application - level forward erasure correction ( al - fec ) codes have become a key component of many content delivery systems .they are widely used as an efficient technique to recover packet losses in internet ( usually caused by congested routers ) or wireless communications ( often caused by a short term fading problem ) .such a network can be regarded as a packet erasure channel ( or equivalently a bit erasure channel , bec ) , characterized by the property that the transmitted data packets are either received without error or erased ( lost ) .packet loss resilience may also be achieved with automatic repeat request ( arq ) techniques ( e.g. , with tcp ) , but : a round trip time ( rtt ) is needed to recover from a loss , which can be a issue for delay - sensitive applications ( e.g. , video - conferencing ) , the return channel may not exist ( e.g. , in case of a unidirectional broadcast network ) , and it does not scale well with the number of receivers in case of multicast or broadcast transmissions .al - fec codes are a key building block of content broadcast technologies such as the flute / alc  protocol stack for the reliable and scalable transmission of files to a potentially huge number of receivers , and the fecframe framework  when dealing with real - time delivery services as in streaming applications .al - fec is now deployed in all systems relying on flute / alc ( e.g. , 3gpp mbms service  or isdb - tmm  ) and sometimes at the link layer as well ( e.g. , the mpe - fec layer of dvb - h systems  ) .  in all of the previous use - cases ( real - time delivery included ) ,only block codes are considered , and the set of source packets is first grouped into blocks where al - fec encoding / decoding is performed . in the present work we introduce an alternative and practical al - fec solution that aims at encompassing both block oriented and sliding window oriented use - cases .random linear codes ( rlc ) are another class of al - fec codes .they are increasingly popular due to their simple yet powerful encoding techniques , in particular in the context of random linear network coding ( rlnc ) where encoding / decoding can be performed at the various network nodes , namely either at intermediate nodes ( e.g. , wifi access point or routers ) or end nodes  . at a source ( sender )node or intermediate node , rlc generates encoded packets ( also called encoded symbols in this paper ) just by linearly combining the available symbols using encoding vectors ( also called coefficients ) randomly selected from a given finite field ( e.g. , ) . in general, the set of available symbols evolves over the time , i.e. , rlc are used as convolutional codes . in , rlc is also utilized in a convolutional manner , but end - to - end ( i.e. , there is no re - encoding within the core network ) , with feedback information from the receiver , which enables to achieve a full reliability when desired .the authors show that the recovery delay for lost packets is in that case independent of the rtt .however the main issue to be considered with rlc is the high decoding complexity , typically a gaussian elimination ( ge ) over a dense linear system .this problem becomes even more pronounced when the number of source symbols involved is large , and/or when the finite field is ( or higher ) so as to improve the erasure recovery capabilities  .  since we believe that rlc can play a key role in network coding systems for the erasure channel  , we have focused on the design of new improved rlc techniques .our goal is to design rlc codes that :  * can be used either as block or convolutional codes ; * can be used with encoding window sizes in 210,000s symbols range , as very large sizes are beneficial to bulk file transfers while small values are useful for real - time contents ; * have excellent erasure recovery performance , and at the same time enable fast encoding / decoding which is essential for devices with limited computational and memory capabilities ; * enable compact signaling ( e.g. , transmitting the full encoding vector does not scale ) ;  in other words , we try to bridge the gap between block and convolutional al - fec codes . with these goals in mind, we have designed the so - called structured random linear codes ( srlc ) .in the present work , as a first step to a complete evaluation , we only focus on use - cases that require only end - to - end encoding ( i.e. , there is a single end point for al - fec encoding / decoding , no matter whether this end is a `` host '' or a `` middlebox '' ) and we evaluate the srlc effectiveness in terms of erasure recovery performance only .the remainder of this paper is organized as follows : section [ sec : related - work ] introduces related work .section [ sec : prop - srlc ] describes the proposed structured random linear codes ( srlc ) in detail .section [ sec : eval ] evaluates the recovery performances of srlc , and we conclude in section [ sec : concl ] .in  batched sparse ( bats ) codes are proposed for file distribution through a communication network where intermediate nodes have coding capabilities  .these codes are designed so as to control the computational and storage requirements at the source , intermediate nodes , and destination , as well as the transmission overhead when transmitting the coding vector .this is made possible by the use of both an outer code ( sender ) , that forms `` batches '' of coded packets , using a specific distribution for defining the number of input packets considered to create each batch , and inner codes ( intermediate nodes ) that perform random linear codings of packets of a given batch .the authors show the good performance of these batch codes , when associated to `` inactivation decoding '' as in . in  , gamma codesare proposed as a family of sparse random linear network codes with outer code like bats codes .the codes also manage `` chunked '' encoded packets . the key idea is to enable the outer code to play as soon as the first chunked packets are recovered , which realizes a joint decoder scheme that coordinates a proper combination of an outer coding and a basic sparse random linear network coding .it was presented that gamma codes can achieve better reception overhead while keeping lower encoding / decoding complexity in fixed block length configuration .these works differ from ours .we designed rlc in order to be flexibly used as either block or convolutional codes , over wide ranges of block / encoding - window sizes .additionally , we do not distinguish outer / inner codes per se , but add a structure to the rlc approach in order to find an appropriate balance between computational complexity and erasure recovery performance .let us now describe the srlc codes , characterized by :  1 .  a mostly sparse binary structure , which reduces the number of symbol xor operations and improves iterative ( it)/structured gaussian elimination ( sge ) decodings  .it is a key feature to favor high speed encoding and decoding ; 2 .  a limited use of non - binary ( over ) coefficients in encoding vectors .it is a key feature to favor a good erasure recovery performance ; 3 .the addition of a dedicated repeat - and - accumulate structure ( as in irregular repeat - accumulate ( ira )  and ldpc - staircase codes ) .it is a second key feature to favor a good erasure recovery performance .  in the following ,we first explain the srlc approach when used in block mode , and later we extend it to the case of convolutional coding .the first idea consists in using both binary and non - binary coefficients .all the examples of this section are for the block mode case , when considering a fixed set of source symbols .[ fig : mix - bin - nonbin ] shows an example parity check matrix , , when the number of source symbols is fixed ( here source symbols ) and both binary and non - binary coefficients are used .the matrix is composed of two parts , the left side and the right side .the columns of correspond to the source symbols from to ( a.k.a .source packets ) , while those of correspond to the repair symbols to be generated ( a.k.a .repair packets ) .each row of represents a constraint ( or equation ) used for instance to generate the repair symbol of the same row . for example , the repair symbol in the first row is generated by '' coefficients are omitted . ] :   three key parameters exist :  * : the source block length ( or encoding window size ) ; * : the density of each `` sparse binary sub - matrix '' , given as the ratio of the number of non - zero coefficients to the total number of coefficients in a sub - matrix : * : the ratio of the number of non - binary columns to ( i.e. , the total number of columns ) in :   should be largely composed of sparse binary parts so that most equations are sparse with binary coefficients , because this is a key for high encoding / decoding speeds .however , a trade - off between speed and erasure recovery performance must be considered since being too sparse and binary negatively impacts the erasure recovery performance .[ fig : avg - ineff - binary - rlc ] shows the average erasure recovery performance of a fully binary rlc with various values as a function of .the performance metric is the `` average decoding inefficiency ratio '' , defined as the ratio of the average number of symbols needed for decoding to complete successfully to : where is called `` decoding overhead '' , and also often expressed as a percentage .the closer to ( achieved with ideal codes ) the ratio , the better . assuming that our target average decoding overhead is set ( arbitrarily ) to 0.1%, we can see in fig .[ fig : avg - ineff - binary - rlc ] that none of the codes achieves the goal , even with binary rlc ( where ) which performs the best .  on the opposite , we see in fig . [ fig : avg - ineff - nonbin_1_40_-rlc ] that adding a few dense non - binary columns ( we used and the same values for ) , the target average decoding overhead is easily achieved with . andadding more non - binary columns easily enables to further improve the decoding performance with smaller values .we will address the question of what are the appropriate \{ , } tuples as a function of in section [ subsec : add - structure ] .  in the srlc design , dense non - binary coefficients are always gathered in columns ( i.e. , assigned to certain symbols ) .the motivation is to enable the use of the high speed structured gaussian elimination optimization  . in this approach , when a stopping set is encountered during it decoding , certain well chosen symbols of the system ( i.e. , corresponding to unknown / non - recovered source symbols ) are logically removed from the linear system .this process enables it decoding to pursue .finally , decoding finishes with a classic gaussian elimination over the removed symbols , and their values are finally re - injected into equations where they were involved . because non - binary coefficients are affected to well identified source symbols , these symbols are immediately logically removed from the linear system .the linear system therefore remains a sparse binary system , not `` polluted '' by non binary coefficients , and most operations consist of fast xor operations over symbols , a key for high speed decoding .adding a structure to codes can be highly beneficial .for instance , the repeat - and - accumulate structure of ira and ldpc - staircase codes significantly improves their performance : because the number of source symbols a repair symbol actually depends on increases with its index , the erasure recovery performance is improved while keeping a sparse system .however , adding this particular structure would make signaling prohibitively complex when the codes are used in convolutional mode .    