# All-photonic one-way quantum repeaters

Daoheng Niu  Cisco Quantum Lab, San Jose, CA 95134, USA  Department of
Physics, The University of Texas at Austin, Austin, TX 78712, USA  Yuxuan
Zhang  Cisco Quantum Lab, San Jose, CA 95134, USA  Department of Physics, The
University of Texas at Austin, Austin, TX 78712, USA  Alireza Shabani  Cisco
Quantum Lab, Los Angeles, CA 90049, USA  Hassan Shapourian  Cisco Quantum Lab,
San Jose, CA 95134, USA

###### Abstract

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 network. 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 technologies and plays a central
role in enabling unconditionally secure communication, distributed quantum
computing, and quantum sensing Kimble (2008); Wehner  _et al._ (2018). Being
an active area of research, the exact requirements and applications of a
large-scale quantum network remain to be better understood. 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 Briegel  _et al._ (1998), and there have been tremendous efforts
over the past decade Munro  _et al._ (2015, 2012); Glaudell  _et al._ (2016);
Muralidharan  _et al._ (2016); Pant  _et al._ (2017); Fowler  _et al._ (2010);
Azuma  _et al._ (2015); Lee  _et al._ (2019); Muralidharan  _et al._ (2014);
Namiki  _et al._ (2016); Muralidharan  _et al._ (2018); Ewert  _et al._
(2016); Ewert and van Loock (2017); Borregaard  _et al._ (2020); Zwerger  _et
al._ (2016). The basic idea is to place a number of repeater stations at
intermediate distances and use quantum correlations in multi-qubit entangled
states to effectively enhance the transmission rate between two distant nodes.

Quantum repeater protocols are generally divided into two categories: The
first category Briegel  _et al._ (1998); Munro  _et al._ (2015) 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 establishing
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 iteration. Two-way
communication limits the performance of these protocols and may necessitate
long-lived quantum memories at repeater stations, although the latter
requirement in principle can be relaxed in all-photonic schemes Azuma  _et
al._ (2015); Lee  _et al._ (2019). The second category of repeater protocols
Muralidharan  _et al._ (2014); Namiki  _et al._ (2016); Muralidharan  _et al._
(2018); Ewert  _et al._ (2016); Ewert and van Loock (2017); Borregaard  _et
al._ (2020) involves sending encoded quantum information in the form of multi-
qubit loss tolerant states which are received and (typically) error corrected
at intermediate 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 network.

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 proposed architecture provides
a general formalism that can be adapted to any Calderbank-Shor-Steane (CSS)
stabilizer code. In particular, one can leverage the remarkable properties
(including large code distance) of the recently developed quantum low-density
parity check (QLDPC) codes Panteleev and Kalachev (2021a, b) in this
formalism. We should contrast our repeater protocol with previous code-
specific protocols such as those based on the quantum parity code (QPC)
Muralidharan  _et al._ (2014); Namiki  _et al._ (2016); Muralidharan  _et al._
(2018); Ewert  _et al._ (2016); Ewert and van Loock (2017); Lee  _et al._
(2019), where a teleportation-based error correction is performed to deal with
erasure and possible operational errors, or other protocols based on tree
graph states Varnava  _et al._ (2006); Zwerger  _et al._ (2016); Borregaard
_et al._ (2020), which can be viewed as teleportation path multiplexers. Our
repeater architecture in short involves 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 information obtained from
measurement outcomes (which also contains loss events) is not processed until
received by the recipient party who performs the error correction Raussendorf
_et al._ (2006); Bolt  _et al._ (2016) 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 advantages. First, the overall performance
is improved over doing error correction at each repeater 111The intuitive
reason behind this fact is that loss events can be corrected not only at the
repeater node where the loss was observed but also through other stabilizers
at neighboring repeater nodes.. 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 quantum and
classical hardware as well as the error correction software were designed for
specific encoding schemes such as the quantum parity code Muralidharan  _et
al._ (2014); Namiki  _et al._ (2016) or tree graph state Borregaard  _et al._
(2020). 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 architecture is effectively carried
out on a one-dimensional cluster state concatenated by the CSS code as
depicted in Fig. 1(c). We derive the condition for a successful transmission
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 Steane (1996) and $[[48,6,8]]$ generalized bicycle code Panteleev and
Kalachev (2021a), and numerically show that their performance is equal or
better than existing protocols while requiring less resources.

Figure 1:  All-photonic quantum repeater architecture (a) Repeater chain,
where $R_{1}$, $R_{2}$, $\cdots$ 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 $X$ stabilizer.
Measurement outcomes 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. 2 for
further information on graph state representation). (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 quantum
code Bolt  _et al._ (2016)).

##  II Results

###  II.1 Quantum repeater protocol

In this section, we introduce our quantum repeater architecture. 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 realization of a quantum code, and repeaters are
placed along the channel to correct errors occurring during the transmission
through a lossy channel. Compared to existing quantum repeater proposals, our
protocol does not require 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 Brendel  _et al._ (1999). 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 location is known but the error type is not. According to our
protocol, the sender encodes the quantum information (logical qubits) in
multi-photon graph states. There are two kinds of qubits in these graph
states: data and ancilla qubits. Data qubits collectively encode the logical
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.2.

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 $X$ and $Z$ stabilizer generators of the CSS code) are prepared by an
RSG and form a (logical) Bell pair by applying a transversal controlled-phase
gate. We note that this transversal gate can be incorporated into the state
generation in the RSG. Next, a transversal controlled-phase gate is applied
between the received logical qubit and the local logical qubit corresponding
to $X$ stabilizer; 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 Knill
_et al._ (2001); Knill (2005); however, the usage of CSS error correcting
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 generation 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 instance, all ancilla qubits belong to the
latter group. The former qubits are subject to the erasure channel with an
overall transmission probability,

$\displaystyle\eta(L)=\eta_{r}10^{-\frac{\alpha_{0}}{10}L},$ (1)

where $\alpha_{0}$ is the signal attenuation rate per unit length in the
optical fiber (which we set to be $0.2$ dB/km and may report as
$e^{-L/L_{\text{att}}}$ with the attenuation length of
$L_{\text{att}}=10/(\alpha_{0}\ln(10))\approx 22$ km), $L$ is the travel
distance, and $\eta_{r}$ denotes the repeater efficiency (or transmittance)
which collectively includes photon-source/detector efficiency, on-chip loss,
and in/out coupling losses. Although the main contributing factor to the
repeater efficiency depends 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 assume that the transmission efficiency
of the latter group of qubits (i.e., internal qubits) is given by
$\sqrt{\eta_{r}}$.

Figure 2:  Graph state construction of CSS codes (a) Tanner graph of the
parity-check matrix $H_{X}$ (or $H_{Z}$) 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 $S_{1,2,3}$ denote the ancilla qubits to
measure stabilizers. (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 $Z$ and $X$
stabilizers, 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.

###  II.2 Measurement-based error correction

As mentioned, we use graph states to implement an all-photonic quantum code. A
graph state (see Ref. Hein  _et al._ (2006) for a detailed review) associated
with graph $G$ of $N$ vertices (i.e., qubits) is defined as a quantum state of
$N$ qubits $\ket{\Psi_{G}}=\prod_{(i,j)\in G}\textsc{cz}_{i,j}\ket{+}^{\otimes
N},$ where subscripts are qubit labels $i,j=1,\cdots,N$, qubit state $\ket{+}$
denotes the eigenstate of $X$ Pauli operator, and $\textsc{cz}_{i,j}$ is a
controlled-phase gate between qubits $i$ and $j$ for every edge $(i,j)$ on
graph $G$. An important property of graph states is that they can be
characterized as stabilizer states with $N$ stabilizer generators, where the
$i$-th stabilizer generator associated with the $i$-th vertex on $G$ is
defined by $P_{i}=X_{i}\bigotimes_{(i,j)\in G}Z_{j}$. In other words, the
$i$-th stabilizer is a product of the $X$ Pauli operator on $i$-th vertex and
$Z$ Pauli 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 $N$ qubits.

A quantum code of distance $d$, denoted by $[[n,k,d]]$, encodes $k$ logical
qubits into $n$ data qubits and is stabilized by $n-k$ Pauli 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 $Z$ Pauli operators. The
stabilizer group associated with $X$ or $Z$ operators 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 $Z$ stabilizers 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 $X$ basis then fixes the value of $Z$ parity checks (See
e.g. Fig. 2(b) for the 7-qubit code). Similarly, one can prepare a graph state
(in Hadamard basis) in terms of the Tanner graph of $X$ operators and fix the
$X$ parity checks by measuring the ancilla qubits.

As mentioned in the previous part, we prepare two graph states associated with
$Z$ and $X$ check operators in each repeater and the receiver and apply
controlled-phase gates to data qubits. This effectively realizes a linear
cluster state of logical qubits (Fig. 1(c)), where we run our error correction
scheme to ensure that the input state from Alice is transmitted to Bob. In the
remainder of this part, we explain the emergent stabilizer operators of the
linear cluster state and discuss our decoding algorithm. We denote the
stabilizer groups and the logical operators of the underlying CSS quantum code
by
$\mathcal{S}_{\sigma}=\\{S_{\sigma,1},S_{\sigma,2},\cdots,S_{\sigma,(n-k)/2}\\}$
and $\mathcal{L}_{\sigma}=\\{\tilde{\sigma}_{1},\cdots,\tilde{\sigma}_{k}\\}$,
respectively, where $\sigma=X,Z$ and tilde is added to distinguish the logical
Pauli operators from the operators acting on the physical qubits.

