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one of the solutions currently discussed in the field of quantum communication @xcite , @xcite , @xcite is a quantum repeater @xcite .
its construction is not so simple as building classic amplifier because of non - cloning theorem which is one of the most basic and important foundations of quantum mechanics .
however , the lack of ability of making perfect copies of quantum information may be compensated with use of teleportation and the phenomenon of entanglement .
the protocol of quantum teleportation may be utilized to amplify the signal i.e. a quantum state during the transfer process .
this approach may be used , e.g. in a fiber , to improve the quality of transferred information . applying the teleportation protocol and the entanglement to amplify the quantum information results with building of so - called quantum repeater ( qr ) .
it should be emphasized that the notion of qr is currently in the center of interest of many researchers @xcite , @xcite .
apart from the theoretical analysis of this subject there are also physical experiments carried out with use of qrs as the elements of quantum networks .
these experiments are being accomplished for the transmission of quantum states in an optical fiber @xcite and also in the air @xcite .
therefore , the progress in the field of quantum computing @xcite , @xcite , @xcite , @xcite , @xcite is strongly connected with the progress of quantum communication because of the need to send information .
the notion of qr is currently discussed as a potential solution to the problems of quantum communication and cryptography ( especially for quantum key distribution ) .
the three generations of qr were presented do far .
the first type utilizes the phenomenon of entanglement and its purification @xcite , @xcite , @xcite .
the second generation @xcite is based on the near - perfectly entangled pairs generation ( however , in this approach still the entanglement purification is used ) .
the third type @xcite , @xcite of qr utilizes the protocol of teleportation .
the quantum information is transmitted from one point to another , so the communication is organized as a one - way scheme .
the solution presented in this chapter belongs to the third generation of qr and it uses so - called one - dit teleportation protocol to correct the errors which may appear during the transfer process . the novelty of described solution consists on using only pauli - z gate for error correction .
more precisely : the result of measurement performed during the teleportation protocol determines the number of pauli - z operations which have to realized as an error correction .
the reminder of this chapter is organized as follows . in section ( [ lbl :
sec : single : dit : teleportation : ms : jw : cn:2016 ] ) there is a quantum teleportation protocol presented and the error correction is performed with use of pauli - z gate .
section ( [ lbl : sec : protocol : quantum : repeater : ms : jw : cn:2016 ] ) describes the realization of qr with use of previously mentioned teleportation protocol .
the interpretation of repeater protocol as a quantum circuit is shown in section ( [ lbl : sec : qc : for : qr : network : ms : jw : cn:2016 ] ) .
the summary and the final conclusions are presented in section ( [ lbl : sec : conclusions : ms : jw : cn:2016 ] ) .
the protocols presented in this section are defined with notions of dits and qudits .
a concept of dit is a generalization of classic bit .
for classic bit there can be distinguished only two states : zero and one . in the case of dit
more states are possible .
the number of these values is symbolized by the letter @xmath0 .
for example , the classic bit is a dit with @xmath1 .
a unit of quantum information is so - called qubit .
a definition of qubit may be presented as : @xmath2 where vectors @xmath3 , @xmath4 stand for the computational base ( in this case the standard base is used ) : @xmath5 , \enspace \vert 1 \rangle = \left [ \begin{array}{c } 0\\ 1\\ \end{array } \right].\ ] ] these vectors represent the classic states zero and one .    by analogy ,
a generalization of qubit is a qudit . in this case
more base states are admissible .
the state of unknown qudit with @xmath0 base states is represented as : @xmath6 the number of base states for a given qudit , expressed as @xmath0 , will be also called a qudit s freedom level .
standard base for qudits requires more vectors e.g. for so - called qutrits ( qudits with @xmath7 ) standard basis vectors  are : @xmath8,\enspace { | 1 \rangle } = \left [ \begin{array}{c } 0 \\ 1 \\ 0 \end{array } \right],\enspace { | 2 \rangle } = \left [ \begin{array}{c } 0 \\ 0 \\ 1 \end{array } \right].\ ] ]    the presented repeater protocol is realized as a single - dit teleportation protocol .
the described protocol differs from other one - dit protocols @xcite , @xcite in this way that only pauli - z gate is used as a correction gate .
the pauli - z ( or just @xmath9 ) gate for qudits may be defined as :    @xmath10    where @xmath11 stands for the root of unity : @xmath12 more precisely , @xmath13 is a @xmath0-th root number @xmath14 of unity .
the symbol @xmath15 represents the imaginary unit .
there are also hadamard gate and cnot gate used .
definitions of these gates for qudits are : @xmath16 where @xmath17 . in the case of cnot gate
the @xmath18 gate may be given as well - this gate is obtained as the hermitian conjugate of cnot operation .
the @xmath18 gate is useful for a quantum teleportation protocol described in this section .
the exemplary matrix form of cnot gate for qutrits ( @xmath7 ) is : @xmath19 where the zeros were replaced by dots to make the notation more legible . in general , the cnot gate for qudits is constructed in a following way : @xmath20 where the symbol @xmath21 stands for matrix direct sum and @xmath22 means the negation operation for qudit with freedom level @xmath0 .
the gates , mentioned above , allow to present the teleportation protocol of an unknown quantum state : @xmath23 the 3-qudit state @xmath24 , where @xmath25 belongs to alice and @xmath26 to bob , after performing the hadamard , cnot and @xmath18 operations on it , due to fig .
