78712,USA3CiscoQuantumLab,LosAngeles,CA90049,USAQuantumrepeateristhekeytechnologyenablerforlong-distanceq...

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All-photonic one-way quantum repeaters Daoheng Niu12Yuxuan Zhang12Alireza Shabani3and Hassan Shapourian1 1Cisco Quantum Lab San Jose CA 95134 USA

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