scientists around the world are striving toward the construction of a scalable quantum computer .theorists can contribute to this effort by developing and optimizing methods for protecting quantum computers against environmental decoherence and systematic hardware imperfections .indeed , the theory of quantum fault tolerance has established that reliable quantum computation is possible if the noise afflicting the computer has suitable properties .the most obvious requirement is that the noise must be sufficiently weak if the noise strength is below a _ threshold of accuracy _ then quantum computing is scalable in principle .recent insights have led to steadily improving estimates of the accuracy threshold , boosting confidence that large - scale quantum computers will be realized eventually .notably , knill proposed a scheme for fault - tolerant quantum computation based on concatenated error - detecting codes and postselection , in which ancilla states are prepared off - line and an ancilla is discarded whenever an error is detected during its preparation .numerical simulations indicate that knill s scheme can tolerate independent stochastic noise with an error probability per gate of about  , and a threshold error rate of order has been rigorously established .unfortunately , the overhead cost of knill s postselection scheme is discouraging because ancillas are rarely accepted .however , knill also proposed a modified version of the scheme with a more moderate overhead cost , which we call the `` fibonacci scheme . ''the fibonacci scheme is based on the observation that a distance-2 code , aside from detecting errors , can also correct errors that occur at known positions in the code block ; thus , in a concatenated code , detected errors at one coding level become _ located _ errors that can be corrected at the next level up .( we call it the fibonacci scheme because the probability of a logical error scales with the coding level like , where is the noise strength and is a fibonacci sequence . )knill s numerical simulations indicate that the fibonacci scheme , too , can tolerate independent stochastic noise with an error rate of about  .  in this paper , we provide a rigorous analysis of the fibonacci scheme .our main result is a lower bound on the accuracy threshold for the fibonacci scheme that is of the same order as the lower bound found earlier for knill s postselection scheme .thus our result proves that a high accuracy threshold can be compatible with a moderate overhead cost , supporting knill s numerical findings .the fibonacci scheme has some features in common with fault - tolerant schemes used in previous proofs of the quantum threshold theorem , but there are also important differences , and correspondingly our new threshold proof has a rather different structure than these previous proofs . as in previous schemes , logical qubits processed by the computer are protected from damage using a quantum code , and encoded gates are realized by gadgets that act on the code blocks and correct the errors . butstandard versions of the quantum threshold theorem are based on concatenated coding and recursive simulations .the code block of a concatenated code is constructed as a hierarchy of codes within codes the code block at level of this hierarchy is built from logical qubits encoded at level of the hierarchy .likewise , in a recursive simulation , fault - tolerant gadgets are constructed as a hierarchy of gadgets within gadgets the gadgets at level are built from gadgets at level .this recursive construction streamlines the analysis ; if the level-1 simulation of an ideal circuit is more reliable than an unprotected `` level-0 '' circuit , then because of the self - similarity of the gadgets , the level-2 simulation will be still more reliable , and so on .in contrast , though the fibonacci scheme uses concatenated coding , the construction of gadgets is not strictly recursive ; rather , the logical controlled - not ( cnot ) gates and encoded measurements are implemented transversally at all levels of the concatenated code .that is , a logical cnot gate is performed by executing many fundamental cnot gates in parallel in a single time step , and encoded measurements are performed by measuring many qubits in parallel in a single time step , followed by classical decoding of the measurement outcomes .errors are corrected by means of quantum teleportation , using encoded bell pairs that _ are _ prepared recursively , and the analysis of the recursive bell pair preparation procedure is the crux of our proof . for the scheme to succeed ,it is essential that the errors in an encoded bell pair are only weakly correlated ; our central task is to formalize a notion of `` weakly correlated '' that makes the proof work , and to show that the encoded bell pairs really have this property at each level of concatenation .a second distinctive feature of the fibonacci scheme , as already mentioned , is that a concatenated error-_detecting _ code is used to correct _ located _ errors .the _ classical _ decoding of a measured code block is performed recursively , starting at the lowest level ; a subblock is flagged if an error is detected in the subblock , and at each level , if an error is detected in a block that has only one flagged subblock , then the flagged subblock is corrected . in principle , flagging detected errors and constructing non - recursive gadgets are independent ideas we could use flagging in a recursive scheme or use non - recursive gadgets without flagging .however , following knill , we have found that combining flagging and non - recursive gadget construction gives an especially high threshold estimate at a moderate overhead cost .this paper is structured as follows . in section [ sec : noise - model ] , we define the correlated `` local stochastic '' noise model that we consider in most of our analysis .( our arguments also apply to uncorrelated pauli noise models , and yield a slightly better lower bound on the accuracy threshold in that case . ) in section [ sec : scheme ] , we formulate the fibonacci scheme for simulating a quantum circuit , and present gadgets for error correction and logical gates based on a concatenated 4-qubit code ; then in section [ sec : decoding ] we define our procedure for decoding the measurement of a code block . in sections [ sec : level - reduction ] and [ sec : css - gadgets ] ,we derive recursion relations that characterize the error correlations in recursively prepared bell pairs and the failure probability of gadgets for operations ; this is the technical heart of our proof . in section [ sec : css - threshold ] , we obtain a lower bound on the accuracy threshold for operations , and we extend it to a complete universal set of fault - tolerant quantum gates in section [ sec : threshold ] .we discuss the overhead cost of the fibonacci scheme in section [ sec : overhead ] .finally , section [ sec : conclusion ] contains our conclusions .we use the term _ location _ to speak of an operation in a quantum circuit that is performed in a single time step ; a location may be a single - qubit or multi - qubit gate , a qubit preparation step , a qubit measurement , or the identity operation in the case of a qubit that is idle during the time step . in a stochastic noise model, we assume that at each circuit location either the ideal operation is executed perfectly or else a fault occurs , and we assign a probability to each _fault path_that is , to each possible set of faulty locations in the circuit .we speak of _ local stochastic noise _ with strength if , for any specified locations in the circuit , the sum of the probabilities of all fault paths with faults at those locations is no larger than .for each fault path , the trace - preserving quantum operation applied at the faulty locations is arbitrary and can be chosen adversarially ; therefore in this noise model the faults can be correlated both temporally and spatially .however , we will exclude `` leakage '' faults in which quantum information escapes from the computational space .  