Having $N-1$ repeater stations between Alice and Bob implies a $(2N+1)$-site
cluster state where graph states in odd (even) layers are used for $Z$ ($X$)
stabilizer measurements (Fig. 1(c)). Each graph state consists of $(3n-k)/2$
qubits where there are $n$ data qubits and $(n-k)/2$ ancilla qubits (Fig.
2(c)). We use superscripts to denote the site number and reserve subscripts to
label data/ancilla qubits. For instance, $X_{q}^{(l)}$ with $q=1,\cdots,n$ and
$X_{a,i}^{(l)}$ with $i=1,\cdots,(n-k)/2$ refer to $X$ Pauli operators on data
qubit $q$ and ancilla qubit $i$ on site $l$, respectively. Following this
notation, we write the on-site graph state stabilizers as

$\displaystyle G^{(l)}_{a,i}=X_{a,i}^{(l)}\bigotimes_{q\in
S_{\sigma^{(l)},i}}Z^{(l)}_{q},$ (2)

and inter-site graph state stabilizers as

$\displaystyle
G^{(l)}_{q}=X^{(l)}_{q}Z^{(l-1)}_{q}Z^{(l+1)}_{q}\bigotimes_{q\in
S_{\sigma^{(l)},i}}Z_{a,i}^{(l)},$ (3)

where $\sigma^{(2j)}=X$ and $\sigma^{(2j+1)}=Z$. This gives overall
$(2N+1)(3n-k)/2$ stabilizer generators. We are measuring all physical qubits
in $X$-basis, whereby the reduced stabilizer group can be constructed by
combining data qubit stabilizers of a given site with the ancilla qubit
stabilizers of its neighbors as shown by the dashed line in Fig. 2(c),
$P^{(l)}_{i}=G^{(l-1)}_{a,i}G^{(l+1)}_{a,i}\bigotimes_{q\in
S_{\sigma^{(l+1)},i}}G^{(l)}_{q}$, which is further simplified into

$\displaystyle P^{(l)}_{i}=X^{(l-1)}_{a,i}X^{(l+1)}_{a,i}\bigotimes_{q\in
S_{\bar{\sigma}^{(l)},i}}X^{(l)}_{q},$ (4)

where we adopt the notation $\bar{\sigma}^{(l)}=H\sigma^{(l)}H$ to denote the
Pauli operators after the Hadamard transformation (e.g., $\bar{X}=Z$ and
$\bar{Z}=X$). The above simplification is due to the fact that
$\mathcal{S}_{X}$ stabilizers commute with $\mathcal{S}_{Z}$ stabilizers;
hence, ancilla qubits associated with $\bar{\sigma}$ stabilizers appear even
number of times and cancel out. We should note that there is no
$X^{(l-1)}_{a,i}$ ($X^{(l+1)}_{a,i}$) in the above formula at the left (right)
boundaries of the cluster state (c.f. Fig. 1(c)). There are $(2N+1)(n-k)$
stabilizer generators associated with $(n-k)$ ancilla qubits per site. Hence,
the logical subspace contains $(2N+1)k$ logical qubits consistent with
$(2N+1)$-site linear cluster state of $k$ logical qubits.

Our goal is to transfer logical qubits across the linear cluster state. To
this end, the necessary logical operators to realize the measurement-based
identity gate for $i$-th logical qubit are given by

$\displaystyle\mathcal{P}_{Z_{i}}$
$\displaystyle=\bigotimes_{l=2k}^{2N}\tilde{X}^{(l)}_{i}=\bigotimes_{l=2k}^{2N}\bigotimes_{q\in\tilde{X}_{i}}X_{q}^{(l)},$
$\displaystyle\mathcal{P}_{X_{i}}$
$\displaystyle=\bigotimes_{l=2k-1}^{2N}\tilde{X}^{(l)}_{i}=\bigotimes_{l=2k-1}^{2N}\bigotimes_{q\in\tilde{X}_{i}}X_{q}^{(l)},$
(5)

which are the logical qubit version of the identity gate in measurement-based
quantum computation Raussendorf  _et al._ (2003).

As mentioned earlier, loss in our protocol is detected during the measurement
process and is viewed as a quantum erasure channel. When it comes to applying
controlled-phase gates, if a qubit is lost at an earlier point, then the gate
will not be active. Mathematically, loss error corresponds to partial tracing
over the missing qubits. For instance, a loss (or erasure) channel for qubit
$a$ is described by the following quantum channel,

$\displaystyle{\cal
D}_{a}(\rho)=\eta_{a}\rho+(1-\eta_{a})\text{tr}_{a}(\rho)\otimes\ket{e}\\!\bra{e},$
(6)

where $\eta_{a}$ is the qubit transmission, $\ket{e}$ denotes an unknown state
outside the computational basis for qubit $a$, which in our case correspond to
an empty (vacuum) state with no photon in either bins. Alternatively, partial
tracing is identical to conjugating with the Pauli group, because
$\text{tr}_{a}(\rho)\otimes\openone_{a}=\frac{1}{4}(\rho+X_{a}\rho
X_{a}+Y_{a}\rho Y_{a}+Z_{a}\rho Z_{a})$. Hence, after a loss event, only
logical and stabilizer operators which commute with the Pauli operators of the
erased qubits remain valid. The loss tolerance is achieved in the following
way: As long as the Pauli operators acting on the erased qubits commute with
the logical operators (5) modulo the stabilizer group (4), a successful
transfer is guaranteed.

Lastly, we propose a simple decoding algorithm to determine the successful
transmission of the input quantum states. First, we note that the stabilizer
group (4) and the logical operators (5) form two disjoint sets associated with
even and odd sites (a.k.a., primal and dual syndrome graphs) which can be
decoded independently. Then, for each set of stabilizers and logical operators
the decoder checks whether or not logical operators can be combined with the
stabilizer group such that they commute with Pauli operators acting on erased
qubits. The details are shown as pseudo codes in Algorithm 1.

input : A list of erased qubits $Q$.

output : Set of corrected qubits.

1 for  _$q\in Q$_ do

2 if  _$|L|\neq 0$_ then

3 for  _$X_{j}\in L$_ do

4 if  _$X_{j}$ acts on $q$_ then

5 Find $s_{i}\in S$ s.t. $s_{i}$ acts on $q$.

6 If $s_{i}$ _exists_ then $X_{j}\leftarrow s_{i}X_{j}$.

7 else Remove $X_{j}$ from $L$.

8

9 Find $R\subseteq S$ s.t. $s_{i}\in R$ acts on $q$.

10 If $|R|=1$ then Remove $s_{i}$ from $S$.

11 else if  _$|R| >1$_ then

12 Construct stabilizer subgroup: $\tilde{R}\leftarrow\\{s_{i}s_{i+1}|s_{i}\in
R\\}$.

13 $S\leftarrow(S\backslash R)\cup\tilde{R}$.

14

15 else return $\emptyset$.

return $L$.

Algorithm 1 Erasure decoder

###  II.3 Performance and resource costs

In this section, we investigate the performance of our proposed repeater
architecture in terms of the effective transmission rate (ETR), denoted by
$\eta_{\text{eff}}$, which is defined by the success probability of receiving
the quantum information at the destination. Obviously, ETR has to outperform
the direct transmission of single photons; however, placing too many repeaters
at short distances from one another is expensive and not a scalable solution.
Along this line, we discuss the trade-off between number of the repeaters and
the ETR. We emphasize that our protocol can be developed for any CSS code.
Intuitively, we look for $[[n,k,d]]$ codes with large code distance $d$, but
we do not want to introduce too much overhead, i.e., we want the number of
transmitted photons per logical qubits, $k/n$, to be small. Lifted product
QLDPC codes Panteleev and Kalachev (2021b, a) with an almost linear distance
$k,d\sim\Theta(n)$ could be a good candidate. However, we want practical
solutions (at least for near-term implementations) where the size of on-site
graph states in the 1D cluster state (c.f., Fig. 1(c)), that is $(3n-k)/2$, is
not too big.

Given the above heuristic considerations, in what follows, we examine the
performance of two example CSS codes: $[[7,1,3]]$ Steane code and $[[48,6,8]]$
generalized bicycle code. The first example represents a minimal quantum error
correcting code and could be a good candidate (albeit with a limited
performance) to consider for near-term experimental realization of our
protocol. The second example is a representative quantum code with a decent
code distance (which implies better performance) and moderate number of
physical qubits. To investigate the performance of each implementation, we
carry out Monte Carlo simulations where a random instance of erasure according
to the loss probability for each type of qubits (data and ancilla) is
generated and we use Algorithm 1 to check if the given instance is correctable
or not. We run this process over many iterations to accumulate statistics and
evaluate the average success probability $\eta_{\text{eff}}$ (See Sec. IV.1
for further details).

Figure 3: Comparison of logical erasure probability of some representative CSS
codes Loss tolerance of the $[[7,1,3]]$ Steane code, $[[48,6,8]]$ generalized
bicycle code and $6\times 6$ toric code (as a reference) are plotted for two
different repeater efficiencies $\eta_{r}=1$ and $0.9$ which are shown as open
and filled circles, respectively. The case of $\eta_{r}=1$ for the $[[7,1,3]]$
code is analytically derived, as given in Eq. (8). Each data point is averaged
over $N_{s}=2\times 10^{6}$ Monte Carlo iterations, and the error bars are
given by $\sqrt{\eta_{\text{eff}}(1-\eta_{\text{eff}})/N_{s}}$. Figure 4:
Effective signal attenuation rate as a function of repeater spacing Using Eq.
(7) the effective decay rate $\alpha_{\text{eff}}$ is obtained by fitting the
data (see the insets for typical fits) and plotted against the distance
between two neighboring quantum repeaters $L_{0}$ for (a) $[[7,1,3]]$ and (b)
$[[48,6,8]]$ codes at different repeater efficiencies $\eta_{r}=$ $0.95$,
$0.9$, and $0.85$. Error bars are not shown as they are comparable with the
data marker size. The inset in (a) shows the effective transmission rate as a
function of total distance $L$ for $(L_{0},\eta_{r})=(6.13\ \text{km},0.95)$
(blue) and $(11.41\ \text{km},0.9)$ (orange). The inset in (b) shows similar
curves for $(7.0\ \text{km},0.95)$ (blue), $(7.42\ \text{km},0.9)$ (orange),
and $(9.41\ \text{km},0.85)$ (green). Each data point is averaged over
$N_{s}=5\times 10^{5}$ Monte Carlo iterations.

We illustrate the loss tolerance of the two codes (without repeaters) in Fig.
3, where for reference we also include the performance of the graph state
implementation of $6\times 6$ toric code (i.e., $[[72,2,6]]$). This
architecture corresponds to Fig. 1(c) with only Alice and Bob and no
repeaters, i.e., three-site cluster state. We consider two cases: One is when
there is no loss other than the channel attenuation $\eta_{r}=1$ (shown as
filled circles), and another case is when $\eta_{r}=0.9$ (shown as open
circles). An immediate observation is that the break-even point for all
encodings in the former case is nearly $0.5$; in other words, they outperform
the direct transmission up to $50\%$ loss rate. The break-even point in the
latter case is decreased as we consider repeater loss but they still perform
better than the direct transition over a wide range of loss rates up to 0.4.
We also observe that the performance of the $[[48,6,8]]$ code is close to or
better than the toric code while the number of transmitted qubits per logical
qubits is much less.