[ lbl : fig : circuit : teleporation : ms : jw : cn:2016 ] , may be described as : @xmath27 where @xmath28 .
the gates cnot and @xmath18 realize the entanglement .
function @xmath29 is expressed as : @xmath30 a correction gate s form depends on the result of measurement performed on the first qudit by alice ( however , both qudits are measured in the standard base what is a consequence of hadamard gate s use in a quantum cirucit 
see fig .
[ lbl : fig : one - way - quantum - repeater : ms : jw : cn:2016 ] ) .
if alice , after the measurement , obtains one of given states @xmath31 then the bob s qudit is characterized by the proper values of probability amplitudes and there is no additional operations needed to correct the obtained state .
this also stands with the accordance to the values of function @xmath29 : @xmath32 , @xmath33 - they equal zero .
if alice , also after the operation of measurement , obtains one of following states @xmath34 then the state of bob
s qudit is @xmath35 the values of @xmath29 function in this case are following : @xmath36 when the @xmath9 gate is used only once , then the correct values of some amplitudes may be recovered . in general
, the result of measurement determines the number of operations @xmath9 which should be performed .
it can be expressed as @xmath37 if @xmath38 stands for the result of measurement performed on the first qudit .
the above description presents the teleportation protocol for an unknown qudit , where @xmath9 gate is needed to correct the final state and only one measurement decides about the way how the correction is done after the teleportation process .    [ cols="^ " , ]     although the suggested number of qudits needed to carry a correct simulation is @xmath39 , it should be emphasized that the phenomenon of entanglement occurs inside of repeaters .
it means that the entanglement is present in a teleportation process and it occurs between three qudits which consist the repeater itself .
there is no entanglement between particular repeaters .
this assumption makes easier the implementation of mentioned quantum phenomena with use of classic computers .
it is also important to know that the repeaters are connected physically ( e.g. optical fiber ) with suitable quantum channel .
if the numeric or symbolic simulation is to be carried out on a classic machine , then only the three - qudit repeater needs to be simulated .
the features of communication channel , which transmits a quantum state , may be simulated apart from the simulation of qr .    ....
-- n : numer of nodes -- i : index of node -- $ \mket{\psi_0}$ : initial state -- h : change history of the transmitted state    h $ \leftarrow$ append(h , $ \mket{\psi_0}$ , 0 ) [ $ \mket{\psi_t}$ , r ] $ \leftarrow$ qrepeater ( $ \mket{\psi_0}$ ) h $ \leftarrow$ append(h , $ \mket{\psi_t}$ , r ) $ \mket{\psi_t}$ $ \leftarrow$
$ z^r \mket{\psi_t}$ while i < n do begin      [ $ \mket{\psi_t}$ , r ] $ \leftarrow$ qrepeater ( $ \mket{\psi_t}$ , r )      ...      other operations
e.g. noise generation      entanglement level measuring and etc .      ...      h $ \leftarrow$ append(h , $ \mket{\psi_t}$ , r )      $ \mket{\psi_t}$ $ \leftarrow$
$ z^r \mket{\psi_t}$          i $ \leftarrow$ i + 1 end $ \mket{\psi_{final}}$ $ \leftarrow$ $ \mket{\psi_t}$ ....    fig .
[ lbl : fig : classical - sim - of - quantum - repeater : ms : jw : cn:2016 ] presents the pseudo - code illustrating the simulation s algorithm for a given state .
the changes are saved in a classic variable @xmath40 which collects the quantum states obtained after the calculations in every node .
the initial state is assigned to variable @xmath41 and the final state to @xmath42 .
the one - dit teleportation protocol with error correction performed by pauli - z gate allows the realization of qr for transmitting the unknown qudit state .
two presented strategies for z - type error correction provide the choice to the user who can run the protocol with the error correction at the end of the whole process or the correction may be performed locally on every qr .
the approach with the correction at the end of transmisssion reduces the use of z gate application .
we use this gate only at the end of the process to complete the communication protocol by correcting the received information .
this solution excludes the need of using pauli - x and hadamard gates , which are utilized in other known protocols for qr realization .
the advantages connected with presented approach are : the influence of possible errors , caused by the imperfect realization of quantum gates , is lower and the construction of qr is easier because only the operation of measurement have to be performed ( the error correction is redundant ) .
the another interesting problem for further research is using quantum error correcting codes on transmitted state .
the presented protocol concerns the pure states , so it would be also important to analyze the quality of transmitted information taking into account the quality of utilized quantum gates and the influence of environment .
we would like to thank for useful discussions with the  _ q - info _ group at the institute of control and computation engineering ( issi ) of the university of zielona gra , poland .
we would like also to thank to anonymous referees for useful comments on the preliminary version of this paper .
the numerical results were done using the hardware and software available at the 
gpu @xmath43-lab  located at the institute of control and computation engineering of the university of zielona gra , poland .
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in this chapter a quantum communication protocol with use of repeaters is presented . 
 the protocol is constructed for qudits i.e. the generalized quantum information units . 
 one - dit teleportation is based on the generalized pauli - z ( phase - flip ) gate s correction . 
 this approach avoids using pauli - x and hadamard gates unlike in other known protocols based on quantum repeaters which were constructed for qubits and qudits . 
 it is also important to mention that the repeaters based on teleportation protocol , described in this chapter , allow a measurement in the standard base ( what simplifies the measurement process ) and the use of teleportation causes only pauli - z operational errors .    
 information transfer , quantum repreater , qudit teleportation protocol