in the fibonacci scheme ,errors are corrected by teleportation , and logical errors occur if the bell measurements in the noisy circuit fail to simulate correctly the bell measurements in the ideal circuit. we will analyze the effectiveness of the scheme by propagating errors forward to the measurements , in order to estimate the failure probability for the measurement of encoded blocks .this error propagation is feasible because qubits are always measured just a few time steps after they are initially prepared .  for each fault path, the operation acting at the faulty locations can be expressed in terms of a set of nontrivial kraus operators , and each kraus operator has a pauli expansion where denotes a tensor product of pauli spin operators that act at the faulty locations . each pauli operator can be propagated forward in time through the ideal circuit .thus , for each fault path and for each time , we obtain a trace - preserving completely positive map acting on the qubits at time , where each kraus operator in also has a pauli expansion . for any qubit labeled will say that the operation causes a `` type - x '' error acting on at time if any pauli operator in the expansion of contains either or acting on ; similarly we say that causes a `` type - z '' error acting on at time if any pauli operator in the expansion of contains either or acting on .thus , if is measured at time , then generates a measurement error with nonzero probability if the measurement is in the z basis and there is a type - x error , or if the measurement is in the x basis and there is a type - z error .we will assume that our quantum computer is equipped to execute elementary operations chosen from the universal set here , and denote the measurement of the single - qubit pauli operators and respectively and denotes the preparation of the single - qubit state , where , , and . to prove our threshold theorem ,first we will show that there is an accuracy threshold for operations in  then _ encoded _ operations can be simulated to any desired accuracy . then we will show that for , by using the reliable encoded operations and the noisy elementary operations and , we can distill accurately encoded copies of and , thus completing a universal set of reliable encoded operations .  in the fibonacci scheme , operations in will be protected using the code , the 4-qubit code concatenated times .( knill actually proposed to concatenate with a 6-qubit code in order to reduce the overhead , but we will analyze a simpler scheme that uses only . )the code has distance 2 and its check operators are and . though the code space is four dimensional , we will use only one of the two encoded qubits to protect the quantum information processed by the computer .the encoded pauli operators acting on this `` logical '' qubit are while the encoded pauli operators acting on the irrelevant `` gauge '' qubit are we do not care about the state of the gauge qubit , and therefore a error acting on the third qubit in the code block is equivalent to a error acting on the fourth qubit the errors differ by the gauge - qubit operator , which commutes with the logical pauli operators .likewise , errors on the first and second qubits are equivalent , as are errors on the first and third qubits or errors on the second and fourth qubits .we will derive our lower bound on by showing that , for , `` gadgets '' realizing encoded operations protected by become highly reliable as increases .how are these gadgets constructed ?first , consider the gadgets protected by ( the case ) .the cnot gadget , shown in fig .[ fig:1 ] , consists of a transversal logical cnot gate , together with error recovery steps on each block that precede and follow the transversal gate .the other gadgets are smaller and more reliable than the cnot gadget , so to obtain a lower bound on the threshold it suffices to study the cnot gadget .inside all gadgets , error recovery is achieved by logical teleportation using bell pairs encoded in .the encoded bell pairs ( 1-bps ) are prepared by using unencoded bell pairs ( 0-bps ) and cnot gates as in fig .[ fig:2 ] . the preparation circuit must be designed carefully to control correlated errors .we do not need to worry about correlated errors in the 0-bps any two - qubit error acting on the bell state is equivalent to a one - qubit error , because is invariant under and .on the other hand , error correlations between the two blocks of the 1-bp might cause trouble , and can arise from a single fault in the transversal cnot gate shown in fig .[ fig:2](b ) .a simple way to suppress correlated errors is to follow the preparation of the 1-bp with the teleportation of both blocks , using two other 1-bps as shown in fig .[ fig:2](c ) . using this circuit ,an error occurs in an output block only if there is a fault in the teleportation of the block ; therefore , errors in different blocks must arise from distinct faults .   the gadget for a cnot gate protected by .a transversal cnot gate , which implements the logical cnot gate acting on two blocks , is preceded and followed by logical teleportations which extract the code s error syndrome .arrows denote cnot gates , with the arrow tip pointing to the target qubit .green and blue triangles pointing to the left denote and respectively , and bell pairs encoded in ( 1-bps ) are prepared as in fig .[ fig:2 ] . , width=313 ]   ( a ) preparation of an unencoded bell pair ( 0-bp ) where green and blue triangles pointing to the right denote and respectively .( b ) a bell pair encoded in ( 1-bp ) is constructed by preparing one block in the logical state , a second block in the logical state , and applying transversal cnot gates .( c ) the two blocks of a 1-bp are teleported , using two other 1-bps , in order to limit correlations between errors in different output blocks . depending on the measurement outcomes ,logical pauli operators ( not shown ) acting on the two output blocks may be needed to complete the teleportations ., width=313 ]  one way to construct the gadget for the cnot gate encoded in would be to replace every elementary cnot gate in fig .[ fig:1 ] by the cnot gadget encoded in . and by repeating this replacement recursively , we could obtain the `` level- '' gadget for the cnot gate encoded in for any .however , in the fibonacci scheme we do _ not _ follow this recursive procedure to construct the higher - level gadgets . instead ,for each the logical cnot gate inside the level- cnot gadget is performed transversally in a single time step just as in fig .[ fig:1 ] , and the measurements of encoded blocks are also transversal .we construct the higher - level gadgets using this non - recursive method because we have found that it yields a significantly better lower bound on the accuracy threshold than a strictly recursive procedure . on the other hand , the level- bell pair ( -bp ) inside the level- cnot gadget _is _ constructed recursively .that is , just as the 1-bp preparation circuit in fig .[ fig:2](c ) , constructed from transversal level-1 cnot gates and measurements , operates on twelve input 0-bps , so the -bp preparation circuit , constructed from level- transversal cnot gates and measurements , acts on twelve input -bps .but this circuit for preparing a -bp still has a flaw that must be addressed for large, each output qubit of the -bp goes unmeasured for many time steps , gradually accumulating errors .let us say that the final step in the -bp preparation circuit , the level- transversal cnot gate acting on the output blocks , occurs in time step .this transversal cnot follows -bp preparations that are completed in time step .thus , there are level- transversal cnot gates in time step , which are preceded by -bp preparations that are completed in time step , and so on .we see that each of the output qubits of the -bp is acted on by cnot gates in consecutive time steps 1 through . but during these steps nothing is being done to correct the errors in these qubits .we can correct this flaw by adding one more ingredient to the scheme ; for we replace the circuit in fig .[ fig:2](c ) , by the circuit in fig .