A simple way to characterize the overall performance of the repeater protocol
is to calculate the effective signal attenuation $\alpha_{\text{eff}}$ as a
function of the repeater spacing. To this end, we fit the ETR by an
exponentially decaying function

$\displaystyle\eta_{\text{eff}}(L_{0},L)\propto
10^{-\frac{\alpha_{\text{eff}}(L_{0})}{10}L},$ (7)

where $L_{0}=L/N$ is the distance between two consecutive repeaters out of
total $(N-1)$ repeaters and plot them in Fig. 4 in which the insets show some
typical fits to the data. As expected, the performance of the repeater
protocol, regardless of the underlying quantum code, degrades as we increase
the repeater spacing. It is worth noting that the effective attenuation of the
$[[7,1,3]]$ code increases almost linearly with $L_{0}$ as opposed to that of
the $[[48,6,8]]$ code. This can be attributed to the large code distance of
the $[[48,6,8]]$ code compared to the $[[7,1,3]]$ code. Furthermore, the
shaded regions in the plots are for reference and indicate where the repeater
scheme is no longer useful as it underperforms the direct transmission, i.e.,
$\alpha_{\text{eff}}>\alpha_{0}=0.2$ dB/km. We observe that the onset repeater
spacing to enter the underperforming regime decreases with decreasing the
repeater efficiency $\eta_{r}$. Another important observation is that as long
as $L_{0}\lesssim 4$ km the $[[48,6,8]]$ code performs in an almost fully
loss-tolerant regime where $\alpha_{\text{eff}}\approx 0$ despite the repeater
photon loss of up to $10\%$.

Figure 5:  Performance of optimized repeater architectures (a) and (b)
Effective transmission rate $\eta_{\text{eff}}$ of a repeater chain based on
$[[7,1,3]]$ and $[[48,6,8]]$ codes after resource optimization (see main text
for details) are plotted for different values of the repeater efficiency
$\eta_{r}$. Inset of panel (a) shows the long distance behavior of
$\eta_{\text{eff}}$ vs $L$ in logarithmic scale which manifestly outperforms
the direct transmission above 200 km by at least three orders of magnitude.
(c) and (d) Optimal number of repeaters per $10$ km as a function of total
distance obtained by minimizing the cost function.

As mentioned before, repeater stations can be placed close to each other to
obtain a total ETR of almost unity. However, this is not a practical approach
for scalability and from the economic standpoint. Therefore, we look for an
optimization scheme to maximize the repeater spacing while not sacrificing the
ETR as much. A natural choice for a cost function to be optimized is the ratio
of the amount of resources used to the overall performance. More explicitly,
we consider the ratio of the number of repeater stations per unit length to
the total ETR. More details are explained in Sec. IV.2. For a given repeater
efficiency $\eta_{r}$ and total distance $L$, the ETR associated with the
optimal repeater spacing is plotted in Fig. 5. We note that the
discontinuities in the optimal values is due to the discrete optimization over
the number of repeaters. As expected, the optimal number of repeaters per 10
km generally increases to compensate the decreasing ETR as the total distance
increases. We also observe that the greater the repeater photon loss rate the
closer the repeater nodes and the lower the ETR.

Ref. |  Characteristics | $\ \ \ \ \eta_{r}\ \ \ \ $ | ${k}/{n}$ | $\eta_{\text{eff}}$ | $L_{0}/L_{\text{att}}(\%)$
---|---|---|---|---|---
Current work |  Measurement-based error correction on CSS QEC graph states. | 0.9 | 8 | 0.6-0.8 | 4-14
Muralidharan _et al._ Muralidharan  _et al._ (2014) |  QPC based on teleportation-based error correction with matter qubits. | 0.9 | 100-200 | 0.6 | 7.5
Ewert _et al._ Ewert  _et al._ (2016) |  All-photonic teleportation error correction based on QPC. | 1 | 100 | 0.78 | 10
Lee _et al._ Lee  _et al._ (2019) |  All-photonic QPC with linear optics. | 0.95 | 400-500 | 0.7 | 4-9
Borregaard _et al._ Borregaard  _et al._ (2020) |  Tree graph state using quantum emitter and 2 auxilliary spin qubits | 0.95 | 200-300 | 0.6 | 13
Rozpędek _et al._ Rozpędek  _et al._ (2021) |  Continuous-variable scheme based on GKP states requiring $18$dB squeezing. | 0.97 | 4-7 | 0.4-0.7 | 1-1.3
Table 1: Comparison with existing one-way protocols. The effective
transmission rate $\eta_{\text{eff}}$ is reported for $10^{3}$-$10^{4}$ km.
Here, $k/n$ denotes the number of photons per logical qubit, $\eta_{r}$ is the
repeater efficiency, and $L_{0}/L_{\text{att}}$ is the ratio of the repeater
spacing to the optical fiber attenuation length.

A few remarks are in order regarding the performance of our protocol compared
to that of the existing protocols. As summarized in Table 1, the $[[48,6,8]]$
code delivers a similar value for ETR at $10^{3}$-$10^{4}$ km total
communication distance while the repeater spacing (in units of the attenuation
length $L_{\text{att}})$ is equal or greater. Furthermore, there is less
constraint on minimum repeater efficiency $\eta_{r}$ and most importantly, an
order of magnitude smaller number of photons per logical qubit $k/n$ than
other discrete-variable protocols. We believe that the performance can even be
improved further by utilizing more efficient QLDPC codes or larger codes with
larger $n$ while keeping $k/n$ fixed.

##  III Discussion

In conclusion, we proposed a general all-photonic one-way architecture for
quantum repeaters which can be applied to any CSS code. We presented a novel
error correction scheme based on measurement-based error correction, where one
needs to only make projective measurements in a fixed basis at each repeater
without further processing the outcomes, and the decoding process is performed
at the destination. An immediate benefit of such error correction scheme is
simplifying typical error correcting tasks involving feedforward processes in
repeater nodes into some fixed operations such as controlled-phase gates and
projective measurements in a fixed basis. Moreover, error correction across
the network has a better performance than the common approach where repeaters
correct errors independently, because a local error can still be corrected
along the repeater chain. We study the performance of our architecture by
using $[[7,1,3]]$ Steane code and $[[48,6,8]]$ generalized bicycle code. As we
showed, the effective transmission rate at thousands of kilometers total
distance is comparable with existing protocols at similar repeater spacings,
while our protocol can work with an order of magnitude smaller number of
photons per logical qubit and tolerate lower repeater efficiencies.

Figure 6:  Fusion-based realization of graph state corresponding to the
$[[7,1,3]]$ Steane code. The resource state is a star graph with 3 to 5
branches where the central vertex (shown as black circles or orange squares)
could be a spin-qubit or photon which is measured during the generation
process. There are two unmeasured data qubits per each vertex which are used
to effectively implement the transversal controlled-phase gate between two
adjacent graph states on the 1D cluster state.

In terms of hardware requirements at repeater stations, our architecture
requires three components: a graph-state generator, a device for performing
controlled-phase gate between the photonic qubits, and single-photon
detectors, all of which are commonly used in previous architectures. Hence, we
do not introduce any extra hardware overhead.

As for the graph-state generator, various architectures based on solid-state
quantum emitters coupled to nanophotonic waveguides have been proposed as
deterministic generators Lindner and Rudolph (2009); Buterakos  _et al._
(2017); Pichler and Zoller (2016); Pichler  _et al._ (2017); Wan  _et al._
(2021); Zhan and Sun (2020); Shapourian and Shabani (2022), and there are ways
to further optimize the graph-state generation algorithm Li  _et al._ (2022).
Deterministic controlled-phase gates can possibly be realized by an array of
(passive) quantum dots coupled to a waveguide Schrinski  _et al._ (2021).
Alternatively, the controlled-phase gate can be replaced by the fusion gates
Browne and Rudolph (2005); Bartolucci  _et al._ (2021), where small resource
graph states are generated per each graph vertex and fusion gates are
performed to construct a larger graph as shown in Fig. 6. Small resource
states can be generated either deterministically based on quantum emitters or
probabilistically using materials nonlinearity Wang  _et al._ (2020). We note
that the fusion gates based on linear optics are probabilistic and a fusion
failure results in an erasure error (which ultimately reduces $\eta_{r}$).
However, this constraint can be alleviated by increasing the success
probability using boosted fusion gates Ewert and van Loock (2014) with
ancillary photons or inner encoding Bartolucci  _et al._ (2021), or completely
circumvented by using a different hardware, e.g., nearly-deterministic memory
assisted Bell-state measurement devices Bhaskar  _et al._ (2020). In this
regard, hybrid approaches to generate arbitrary graph states Hilaire  _et al._
(2022) can also be useful. All in all, we envision a minimal hardware
architecture based on few active elements where we use delay lines to generate
time-multiplexed entangled photons. We postpone a more detailed discussion on
hardware designs and analysis of secret-key rate using a quantum key
distribution scheme to a future work.

An important property of our architecture is being all-photonic where the
major error is the qubit erasure which is handled more efficiently compared to
the Pauli errors. This fact can be further leveraged by constructing quantum
codes with a larger code distance and higher erasure threshold to design even
more efficient and loss-tolerant protocols. In a typical CSS code
$(C_{X},C_{Z})$ corresponding to $X$ (primal graph) and $Z$ (dual graph)
stabilizers, respectively, we want both $C_{X}$ and $C_{Z}$ codes to have a
large code distance. However, our setup is asymmetric in the sense that only
layers associated with primal graph ($C_{X}$) are subject to the channel loss.
Therefore, it would be interesting to explore moderate-size CSS codes with a
large code distance in one code (e.g., $C_{X}$) while the other code (e.g.,
$C_{Z}$) distance is not too large.

Using the stabilizer formalism, we have derived the criterion for a successful
transmission of logical qubits based on which we presented a simple quantum
decoder. In principle, more efficient decoders Delfosse and Zémor (2020);
Connolly  _et al._ (2022) may not only improve the performance but also enable
exploring many more codes and comparing their performance as a repeater
protocol.

Last but not least, the proposed quantum repeater architecture in this paper
permits concatenation with other encoding schemes. For instance, one could
replace each vertex in the graph state by a small tree graph to further
enhance the performance. Moreover, similar architectures might be designed for
continuous-variable encodings where the graph vertices represent squeezed
states or the Gottesman-Kitaev-Preskill (GKP) bosonic code, as it has been
done for conventional repeater protocols Fukui  _et al._ (2021); Rozpędek  _et
al._ (2021); Raveendran  _et al._ (2022); Note (5). A unique advantage of the
latter scheme is that deterministic fusion gates can be constructed in terms
of beam splitters and homodyne measurements. 55footnotetext: While such
protocols feature no need for a quantum memory, their realization is largely
hindered by the difficulty of generating GKP states in the optical domain with
large squeezing values.

##  IV Methods

###  IV.1 Monte Carlo Simulation

As discussed in Sec. II.3, the performance of the quantum repeaters is
determined by the transmission rate of the quantum information, which is
defined as the average probability of the encoded logical qubit being
successfully transmitted. We run Monte-Carlo simulations to evaluate the
average probability given the number of repeaters. There is however some
simplifications when the repeater efficiency is perfect, i.e., when
$\eta_{r}=1$; in this case, the expression for the ETR is given by
$\eta_{\text{eff}}=(\eta_{\text{eff},1})^{N}$ where $\eta_{\text{eff},1}$ is
the ETR of directly transmitting the code. This is because no loss for ancilla
qubits implies measuring the stabilizers perfectly so each layer in Fig. 2
operates independently. As a result, the overall ETR is given by the product
of individual success probabilities.