in order to solve this problem ,we propose to add a single accumulative row to create , defined as the `` heavy repair symbol '' , and to make all repair symbols depend upon .[ fig : srlc - block - example ] shows an example of the proposed srlc in block mode . in this example , is generated by :   this simple structure enables a compact signaling to enable a receiver to determine the relationships between source and repair symbols .this is accomplished , in block mode , by the knowledge of the matrix generation algorithms ( specified in a non ambiguous way in the specifications ) , plus the \{ , , } tuple that is sent once at encoder / decoder synchronization time .because the full encoding vector is not sent along with repair packets ( it is useless if is known ) , the approach can scale with very large values .we will see in section [ subsec : appli - convol ] how to perform signaling when srlc are used in a sliding window mode .let us now determine the most appropriate values for the \{ , } tuple for a given and target average performance .of course :  * should be as small as possible to reduce computation complexity , and * should be as small as possible so that it decoding performs well .  to find appropriate values , we did as described in algorithm [ algo : find - value ] .we set the target average overhead to ( same value as in fig .[ fig : avg - ineff - nonbin_1_40_-rlc ] ) plus a security margin ( set to ) so as to accommodate some fluctuations during the optimization process . as a result, we obtain a table of \{ , } tuples , with an entry for each value .note that this table ( not reproduced here ) does not need to be sent to the receiver(s ) as the \{ , , } tuple is communicated at synchronization time to the receiver(s ) .this provides additional flexibility since the the target code performance may be changed dynamically for the following transfers , at the discretion of the sender . ; / * for instance * / ; / * for instance * / / * first of all , find if is set to * / ; ;  ; ; break ;  / * then find smallest for the selected * / ;  ; ; break ;   ;    ; / * reset to zero as well * / ; / * reset to zero * / ; ; ;  ;   ; ; ; / * non - binary col . ,choose coeff randomly */ ; ; / * binary column , choose 0 or 1 randomly * / ; ; ; ; ;  convolutional coding is appropriate to situations where a fully or partially reliable delivery of continuous data flows is needed , especially when these data flows feature real - time constraints , as in  .srlc can then be used as a convolutional code , in a systematic way ( i.e. , source symbols are sent on the network ) , as described in algorithm [ algo : convol ] . the way the encoding windowis managed ( i.e. , how to set the encoding window start and the number of source symbols in the window ) is a key aspect that depends on the protocol in use .[ fig : srlc - convol - example ] illustrates the use of srlc in the simple sliding window mode . herethe encoding window has a fixed size , , and slides in a regular way over the source symbol flow . the target code rate ( is such that one repair symbol is sent after two source symbols .the only exception is at session start : the encoder waits for source symbols to be available , and then generates two repair symbols , including a heavy repair one , ( i.e. , the xor sum from to ) .then , after sending two more source symbols , and , the srlc encoder considers the union of the encoding windows since the previous repair computation ( i.e. , from to ) and generates a new repair symbol , . accumulates the current heavy repair symbol , ( i.e. , the xor sum from to ) to the encoding vector : here also , the encoding vector is set according to the \{ , , } tuple , using pre - calculated tables as described in section [ subsec : add - structure ] , and a pseudo - random number generator ( prng ) that can be seeded by a specific value communicated to the receiver .note that the repair symbol identifier may be used as a seed .  from a signaling point of view, we can assume that the \{ , , } tuple and all the algorithms are known by both ends .in that case , it is sufficient for the sender to let the receiver know the union of the encoding windows considered ( e.g. , from to in the case of ) , the repair symbol identifier , along with the prng seed ( if different from the repair symbol identifier ) .this is all the srlc decoder needs to know to generate the constraint equation associated to this repair symbol , even for large encoding window sizes .  in practice ,the heavy repair symbols are transmitted periodically in order to remove the long term dependencies they create .this is useful if past source symbols remain impossible to recover by a given receiver ( who for instance joined the session late ) .in this section we evaluate the srlc erasure recovery performance both in block and convolutional modes .   all the tests are carried out with the performance evaluation tools provided by our ` openfec.org ` project  and a modified version of the ` kodo ` library  for the codec implementation .we use the pre - calculated values for \{ , } ( see algorithm [ algo : find - value ] ) and we choose in all tests. however srlc is by nature rateless and the actual code rate is of little importance ( i.e. , the decoding overhead does not depend on the code rate ) . because we do not want to define any specific channel model ( e.g. , the two transition probabilities of a gilbert model ) , in all tests we assume the source and repair symbols are transmitted in a fully random order , which means that only the packet loss rate is of importance .finally we assume that gaussian elimination decoding is used for maximum performance , rather than it decoding ( we do not consider decoding speeds in this work ) .let us focus on the srlc in block mode .we measure both the average inefficiency ratio as a function of and the decoding failure probability as a function of the number of received symbols in addition to ( in both cases decoding is said to fail as soon as at least one erased source symbol can not be recovered ) .the first goal of tests is to demonstrate the efficiency of the use of a heavy repair symbol .the second goal is to assess the performance of srlc codes .however , due to the space limitations , we only show the results when is small , from 50 to 500 symbols .[ fig : avg - ineff - comp - heavyrepair ] compares the two options for or .we see the benefits of using the heavy repair symbol , especially when is small , on average .let us look at fig .[ fig : srlc - decfail ] , when and . in both cases ,the decoding failure probability curves are similar when the number of received symbols is only slightly higher than ( i.e. , for low overheads ) .however we clearly see a difference when the overhead is higher , meaning that there is a significant number of tests where decoding fails without any heavy repair symbol : extra symbols need to be received ( % overhead ) for the decoding failure probability to go below . on the opposite ,the full featured srlc solution reaches a decoding failure probability lower than with 209 symbols only ( a % overhead ) .[ fig : srlc - decfail]-(b ) also confirms the excellent recovery performance of srlc codes , not only on average , but also when looking precisely at the decoding failure probability .let us now consider srlc in convolutional mode .since we are focusing on real - time flows , like video / audio real - time streaming systems , a full reliability is not necessarily required ( this is different from typical use in block mode , for file transfer applications ). therefore we measure the srlc average source packet loss ratio ( once decoding is finished ) as a function of packet loss probability , and compare it with those of binary rlc ( i.e. , ) and of rlc over ( all coefficients are randomly chosen in ) .additionally , to make the comparison more visible , we measure the decoding failure probability of the three codes .  the performance results for the transmission of source symbols in total and a window of size symbols ,are shown in fig .[ fig : srlc - srcloss - convol ] .we see that srlc performs the best , even when compared to rlc over , which is exceptional .the performance results for a larger encoding window , of size symbols , are shown in fig .[ fig : srcloss - k100 ] .we see that all the average loss ratios improve when compared to the case , because a larger encoding window size offers better protection .therefore there is no significant differences among the three codes especially on average .however , when looking at the decoding failure probability in fig .[ fig : decfail - k100 ] , the srlc performance pronouncedly becomes worse than that of rlc codes over .one reason is that srlc uses the table optimized for the block mode case .a new table should be calculated for the convolutional case .this work introduces the srlc codes , an end - to - end al - fec solution that is sufficiently flexible to be applied in block mode and convolutional mode . in order to enable excellent erasure recovery performance as well as fast encoding and decoding speeds ,these codes have been designed in a manner that favors a mostly sparse and binary structure , with some well chosen non binary coefficients , plus a heavy binary row . additionally , the design is such that it facilitates an efficient signaling , the parameters exchanged to synchronize encoder and decoders being kept to a minimum .these considerations make srlc codes a very practical solution , no matter the block or encoding window size : small , medium or large .our evaluation of their erasure recovery performance confirms the benefits in future works we will analyze the encoding and decoding complexity ( similarly the associated speeds ) of srlc codes .we will also further optimize the value of the code internal parameters when used in convolutional mode , both in a fixed - size configuration and in elastic window configuration ( e.g. , as in  ) .t. paila , r. walsh , m. luby , v. roca , and r. lehtonen , `` flute - file delivery over unidirectional transport '' , ietf rmt working group , request for comments , rfc 6726 ( `` standards track / proposed standard '' ) , nov .a. yamada , h. matsuoka , t. ohya , r. kitahara , j. hagiwara , and t. morizumi , `` overview of isdb - tmm services and technologies '' , ieee international symposium on broadband multimedia systems and broadcasting ( bmsm11 ) , june 2011 .tournoux , e. lochin , j. lacan , a. bouabdallah , and v. roca , `` on the fly erasure coding for time - constrained applications '' , ieee transactions on multimedia , vol .797 - 812 , aug .s. nazir , d. vunkobratovi ' and v. stankovi , `` peformance evaluation of raptor and random linear codes for h.264/avc video transmission over dvb - h networks '' , ieee international conference on acoustics , speech and signal processing ( icassp11 ) , may 2011 .v. roca , m. cunche , c. thienot , j. detchart and j. lacan , `` rs+ldpc - staircase codes for the erasure channel : standards , usage and performance '' , ieee international conference on mobile computing , networking and communications ( wimob13 ) , nov . 2013 .v. roca , c. neumann and d. furodet , `` low density parity check ( ldpc ) staircase and triangle forward error correction ( fec ) schemes '' , ietf rmt working group , rfc 5170 ( `` standards track / proposed standard '' ) , june 2008 .