[ fig:3 ] . in this modified circuit ,after a -bp is prepared , we teleport every output subblock by using additional -bps . in the modified -bp preparation circuit , then , the last step is a -bp preparation acting on the subblocks . in turn ,the last step in the -bp preparation is a -bp preparation , the last step in the -bp preparation is a -bp preparation , and so on .thus , the output qubits from the circuit actually emerge directly from 1-bps , and there is no accumulation of error to worry about .( a ) the complete preparation circuit for a -bp where .it is identical to the circuit shown in fig .[ fig:2](c ) , except that a final teleportation of each level- subblock has been added .( b ) the teleportation circuit ., width=309 ]  to determine the effectiveness of this scheme , where errors are corrected by logical teleportation , we need to estimate the probability of failure when a code block is measured .measurements are performed transversally to measure the encoded , we measure all qubits in the block simultaneously in the x basis , and to measure the encoded , we measure all qubits in the block simultaneously in the z basis. then these measurement outcomes are decoded using a classical recursive algorithm that is assumed to be noiseless .first , the subblocks are decoded , then the subblocks , _etc_. , up to the final block . at each level ,the decoding of a block makes use of the error syndrome computed in the previous step during the decoding of its four subblocks . specifically , whenever we detect an error in a subblock for , we _ flag _ this subblock to indicate that the decoding of the subblock has an enhanced probability of failure .then , if a block has a nontrivial error syndrome and a single flagged subblock , we decode the block under the assumption that the flagged subblock was decoded incorrectly .  at the top level, an error might be detected in the decoding of the final block that can not be corrected ( either because no subblock is flagged , or because more than one subblock is flagged ) .if such an uncorrectable error is found at the top level in the recursive preparation of an encoded bell pair , then the preparation is aborted . on the other hand ,if such an uncorrectable error is detected at a measurement s top level in the teleportation step inside a gadget , then the code has failed to protect the measurement effectively .we will prove that , for sufficiently weak noise , the probability of detecting an error in the final decoding step of a block decreases rapidly with increasing .thus the overhead cost due to aborting and restarting the preparation of encoded bell pairs is moderate ; the preparation of a -bps is rarely aborted once is large .furthermore , the probability of a logical error inside a level- gadget becomes arbitrarily small for sufficiently large .now we will specify in more detail the ( noiseless ) classical algorithm for decoding measurements of code blocks , and we will discuss the effectiveness of this algorithm . under appropriate assumptions about the noise in the measurements , we would like to estimate how the probability of error and the probability of flagging propagates from level to level as the block is recursively decoded .  at each level of the recursive decoding ,we are to interpret a 4-bit string corresponding to the measured eigenvalues of the encoded or the encoded in the four subblocks of a block , where some of the bits might be flagged .the decoder knows which bits are flagged , but of course it does not know which bits have errors .we may consider a joint distribution that governs the probability of error and the probability of flagging for the four bits .for each measurement in the fibonacci simulation , and for each level of the concatenated code , this distribution is completely determined in principle by the decoding procedure at lower levels and by the noise model .that is , if we know the probability of each fault path and the noise operation associated with each fault path , we can propagate the errors forward ( as described in section [ sec : noise - model ] ) to determine an error model for the measured block . then using this error model and the decoding procedure , we can assign a probability ( or at least an upper bound on the probability ) for each nontrivial pattern of flagging and errors at each decoding level .  to prove our threshold theorem, we will need to show that the decoding errors at each concatenation level are only weakly correlated . a suitable notion of weakly correlated , which we call `` quasi - independence, '' is defined as follows :  [ def:1 ] let be a probability distribution for a set of bits , where the -bit binary vector specifies the positions of bits with errors , and the -bit binary vector specifies the positions of flagged bits .let and denote disjoint subsets of containing bits and bits respectively , and let denote a subset of containing bits .we say that the distribution is _ quasi - independent _ with strength if , for any choice of , , and , the probability that all bits in are unflagged with errors , all bits in are flagged , and all bits in are flagged with errors is no larger than .we say that is the error strength for unflagged bits , is the error strength for flagged bits , and is the flagging strength .  when applied to the decoding of , the errors of interest are errors ; hence quasi - independence constrains how errors are correlated with one another .likewise , when applied to the decoding of , quasi - independence constrains how the errors are correlated with one another .but we will not need to know much about how errors are correlated with errors to analyze the failure probability for measurements of encoded blocks .now we would like to show that when we recursively decode a block , if the errors are quasi - independent at level they will also be quasi - independent at level ; furthermore , we wish to estimate the error strengths at level given an estimate of the error strengths at level .first , let us formulate precisely our algorithm for flagging a block :   if no bit in the block is flagged , then we flag the block whenever the syndrome is non - trivial , if exactly one bit in the block is flagged , then we _ never _ flag the block , if two or more bits in the block are flagged , then we flag the block if errors on a pair of flagged bits can cause a logical error .  in case , we flag the block because we have detected an error that we do not know how to correct . in case , rather than flagging the block , we perform error correction ( in the event of a nontrivial syndrome ) under the assumption that the flagged bit has an error . in case , we do not flag the block if errors at the flagged bits would affect only the gauge qubit , not the logical qubit .  given this flagging algorithm and assuming quasi - independent errors on the bits , we can see that the blocks also have quasi - independent errors .we may define a function , which we call decode , that maps upper bounds on the error strengths for the bits to upper bounds on the error strengths for the blocks .this function is specified in table [ func : decode ] . in the rest of this section , we explain how the function decode is derived .  [ cols="<",options="header " , ]now we have shown that , if is sufficiently small , gates can be simulated with arbitrarily good accuracy . to complete our proof of a quantum accuracy threshold theorem based on the fibonacci scheme , we must extend this protected gate set to a universal set .it suffices to augment the highly reliable encoded gates with highly reliable encoded versions of the single - qubit preparations and .using the reliable encoded operations , high - fidelity encoded copies of and can be _ distilled _ starting with sufficiently accurate noisy copies .the distillation protocol works if encoded copies of can be prepared such that the probability of a logical error is below  ; the distillation threshold for states is even higher .one way to prepare encoded copies of is by `` teleportation into the code block ''first , we prepare a bell pair encoded using , then we recursively decode one of the two encoded blocks to a single qubit , and finally we perform a bell measurement on the decoded qubit and an input qubit prepared in the state .the level- encoded bell pair can be prepared reliably if is below the threshold and is large .but the teleportation procedure could fail either because there is an error in the recursively decoded state , or because a fault occurs in the preparation of or in the two - qubit bell - measurement circuit .let us consider the probability of error in the recursive decoding of a block to a single qubit .a level-1 block can be decoded to a qubit using a circuit that contains three cnot gates . in the first decoding step we use level- cnot gates to decode the block to a block , in the second decoding step we use level- cnot gates to decode the block to a block , and so on , completing the decoding to a single qubit after steps .each encoded cnot gate is followed by error - correcting teleportation steps acting on both output blocks .we abort the `` teleportation into the block '' protocol if , at any stage of decoding the block , a flag is raised in the top - level decoding of a measurement inside any error - correcting teleportation .