It is possible to derive a closed-form expression for the success probability
$\eta_{\text{eff},1}$ as a function of the photon direct transmission rate
$\eta$ for small codes using basic combinatorics. For example, when
considering the $[[7,1,3]]$ $\eta_{\text{eff},1}$ takes a polynomial form of
order seven as follows

$\displaystyle\eta_{\text{eff},1}=\sum_{j=3}^{7}a_{j}\eta^{j}(1-\eta)^{7-j},$
(8)

where the coefficients are $(a_{3},\cdots,a_{7})=(7,28,21,7,1)$. The above
expression is plotted as the solid blue curve Fig. 3 which matches the
numerical data points shown as filled circles.

For more complex quantum codes or when we consider finite repeater efficiency
$\eta_{r}<1$, a closed-form expression cannot be easily obtained. Hence, we
resort to numerical simulations. To this end, we perform Monte Carlo
simulations and run our decoder on a $2N$-layer syndrome graph to determine
how many logical qubits are successfully transmitted. We note that the
syndrome graph is decomposed into two $N$-layer subgraphs, a.k.a. primal and
dual, associated with $X$ and $Z$ stabilizers as expected for any CSS code
(see Sec. II.2 for more details). Thus, each iteration involves psuedo-
randomly generating a set of lost photons, where the loss events are
independent and their probability depends on whether the qubit type is data or
ancilla on primal or dual subgraph. As explained in Sec. II.1, the ancilla
qubits on all layers as well as data qubits on even layers are not transmitted
through the channel. As a result, we randomly generate loss events with
probability $1-\sqrt{\eta_{r}}\approx(1-\eta_{r})/2$ uniformly for all qubits
on the dual syndrome subgraph (i.e., data qubits on even sites and ancilla
qubits on odd sites) where the inter-site stabilizers are based on $H_{Z}$
parity-check matrix of the input CSS code. In contrast, we randomly generate
loss events with probability $\eta$ in Eq. (1) for data qubits on odd sites
and with probability $1-\sqrt{\eta_{r}}$ for ancilla qubits on even sites to
form the primal syndrome subgraph where the inter-site stabilizers are based
on $H_{X}$ parity-check matrix.

Finally, the overall ETR $\eta_{\text{eff}}$ is found by multiplying the
average success probability of the two syndrome graphs. This process
constitutes how we generate the raw data; i.e., $\eta_{\text{eff}}$ as a
function of number of repeaters, repeater efficiency $\eta_{r}$, and the
channel loss rate. To study the distance dependence of ETR and optimize the
repeater spacing, we translate channel loss to distance via Eq. (1) and
interpolate or extrapolate the data using Eq. (7) to obtain
$\eta_{\text{eff}}$ for arbitrary distances. We justify this approximation
numerically in Fig. 4.

###  IV.2 Cost function

In this section, we discuss how to define a reasonable cost function which
takes into account both the performance and the resource cost in our repeater
architecture. For a fixed total distance $L$, as we increase the number of
repeaters $N$, the repeater spacing $L_{0}$ is decreased, and the channel
transmission rate $\eta$ approaches unity; as a result, the total ETR
$\eta_{\text{eff}}$ saturates to its maximum value bounded by the repeater
efficiency. However, this is not a cost-effective approach as quantum
repeaters are costly resources. Therefore, we consider the trade-off between
the performance and the resource cost. For this purpose, a quantitative
measure of performance is simply the ETR. Moreover, the resource cost can be
taken into account in terms of the number of repeaters per unit distance and
the number of physical qubits in the deployed quantum code. This naturally
leads to the following quantity for the cost function,

$\displaystyle
C=\frac{N/L}{\eta_{\text{eff}}}\cdot\frac{n_{\text{physical}}}{n_{\text{logical}}},$
(9)

where $n_{\text{physical}}=n$, and $n_{\text{logical}}=k$ are the total number
of physical qubits and the total number of logical qubits in a $[[n,k,d]]$ CSS
code. We note that in our simulations we optimize the number of repeaters
using the same code; hence, the second factor in $C$ is fixed in practice and
the cost function only depends on the first factor.

With the definition of cost function, we explore the optimized quantum
repeaters arrangement over a wide range of total distances, from $L=10$ km to
$L=10^{4}$ km. For a given $L$, we calculate the cost function over a range of
numbers of repeaters from $1$ to $N_{\text{max}}$, where $N_{\text{max}}$ is
chosen to be large enough so that the global minimum is contained. The
resulting optimized number of repeaters $N_{\text{opt}}$ and the corresponding
ETR as a function of total distance are plotted in Fig. 5. To evaluate
$\eta_{\text{eff}}$ for arbitrary values of $N$ and $L_{0}$, we run Monte
Carlo simulations on a 2D grid within the range $2\leq N\leq 30$ and
$0<\eta(L_{0})<1$, use the exponential ansatz (7) to extract
$\alpha_{\text{eff}}(L_{0})$ (along with the proportionality coefficient) as a
function of $L_{0}$, and interpolate the corresponding value of effective
decay rate. Finally, we plug in $N=L/L_{0}$ into Eq. (7), obtain the ETR as
well as the corresponding value for the cost function (9).

## Data and Code Availability

All the data presented in this paper is the result of numerical simulations.
The code used to generate this data is available upon request.

## Acknowledgements

The authors acknowledge insightful discussions with Bing Qi, Galan Moody,
Stefano Paesani, and Stephen DiAadamo.

## Competing interests

The authors declare no competing interests.

## Author Contribution

HS conceived the project and designed the proposed repeater architectures. HS
and DN designed and implemented the measurement-based error correction
protocol and the Monte-Carlo simulations. YZ assisted with numerical
implementation. HS and DN wrote the manuscript with inputs from all the
coauthors.

##  Appendix A QLPDC codes

In this appendix, we briefly review the general properties of QLPDC codes and
discuss the $[[48,6,8]]$ generalized bicycle code Panteleev and Kalachev
(2021a) which is used as an example in our repeater protocol.

A stabilizer quantum code $[[n,k,d]]$ encodes $k$ logical qubits into $n$
($n>k$) data qubits and is equipped with $(n-k)$ stabilizer generators,
represented by the rows of the parity check binary matrix $H=(H_{X}|H_{Z})$
where the columns are associated with $X$ and $Z$ Pauli operators acting on
data qubits. A code is further characterized by the code distance $d$ which is
related to its ability to correct errors. In our context, we use quantum codes
to correct the qubit erasure error. For example, $d-1$ is the maximum number
of erasure errors, which is guaranteed to be correctable. One specific type of
quantum codes is the CSS codes, which we use in our quantum repeater
architecture. Such codes are characterized by two classical linear codes
$(C_{X},C_{Z})$ associated with $X$ and $Z$ stabilizers and their
corresponding parity check matrices $H_{X}$ and $H_{Z}$. In other words, the
parity-check matrix of a CSS code takes the following form,

$\displaystyle H=\begin{bmatrix}H_{X}&\\\ &H_{Z}\end{bmatrix}.$ (10)

The commutation relation is equivalent to the binary identity
$H_{X}H_{Z}^{T}=0$. The number of logical qubits is found by
$k=n-\text{rk}(H_{X})-\text{rk}(H_{Z})$, where one usually imposes the
condition $\text{rk}(H_{X})=\text{rk}(H_{Z})=r$, such that $k=n-2r$. We use
$\tilde{X}_{i}$ and $\tilde{Z}_{i}$ to denote the logical operator acting on
the $i$-th logical qubit and represent them as rows of two binary matrices
$L_{X}$ and $L_{Z}$, respectively. The fact that logical operators commute
with the stabilizer generators while those acting on the same logical qubit
anti-commute implies the following binary matrix identities

$\displaystyle L_{\sigma}H_{\overline{\sigma}}=0,\quad
L_{\sigma}L_{\sigma^{\prime}}^{T}=\delta_{\overline{\sigma}\sigma^{\prime}}\openone_{k},$
(11)

where $\sigma=X,Z$ denotes Pauli operators and $\overline{\sigma}=H\sigma H$
after the Hadamard transformation. Given a code with a parity-check matrix
$H$, we use the recipe outlined in Refs. Gottesman (1997); Nielsen and Chuang
(2010) to find the logical operators based on an ansatz satisfying the above
commutation relations. In short, we obtain the logical operators by first
transforming $H$ into the standard form via Gaussian elimination and then
solving for the ansatz.

A quantum low density parity check (QLDPC) code is a stabilizer code defined
by a sparse binary parity-check matrix, where the sparsity implies that the
weights of all rows and columns in $H$ are upper bounded by some universal
constant as the code length $n$ grows in an infinite family of codes. We use
the $[[48,6,8]]$ code in our repeater protocol, which belongs to a family of
codes called the generalized bicycle (GB) codes. The GB codes provide a
general ansatz for CSS codes Kovalev and Pryadko (2013), where the parity-
check matrices are defined by $H_{X}=[A,B]$ and
$H_{Z}=[B^{\text{T}},A^{\text{T}}]$ in terms of two binary $\ell\times\ell$
circulant matrices $A$ and $B$ which always commute. We note that the
commutation relation $H_{X}H^{T}_{Z}=0$ is manifestly satisfied since $A$ and
$B$ commute. A binary circulant matrix $A$ is represented as
$A=a_{0}I+a_{1}P+...+a_{\ell-1}P^{\ell-1}$ where $I$ is the $\ell\times\ell$
identity matrix and $P$ is the permutation matrix, i.e., the right cyclic
shift by one position $P=[P_{ij}]_{\ell\times\ell}$ with
$P_{ij}=\delta_{i-1,j}$ where $\delta_{ij}$ is the Kronecker delta function. A
circulant matrix can alternatively be represented in a polynomial form
$a(x)=a_{0}+a_{1}x+...+a_{l-1}x^{l-1}$. An important property of circulant
matrices is that their rank is determined algebraically as $\ell-\deg g(x)$
where $\deg g(x)$ is the degree of the polynomial
$g(x)=\gcd(a(x),x^{\ell}-1)$. Therefore, the dimension of a GB code defined by
$a(x)$ and $b(x)$ is given by $k=2\deg g(x)$ where
$g(x)=\gcd(a(x),b(x),x^{\ell}-1)$ because
$\text{rk}(H_{X})=\text{rk}(H_{Z})=n-\deg g(x)$. As explained in Ref.
Panteleev and Kalachev (2021a), possible values of the dimension $k$ of the GB
codes with a fixed circulant size $\ell$ correspond to the degree of $g(x)$,
that is given by all possible factors of the polynomial $x^{\ell}-1$. Hence,
to produce a QLDPC code, one needs to find low-weight polynomials $a(x)$ and
$b(x)$. This can be done via an exhaustive search over all polynomials of the
given weight when $\ell$ is relatively small or done via a random search where
a high success probability is guaranteed.