Abstract: several types of al - fec ( application - level fec ) codes for the packet erasure channel exist . random linear codes ( rlc ) , where redundancy packets consist of random linear combinations of source packets over a certain finite field , are a simple yet efficient coding technique , for instance massively used for network coding applications . however the price to pay is a high encoding and decoding complexity , especially when working on , which seriously limits the number of packets in the encoding window . on the opposite , structured block codes have been designed for situations where the set of source packets is known in advance , for instance with file transfer applications . here the encoding and decoding complexity is controlled , even for huge block sizes , thanks to the sparse nature of the code and advanced decoding techniques that exploit this sparseness ( e.g. , structured gaussian elimination ) . but their design also prevents their use in convolutional use - cases featuring an encoding window that slides over a continuous set of incoming packets .  in this work we try to bridge the gap between these two code classes , bringing some structure to rlc codes in order to enlarge the use - cases where they can be efficiently used : in convolutional mode ( as any rlc code ) , but also in block mode with either tiny , medium or large block sizes . we also demonstrate how to design compact signaling for these codes ( for encoder / decoder synchronization ) , which is an essential practical aspect .


the most important obstacle in quantum information processing is decoherence .it causes a quantum computer to lose its quantum properties destroying its performance advantages over a classical computer .the unavoidable interaction between the open quantum processor and its environment corrupts the information stored in the system and causes errors that may lead to wrong outputs . in general, environments may be very complex systems characterized by many uncontrollable degrees of freedom .a useful active strategy to defend quantum coherence of a processing against environmental noise is that of quantum error correcting codes ( qecc ) perimeter , knill97 , calderbank97 where , in analogy to classical information theory , quantum information is stabilized by using redundant encoding and measurements .  the formal mathematical description of the qubit - environment interaction is often given in terms of quantum channels .quantum error correction is usually developed under the assumption of i.i.d .( identically and independently distributed ) errors .these error models are characterized by memoryless communication channels such that -channel uses is given by .in such cases of complete independent decoherence , qubits interact with their own environments which do not interact with each other .however , in actual physical situations , qubits may interact with a common environment which unavoidably introduces correlations in the noise .for instance , there are situations where qubits in a ion trap set - up are collectively coupled to their vibrational modes . in other situations , different qubits in a quantum dot design are coupled to the same lattice , thus interacting with a common thermal bath of phonons .the exchange of bosons between qubits causes spatial and temporal correlations that violate the condition of error independence .memory effects introduce correlations among channel uses with the consequence that .recent studies try to characterize the effect of correlations on the performance of qeccs .it appears that correlations may have negative or positive impact on qeccs depending on the features of the error model being considered .furthermore , the noise may be asymmetric .most of the quantum computing devices are characterized by relaxation times ( ) that are one - two orders of magnitude larger than the corresponding dephasing times ( ) .relaxation leads to both bit - flip and phase - flip errors , whereas dephasing ( loss of phase coherence , phase - shifting ) only leads to phase - flip errors .such asymmetry between and translates to an asymmetry in the occurrence probability of bit - flip ( ) and phase - flip errors ( ) .the ratio is known as the channel asymmetry .quantum error correction schemes should be designed in such a way that no resources ( time and qubits ) are wasted in attempting to detect and correct errors that may be relatively unlikely to occur .quantum codes should be designed in order to exploit this asymmetry and provide better performance by neglecting the correction of less probable errors .indeed , examples of efficient quantum error - correcting codes ( for instance , asymmetric stabilizer csscodes ) taking advantage of this asymmetry are given by families of codes of the calderbank - shor - steane ( css ) type .following these lines of investigations , in this article we study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors . specifically , we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five and seven - qubit stabilizer codes .we characterize the performance of the codes by means of the entanglement fidelity  as function of the error probability and degree of memory ( correlations ) .we show that the performance of both codes is lowered in the presence of correlations and error probability threshold values for code effectiveness are computed vs. the degree of memory .the error correction schemes here considered only work for low values of . furthermore , we uncover that the asymmetry in the error probabilities does not affect the performance of the five - qubit code while it does affect the performance of the seven - qubit code which results less effective when considering correlated and symmetric depolarizing errors , but more effective for correlated and asymmetric errors .the layout of the paper is as follows . in sectionii , we consider a depolarizing noisy quantum memory channel characterized by symmetric error probabilities and qec is performed via the \right ] ] as function of the error probability and degree of memory . in section iii ,qec is performed via the \right ] ] and \right ] } \left ( \mu \text { , } p\right ) ] stabilizer code . the performance of quantum error correcting codes is quantified by means of the entanglement fidelity \right ] } \left ( \mu \text { , } p\right ) ] denotes the error probability ,  ] code is the smallest single - error correcting quantum code . of all qeccs that encode qubit of data and correct all single - qubit errors , the \right ] ] of group generators is given by , \right ] } \overset{% \text{def}}{=}\left\ { x^{1}z^{2}z^{3}x^{4}\text { , } x^{2}z^{3}z^{4}x^{5}\text{% , } x^{1}x^{3}z^{4}z^{5}\text { , } z^{1}x^{2}x^{4}z^{5}\right\ } \text{.}\]]the distance of the code is and therefore the weight of the smallest error  that can not be detected by the code is .finally , we recall that it is a non - degenerate code since the smallest weight for elements of ( other than identity ) is and therefore it is greater than the distance .the encoding for the \right ] ] is the check matrix for the five - qubit code , \right ] } \overset{\text{def}}{=}\left (  \begin{array}{ccccccccccc } 1 & 1 & 0 & 0 & 0 & | & 0 & 0 & 1 & 0 & 1 \\  0 & 1 & 1 & 0 & 0 & | & 1 & 0 & 0 & 1 & 0 \\  0 & 0 & 1 & 1 & 0 & | & 0 & 1 & 0 & 0 & 1 \\  0 & 0 & 0 & 1 & 1 & | & 1 & 0 & 1 & 0 & 0% \end{array}% \right ) \text { , }  \label{check1}\]]and is the vector in the -dimensional binary vector space corresponding to the error operator .the set of correctable error operators is given by, the cardinality of defining the channel in ( [ nota ] ) equals .all weight zero and one error operators satisfy the error correction conditions , , and , with .the two sixteen - dimensional orthogonal subspaces and of generated by the action of on and are given by, , , ... , and, .notice that with , and , .therefore , it follows that .the recovery superoperator with , ..., is defined as , the unitary operator is such that for . notice that ( see appendix a for the explicit expressions of recovery operators), that is a trace preserving quantum operation , , since with , ... , and is an