( we assume that sufficiently many encoded bell pairs are prepared in parallel that with high probability at least one ancilla preparation is accepted . )a flag could be raised in either the or measurement for either output block of any cnot gate at any level , so the total probability of a flag during decoding is a logical error , too , could occur in either the or measurement for either output block of any cnot gate at any level , or it might occur in one of the three level-0 cnot gates in the final stage of decoding , so the joint probability of acceptance and a logical error in the decoded qubit is    we find then , that the probability of failure for the teleportation into the code block is the second term is an upper bound on the probability of error in the decoded qubit conditioned on acceptance , and the first term is an upper bound on the probability of a fault in the level-0 teleportation circuit ( the preparation , one cnot gate , and two single - qubit measurements ) .it is possible by using our recursion equations to prove that if the fundamental noise strength is below the accuracy threshold , then , for all , is bounded above by for local stochastic noise and for independent depolarizing noise .thus , in both cases encoded ancilla states can be prepared with a probability of error below the distillation threshold , and robust universal quantum computation is possible .alternatively , we can bypass the need to bound for all by noting that for we have for local stochastic noise and for independent depolarizing noise .it was proven previously that , for a scheme based on concatenation of the 9-qubit bacon - shor code , if the strength of local stochastic noise is below the threshold , then for any a block can be decoded with error probability at most  .therefore we can concatenate the fibonacci scheme at level with the bacon - shor scheme , and perform teleportation into the code block with a total probability of a logical error no larger than for local stochastic noise and for independent depolarizing noise , still below the distillation threshold .we conclude , then , that for the fibonacci scheme the accuracy threshold is also a threshold for universal quantum computation . in summary, we have proved :  [ quantum accuracy threshold theorem for the fibonacci scheme ] [ theorem:1 ] an ideal quantum circuit can be accurately simulated by a noisy quantum circuit using the fibonacci scheme for fault - tolerant quantum computation if the fundamental noise strength is no larger than , where for local stochastic noise and for independent depolarizing noise .the ratio of the size of the noisy circuit to the size of the ideal circuit is bounded above by a polynomial in the logarithm of as specified in section [ sec : overhead ] below .the overhead factor is polylogarithmic in because , although the number of locations inside a level- gadget grows approximately exponentially with the level of concatenation , the probability of a logical error in a level- gadget decreases _double exponentially _ with , if is less than the threshold value .  for the fibonacci scheme ,the probability of a ( flagged ) logical error at level scales with according to eq .( [ flagged - failure - scaling ] ) : we can simulate a circuit of size with reasonable accuracy if the probability of a logical error per gate is of order , or if the size of the largest level- gadget is bounded above by ( where is a constant ) , then the overhead factor can be expressed as this is polylogarithmic scaling as asserted in theorem 1 .  to examine in more detail how the gadget size depends on , we must take into account that our procedure for preparing a -bp uses _ postselection _ : the -bp is rejected if a flag is raised at the top level in either one of the two block teleportations or in any one of the eight subblock teleportations .thus the probability of rejection scales with like the probability of a top - level flag in a level- block , which is according to eqs .( [ flag - strength - scaling ] ) and ( [ flagged - failure - scaling ] ) . for the purpose of estimating the overhead, we may imagine that a constant number ( independent of ) of -bp preparations are executed in parallel , where the constant is chosen to be large enough so that the probability that all of the -bps are rejected is sufficiently small .for example , for each -bp used in the level- cnot gadget , suppose that -bps are prepared in parallel .the cnot gadget may fail because , in one of the -bp preparations contained inside the gadget , all of the -bps are rejected this occurs with probability .the probability of a logical error in the cnot gadget arising from other sources is ; thus the probability of failure due to the rejection of all -bps is comparable to other contributions to the failure probability if we choose   similarly , for each input -bp used in the preparation of a level- bp , suppose that -bps are prepared in parallel .then the probability that all -bps are rejected is . in this case, should be large enough that the probability of failure due to rejection of all the -bps is comparable to the probability of an unflagged level- logical error in an accepted -bp , which is according to eq .( [ unflagged - error - scaling ] ). therefore we choose the -bp preparation circuit ( for ) shown in fig .[ fig:3 ] uses as input 12 -bps , plus another 8 -bps for the final subblock teleportations . for our overhead estimate, we replace these 20 -bp preparations by -bp preparations .in addition , the -bp preparation circuit contains 28 level- transversal logical cnot gates ( to prepare three level- bell pairs , for the two block teleportations , and for the eight subblock teleportations ) plus 32 level- transversal measurements ( for the two block teleportations and for the eight subblock teleportations ) .more generally , suppose that the preparation circuit for a -bp ( ) uses -bps , level- cnot gates and level- measurements .furthermore , suppose that the logical cnot gates and measurements are transversal , so that a level- logical cnot uses fundamental two - qubit cnot gates and a level- logical measurement uses fundamental single - qubit measurements , where is the block size of the code that is concatenated times ( for the scheme we have analyzed ) .let denote the number of -bp preparations , denote the number of level-1 cnot gates , and denote the number of level-1 measurements contained in a level- bp preparation .these quantities obey the recursion relations : solving these recursion relations , we find assuming .these expressions illustrate how the number of locations inside a level- gadget grows roughly exponentially with , at least for the case of the gadgets .the distillation protocols that prepare high - fidelity encoded ancillas for non - clifford gates use gadgets , and require a number of rounds of that grows linearly with ; thus the circuit size for the non - clifford gadgets can also be bounded above by an exponential in .an important feature of the ( non - recursive ) fibonacci scheme , not shared by the strictly recursive schemes analyzed in , is that although the size of the noisy circuit is larger than the size of the ideal circuit by a factor polylogarithmic in , the `` quantum depth '' of the noisy circuit is larger than the depth of the ideal circuit by only a constant factor .