The $[[48,6,8]]$ code is characterized by $\ell=24$ and the following
circulant matrices Panteleev and Kalachev (2021b):
$a(x)=1+x^{2}+x^{8}+x^{15}$, $b(x)=1+x^{2}+x^{12}+x^{17}$.

## References

  * Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
  * Wehner  _et al._ (2018) S. Wehner, D. Elkouss, and R. Hanson, Science 362, eaam9288 (2018).
  * Briegel  _et al._ (1998) H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).
  * Munro  _et al._ (2015) W. J. Munro, K. Azuma, K. Tamaki, and K. Nemoto, IEEE Journal of Selected Topics in Quantum Electronics 21, 78 (2015).
  * Munro  _et al._ (2012) W. J. Munro, A. M. Stephens, S. J. Devitt, K. A. Harrison, and K. Nemoto, Nature Photonics 6, 777 (2012).
  * Glaudell  _et al._ (2016) A. N. Glaudell, E. Waks, and J. M. Taylor, New Journal of Physics 18, 093008 (2016).
  * Muralidharan  _et al._ (2016) S. Muralidharan, L. Li, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Scientific reports 6, 1 (2016).
  * Pant  _et al._ (2017) M. Pant, H. Krovi, D. Englund, and S. Guha, Phys. Rev. A 95, 012304 (2017).
  * Fowler  _et al._ (2010) A. G. Fowler, D. S. Wang, C. D. Hill, T. D. Ladd, R. Van Meter, and L. C. L. Hollenberg, Phys. Rev. Lett. 104, 180503 (2010).
  * Azuma  _et al._ (2015) K. Azuma, K. Tamaki, and H.-K. Lo, Nature communications 6, 1 (2015).
  * Lee  _et al._ (2019) S.-W. Lee, T. C. Ralph, and H. Jeong, Phys. Rev. A 100, 052303 (2019).
  * Muralidharan  _et al._ (2014) S. Muralidharan, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, Phys. Rev. Lett. 112, 250501 (2014).
  * Namiki  _et al._ (2016) R. Namiki, L. Jiang, J. Kim, and N. Lütkenhaus, Phys. Rev. A 94, 052304 (2016).
  * Muralidharan  _et al._ (2018) S. Muralidharan, C.-L. Zou, L. Li, and L. Jiang, Phys. Rev. A 97, 052316 (2018).
  * Ewert  _et al._ (2016) F. Ewert, M. Bergmann, and P. van Loock, Phys. Rev. Lett. 117, 210501 (2016).
  * Ewert and van Loock (2017) F. Ewert and P. van Loock, Phys. Rev. A 95, 012327 (2017).
  * Borregaard  _et al._ (2020) J. Borregaard, H. Pichler, T. Schröder, M. D. Lukin, P. Lodahl, and A. S. Sørensen, Physical Review X 10, 021071 (2020).
  * Zwerger  _et al._ (2016) M. Zwerger, H. J. Briegel, and W. Dür, Appl. Phys. B 122, 50 (2016).
  * Panteleev and Kalachev (2021a) P. Panteleev and G. Kalachev, Quantum 5, 585 (2021a).
  * Panteleev and Kalachev (2021b) P. Panteleev and G. Kalachev, IEEE Transactions on Information Theory 68, 213 (2021b).
  * Varnava  _et al._ (2006) M. Varnava, D. E. Browne, and T. Rudolph, Phys. Rev. Lett. 97, 120501 (2006).
  * Raussendorf  _et al._ (2006) R. Raussendorf, J. Harrington, and K. Goyal, Annals of Physics 321, 2242 (2006).
  * Bolt  _et al._ (2016) A. Bolt, G. Duclos-Cianci, D. Poulin, and T. M. Stace, Phys. Rev. Lett. 117, 070501 (2016).
  * Note (1) The intuitive reason behind this fact is that loss events can be corrected not only at the repeater node where the loss was observed but also through other stabilizers at neighboring repeater nodes.
  * Steane (1996) A. M. Steane, Phys. Rev. Lett. 77, 793 (1996).
  * Brendel  _et al._ (1999) J. Brendel, N. Gisin, W. Tittel, and H. Zbinden, Phys. Rev. Lett. 82, 2594 (1999).
  * Knill  _et al._ (2001) E. Knill, R. Laflamme, and G. J. Milburn, nature 409, 46 (2001).
  * Knill (2005) E. Knill, Nature 434, 39 (2005).
  * Hein  _et al._ (2006) M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. V. den Nest, and H.-J. Briegel, arXiv:quant-ph/0602096 (2006).
  * Raussendorf  _et al._ (2003) R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003).
  * Rozpędek  _et al._ (2021) F. Rozpędek, K. Noh, Q. Xu, S. Guha, and L. Jiang, npj Quantum Information 7 (2021), 10.1038/s41534-021-00438-7.
  * Lindner and Rudolph (2009) N. H. Lindner and T. Rudolph, Phys. Rev. Lett. 103, 113602 (2009).
  * Buterakos  _et al._ (2017) D. Buterakos, E. Barnes, and S. E. Economou, Phys. Rev. X 7, 041023 (2017).
  * Pichler and Zoller (2016) H. Pichler and P. Zoller, Phys. Rev. Lett. 116, 093601 (2016).
  * Pichler  _et al._ (2017) H. Pichler, S. Choi, P. Zoller, and M. D. Lukin, Proceedings of the National Academy of Sciences 114, 11362 (2017).
  * Wan  _et al._ (2021) K. Wan, S. Choi, I. H. Kim, N. Shutty, and P. Hayden, PRX Quantum 2, 040345 (2021).
  * Zhan and Sun (2020) Y. Zhan and S. Sun, Phys. Rev. Lett. 125, 223601 (2020).
  * Shapourian and Shabani (2022) H. Shapourian and A. Shabani, arXiv:2206.11307 (2022), https://arxiv.org/abs/2206.11307.
  * Li  _et al._ (2022) B. Li, S. E. Economou, and E. Barnes, npj Quantum Information 8, 1 (2022).
  * Schrinski  _et al._ (2021) B. Schrinski, M. Lamaison, and A. S. Sørensen, arXiv:2112.11328 (2021).
  * Browne and Rudolph (2005) D. E. Browne and T. Rudolph, Phys. Rev. Lett. 95, 010501 (2005).
  * Bartolucci  _et al._ (2021) S. Bartolucci, P. Birchall, H. Bombin, H. Cable, C. Dawson, M. Gimeno-Segovia, E. Johnston, K. Kieling, N. Nickerson, M. Pant, _et al._ , arXiv:2101.09310 (2021).
  * Wang  _et al._ (2020) J. Wang, F. Sciarrino, A. Laing, and M. G. Thompson, Nature Photonics 14, 273 (2020).
  * Ewert and van Loock (2014) F. Ewert and P. van Loock, Phys. Rev. Lett. 113, 140403 (2014).
  * Bhaskar  _et al._ (2020) M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, _et al._ , Nature 580, 60 (2020).
  * Hilaire  _et al._ (2022) P. Hilaire, L. Vidro, H. S. Eisenberg, and S. E. Economou, arXiv:2205.09750 (2022).
  * Delfosse and Zémor (2020) N. Delfosse and G. Zémor, Phys. Rev. Research 2, 033042 (2020).
  * Connolly  _et al._ (2022) N. Connolly, V. Londe, A. Leverrier, and N. Delfosse, arXiv:2208.01002 (2022).
  * Fukui  _et al._ (2021) K. Fukui, R. N. Alexander, and P. van Loock, Phys. Rev. Research 3, 033118 (2021).
  * Raveendran  _et al._ (2022) N. Raveendran, N. Rengaswamy, F. Rozpędek, A. Raina, L. Jiang, and B. Vasić, Quantum 6, 767 (2022).
  * Note (5) While such protocols feature no need for a quantum memory, their realization is largely hindered by the difficulty of generating GKP states in the optical domain with large squeezing values.
  * Gottesman (1997) D. Gottesman,  _Stabilizer codes and quantum error correction_ (California Institute of Technology, 1997).
  * Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge Univ. Pr., Cambridge, UK, 2010).
  * Kovalev and Pryadko (2013) A. A. Kovalev and L. P. Pryadko, Phys. Rev. A 88, 012311 (2013).


# On non-Abelian Lie Bracket of Generalized
Covariant Hamilton Systems

Gen WANG Department of Mathematics, Zhejiang Normal University, ZheJiang,
Jinhua 321004, China. Email<EMAIL_ADDRESS>
###### Abstract

This is a theoretical paper, which is a continuation of [1], it considers the
non-abelian Lie algebra $\mathcal{G}$ of Lie groups for
$\left[{{X}_{i}},{{X}_{j}}\right]=c_{ij}^{k}{{X}_{k}}\in\mathcal{G}$ on the
foundation of the GCHS, where $c_{ij}^{k}\in{{C}^{\infty}}\left(U,R\right)$
are the structure constants. The GPWB 1 is nonlinear bracket applying to the
non-Euclidean space, the second order (2,0) form antisymmetric curvature
tensor ${{F}_{ij}}=c_{ij}^{k}{{D}_{k}}$, and Qsu quantity
${{q}_{i}}=w_{i}^{k}{{D}_{k}}$ are accordingly obtained by using the non-
abelian Lie bracket. The GCHS
$\left\\{H,f\right\\}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ holds for the
non-symplectic vector field $X_{H}^{M}\in\mathcal{G}$ and
$f\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ that implies the covariant
evolution equation consists of two parts, NGHS and W dynamics along with the
second order invariant operator
$\frac{{{\mathcal{D}}^{2}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}+2w\frac{d}{dt}+\beta$.

111Key words: GCHS, non-abelian Lie algebra, W Dynamics, GPWB, curvature
tensor, Qsu quantity.

##  1 Preliminaries

###  1.1 Non-Abelian Lie algebra

A non-abelian group is a group $\left(G,*\right)$ in which there exists at
least one pair of elements $a$ and $b$ of $G$, such that $a*b\neq b*a$. Most
of the interesting Lie groups are non-abelian. A Lie algebra is a vector space
${\mathfrak{g}}$ together with a non-associative, alternating bilinear map
${\displaystyle{\mathfrak{g}}\times{\mathfrak{g}}\rightarrow{\mathfrak{g}};(x,y)\mapsto[x,y]}$,
called the Lie bracket, satisfying the Jacobi identity. For any associative
algebra $L$ with multiplication $*$, The Lie bracket of two elements of Lie
algebra is defined to be their commutator in $L$: $[a,b]=a*b-b*a\neq 0$. The
associativity of the multiplication $*$ in $L$ implies the Jacobi identity of
the commutator in Lie bracket.