orthonormal basis for .finally , the action of this recovery operation on the map in ( [ nota ] ) yields,  _ entanglement fidelity_. entanglement fidelity is a useful performance measure of the efficiency of quantum error correcting codes .it is a quantity that keeps track of how well the state and entanglement of a subsystem of a larger system are stored , without requiring the knowledge of the complete state or dynamics of the larger system .more precisely , the entanglement fidelity is defined for a mixed state in terms of a purification to a reference system .the purification encodes all of the information in .entanglement fidelity is a measure of how well the channel preserves the entanglement of the state with its reference system .the entanglement fidelity is defined as follows , is any purification of , is the identity map on and is the evolution operator extended to the space , space on which has been purified .if the quantum operation is written in terms of its kraus error operators as , , then it can be shown that , expression for the entanglement fidelity is very useful for explicit calculations . finally , assuming that choosing a purification described by a maximally entangled unit vector for the mixed state , we obtain expression in ( [ nfi ] ) represents the entanglement fidelity when no error correction is performed on the noisy channel in ( [ pla1 ] ) .  herewe want to describe the action of in ( [ r5 ] ) restricted to the code subspace .note that the recovery operators can be expressed as, . recalling that , it turns out that, now need to compute the matrix representation _ { |\mathcal{c}} ] becomes, _ { |\mathcal{c}}=\left (  \begin{array}{cc } \left\langle 0_{l}|a_{l}^{\prime \dagger } a_{k}^{\prime } |0_{l}\right\rangle & 0 \\  0 & \left\langle 1_{l}|a_{l}^{\prime \dagger } a_{k}^{\prime } |1_{l}\right\rangle% \end{array}% \right ) = \sqrt{\tilde{p}_{l}}\delta _ { lk}\left (  \begin{array}{cc } 1 & 0 \\  0 & 1% \end{array}% \right ) \text{,}\]]while for any pair with , , ... , and , it follows that, conclude that the only matrices _ { |% \mathcal{c}} ] defined as, \right ] } \left ( \mu \text { , } % p\right ) \overset{\text{def}}{=}\mathcal{f}^{\left [ \left [ 5,1,3\right ] % \right ] } \left ( \frac{1}{2}i_{2\times 2}\text { , } \mathcal{r\circ } \lambda^{(5)}\right ) = \frac{1}{\left ( 2\right ) ^{2}}\sum_{k=0}^{2^{10}-1}\sum% \limits_{l=1}^{16}\left\vert \text{tr}\left ( \left [ r_{l}a_{k}^{\prime } % \right ] _ { |\mathcal{c}}\right ) \right\vert ^{2}\text { , }  \label{fidel}\]]results, \right ] } \left ( \mu \text { , } % p\right ) = \sum_{m=0}^{15}\tilde{p}_{m}=p_{00}^{4}p_{0}+3\left [ 2p_{00}^{3}p_{10}p_{0}+3p_{00}^{2}p_{01}p_{10}p_{0}\right ] \text{. } \label{sym5}\]]notice that the expression in ( [ fidel ] ) represents the entanglement fidelity after the error correction scheme provided by the five - qubit code is performed on the noisy channel . the explicit expression for \right ] } \left ( \mu \text { , } % p\right ) ] .the effectiveness parametric region \right ] } ] defining the parametric region where qec is effective is plotted in figure .furthermore , we point out that the presence of correlations in symmetric depolarizing errors does not improve the performance of the five - qubit code since \right ] } \left ( \mu \text { , } p\right ) \leq \mathcal{f}^{\left [ \left [ 5,1,3\right ] % \right ] } \left ( 0\text { , } p\right ) ] .finally , the plots of \right ] } \left ( \mu \text { , } p\right ) ] vs. with ( for , the error correction scheme is not effective anymore ) for ( thick solid line ) , ( thin solid line ) and ( dashed line).,scaledwidth=40.0% ]in this section , we consider a depolarizing noisy quantum memory channel with symmetric error probabilities and qec is performed via the \right ] ] as function of the error probability and degree of memory ._ error model_. consider seven qubits and correlated errors in a depolarizing quantum channel , \text{,}\]]where , , , are the pauli operators and the coefficients with ,  satisfying the normalization condition, the depolarizing channel , coefficients are explicitly defined in ( [ conditional ] ) ._ error operators_. in an explicit way , the depolarizing channel can be written as, are the enlarged error operators acting on the seven qubit quantum states .the cardinality of the error operators defining is and is obtained by noticing that, is the cardinality of weight- error operators in ( [ jfk ] ) .  _encoding_. the calderbank - shor - steane ( css ) codes are constructed from two classical binary codes and that have the following properties : 1 ) and are  ] codes , respectively ; 2 ) ; 3 ) and ( the dual code of ) are both -error correcting codes . for instance , in case of the seven - qubit code , the two classical codes are the  ] binary simplex code .the dual code is the  ] -css code encodes qubit in qubits . the cardinality of its stabilizer group is and the set \right ] } ] code is given by , \text { , }  \label{code7}\]]and, \text{.}\]]__recovery operators__. recall that errors with non - vanishing error syndrome are detectable and that a set of invertible error operators is correctable if the set given by detectable .it is straightforward , though tedious , to check that, is the error syndrome of the error operator ( see appendix b for their explicit expressions ) defined as , \right ] } v_{a_{k}^{\prime } } \text{.}\]]the quantity \right ] } ] -code has distance and therefore all errors with , of weight less than satisfy the relation, at least one error of weight exists that violates it .it is straightforward , though tedious , to check that all - and -qubit error operators satisfy this equation ( therefore , they are detectable ) .instead , there are -qubit errors that do not satisfy ( [ de ] ) . for instance, the error operator is such that .the \right ] ] assuming to correct the set of errors in ( [ set1 ] ) ; second , we will compute \right ] } \left ( \mu \text { , } % p\right ) ] of each with , ... , and , ... , where, _ { |\mathcal{c}}\overset{\text{def}}{=}% \left (  \begin{array}{cc } \left\langle 0_{l}|r_{l+1}a_{k}^{\prime } |0_{l}\right\rangle  & \left\langle 0_{l}|r_{l+1}a_{k}^{\prime } |1_{l}\right\rangle  \\  \left\langle 1_{l}|r_{l+1}a_{k}^{\prime } |0_{l}\right\rangle  & \left\langle 1_{l}|r_{l+1}a_{k}^{\prime } |1_{l}\right\rangle  \end{array}% \right ) \text{.}\]]for , , ... , , we note that _ { |\mathcal{c}} ] with non - vanishing trace are given by _ { |\mathcal{c}} ] vs. for with ( for ,the error correction scheme is not effective anymore ) ( thick solid line ) , ( thin solid line ) and ( dashed line).,scaledwidth=40.0% ]  therefore , the entanglement fidelity \right ] } \left ( \mu \text { , } p\right ) ] is given in appendix b), \right ] } \left ( \mu  \text { , } p\right ) = p_{00}^{6}p_{0}+6p_{00}^{5}p_{10}p_{0}+15p_{00}^{4}p_{01}p_{10}p_{0}+6p_{00}^{4}p_{10}^{2}p_{0}+24p_{00}^{3}p_{01}p_{10}^{2}p_{0}+12p_{00}^{2}p_{01}^{2}p_{10}^{2}p_{0}% \text{. }  \label{s1}\]]note that for arbitrary degree of memory , % \right ] } \left ( \mu \text { , } p\right ) = 1\text { and , } \lim_{p\rightarrow 1}% \mathcal{f}_{\text{set-}1}^{\left [ \left [ 7,1,3\right ] \right ] } \left ( \mu  \text { , } p\right ) = 0\text{,}\]]and for vanishing memory parameter , \right ] } \left ( 0% \text { , } p\right ) = \frac{4}{3}p^{7}-\frac{35}{3}p^{6}+\frac{112}{3}p^{5}-% \frac{175}{3}p^{4}+\frac{140}{3}p^{3}-\frac{49}{3}p^{2}+1\text{.}\]]we emphasize that the presence of correlations in symmetric depolarizing errors does not improve the performance of the seven - qubit code since \right ] } \left ( \mu  \text { , } p\right ) \leq \mathcal{f}_{\text{set-}1}^{\left [ \left [ 7,1,3\right ] \right ] } \left ( 0\text { , } p\right ) ] where the correction scheme is effective, \right ] } \overset{% \text{def}}{=}\left\ { \left ( \mu \text { , } p\right ) \in \left [ 0\text { , } 1% \right ] \times \left [ 0\text { , } 1\right ] : \mathcal{p}_{\text{set-}1}^{\left [ % \left [ 7,1,3\right ] \right ] } \left ( \mu \text { , } p\right ) < p\right\ } \text{.