( that is , the number of time steps in the simulation is larger than the number of time steps in the ideal computation by a constant factor , _ if _ we neglect the time required for classical processing of measurement outcomes . )  note that the preparation of bell pairs and the distillation of ancilla states are performed off - line and hence do not contribute to the simulation s depth .a transversal level- logical cnot gate can be executed in a single time step for any value of .the quantum measurements comprising the transversal measurement of a level- block can also be performed in a single time step , but to determine the outcome of the encoded measurement classical processing of the measurement outcomes is required , which can be executed by a classical circuit with depth linear in .  whenever a non - clifford gate is teleported , the outcome of the encoded measurement is needed to determine what clifford gate should be applied in the next step .if classical processing is much faster than quantum processing , then we may neglect the time needed for the classical decoding of measured blocks ; under this assumption , non - clifford gates as well as clifford gates can be simulated in constant depth .we will not present here a more detailed analysis of the resource requirements for the fibonacci scheme , but we note that estimates based on numerical simulations are discussed in the supplementary information in ref .  .we can expect the accuracy of quantum computing hardware to improve steadily .even so , overcoming the limiting effects of noise will pose a challenge to the builders of quantum computers well into the future , and fault - tolerant simulations will be needed to operate large - scale quantum computers reliably .there are two central questions about fault - tolerant schemes : how much noise can be tolerated and how does the overhead cost depend on the strength of the noise ?much can be learned about the answers to these questions through numerical studies of fault - tolerant circuits .but it is also useful to prove rigorous statements about the performance of fault - tolerant methods .for one thing , a long computation might fail due to rare noise fluctuations ; the probability of rare events could be seriously underestimated in numerical simulations , so that rigorous upper bounds on the probability are valuable .furthermore , rigorous analysis can deepen our conceptual understanding regarding which approaches to fault tolerance are most effective , and inspire new proposals for fault - tolerant circuit design .this paper , like the earlier paper , provides a rigorous analysis of a clever approach to quantum fault tolerance introduced by knill ; in we analyzed knill s `` postselection scheme '' and here we have analyzed his `` fibonacci scheme , '' which has a much more favorable overhead cost than the postselection scheme . in each case we have established a rigorous lower bound on the quantum accuracy threshold that is significantly higher than the previously proven lower bounds for other fault - tolerant schemes , though still about an order of magnitude below knill s numerical estimates .( high numerical estimates of the accuracy threshold have also been found in , though for that scheme a fully rigorous analysis has not yet been done . )  for the fibonacci scheme , our rigorous lower bound on the threshold noise strength is for local stochastic noise and for independent depolarizing noise .our previously derived lower bound for the postselection scheme was , but that result applies to a modified noise model with weaker noise correlations than local stochastic noise and stronger noise correlations than independent depolarizing noise .all of these results notably improve on the lower bound for local stochastic noise found earlier for the concatenated bacon - shor scheme .our proof of the threshold theorem uses some new methods that might be fruitfully applied elsewhere .we have introduced a new wrinkle on `` level reduction '' by characterizing the noise in recursively prepared bell pairs using a hierarchical effective circuit that produces the same output as the actual bell - pair preparation circuit .we have also developed tools for analyzing the recursive decoding of concatenated code blocks assisted by flags that point to probable error locations . in a separate paper , we have shown that flagging can be helpfully invoked in schemes designed to protect against highly biased noise models , where type - z errors are far more likely than type - x errors .this paper supersedes an earlier discussion where a recursive scheme which used flagging was studied .the analysis reported in was flawed , and the threshold estimate derived there was incorrect .we are grateful to daniel gottesman , david divincenzo , and barbara terhal for helpful discussions and comments .this research is supported in part by doe under grant no .de - fg03 - 92-er40701 , nsf under grant no .phy-0456720 , and nsa under aro contract no .w911nf-05 - 1 - 0294 .

Abstract: we rigorously analyze knill s fibonacci scheme for fault - tolerant quantum computation , which is based on the recursive preparation of bell states protected by a concatenated error - detecting code . we prove lower bounds on the threshold fault rate of for adversarial local stochastic noise , and for independent depolarizing noise . in contrast to other schemes with comparable proved accuracy thresholds , the fibonacci scheme has a significantly reduced overhead cost because it uses postselection far more sparingly .


one of the many skills involved in learning how to play the clarinet is to control the attack of a new note .tonguing is an important aspect of a clear and precise attack , but the evolution of the mouth pressure during the first instants of the note is also seen to affect the attack considerably .moreover , in some particular situations , tonguing may not be involved in starting a new note .hence there is both scientific and practical interest in the question : what combinations of tonguing and evolution of the blowing pressure produces sharp and precise attacks ?  as a self - sustained musical instrument ,the clarinet can be seen as a dynamic system in which the oscillation is controlled by input parameters from the musician .two of the most important are the blowing pressure and the lip force upon the reed .models predict the range of parameter values that allow for the production of a musical note .other useful predictions are the dependence of amplitude of oscillation on these two parameters , period doubling bifurcation points , or the parameter regions where the reed touches the lay of the mouthpiece . .more complex models exist that can simulate the reed oscillation in time - domain or harmonic balance methods .they provide more accurate predictions but their complexity makes it hard to grasp the causality relation between parameters and consequences in oscillatory behaviour .  in previous studies in which mouth pressure was gradually increased at constant rates , oscillations appeared at a much higher mouth pressure threshold than that predicted assuming a constant mouth pressure .high thresholds were observed in an artificially blown clarinet and even higher in numerical simulations .analytical reasoning  based on dynamic bifurcation theory predicts a delay in the threshold of oscillation for a linearly increasing mouth pressure , but the exact value of mouth pressure at which it occurs is only valid for simulations performed with very high precision .the threshold observed with normal precision simulations can only be explained with a modified theory using stochastic perturbations .this article extends previous studies by the present authors by switching the focus from the threshold of oscillation to a complete description of the amplitude of oscillation .a simplified model of a note attack is a constant increase in the mouth pressure ( as used in previous articles ) which ceases increasing and then remains constant at a defined value .the effect of ceasing the pressure increase is studied analytically to develop a full recipe for estimating the envelope of the attack .this recipe is then explored by comparing to actual simulations of the raman model .  