######  Definition 1 ([4]).

A Lie algebra is a vector space ${\mathfrak{g}}$ over some field $F$ together
with a binary operation
$[\cdot,\cdot]:{\mathfrak{g}}\times{\mathfrak{g}}\to{\mathfrak{g}}$ called the
Lie bracket that satisfies the following axioms:

(i)

Bilinearity, $[ax+by,z]=a[x,z]+b[y,z]$ for all scalars $a,b\in F$ and all
elements $x,y,z\in{\mathfrak{g}}$.

(ii)

The Jacobi identity, $[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0$, for all
$x,y,z\in{\mathfrak{g}}$.

(iii)

Anticommutativity, ${\displaystyle[x,y]=-[y,x]}$, for all elements
$x,y\in{\mathfrak{g}}$.

It is customary to express a Lie algebra in lower-case fraktur, like
${\mathfrak{g}}$. If a Lie algebra is associated with a Lie group, then the
spelling of the Lie algebra is the same as that Lie group.

######  Lemma 1 ([2, 4]).

Let ${{X}_{1}},\cdots,{{X}_{n}}$ be a basis of Lie algebra $\mathcal{G}$ of
Lie groups, $\left[{{X}_{i}},{{X}_{j}}\right]\in\mathcal{G}$, hence there are
structural constants $c_{ij}^{k}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$
such that

$\left[{{X}_{i}},{{X}_{j}}\right]=c_{ij}^{k}{{X}_{k}}$

satisfying the following properties

  1. 1.

$c_{ij}^{k}+c_{ji}^{k}=0$

  2. 2.

$c_{ij}^{r}c_{rk}^{s}+c_{jk}^{r}c_{ri}^{s}+c_{ki}^{r}c_{rj}^{s}=0$

Suppose that
$\frac{\partial}{\partial{{x}^{1}}},\cdots,\frac{\partial}{\partial{{x}^{n}}}$
is the basis of Lie algebra $\mathcal{G}$ on the Euclidean space
${{\mathbb{R}}^{n}}$, then it yields $\left[{{X}_{i}},{{X}_{j}}\right]=0$,
where ${{X}_{i}}=\frac{\partial}{\partial{{x}^{i}}}\in\mathcal{G}$,
subsequently, structure constant
$c_{ij}^{k}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ of Lie groups hold
$c_{ij}^{k}=0$. As explained, the GCHS on Poisson-W manifold
$\left(P,W,\left\\{,\right\\}\right)$ is only for the non-Euclidean space, it
implies that Poisson-W manifold $\left(P,W,\left\\{,\right\\}\right)$ should
be built upon non-abelian group. In the GPWB, we will consider that
$\left[{{X}_{i}},{{X}_{j}}\right]\neq 0$ for the basis
${{X}_{i}}\in\mathcal{G}$. On the basis of lemma 1, we can obtain
$\left[{{X}_{i}},{{X}_{j}}\right]\chi=c_{ij}^{r}{{A}_{r}}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$,
where ${{A}_{r}}$ is structural derivative.

The algebraic definition for the Lie derivative of a tensor field follows from
the following theorems

######  Theorem 1 ([6]).

Let $M$ be $n$-dimensional $C^{\infty}$ manifold,
$X\in{{C}^{\infty}}\left(TM\right)$, let
${{L}_{X}}:{{C}^{\infty}}\left({{\otimes}^{r,s}}TM\right)\to{{C}^{\infty}}\left({{\otimes}^{r,s}}TM\right)$,
$\theta\mapsto{{L}_{X}}\theta$ satisfy

  1. 1.

${{L}_{X}}f=Xf,f\in{{C}^{\infty}}\left(M,\mathbb{R}\right)={{C}^{\infty}}\left({{\otimes}^{0,0}}TM\right),$

  2. 2.

${{L}_{X}}Y=\left[X,Y\right],Y\in{{C}^{\infty}}\left(TM\right)={{C}^{\infty}}\left({{\otimes}^{1,0}}TM\right)$

######  Theorem 2 ([8], Poincaré’s lemma).

Let $\mathcal{O}$ be a ball in ${{\mathbb{R}}^{m}}$ and $F$ a $k$-form such
that $dF=0$. Then there is a $k-1$ form $g$ on $\mathcal{O}$ such that $F=dg$.

###  1.2  Generalized Covariant Hamilton Systems

Let the cotangent bundle ${{T}^{*}}Q$ is the phase space of a configuration
space $Q$. There is an intrinsic symplectic structure on ${{T}^{*}}Q$ that can
be described in various equivalent ways. Assume first that $Q$ is
$n$-dimensional, and pick local coordinates $(q^{1},\cdots,q^{n})$ on $Q$.
Since $(dq^{1},\cdots,dq^{n})$ is a basis of ${{T}^{*}}_{q}Q$, we can write
any $\alpha\in{{T}^{*}}_{q}Q$ as $\alpha=p_{i}dq^{i}$. This procedure defines
induced local coordinates $(q^{1},\cdots,q^{n},p_{1},\cdots,p_{n})$ on
${{T}^{*}}Q$. Define the canonical symplectic form on ${{T}^{*}}Q$ by

$\Omega=d{{p}_{i}}\wedge d{{q}^{i}}$ (1)

This defines a two-form $\Omega$, which is clearly closed, and in addition, it
can be checked to be independent of the choice of coordinates
$(q^{1},\cdots,q^{n})$.

The GCHS has generalized the GHS to the more general form [1]

$\frac{\mathcal{D}x}{dt}={\dot{x}}+wx=W\left(x\right)DH\left(x\right),~{}~{}~{}x\in{{\mathbb{R}}^{m}}$
(2)

and greatly opened up a new insight to study the Hamiltonian system with extra
stucture. The GCHS includes the NGHS
${\dot{x}}=\frac{dx}{dt}=J\left(x\right)DH\left(x\right)\in{{C}^{\infty}}\left(TM\right)$
and W dynamics
$w=-\widehat{W}H=\left\\{H,1\right\\}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$,
The GCHS is compatible with the GPWB

$\left\\{f,g\right\\}={{X}_{f}}g+{{X}_{\chi}}\left(f,g\right)\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$

Based on the GCHS and the GPWB, we obtained the covariant evolution as a
general dynamical system
$\frac{\mathcal{D}}{dt}f=\left\\{H,f\right\\}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$,
the GCHS $\left\\{H,f\right\\}=\Omega\left(X_{H}^{M},X_{f}^{M}\right)$ is also
supported by a symplectic 2-form
$\Omega\in{{C}^{\infty}}\left({{\bigwedge}^{2}}{{T}^{*}}M\right)$ on cotangent
bundle ${{T}^{*}}M$ along with non-symplectic vector field $X_{H}^{M}$ which
is supported by ${{i}_{X_{f}^{M}}}\Omega=df\wedge d\chi$ which is not always
zero based on the theorem 2 along with equation (1).

Conservative force, a particle of mass $m$ moves under the influence of a
conservative force derived from the gradient $\nabla$ of a scalar potential,
the force is given by $F=-\nabla V(\mathbf{r})$, where $V$ is potential
energy. According to the covariant canonical equations and momentum theorem
$\frac{d}{dt}p=F$, we can obtain the following corollary

######  Corollary 1.

the force field $F$ on the Poisson-W manifold
$\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$ is shown as
$F=\frac{d}{dt}p=-DH$, its components is $F_{k}={\dot{p}_{k}}=-{{D}_{k}}H$,
where $D=\nabla+A,~{}~{}A=\nabla\chi$.

Hence the GCHS can be naturally rewritten in the form
$\left\\{H,f\right\\}={{J}_{ij}}{{D}_{i}}H{{D}_{j}}f={{F}_{i}}{{J}_{ji}}{{D}_{j}}f$,
in which the ${F}_{i}=-{{D}_{i}}H$ is the components of force $F$. Based on
the GPWB, theoretically, we have the conservation of energy given by
$\left\\{H,H\right\\}={{J}_{ij}}{{D}_{i}}H{{D}_{j}}H={{J}_{ij}}{{F}_{i}}{{F}_{j}}=0$,
and the GCHS is reexpressed as
$\overset{\cdot}{\mathop{{{x}_{i}}}}\,={{J}_{ij}}{{D}_{j}}H=-{{J}_{ij}}{{F}_{j}}={{J}_{ji}}{{F}_{j}}$.

######  Corollary 2.

There is an invariance on the Poisson-W manifold
$\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$ such that

$\frac{{{\mathcal{D}}^{2}}}{d{{t}^{2}}}f=\ddot{f}+2w\dot{f}+\beta f$

for all $f\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$, where
$\frac{df}{dt}=\dot{f},\frac{{{d}^{2}}f}{d{{t}^{2}}}=\ddot{f}$, and
$\beta={{w}^{2}}+\frac{d}{dt}w$.

Thusly, the second order covariant derivative of time
$\frac{{{\mathcal{D}}^{2}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}+2w\frac{d}{dt}+\beta$
is invariant mathematical structure on the Poisson-W manifold. It also can be
shown in the form

$\frac{{{\mathcal{D}}^{2}}f}{d{{t}^{2}}}=\frac{\mathcal{D}}{dt}\left\\{H,f\right\\}=\frac{d}{dt}\left({{D}_{i}}f\dot{x_{i}}\right)+w{{D}_{i}}f\dot{x_{i}}$

where NGHS is $\dot{x_{i}}={{J}_{ij}}{{D}_{j}}H$. In order to prove the
practical effect of corollary 2, we can put it to one of the covariant
canonical Hamilton’s equations,

${{\overset{\circ}{\mathop{F}}\,}_{k}}=\frac{\mathcal{D}{{p}_{k}}}{dt}=-{{D}_{k}}H+{{p}_{k}}w$
(3)

Then we act the CTO on covariant force ${{\overset{\circ}{\mathop{F}}\,}_{k}}$
which leads to the consequence

$\displaystyle\frac{\mathcal{D}}{dt}\frac{\mathcal{D}{{p}_{k}}}{dt}$
$\displaystyle=-\frac{\mathcal{D}}{dt}{{D}_{k}}H+\frac{\mathcal{D}}{dt}\left({{p}_{k}}w\right)$
$\displaystyle=-\frac{d}{dt}{{D}_{k}}H-w{{D}_{k}}H+\frac{d}{dt}\left({{p}_{k}}w\right)+{{p}_{k}}{{w}^{2}}$
$\displaystyle=\frac{{{d}^{2}}}{d{{t}^{2}}}{{p}_{k}}+2w\frac{d}{dt}{{p}_{k}}+{{p}_{k}}\beta$

Obviously, we can draw a conclusion that second order form of covariant
canonical Hamilton’s equations with respect to time $t$ holds invariance.