}\]]furthermore , it turns out that \right ] } \left ( \mu \text { , } p\right ) \leq \mathcal{f}^{\left [ % \left [ 5,1,3\right ] \right ] } \left ( \mu \text { , } p\right ) ] where the area of the parametric region \right ] } ] ( see figure ) .the plots of % \right ] } \left ( \mu \text { , } p\right ) ] appears in appendix b. in absence of correlations and considering symmetric error probabilities , the two entanglement fidelities are the same .therefore , we conclude that in the presence of memory effects , it does matter which set of errors we choose to correct , even limiting our analysis to symmetric error probabilities .we will see that the freedom of such choice becomes even more important when combining memory effects and asymmetric error probabilities .for the seven - qubit code applied for the correction of correlated depolarizing errors in set- , it turns out that for increasing values of the memory parameter , the maximum values of the errors probabilities for which the correction scheme is effective decrease .for instance , to corresponds a threshold while to corresponds .  in conclusion, it follows that in the presence of correlated and symmetric depolarizing errors , the performances of both the five and the seven - qubit quantum codes are lowered .furthermore , the five - qubit code is characterized by a parametric region ( where its correction scheme is effective ) that is larger than the one provided by the seven - qubit code ( for both selected sets of correctable errors ) .furthermore , in the parametric region where both error correction schemes are effective , the five - qubit code outperforms the seven - qubit code .  in the next section, we will discover that the situation is slightly different when considering asymmetries and memory effects in depolarizing channels .symmetric case : threshold curves \right ] } \left ( p\right ) ] ( thin solid line ) and \right ] } \left ( p\right ) ] and % \right ] ] becomes, \right ] } \left ( \mu \text { , } p\right ) & = & p_{00}^{4}p_{0}+\left [ p_{00}^{3}p_{10}p_{0}+3p_{00}^{2}p_{01}p_{10}p_{0}+p_{00}^{3}p_{01}p_{1}% \right ] +  \notag \\ & &  \notag \\ & & \left [ p_{00}^{3}p_{20}p_{0}+3p_{00}^{2}p_{02}p_{20}p_{0}+p_{00}^{3}p_{02}p_{2}% \right ] + \left [ p_{00}^{3}p_{30}p_{0}+3p_{00}^{2}p_{03}p_{30}p_{0}+p_{00}^{3}p_{03}p_{3}% \right ] \text { , }  \label{asym5}\end{aligned}\]]where, some straightforward algebra , \right ] } \left ( \mu \text { , } p\right ) ] ( dashed line ) and \right ] } \left ( p\right ) ] evaluated assuming to correct the set- of error operators becomes, \right ] } \left ( \mu \text { , } p\right )  & = & p_{00}^{6}p_{0}+2p_{00}^{5}p_{01}\left ( p_{1}+p_{2}+p_{3}\right ) + 5p_{00}^{4}p_{01}p_{0}\left ( p_{10}+p_{20}+p_{30}\right ) + p_{00}^{4}p_{0}\left ( 2p_{30}p_{33}+p_{30}^{2}+3p_{10}p_{31}\right ) +  \notag \\ & &  \notag \\ & & + p_{00}^{3}p_{01}p_{30}p_{0}\left ( 8p_{30}+4p_{33}+12p_{10}\right ) + 6p_{00}^{2}p_{01}^{2}p_{30}p_{0}\left ( p_{30}+p_{10}\right ) \text { , } \label{xx}\end{aligned}\]]where, that for , \right ] } \left ( \mu \text { , } % p\right ) ] . in absence of correlations , the entanglement fidelity \right ] } ] is given in appendix c. we point out that in the absence of correlations but with asymmetric error probabilities , the seven - qubit code can outperforms the five - qubit code, \right ] } \left ( 0% \text { , } p\right ) \geq \mathcal{f}_{\text{asymmetric}}^{\left [ \left [ 5,1,3% \right ] \right ] } \left ( 0\text { , } p\right ) \equiv\mathcal{f}_{\text{% symmetric}}^{\left [ \left [ 5,1,3\right ] \right ] } \left ( 0\text { , } p\right )  \text{.}\]]in figure , we plot the threshold curves \right ] } \left ( p\right ) ] versus in the case case of asymmetric error probabilities .asymmetries in the error probabilities enlarge the parametric regions where the seven - qubit code is effective for error correction . furthermore , comparing the performances of such codes on a common region where they are both effective , the seven - qubit code turns out to outperform the five - qubit code in the presence of asymmetries and correlations . in figure , we plot \right ] } \left ( \mu \text { , } % p\right ) ] and \right ] } \left ( \mu \text { , } p\right ) ] ( dashed line ) , \right ] } = % \mathcal{f}_{\text{symmetric}}^{\left [ \left [ 5,1,3\right ] \right ] } \left (  \protect\mu \text { , } p\right ) ] ( thick solid line ) vs. with ( where both error correction schemes are effective ) for with , and ., scaledwidth=40.0% ]in this article , we have studied the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors . specifically , we considered the depolarizing noisy quantum memory channel and performed quantum error correction via the five and seven - qubit stabilizer codes .we have shown that memory effects in the error models combined with asymmetries in the error probabilities can produce relevant changes in the performances of quantum error correction schemes by qualitatively affecting the threshold error probability values for which the codes are effective . in summary, we have uncovered the following findings :  1 .  in the presence of correlated and symmetric depolarizing errors , the performances of both thefive and the seven - qubit quantum stabilizer codes are lowered for fixed values of the degree of memory .furthermore , such error correction schemes only work for low values of .2 .  in the presence of correlated and symmetric depolarizing errors ,the five - qubit code is characterized by a parametric region ( where its correction scheme is effective ) that is larger than the one provided by the seven - qubit code .furthermore , in the parametric region where both error correction schemes are effective , the five - qubit code outperforms the seven - qubit code .3 .  the asymmetry in the error probabilities does not affect the performance of the five - qubit code quantified in terms of its entanglement fidelity .on the contrary , it does affect the performance of the seven - qubit code which is less effective when considering correlated and symmetric depolarizing errors .this peculiar effect is rooted in the stabilizer structure of the css seven - qubit code : it is a consequence of the freedom in selecting the set of correctable error operators even after the stabilizer generators have been specified .the performance of the seven - qubit code significantly improves when considering correlated and asymmetric depolarizing errors .furthermore , in such case it is also characterized by higher ( than the one provided by the five - qubit code ) error probability threshold values .this result confirms that in order to make effective use of quantum error correction , a physical implementation of a channel with a sufficiently low error probability , as well as a code with sufficiently high threshold probability is needed knill05 .we conclude that in order to optimize the seven - qubit code performance , it is very important to know the experimental details of the physical implementation of the quantum memory channel being considered . furthermore , in order to make effective use of quantum error correction , a more detailed analysis of the physical noise models for various qubit implementations is needed .this requirement , as we have shown , becomes even more pressing when dealing with noise models where memory effects are combined with asymmetries in the error probabilities .c. c. thanks c. lupo and l. memarzadeh for useful comments .c. c. is also grateful to v. aggarwal and r. calderbank for their kind hospitality and useful discussions during his visit at the program in applied and computational mathematics at princeton university .this work was supported by the european community s seventh framework program under grant agreement 213681 ( corner project ; fp7/2007 - 2013 ) .in this appendix , we briefly discuss few technical details omitted in the manuscript concerning the application of the five - qubit code to the depolarizing memory channel with both symmetric and asymmetric error probabilities ._ error operators_. the weight zero and one quantum error operators in in ( [ nota ] ) are given by, the coefficients with , ... , results,, errors__. recall that an error operator is detectable by the code