$\left\\{\begin{matrix}{{a}_{i}}=\frac{{{\mathcal{D}}^{2}}{{x}_{i}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}{{x}_{i}}+2w\frac{d}{dt}{{x}_{i}}+{{x}_{i}}\beta\\\
\frac{{{\mathcal{D}}^{2}}{{p}_{k}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}{{p}_{k}}+2w\frac{d}{dt}{{p}_{k}}+{{p}_{k}}\beta\\\
\end{matrix}\right.$

Hence we can regard acceleration flow
${{a}_{i}}=\frac{{{\mathcal{D}}^{2}}{{x}_{i}}}{d{{t}^{2}}}=\frac{{{d}^{2}}}{d{{t}^{2}}}{{x}_{i}}+2w\frac{d}{dt}{{x}_{i}}+{{x}_{i}}\beta$
as general geodesic equation.

###  1.3 Generalized Poisson-W bracket

######  Definition 2 ([1]).

Let a vector field
${{X}_{f}}={{J}_{ij}}{{\partial}_{i}}f{{\partial}_{j}}\in{{T}_{p}}M$ be given
on the Poisson-W manifold, so there is a bundle map

${{T}_{f}}M:f\in{{C}^{\infty}}\left(M,\mathbb{R}\right)\to
X_{f}^{M}={{X}_{f}}+f{{X}_{\chi}}$

for all
${{X}_{f}},{{X}_{\chi}}={{J}_{ij}}{{A}_{i}}{{\partial}_{j}}\in{{T}_{p}}M$.
${X_{f}^{M}}$ is non-symplectic vector field, the space denoted by
$\left(Z_{N},\Omega\right)$.

Analytic expression of the GPWB
$\left\\{f,g\right\\}={{D}^{T}}fJDg={{J}_{ij}}{{D}_{i}}f{{D}_{j}}g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$
with the vector operator $D=\nabla+A$,

$\left\\{f,g\right\\}={{D}^{T}}fJDg={{\left\\{f,g\right\\}}_{GHS}}+f{{\left\\{\chi,g\right\\}}_{GHS}}-g{{\left\\{\chi,f\right\\}}_{GHS}}$

for all $f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ [see [1] for more
details].

######  Theorem 3 ([1]).

The generalized Poisson-W bracket of two functions
$f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ is shown as

$\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GHS}}+{{X}_{\chi}}\left(f,g\right)$

where $\left\\{f,g\right\\}=-\left\\{g,f\right\\}$ is skew-symmetric, and
${{X}_{\chi}}\left(f,g\right)=f{{X}_{\chi}}g-g{{X}_{\chi}}f=-{{X}_{\chi}}\left(g,f\right)$
is together defined.

With the support of theorem 1, the GPWB can be shifted to the form

$\left\\{f,g\right\\}={{\mathcal{L}}_{X_{f}^{M}}}g-{{\mathcal{L}}_{g{{X}_{\chi}}}}f$

this is very easily proved. Hence the identity holds
${{\mathcal{L}}_{X_{f}^{M}}}g={{\mathcal{L}}_{g{{X}_{\chi}}}}f$ if
$\left\\{f,g\right\\}=0$.

######  Theorem 4 ([1]).

For all $f,g,h\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$,
$\lambda,\mu\in\mathbb{R}$, the GPWB has the following important properties

  1. 1.

Symmetry: $\left\\{f,g\right\\}=-\left\\{g,f\right\\}$.

  2. 2.

Bilinearity: $\left\\{\lambda f+\mu
g,h\right\\}=\lambda\left\\{f,h\right\\}+\mu\left\\{g,h\right\\}$.

  3. 3.

GJI:
$\left\\{f,\left\\{g,h\right\\}\right\\}+\left\\{g,\left\\{h,f\right\\}\right\\}+\left\\{h,\left\\{f,g\right\\}\right\\}=0$.

  4. 4.

Generalized Leibnitz identity:
${{\left\\{fg,h\right\\}}}={{\left\\{fg,h\right\\}}_{GHS}}+{{X}_{\chi}}\left(fg,h\right)$.

  5. 5.

Non degeneracy: if for all $F$, $\left\\{f,g\right\\}=0$, then
${{\left\\{f,g\right\\}}_{GHS}}={{X}_{\chi}}\left(g,f\right)$.

######  Theorem 5.

Let $\left(U,{{x}^{i}}\right)$ be a given local coordinates on the Poisson-W
manifold $\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$ together with
$[\cdot,\cdot]:{\mathfrak{g}}\times{\mathfrak{g}}\to{\mathfrak{g}}$ such that

$\displaystyle{{\mathcal{L}}_{X_{f}^{M}}}X_{g}^{M}=\left[X_{f}^{M},X_{g}^{M}\right]\left|{}_{U}\right.=\left[{{X}_{f}},{{X}_{g}}\right]+f\left[{{X}_{\chi}},{{X}_{g}}\right]$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+g\left[{{X}_{f}},{{X}_{\chi}}\right]+\left(2{{X}_{f}}g+{{X}_{\chi}}\left(f,g\right)\right){{X}_{\chi}}$

for all $X_{f}^{M},X_{g}^{M}\in\mathfrak{g}$ and
$f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$.

###### Proof.

Let $\left(U,{{x}^{i}}\right)$ be a given local coordinates on the Poisson-W
manifold $\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$. According to the
definition 1 and theorem 1, Given a function
$K\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$, and
$X_{f}^{M},X_{g}^{M}\in\mathfrak{g}$, with the help of the theorem 3, then it
follows the Lie bracket

$\displaystyle\left[X_{f}^{M},X_{g}^{M}\right]\left|{}_{U}\right.K$
$\displaystyle=X_{f}^{M}X_{g}^{M}K-X_{g}^{M}X_{f}^{M}K$
$\displaystyle=\left({{X}_{f}}+f{{X}_{\chi}}\right)\left({{X}_{g}}+g{{X}_{\chi}}\right)K-\left({{X}_{g}}+g{{X}_{\chi}}\right)\left({{X}_{f}}+f{{X}_{\chi}}\right)K$
$\displaystyle=\left[{{X}_{f}},{{X}_{g}}\right]K+f\left[{{X}_{\chi}},{{X}_{g}}\right]K+g\left[{{X}_{f}},{{X}_{\chi}}\right]K$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+\left(2{{X}_{f}}g+{{X}_{\chi}}\left(f,g\right)\right){{X}_{\chi}}K$

then we prove the theorem.

∎

Subsequently, on the basis of theorem 5, we can naturally verify the
definition 1 one by one. $X_{f}^{M}\in\mathfrak{g}$ is generating vector
function derived by the function $f$, hence taken a condition $f=g$ for
alternativity, then $\left[X_{f}^{M},X_{f}^{M}\right]=0$ apparently. Clearly,
anticommutativity is automatically satisfied as
$\left[X_{f}^{M},X_{g}^{M}\right]\left|{}_{U}\right.=-\left[X_{g}^{M},X_{f}^{M}\right]\left|{}_{U}\right.$,
bilinearity and the Jacobi identity are obvious,

Operationally, taking the Hamiltonian $H$ into the account which leads to the
result

$\left[X_{f}^{M},X_{g}^{M}\right]H=\left[{{X}_{f}},{{X}_{g}}\right]H-2w{{X}_{f}}g-{\widehat{f}_{X}}\left(w,\chi,H\right)\left(f,g\right)$

where
${\widehat{f}_{X}}\left(w,\chi,H\right)=w{{X}_{\chi}}+{{X}_{w}}+{{X}_{\chi}}{{X}_{H}}$,
owing to the GPWB supported by the structural function $\chi$ or the
structural derivative $A_{i}$, hence it implies that

$\left[X_{f}^{M},X_{g}^{M}\right]H-\left[{{X}_{f}},{{X}_{g}}\right]H=2w{{X}_{f}}g+{\widehat{f}_{X}}\left(w,\chi,H\right)\left(f,g\right)$

##  2 Main Results

We will take the lemma 1 on the non-abelian Lie groups to analyze and discuss
the GCHS and its extensions. Consequently, the commutating relation of
operator [1] ${{D}_{i}}=\partial_{i}+A_{i}$ is

$\left[{{D}_{i}},{{D}_{j}}\right]={{F}_{ij}}=\left[{{\partial}_{i}},{{\partial}_{j}}\right]+{{\partial}_{i}}{{A}_{j}}-{{\partial}_{j}}{{A}_{i}}$

Generally, called the second order (2,0) form curvature tensor. Owing to the
consideration of lemma 1, then we obtain
${{F}_{ij}}g=\left[{{D}_{i}},{{D}_{j}}\right]g=c_{ij}^{k}\left({{\partial}_{k}}g+g{{\partial}_{k}}\chi\right)$
holds for all $g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ on the non-Abel
group, it can be simplified as ${{F}_{ij}}=c_{ij}^{k}{{D}_{k}}$, hence
${{q}_{i}}={{F}_{ij}}{{v}^{j}}=w_{i}^{k}{{D}_{k}}$, where
${{v}^{j}}c_{ij}^{k}=w_{i}^{k}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$,

######  Claim 1.

Let ${{X}_{i}}=\frac{\partial}{\partial{{x}^{i}}}\in\mathcal{G}$ be the basis
on the non-abelian Lie groups, then two operators hold

${{F}_{ij}}=c_{ij}^{k}{{D}_{k}},~{}~{}~{}{{q}_{i}}=w_{i}^{k}{{D}_{k}}$

where
${{v}^{j}}c_{ij}^{k}=w_{i}^{k}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$.
${{F}_{ij}}$ is the second order (2,0) form antisymmetric curvature tensor,
and ${{q}_{i}}$ is calles Qsu quantity.

It subsequently leads to
${{t}_{k}}={{q}_{k}}H=w_{k}^{j}{{D}_{j}}H=-w_{k}^{j}{{F}_{j}}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$.
Plugging the non-Abel condition into the corollary 1, then it gives the
equation

$u_{kj}={{D}_{j}}{{F}_{k}}-{{D}_{k}}{{F}_{j}}=c_{kj}^{i}{{F}_{i}}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$

W dynamics can be reshown as
$w={{X}_{H}}\chi={{J}_{ji}}{{F}_{i}}{{\partial}_{j}}\chi$, therefore, we can
obtain the below corollary

######  Corollary 3.

Based on lemma 1 and corollary 1, one has

${{t}_{k}}={{q}_{k}}H=w_{k}^{j}{{D}_{j}}H=-w_{k}^{j}{{F}_{j}}$
$u_{kj}={{D}_{j}}{{F}_{k}}-{{D}_{k}}{{F}_{j}}=c_{kj}^{i}{{F}_{i}}$

where $u_{jk}=-u_{kj}$.

######  Theorem 6.

Let the non-abelian Lie groups be given on the Poisson-W manifold
$\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$ along with the Hamiltonian $H$
such that

$\left(\begin{matrix}{{u}_{ji}}\\\ {{t}_{i}}\\\
\end{matrix}\right)=\left(\begin{matrix}{{F}_{ij}}\\\ {{q}_{i}}\\\
\end{matrix}\right)H$

is called reciprocal force equations.