if and only if some where is the projector on the code space .on the contrary , a set of error operators is correctable if and only if any pair of error operators in where define a positive semi - definite hermitian matrix .we emphasize that the notion of _ correctability _ depends on all the errors in the set under consideration and , unlike _ detectability _ , can not be applied to individual errors .for invertible error operators ( such as the ones considered here ) , there is a simple relationship between detectability and correctability .a set is correctable if and only if the operators in the set are detectable .it would be awfully tedious to identify either detectable errors or sets of correctable errors by means of ( [ aaa ] ) and ( [ aa ] ) for the five and seven - qubit codes characterized by the codewords in ( [ code5 ] ) and ( [ code7 ] ) , respectively .however , the quantum stabilizer formalism allows to simplify such task .this is a consequence of the fact that by means of such formalism it is sufficient to study the effect of the error operators on the generators of the stabilizer and not on the codewords themselves . in our work , we have made use of the stabilizer formalism together with the simple relationship between detectability and correctability for invertible error operators s in order to identify sets of correctable and detectable errors .it is known that errors with non - vanishing error syndrome are detectable .it is straightforward to check that, is the error syndrome of the error operator defined as, \right ] } v_{a_{k}^{\prime } } \text{.}\]]the quantity \right ] } ] in ( [ sym5 ] ) is given by, \right ] } \left ( \mu \text { , } % p\right ) & = & \mu ^{4}\left ( 4p^{5}-7p^{4}+3p^{3}\right ) + \mu ^{3}\left ( -16p^{5}+36p^{4}-26p^{3}+6p^{2}\right ) +  \notag \\ & &  \notag \\ & & + \mu ^{2}\left ( 24p^{5}-66p^{4}+63p^{3}-24p^{2}+3p\right ) + \mu \left ( -16p^{5}+52p^{4}-60p^{3}+28p^{2}-4p\right ) +  \notag \\ & &  \notag \\ & & + \left ( 4p^{5}-15p^{4}+20p^{3}-10p^{2}+1\right ) \text{. }  \label{use21}\end{aligned}\ ] ]in this appendix , we briefly discuss few technical details omitted in the manuscript concerning the application of the seven - qubit code to the depolarizing memory channel with both symmetric and asymmetric error probabilities ._ error operators _ ( set- ) .the set of correctable error operators ( set- ) is explicitly defined by error operators . the only weight- error operator is given by, weight- error ( correctable ) operators are given by, , the weight- ( correctable ) error operators are,,  _ detectable errors _ ( set- ) . _ _ _ _ for the sake of completeness , we show in an explicit way that these errors are detectable .considering , the error syndrome of weight- and weight- error operators is given by, , for the error syndrome of weight- error operators is given by,, these errors have non - vanishing error syndromes , they are detectable . as a side remark , we point out that following the above - mentioned line of reasoning , it can be shown that , with , .  _ entanglement fidelity _ ( set- ) . the explicit expression for \right ] } \left ( \mu  \text { , } p\right ) ] in ( [ s2 ] )is given by, % \right ] } \left ( \mu \text { , } p\right ) & = & \mu ^{6}\left ( \frac{4}{3}% p^{7}+p^{6}-5p^{5}+\frac{8}{3}p^{4}\right ) + \mu ^{5}\left ( -8p^{7}+\frac{20}{% 3}p^{6}+16p^{5}-\frac{64}{3}p^{4}+\frac{20}{3}p^{3}\right ) +  \notag \\ & &  \notag \\ & & \mu ^{4}\left ( \allowbreak 20p^{7}-\frac{145}{3}p^{6}+\frac{70}{3}% p^{5}+23p^{4}-\frac{70}{3}p^{3}+\frac{16}{3}p^{2}\allowbreak \right ) +  \notag \\ & &  \notag \\ & & \mu ^{3}\left ( -\frac{80}{3}p^{7}+\frac{320}{3}p^{6}-\frac{460}{3}p^{5}+% \frac{272}{3}p^{4}-\frac{40}{3}p^{3}-\frac{16}{3}\allowbreak p^{2}+\frac{4}{3% } p\right ) +  \notag \\ & &  \notag \\ & & \mu ^{2}\left ( \allowbreak 20p^{7}-\frac{335}{3}p^{6}+235p^{5}-\frac{710}{3% } p^{4}+\frac{350}{3}p^{3}-25\allowbreak p^{2}+\frac{5}{3}p\right ) +  \notag \\ & &  \notag \\ & & \mu \left ( -8p^{7}+\frac{172}{3}p^{6}-\frac{460}{3}p^{5}+200p^{4}-\frac{400% } { 3}p^{3}+\frac{124}{3}\allowbreak p^{2}-4p\right ) +  \notag \\ & &  \notag \\ & & \left ( \frac{4}{3}p^{7}-\frac{35}{3}p^{6}+\frac{112}{3}p^{5}-\frac{175}{3}% p^{4}+\frac{140}{3}p^{3}-\frac{49}{3}p^{2}+1\right ) \text{.}\end{aligned}\ ] ]_ entanglement fidelity_. substituting ( [ x ] ) in ( [ xx ] ) the explicit expression for \right ] } \left ( \mu \text { , } p\right ) $ ] becomes, \right ] } \left ( \mu \text { , } p\right ) = \mathcal{a}_{6}\left ( \mu \text { , } p\right ) + \mathcal{% a}_{5}\left ( \mu \text { , } p\right ) + \mathcal{a}_{4}\left ( \mu \text { , } % p\right ) + \mathcal{a}_{3}\left ( \mu \text { , } p\right ) + \mathcal{a}_{2}\left ( \mu \text { , } p\right ) + \mathcal{a}_{1}\left ( \mu \text { , } p\right ) + \mathcal{% a}_{0}\left ( \mu \text { , } p\right ) \text{. }  \label{asymg}\]]the quantities , , and are given by, \text { , }  \notag \\ & &  \notag \\ & &  \notag \\\mathcal{a}_{5}\left ( \mu \text { , } p\right ) & = & \text { } \mu ^{5}\left [  \begin{array}{c } \left ( -36p^{7}+90p^{6}-74p^{5}+20p^{4}\right ) + \alpha _ { z}\left ( -30p^{6}+70p^{5}-52p^{4}+12p^{3}\right ) + \\  \\  + \left ( \alpha _ { z}^{2}+\alpha _ { x}\alpha _ { z}\right ) \left ( 126p^{7}\allowbreak -330p^{6}+300p^{5}-108p^{4}+12p^{3}\right)% \end{array}% \right ] \text { , }  \notag \\ & &  \notag \\ & &  \notag \\ \mathcal{a}_{4}\left ( \mu \text { , } p\right ) & = & \text { } \mu ^{4}\left [  \begin{array}{c } \left ( 90p^{7}-285\allowbreak p^{6}+330p^{5}-165p^{4}+30\allowbreak p^{3}\right ) + \alpha _ { z}\left ( 60p^{6}\allowbreak -180p^{5}+192p^{4}-84p^{3}+12p^{2}\allowbreak \right ) + \\  \\  + \left ( \alpha _ { z}^{2}+\alpha _ { x}\allowbreak \alpha _ { z}\right ) \left ( -315p^{7}+975p^{6}-\allowbreak 1110p^{5}+558p^{4}-114p^{3}+6p^{2}\right)% \end{array}% \right ] \text { , }  \notag \\ & &  \notag \\ & &  \notag \\ \mathcal{a}_{3}\left ( \mu \text { , } p\right ) & = & \mu ^{3}\left [  \begin{array}{c } \left ( -120p^{7}+460p^{6}-680p^{5}+\allowbreak 480p^{4}-160p^{3}+\allowbreak 20p^{2}\right ) + \\  \\  + \alpha _ { z}\left ( -60p^{6}+220p^{5}-304p^{4}+192p^{3}-52p^{2}+4\allowbreak p\right ) + \\  \\  + \left ( \alpha _ { z}^{2}+\alpha _ { x}\alpha _ { z}\right ) \left ( 420p^{7}-1500p^{6}+2040\allowbreak p^{5}-1296p^{4}+372p^{3}-36p^{2}\right)% \end{array}% \right ] \text{,}\end{aligned}\]]while , and are , \text { , }  \notag \\ & &  \notag \\ & &  \notag \\ \mathcal{a}_{1}\left ( \mu \text { , } p\right ) & = & \text { } \mu \left [  \begin{array}{c } \left ( -36p^{7}+186p^{6}-390p^{5}+420p^{4}-240p^{3}+66p^{2}-6\allowbreak p\right ) + \\  \\  + \alpha _ { z}\left ( -6\allowbreak p^{6}+30p^{5}-60p^{4}+60p^{3}-30p^{2}+6p\right ) + \\  \\  + \left ( \alpha _ { z}^{2}+\alpha _ { x}\alpha _ { z}\right ) \left ( \allowbreak 126p^{7}-570p^{6}+\allowbreak 1020\allowbreak p^{5}-900p^{4}+390p^{3}-66p^{2}\right)% \end{array}% \right ] \text { , }  \notag \\ & &  \notag \\ & &  \notag \\ \mathcal{a}_{0}\left ( \mu \text { , } p\right ) & = & \text { } \left ( 1-p\right ) ^{7}+7p\left ( 1-p\right ) ^{6}+21p^{2}\left ( 1-p\right ) ^{5}\left [ \alpha _ { z}^{2}+\alpha _ { x}\alpha _ { z}\right ] \text{.}\end{aligned}\ ] ]

Abstract: we study the performance of common quantum stabilizer codes in the presence of asymmetric and correlated errors . specifically , we consider the depolarizing noisy quantum memory channel and perform quantum error correction via the five and seven - qubit stabilizer codes . we characterize these codes by means of the entanglement fidelity as function of the error probability and the degree of memory . we show that their performances are lowered by the presence of correlations and we compute the error probability threshold values for codes effectiveness . furthermore , we uncover that the asymmetry in the error probabilities does not affect the performance of the five - qubit code while it does affect the performance of the seven - qubit code which results less effective when considering correlated and symmetric depolarizing errors but more effective for correlated and asymmetric errors .


nonlinear evolution equations that form a singularity in finite time are ubiquitous in nature .applications are broad appearing in models from chemistry , physics , biology , rocket , and combustion engineering .solid fuel ignition models attempt to approximate an explosive event , identified as the rate of change of the temperature increasing without bound and forming a singularity in finite time .the formation of the singularity equates to ignition in the combustor .moreover , the finite time ignition is triggered if the temperature reaches a critical threshold .the sophistication of combustion models can be reduced considerably if one examines asymptotically close to the ignition time . in such cases , the equation for the temperature decouples from the chemistry and mass - species fraction .the final result is a highly nonlinear differential equation , in particular , where dirichlet boundary conditions specified on the boundary and for the source term , , is a highly nonlinear function .it is positive and strictly increasing for most importantly , it tends to infinity when the degeneracy function and models particular heat transportation characteristics within the domain .while it remains positive within the domain it may vanish or become singular on a countable subset of points on the boundary .if the degeneracy function vanishes this is seen as a defect in the transportation of heat , while if it becomes singular it is a defect in the diffusion of heat . in either case, the degeneracy indicates a defect at a particular location of the combustor or , common to combustion engine designs , a special material concentration point .a sufficient condition for the singularity to form in the temporal derivative is that as in  . in such cases , the solution is said to quench at the _ quenching time _ over the domain the set is called the _ quenching location set , _ and in this study involves a single point in the domain . on one hand ,given an initial condition and domain shape there is no guarantee that the solution will quench .one the other hand , it is known that for a piecewise smooth and connected domain there does exists a critical size for which the solution will always quench for all sizes larger .this was originally observed and proven in the one - dimensional case by kawarada for a particular and then was extended to general source terms by levine and montgomery . in higher dimensions ,the results are not nearly as precise and finding estimates to the critical size that quenching persists for a particular shape is problematic .fortunately , there are known theoretical estimates for rectangular domains . numerical approximations have been employed to extend and resolve these estimates further . in this paper , stemming from the creation of lower and upper bound solutions to the singular problem , theoretical estimates of the critical quenching domain are established in 2 for arbitrary domains .this provides an avenue of creating novel estimates for arbitrary domains , in particular when the domain shape is convex and will be used to provide additional verification of the numerical algorithm in the experiments of 6 .this paper is also interested in the development and analysis of an adaptive splitting algorithm that can accurately be used to explore the singular problem posed over an elliptical domain . to date , numerical explorations have primarily focused on rectangular domains while circular and elliptical geometries may be more applicable to realistic applications .in addition , the analysis of 2 provides valuable estimates to the critical quenching domains and are used to further validate the numerical scheme in the experiments of 6 .moreover , the intricate numerical methodologies , such as splitting , adaptation , and nonuniform grids requires extensions to such geometric considerations .indeed , after careful design , the numerical analysis shows that the numerical scheme is indeed reliable , accurate , monotonically increases toward a steady state or quenching , and is weakly stable .the paper is organized as follows . in the following section theoretical estimates for quenching domains for piecewise smooth ,connected domains are given .it is evident that these results can be extended to blow - up problems . in addition , as a corollary to the theorems , quenching time estimates can be established for arbitrary domain shapes and sizes . in 3 the study begins its focus on elliptical domains .the equations are then written in elliptical coordinates .hence , the computation is designed over the rectangular grid generated by the coordinate transformation .appropriate boundary conditions are then established . in 4 the adaptive , second order , splitting algorithm is given .the numerical analysis of the algorithm is in 5 . in 6 numerous numerical experiments are offered , namely a statistical analysis of introduced errors , critical quenching domain calculations for elliptical domains of certain ratios of minor to major axes , and experimentally studying the effect of the mathematical degeneracy on the solution s fundamental characteristics .the paper is then concluded in 7 .for singular , reaction - diffusion equations of the quenching type one of most interesting features is the existence of a critical quenching domain size for a particular domain shape . as one might expect , showing that this is indeed the case for arbitrary domains is a difficult task , especially in higher dimensions .recently , in , it has been shown that for connected piecewise smooth domains there exists a critical quenching domain size .in realistic domain shapes , this novel result indicates this phenomena will persist , however , theoretical estimates of the critical size are lacking .even so , reliable theoretical estimates have been established for rectangular domains .these estimates have been verified numerically in a number of resources .the use of upper and lower solutions to differential equations has a long history in analysis of blow - up problems .the fundamental idea can be traced back to .consider the solution of ( [ introduction : generaleq1])-([introduction : generaleq2 ] ) .let be a time - dependent lower bound of then if in finite time then the quenching singularity forms in at a quenching time of therefore , if one can establish a time - dependent lower bound of that approaches a value of one in finite time then will quench .similarly , let be a time - dependent upper bound of then if for all time then clearly will not quench .a _ lower solution _ of ( [ introduction : generaleq1])-([introduction : generaleq2 ] ) is a function that satisfies   an _ upper solution _ of ( [ introduction : generaleq1])-([introduction : generaleq2 ] ) is a function that satisfies   in this light , upper and lower solutions provide a valid tool of analysis for such problems . this idea is utilized in creating estimates to the critical size for arbitrary , connected , piecewise smooth domain shapes .consider nagumo s lemma ( , 24 ) ,  let satisfy ( [ introduction : generaleq1])-([introduction : generaleq2 ] ) with reactive term .if is a lower solution then for and similarly , if is an upper solution then for and   clearly , as stated earlier , if the lower solution approaches one in the domain then will quench . given the shape of the domain there exists a critical size for which the lower solution approaches a value of one .thus the lower solution s critical quenching domain size is a lower bound to the critical quenching domain size for on the other hand , if the upper solution does not quench , that is approach a value of one , then neither does hence , the upper solution s critical quenching domain size is an upper bound to the critical quenching domain for   define as the maximal rectangle that is a subset of similarly , define as the minimal rectangle that is a subset of .two theorems are now established .let where satisfies then is a lower solution .  since then by definition , on and is zero initially .let be a solution to then restricted to the domain is an upper solution to   since and is monotonically increasing then it is assured that on thus on   the above theorems imply that the quenching time can be bounded by that of quenching times found for rectangular domains .this offers reliable bounds for convex domains , hence providing much needed direction and insight to verify numerical calculations of quenching domains .it seems plausible that sufficient improvements can also be made for non - convex domains .the bounds provide a valid check to the consistency of quenching computations .while the application here are motivated by quenching problems , the above results are easily transferrable to other singular problems , in particular blow - up problems .these theorems will be used to help verify the numerical experiments of 6 . to illustrate the theorems ,consider an elliptical geometry , that is , for some positive constants and .then the maximal inscribed rectangle is similarly , the minimum rectangle that can be inscribed in is let for these rectangular shapes the critical quenching domain has been established .hence , theorem 2.2 can be used to offer theoretical bounds for particular ratios of minor and major axes .the results are offered in table [ tablequenchingboundstheory ] ..estimated quenching domains for ellipses with minor and major axis of and respectively .for particular ratios , the third column bounds the critical quenching area of the ellipse .the last column shows the theoretical bounds for the critical product , which is related to third column by a multiple of . as a comparison, the second column shows what the known critical quenching area of a rectangle with length to width ratio of for brevity , quenching time bounds are not included . [ cols="<,>,^,^",options="header " , ]    * _ experiment 3 ._ * the mathematical degeneracy encompasses interesting physical phenomena in addition to an intellectual interesting nuance to such singular partial differential equations . in the case of rectangular domains with dirichlet boundary conditions, it has been observed that the quenching location shifts toward the degeneracy , that is , the location for which on the boundary .experimentally , it is observed that the quenching location is shifted .consequently , a mathematical proof has yet to be established and is recommended as a novel undertaking .  