###### Proof.

On the basis of lemma 1, the claim 1 and the corollary 3, the proof can be
directly derived by the following

$\left(\begin{matrix}{{F}_{ij}}\\\ {{q}_{i}}\\\
\end{matrix}\right)H=\left(\begin{matrix}c_{ij}^{k}\\\ w_{i}^{k}\\\
\end{matrix}\right){{D}_{k}}H=-\left(\begin{matrix}c_{ij}^{k}\\\ w_{i}^{k}\\\
\end{matrix}\right){{F}_{k}}=\left(\begin{matrix}{{u}_{ji}}\\\ {{t}_{i}}\\\
\end{matrix}\right)$

in which the corollary 1 has been plugged through the proof.

∎

within the framework of the GCHS, there are two second order of the GCHS with
respect to time and coordinates
$\frac{\mathcal{D}}{dt}\frac{\mathcal{D}{{x}_{k}}}{dt}$,
${{D}_{i}}{{D}_{j}}\frac{\mathcal{D}{{x}_{k}}}{dt}$, the former one is about
the acceleration flow, meanwhile the latter is about curvature.
${{D}_{k}}\frac{\mathcal{D}{{x}_{k}}}{dt}$ is also the considerable factors.

######  Theorem 7.

Let vector field $v={{v}_{i}}\frac{\partial}{\partial{{x}^{i}}}\in{{T}_{p}}M$
be on the Poisson-W manifold $\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$,
then dot product between $D=\nabla+A$ and the GCHS is given by

$D\cdot\frac{\mathcal{D}x}{dt}=\nabla\cdot v+\left(x\cdot D+2\right)w$

where $w=-\widehat{W}H=\left\\{H,1\right\\}$ is W dynamics.

###### Proof.

Based on the equation (2), let $D_{l}$ act on the component expression of GCHS
and obtain

$\displaystyle{{D}_{l}}\frac{\mathcal{D}{{x}_{k}}}{dt}$
$\displaystyle={{D}_{l}}{\dot{x_{k}}}+{{D}_{l}}\left(w{{x}_{k}}\right)={{D}_{l}}{\dot{x_{k}}}+{{x}_{k}}{{D}_{l}}w+w{{\delta}_{lk}}$
(4)
$\displaystyle={{\partial}_{l}}{\dot{x_{k}}}+{{A}_{l}}{\dot{x_{k}}}+{{x}_{k}}{{\partial}_{l}}w+{{x}_{k}}{{A}_{l}}w+w{{\delta}_{lk}}$

If $l=k$ holds for the (4), then the consequence will be accordingly obtained

${{D}_{k}}\frac{\mathcal{D}{{x}_{k}}}{dt}=\nabla\cdot v+\left(x\cdot
D+2\right)w$ (5)

where the divergence of vector $v$ is $\nabla\cdot
v={{\partial}_{k}}{\dot{x_{k}}}$. ∎

Apparently, this equation of theorem 2 simultaneously contains two kinetic
quantities $v,w$. Similarly, we can calculate the following corollary

######  Corollary 4.

An identity is given

${{\partial}_{kj}}{{W}_{jk}}=\nabla\cdot v+\left(x\cdot\widehat{D}+3\right)w$

where ${{\partial}_{kj}}={{D}_{k}}{{F}_{j}}+{{F}_{j}}{{D}_{k}}$,
${{F}_{j}}=-{{D}_{j}}H$ is generalized force.

Obviously, this is a second order differential equation, If the divergence of
vector field $v$ is 0, namely $\nabla\cdot v=0$, then it leads to the result
${{\partial}_{kj}}{{W}_{jk}}=\left(x\cdot\widehat{D}+3\right)w$, this is only
connected to the W Dynamics.

######  Theorem 8.

on the Poisson-W manifold $\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$
along with $\frac{\mathcal{D}}{dt}{{x}^{i}}=\left\\{H,{{x}^{i}}\right\\}$ such
that

$\left[{{D}_{i}},{{D}_{j}}\right]\frac{\mathcal{D}}{dt}{{x}^{i}}=c_{ij}^{i}\left(\nabla\cdot
v+w\right)+\vartheta_{j}^{k}{{D}_{k}}w+w_{j}^{i}{{A}_{i}}$

where
$\vartheta_{j}^{k}={{x}^{i}}c_{ij}^{k}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$.

###### Proof.

By substituting the claim 1 and the GCHS (2) into together, one can calculate
the equation below

$\displaystyle\left[{{D}_{i}},{{D}_{j}}\right]\frac{\mathcal{D}}{dt}{{x}^{i}}$
$\displaystyle=c_{ij}^{k}{{\partial}_{k}}{{v}^{i}}+{{v}^{i}}c_{ij}^{k}{{A}_{k}}+wc_{ij}^{k}{{\partial}_{k}}{{x}^{i}}+{{x}^{i}}c_{ij}^{k}{{\partial}_{k}}w+w{{x}^{i}}c_{ij}^{k}{{A}_{k}}$
$\displaystyle=c_{ij}^{k}{{\partial}_{k}}{{v}^{i}}+w_{j}^{k}{{A}_{k}}+wc_{ij}^{k}\delta_{k}^{i}+{{x}^{i}}c_{ij}^{k}{{\partial}_{k}}w+w{{x}^{i}}c_{ij}^{k}{{A}_{k}}$
(6)
$\displaystyle=c_{ij}^{i}\left({{\partial}_{i}}{{v}^{i}}+w\right)+\vartheta_{j}^{k}{{D}_{k}}w+w_{j}^{i}{{A}_{i}}$

where $\vartheta_{j}^{k}={{x}^{i}}c_{ij}^{k}$. Using the claim 1, the equation
(8) can be rewritten as

$\left[{{D}_{i}},{{D}_{j}}\right]\frac{\mathcal{D}}{dt}{{x}^{i}}=c_{ij}^{i}\left(\nabla\cdot
v+w\right)+{{q}_{j}}+\vartheta_{j}^{k}{{D}_{k}}w-w_{j}^{i}{{\partial}_{i}}$

∎

As a result of theorem 8, we can deduce the general operator shown as

$\left[{{D}_{i}},{{D}_{j}}\right]\frac{\mathcal{D}}{dt}=F_{ij}\frac{\mathcal{D}}{dt}+wc_{ij}^{k}{{\partial}_{k}}$

for all function in the ${{C}^{\infty}}\left(M,\mathbb{R}\right)$.

######  Theorem 9.

Let covariant force
${{\overset{\circ}{\mathop{F}}\,}_{k}}=\frac{\mathcal{D}{{p}_{k}}}{dt}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$
be specified on the Poisson-W manifold
$\left(P,W,\left\\{\cdot,\cdot\right\\}\right)$ along with the non-Abel group
such that

${{f}_{kj}}={{D}_{j}}{{\overset{\circ}{\mathop{F}}\,}_{k}}-{{D}_{k}}\overset{{}^{\circ}}{\mathop{F}}\,{{}_{j}}=u_{jk}+\left({{c}_{jk}}+{{L}_{kj}}+{{L}_{kj}}\chi\right)w$

is called reciprocal tensor. where
${{c}_{jk}}={{\xi}_{jk}}-{{\xi}_{kj}},~{}~{}{{L}_{kj}}={{p}_{k}}{{\partial}_{j}}-{{p}_{j}}{{\partial}_{k}}$,
and antisymmetric
${{f}_{kj}}=-{{f}_{jk}}\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$.

###### Proof.

The covariant force is shown by equation (3), then acting the covariant
derivative ${{D}_{j}}$ on it, we derive the equation

$\displaystyle{{D}_{j}}{{\overset{\circ}{\mathop{F}}\,}_{k}}$
$\displaystyle={{D}_{j}}\frac{\mathcal{D}{{p}_{k}}}{dt}=-{{D}_{j}}{{D}_{k}}H+{{D}_{j}}\left({{p}_{k}}w\right)$
$\displaystyle=-{{D}_{j}}{{D}_{k}}H+w{{D}_{j}}{{p}_{k}}+{{p}_{k}}{{D}_{j}}w-{{A}_{j}}{{p}_{k}}w$

Similarly, we have
${{D}_{k}}{{\overset{\circ}{\mathop{F}}\,}_{j}}=-{{D}_{k}}{{D}_{j}}H+{{p}_{j}}{{D}_{k}}w+w{{D}_{k}}{{p}_{j}}-{{A}_{k}}{{p}_{j}}w$,
we can derive the difference by subtracting the two equations above

$\displaystyle{{f}_{kj}}$
$\displaystyle={{D}_{j}}{{\overset{\circ}{\mathop{F}}\,}_{k}}-{{D}_{k}}\overset{{}^{\circ}}{\mathop{F}}\,{{}_{j}}$
$\displaystyle={{F}_{kj}}H+w\left({{\xi}_{jk}}-{{\xi}_{kj}}\right)+\left({{p}_{k}}{{D}_{j}}w-{{p}_{j}}{{D}_{k}}w\right)$
$\displaystyle=c_{jk}^{i}{{F}_{i}}+w{{c}_{jk}}+\left({{p}_{k}}{{D}_{j}}w-{{p}_{j}}{{D}_{k}}w\right)$
$\displaystyle=u_{jk}+\left({{c}_{jk}}+{{L}_{kj}}+{{L}_{kj}}\chi\right)w$

since then the corollary 3 and antisymmetric (2,0) tensor ${{F}_{ij}}$ have
been used for the proof. ∎

According to the definition of ${{\xi}_{jk}}$ in [1], naturally, it leads to
the simple expression

${{f}_{kj}}=u_{jk}+\left({{L}_{kj}}+{{L}_{kj}}\chi\right)w$

of reciprocal tensor at $j\neq k$.

## References

  * [1] G Wang, D Y Peng . A Study of Generalized Covariant Hamilton Systems On Poisson-W manifold[J]. arXiv:1710.10597v4.
  * [2] Jost, Jürgen. Riemannian geometry and geometric analysis[M]. Springer Science, Business Media, 2008, 17-20.
  * [3] Marsden J E, Ratiu T S. Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems[M]. Springer Science , Business Media, 2013, 61-85,157-165.
  * [4] Jianwei Zhou. A lecture on differential geometry[M]. (2010): 211-243.
  * [5] Stephen Wiggins. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. Springer, 2003, [197-225].
  * [6] Weinstein A. Lectures on symplectic manifolds[M]. American Mathematical Soc., 1977.15-20.
  * [7] Da Silva A C, Da Salva A C. Lectures on symplectic geometry[M]. Berlin: Springer, 2001.
  * [8] Meyer K, Hall G, Offin D. Introduction to Hamiltonian dynamical systems and the N-body problem[M]. Springer Science, Business Media, 2008.