in this light, the effect of a mathematical degeneracy is examined here for two situations .if at a boundary point , then this amounts to a degeneracy in the transportation of heat , as the coefficient of vanishes . in this case , as a result of theorem 5.3 and lemma 5.4 , the maximum temporal step is more restricted .in contrast , if at a boundary point , then this amount to a degeneracy in the diffusion of heat , as the coefficient of the laplacian vanishes .in such case , the maximal temporal step is larger than without a degeneracy .both situations are considered here .let where  is the maximum value of restricted to , and is chosen on the projection of plane s normal onto the plane is chosen such that it is perpendicular to the tangent line at the point of degeneracy .a major and minor axes of and , respectively , is used throughout the discussion .other ratios were explored and share similar observations .let and in such case , the defect contributes to the overall size of the source function in the numerical computation .hence , the quenching time is anticipated to be lower compared to the degenerate - free case . in fact , for the particular domain shape and size the quenching times for the degenerate and degenerate free cases are determined to be and respectively , in line with this conjecture .it has been observed numerically that for rectangular domains the quenching location shifts toward the degeneracy consider the cases for the computation utilizes fine fixed uniform grids to determine the quenching location .remarkably , for a degeneracy located at on the boundary , the quenching location is the same distance in the inward normal direction for each computation ! in other words , the quenching location for each case of forms an inner annulus , which is not necessarily an ellipse . more precisely , the quenching location is where is the degeneracy location and is a fixed constant .clearly this observation depends on the type of degeneracy , however it indicates that the curvature of the boundary at the point of degeneracy plays a delicate role in influencing the quenching location .figure [ example3loc](a ) shows the elliptical boundary and the corresponding quenching locations for each value of the dashed line is the corresponding ellipse with the same shape as the boundary and passing through the quenching locations for and figure [ example3sol](a ) shows a contour plot of the solution in the iteration just prior to quenching in the case .  )is shown . a dashed inner elliptical boundary that passes through the and quenching locationsis given for scale .the annulus given by a fixed distance from the outer boundary is given , for which the quenching locations closely follow .the number located at each enclosed circle pairs the corresponding degeneracy with the quenching location for ,title="fig : " ] ) is shown .a dashed inner elliptical boundary that passes through the and quenching locations is given for scale .the annulus given by a fixed distance from the outer boundary is given , for which the quenching locations closely follow .the number located at each enclosed circle pairs the corresponding degeneracy with the quenching location for ,title="fig : " ]  let again , the single defect is located on the boundary in increments of at these locations is singular , effectively lessening the diffusive effects .hence , it is still expected that the quenching time will be less than the degenerate - free case .however , it will not be more than the previous case of the defect in the transportation of heat .this is because the size of the source term is lessened . for this particular shape , the quenching timeis calculated to be   interestingly , a similar pattern emerges in this case .again , the quenching locations closely fall on an inner annulus constructed by tracing a path a distance from the outer boundary . however , the quenching location shifts away from the degeneracy rather than toward it . these results are shown in figure [ example3loc](b ) . a contour plot of the solution just prior to quenching is also shown in figure [ example3sol](b ) .the quenching location is also identified by a dot in each of the plots.,title="fig : " ]  the quenching location is also identified by a dot in each of the plots.,title="fig : " ]this paper designs , implements , and analyzes a temporally adaptive splitting scheme that utilizes nonuniform grids for singular reaction diffusion equations with a nonlinear absorption term .the multi - faceted quenching phenomena presents challenges to resolving dynamics close to the onset of quenching , in determining the effect of defects on the quenching location and dynamics of the solution , and in determining critical quenching sizes for particular domain shapes .this paper presents improvements to our understanding in all of these categories adding interesting general results while , at the heart of the paper , in application to elliptical domains .determining theoretical estimates to the critical quenching domain size for a given shape is a delicate and difficult task .this is evidenced by the literature devotion to existence theorems of critical quenching domain , rather than estimates of the size . only for rectangular domainssuch estimates have surfaced theoretically and numerically .this focus is not wasted , rather is a blessing to exploring other domain shapes . in particular , the acquired knowledge of critical quenching domains for rectangular domains can be used to construct lower and upper solutions for quenching problems posed over arbitrary piecewise smooth domains .this is then used to construct theoretical estimates to the critical quenching domain sizes . in thisregard , this result can be extended to blow - up problems as well .additionally , through comparable techniques it is our belief that sharp estimates can be established and is a target of our future research .the governing equations are posed over an elliptical domain and the numerical scheme is subsequently developed .first , a coordinate transformation maps the elliptical domain into a rectangular one .this requires an additional boundary condition , which is realized through investigating certain translational symmetries in the coordinate transformation .the proposed boundary condition can be implemented in a straight - forward way in the spatial discretization .the diffusion operators are discretized using nonuniform central differences .the current study considers fixed grids , however the grids are developed through an adaptive procedure called exponentially evolving grids .the numerical solution at any iterate provides information in order to develop a least squares approximation to the solution s temporal derivative .the collocation function is chosen such that certain properties are available , namely the finest resolution is placed near the location of the solution s extrema .experiments suggest this approach develops smooth mapping functions from the logical to the physical spaces , indicating a second order accuracy .future studies are under way to better understand this type of adaptation and , hence , is reserved for a later follow - up paper .the ensuing numerical analysis indicates that provided certain criteria are met , that the numerical solution monotonically increases and remains positive throughout the computation .a general theorem is provided that can be used to establish this result for any problem that its semi - discretization is of the form of ( [ goveqs : mol ] ) .hence , while the primary discussion is focused on quenching problems over elliptical domains , the theorem makes only use of the form of the semi - discretization and provides criteria for the involved matrices .weak stability is then established in the von neumann sense in the . while this lacks the full nonlinear stability analysis the experiments provide a novel way of quantifying the effects of error throughout a computation in a statistical sense .it is true that for nonlinear problems , the stability analysis of any developed scheme is a difficult task . often , weak stability is all that is offered .however , the cumulative effect of error can be statistically analyzed using monte carlo methods .it is our belief that this methodology could be used as an alternative that offers valid experimental and statistical evidence toward the stability of the numerical scheme . in this light ,the experiments indicate remarkable stability to continued perturbations throughout the computation .defects to the transportation and diffusion of heat are physically motivated in addition to mathematically interesting to their effect on the solution to the partial differential equation . here , the scheme is used to analyze the effect of such defects .it has been observed that in the context of rectangular domains defects in the transportation of heat results in movement of the quenching location toward the defect .this phenomena is observed for elliptical domains . to our knowledge , no such study has been presented on the effect of defects in the diffusion of heat .interestingly , the quenching location shifts away from the defect ! while these results have been established numerically it has yet to be seen theoretically and is left as an open problem .m.a . beauregard and q. sheng , _ a fully adaptive method to approximate reaction diffusion equations of the quenching type over circular domains _ , numer .meth partial diff .eqns , doi 10.1002/num.21820 , 2013 .

Abstract: solid fuel ignition models , for which the dynamics of the temperature is independent of the single - species mass fraction , attempt to follow the dynamics of an explosive event . such models may take the form of singular , degenerate , reaction - diffusion equations of the quenching type , that is , the temporal derivative blows up in finite time while the solution remains bounded . theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies . quenching domains are known to exist for piecewise smooth boundaries , but often lack theoretical estimates . rectangular geometries have been primarily studied . here , this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise , smooth , connected geometries . elliptical domains are of primary interest and a peaceman - rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids . rigorous numerical analysis ensures numerical solution monotonicity , positivity , and linear stability of the proposed algorithm . simulation experiments are provided to illustrate the accomplishments .   nonlinear evolution equations , quenching , splitting , elliptical domains , finite difference