in section [ sec : clari - theo ] , the model of the clarinet used in this work is briefly presented , as well as some of its known properties .the remaining of this section provides a brief overview of the key concepts that are needed for the present article ( most of these concepts are described with more details in two articles by the authors ) .section [ sec : envel - after - dynam ] describes the calculation of the envelope of the oscillations relative to the invariant curve , firstly in an ideal case with infinite precision , then with limited precision or noise ( section [ sec : traj - syst - affect ] ) . to some extent , these methods were already employed in previous articles to determine a dynamic threshold of oscillation . herethey are extended to calculate the envelope before this threshold is reached .section [ sec : interr - ramps - contr ] presents a method to take into account a discontinuity in the time derivative of the mouth pressure . in section [ sec : examples ] , the models are applied to particular examples and simulations , analysing the consequences in terms of expected evolution of the sound .a list of the symbols used in this article is provided in appendix [ sec : notation - table ] .for an elementary analysis , the clarinet can be described using a version of the lossless raman model , originally used for the bowed string .the system is described by two state variables and , made non - dimensional by dividing them respectively by the minimum pressure that closes the reed in steady - state , and the maximum flow allowed by the reed valve .a non - linear function relates the pressure difference between the mouth and the mouthpiece ( , where is the mouth pressure ) to the volume of air that flows past the reed ( ) .the derivation of this formula is given for instance by chaigne and kergomard  .[ nonlin_carac_2eq_ad]f(p)= ( p - 1 ) p < 0 ; + ( 1-p ) p ; + 0 p > 1.[carnl_beatreed ]  the control parameters of the system are the mouth pressure and the embouchre parameter . is related the lip force via the opening area of the reed at rest and is proportional to the characteristic impedance at the resonator input .three examples of the function ( fig .[ rep_fre ] ) show that smaller values of bring the characteristic function closer to that of a stopped pipe ( ) .increasing shifts the curve along the -axis .the reed - mouthpiece system drives the resonator .it is linked to it by the acoustic variables and found in eq . . for a time - domain descriptionit is usually simpler to describe the resonator using two non - dimensional traveling wave variables and , respectively the outgoing and incoming pressure waves : the incoming wave at the bore input is the opposite of the delayed outgoing wave , since no losses in the propagation or reflection are considered and are usually written respectively as and .the latter form is used in this article for conciseness . ] . in practice, only one value of is calculated in each round - trip of the wave , with a duration of , where is the resonator length and the speed of sound .all the variables can thus be discretized , meaning the value of a variable at time .the behaviour of the whole instrument then can be described in a single iterative equation :    function can be obtained by replacing and in function with eq . .an explicit formulation for is given by taillard _ , for .[ rep_gre ] shows that the change from coordinates to can be performed graphically as a mirror about the axis and a rotation about the origin . like , also depends on the control parameters and . to keep the notation simple ,the parameters will be omitted when constant . will be included as an argument to the function when it varies with time .  in most works on the clarinet , functions and studied in a static - parameter regime , referring to a case where the instrument is blown at a constant pressure with a constant force applied on the lip .this article focuses on a case where the mouth pressure varies over time , a situation is referred to hereafter as dynamic - parameter regime , or simply dynamic regime .the graphics of fig . [ rep_gre ]thus change over time .   in a static - parameter regime , there is a value of establishing the transition between non - oscillating and oscillating solutions .this is called the _static oscillation threshold_. above this value , the clarinet system can oscillate , and will indeed oscillate for most initial values . however , for particular sets of initial conditions ( in the scope of this paper sets of and ) , the solution is non - oscillating .these sets correspond to the `` invariant manifolds '' .if does not vary with time , the invariant manifold is called a fixed point , as the variable will remain constant ( ) .the fixed point can be found by solving :     is a function of , .when varies with time , the invariant manifold can not correspond to a single fixed point , but is also time - dependent , corresponding to an `` invariant curve '' . perhaps surprisingly , it is not the set of values .the invariant curve is defined as the set of values ( , ) such that during the planned time - variation of , this set of values will always be followed , independently of the particular value the system is initiated in .the following equation is a defining condition for this curve :   a method for calculating the invariant curve for the clarinet system is given in a previous article . in appendix[ sec : perturb ] simpler expressions for the invariant curve are given by using the characteristic curve expressed as instead of function .the invariant curve depends on how the parameter varies in time , i. e. , it is different for different rates of variation of ( different values ) .   in both static and dynamic cases ,the non - oscillating solutions can be either stable or unstable , depending on the behaviour of the system initialized close to the invariant manifold .if initialized with a value close to a stable invariant manifold , the state variable will approach it exponentially .conversely , the state variable is repelled exponentially by an unstable manifold while in its vicinity .the distance to a fixed point ( in a static - parameter case and while is sufficiently close to the fixed point ) is an exponential function of time ( expressed as iteration number ) : ^n . \label{sol_stat}\ ] ] where is the derivative of the iterative function at the fixed point .when this value exceeds , the fixed point is unstable and the oscillation grows . due to the non - linear nature of the system ,the oscillation can not grow forever , of course , and it stabilises in a periodic solution .  ,solution of eq .  for different values of its initial value . from left to right : , and .exponential envelope deduced from the function .the following parameters are used : ( constant ) and . ]  in a static - parameter context , the oscillation would eventually stabilise in an oscillatory regime between values given by the 2-branch part of the static bifurcation diagram ( an extensive discussion is given by taillard _ ) .  for time - varying parameters ,the evolution of the system can be interpreted as a dynamic bifurcation diagram . in this case, it is observed that the system still follows closely the invariant curve even after it becomes unstable ( see for instance fig .[ zoom1 ] ) . eventually an oscillation appears at a value of mouth pressure much higher than the static oscillation threshold , so that we speak of a _bifurcation delay_. the new threshold is called the _ dynamic oscillation threshold_. above this threshold , a periodic regime is established whose amplitude is given approximately by the 2-branch part of the static bifurcation diagram .  the article focuses on providing the necessary elements to calculate the amplitude envelopes in different conditions , including when the time - variation of a parameter abruptly changes rate .the duration of the transient is mainly characterized by two aspects :  * the time constant of the exponential approach or departure from the invariant manifold , which is proportional to , as shown by eq .  * the value of the initial condition of , orhow far it is from the invariant curve or the fixed point .figure [ figeda : extrasci1artrec1 ] illustrates how , for a similar exponential time constant ( and parameters that are constant in time ) , it is possible to obtain very different transient times by changing the value of the initial conditions .these are important results for understanding the behaviour of the system in a situation where the parameters change .the differences in dynamic parameter contexts are :  * if the parameter starts increasing at a value below the static oscillation threshold , the system will first undergo an approach to the invariant curve , and only beyond this value will it start the departure phase .in fact the approach can be so dramatic that a visible oscillation is only observed far beyond the static threshold . *the exponential time - constant varies throughout the growth of the parameter , but it is not simply given by at each time .  in realistic experimental situations , however , stochastic fluctuations prevent the system from coming too close to the invariant curve in the approach phase , and this can reduce the bifurcation delay .this section provides a method to describe the oscillation amplitude in the particular case of a clarinet model system in which the blowing pressure parameter increases with time at a small constant nondimensional rate :  [ dynsys_ppar ] x_n = g(x_n-1,_n)[dynsys_ppar_a ] + _ n = n+_0 .first , the case with an arbitrarily high precision is analysed . is the state variable of the system described in section [ sec : elementary - modelre ] . with the knowledge of and its previous value remaining variables of the system can be calculated . in is shown that during a significant part of a slow transient , is close to the invariant curve described above .as seen in the previous section , for a constant parameter , the envelope is well described by an exponential envelope ( eq . ) , as long as the state variable remains sufficiently close to the fixed point so that function is well approximated by its tangent line .  ) .( dashed black line ) invariant curve .( blue line ) curve of fixed points . , and . ]  fig .[ zoom1 ] suggests that when the parameter varies over time , follows more closely the invariant curve than the curve of fixed points ( ) . instead of following the distance to the fixed point as in eq ., a new variable is therefore defined :   note that , when the parameter is constant , the definition reverts to of eq ., as can be verified by substituting in the perturbation approximation to ( see appendix [ sec : perturb ] , eq . ) .  for small amplitudes , the function in eq .can be expanded as a first - order taylor series around the invariant curve .the advantage of switching to this description is that future values of the oscillation amplitude can be approximated using a simple function relating to an initial iteration :    eq .is the equivalent to eq .  for variable parameters ( see for details ) .function is defined by :    in the applications shown in this article , is always used as a definite integral . as a consequence the integration constant , orone of the bounds of the integral can be defined arbitrarily . is used in this article as a reference point close to the minimum amplitude , although for the minimum is attained at a slightly lower pressure .the discrete equivalent of eq .is :   the `` product form ''  highlights that when the magnitude of is smaller than in modulus , which happens before the static threshold is reached , approaches the invariant curve . beyond this threshold , moves away from the invariant curve , but initially at a very slow pace , because the logarithm remains close to .although is not easy to calculate analytically , for small values of the increase rate , the derivative can be approximated by its value at the fixed point , and the integral written in the form :    the error in committed in this approximation is observed to be smaller than ( the difference between and in fig . [fig : wn05 ] is much smaller than ) .  for the clarinet model ,the expressions involved in the calculation of the derivative are too complicated if function is used in its explicit form .however , they can be obtained in a simple form ( see appendix [ sec : perturb ] , eq . ) from the definition of in coordinates , providing simpler expressions for a numerical calculation of the integral . in the rest of this paper we use the approximate form .calculated using eq .( dashed line ) and approximately using eq .( solid line ) . , . ]the predicted amplitude is calculated as a distance to the invariant curve :   at iteration , .the graphic in fig .[ fig : wn05 ] can be used to predict the qualitative behavior of the system : starting at a value , the distance to the invariant curve is a monotonic function of . whenever , the amplitude is smaller than the starting value .conversely , when the amplitude is higher . corresponds to the _ dynamic oscillation threshold _ , as defined in .the curve described by eq .is often a good approximation of the envelope for most of the range of the growth parameter , except for large values of , which typically arise in 2 situations :  * in the beginning of the transient , where the iterate can be far from the invariant curve , depending on the initial conditions .note that the invariant curve usually diverges for small values of , so that even for reasonable values of , the amplitude can be very large .the region where this curve diverges depends on , but is usually well below the static threshold ( see appendix [ sec : pert - terms ] ) .* at the end of the transient , where finally escapes from the invariant curve .  in practice these two situations can be avoided by carefully choosing the time interval of interest .for example , a few initial iterations may be calculated exactly using the recursive relation until they become sufficiently close to the invariant curve . in the end of the transientthe envelope would not be valid for other reasons , in particular because the linear approximation in eq .is not valid ( otherwise the envelope would grow indefinitely ) .the prediction is valid until a few ( 3 or 4 ) iterations before the envelope starts stabilising in the oscillating branch of the bifurcation diagram .the curve in eq. often reaches very small values if the value of is sufficiently small . as a quick example of application , consider a simulation started at a value of close to . for this case , fig .[ fig : wn05 ] shows that the value of the amplitude at is is the difference between the minimum of ( at ) and the starting value of . for ,this means that the minimum amplitude will be . reducing the increase rate by a factor of ten ( )brings the minimum amplitude down to the suprisingly low value of . in general , the minimum amplitude reached by the system can be roughly calculated with :   a few remarks are suggested by these extremely low values .firstly , extremely low values can not be computed using ordinary machine precision . in this article , the calculations are performed with a _ python _library ( _ mpmath _ ) that simulates arbitrary precision in an ordinary machine .[ dyn_bif_digramartrecno ] shows how three different values of the precision produce very different envelopes . for certain values of errors are many orders of magnitude higher than the precision of the calculations . beyond a certain value of the precision ,the envelope is not greatly affected , only producing `` microscopic '' errors , which are of the same magnitude as the precision . in practice ,the precision required to simulate the system should be higher than the minimum amplitude reached by the system .this ensures that the difference between the simulation and the exact system never exceeds , otherwise larger differences are expected because of the change in dynamic threshold .  .dynamic diagram is obtained with and , from numerical simulations performed with three different numerical precisions : , and . ]second , even if the simulations are performed using correct precision , the amplitudes can only be seen relative to an accurately calculated invariant curve .the estimation of requires a precision so that it can be used as an accurate reference for determining the amplitudes .  in this paper , the invariant curveis calculated approximately using a perturbation series ( see appendix [ sec : perturb ] ) , whose precision depends on the number of perturbation terms . assuming that the perturbation terms all have the same magnitude ( which as shown in fig .[ fig : icterms ] is true for ) , the biggest influence in precision comes from the powers of that multiply each term in eq . .using this simple reasoning , a number of terms is required for an invariant curve with precision :    returning to the previous example , for , perturbation terms are required to observe correctly the envelope at very low amplitudes , whereas for the number of terms is .however , even though the invariant curve requires a lengthy calculation in order to serve as a reference for the observation of , a direct estimation can be obtained with a much cruder approximation of the invariant curve , as shown below .note that the previous argument is typically valid for high values of . for low values , some of the perturbation terms can reach values higher than 1 , especially for high values of .the argument seems valid in general above the static threshold ( see appendix [ sec : pert - terms ] and fig . [fig : icterms ] ) .  in real systems ,the problem of precision does not apply .however , experimental systems are very often affected by noise from different sources .the major source of noise in the clarinet is turbulence , which can not be avoided even with a very precise control of the pressure .noisy situations , as well as finite precision situations , can be analysed introducing a stochastic variable in the iterative system ( eq . )if numerical simulations are run with a precision coarser than the calculated through eq . , the previous formul must be extended .the limited precision ( i.e. below ) used in simulations is modelled as a stochastic variable ( with a standard deviation of ) in the system .this case is studied in  . a `` squared average '' trajectory is described by :    the two terms of the right - hand side of eq .are functions of the parameter .the term labeled corresponds to the approximation of the trajectory in the absence of noise , the same as in eq .  . is the expected value of the additional distance to the invariant curve due to the presence of noise . in practice , when the noise level is sufficiently high or the precision low ( relative to the estimation of eq . ) , only the term is relevant , i.e. the trajectory of the system is described by : with    for sufficiently small , since by definition and considering eq ., it can be deduced that for close to ( keeping in mind that depends exponentially on with small , in this article , and the minimum of and are close to ) and is negligible for all remaining values of .this allows a simplification of the expression for as described below .  according to the shape of ( see fig .[ fig : wn05 ] ) , a second - order taylor expansion of around the static oscillation threshold is used to simplify its expression ( for details , see appendix [ secan : itilde ] ) :    using approximation , the expression of can be simplified to :    details of the calculations of the simplified expression are given in appendix [ sec : detailbn ] .this amplitude does not depend on the starting amplitude , and is also independent of the starting value of .it is interesting to notice that according to , expression  can also be written :    in this form , eq .shows that , in the presence of noise and far beyond the static threshold , the envelope followed by the system has the same shape as without noise , but with a different amplitude , i.e in this case we have :    where is a constant deduced from eq . .as a remark , a different calculation with similar objectives is made in a previous article  to determine the dynamic thresholds in presence of noise .the approximation was used formally to integrate in eq . .the result is an explicit expression for , and therefore of the dynamic oscillation threshold . here, is numerically integrated , keeping its precise expression given by eq . .this leads to a better estimation of the envelope , but that envelope does not have an analytic expression .this section describes the behaviour of the system for an example profile consisting of a limited linear growth of the parameter at a constant rate followed by a constant value for an indefinite period of time .the parameter is therefore formally defined as :  [ gamma_prof]_n= n+_0  n m + _ m  n > m.    due to the change in increase rate at the growth phase and the static phase are studied independently . an amplitude envelope computed for the growth phase and another for the static phase .the two envelopes are connected at since the initial value is deduced from .the method is described in the next sections and summarised in fig .[ fig : algo ] .as explained in section [ sec : envel - after - dynam ] , the first few ( ) iterations must usually be performed manually .these correspond to an `` approach phase '' that brings the system close enough to the invariant curve so that the assumption of linearity is valid . at iteration the state of the system is given by :   the number of iterations required for the approach phase depends on the starting value of and the increase rate . in practicethe state of the system is simulated iteratively until it reaches an amplitude .a complication to this view arises when goes through a superstable point defined by . at this pointthe iterations can approach arbitrarily the invariant curve .although this situation can be analysed under some simplifying assumptions , this is not done in this article , and the reader is referred to baesens for a detailed description of this case or to bergeot in the context of the clarinet . a simple way of circumventing this problem is to force to bring beyond the super - stable point .a few explicit iterations ( usually less than 5 ) allow the calculation of the amplitude .iteration is used as a safe starting point for the analytic determination of the envelope .  when the simulations are performed with a lower precision than that required for simulating the exact system ( see eq . ) , the initial value does not affect the growth of oscillations . in this case , an average squared amplitude is given by eq .starting from .therefore , the envelope is given by :    this approximation is valid for , which is the usual region of interest . below the oscillations are mostly random , with an average level that remains close to the standard deviation of the stochastic perturbation .   at , becomes constant and the oscillation undergoes an exponential growth ( provided that ) , given by eq .  where the initial value is replaced by the value deduced from the previous study of the growing phase :  ^{(n - m)}\right|.\ ] ]  the starting amplitude for the static phase is given by continuity of : due to the change in invariant manifold from the invariant curve to .  as a remark , when the amplitude is sufficiently small ( i. e. ), the starting amplitude can be given simply by the difference between the invariant curve and the curve of fixed points :    in such a situation , the transient time is roughly given by the time until the slope discontinuity in the blowing pressure profile , plus a delay corresponding to the time needed for the oscillations to grow from ( independently of ) to the final amplitude .since the starting amplitude and the exponential coefficient ( in eq . )are independent of the slope of the growth phase , so is the duration of the transient resulting from the interruption in the growth .this matches observations on real instruments blown artificially .  in any case, the oscillation usually starts very close to the fixed point .this ensures that the linear approximation is valid on a large part of the transient ( see fig .[ figeda : extrasci1artrec1 ] ) .a few examples of simulations are presented in this section , together with predictions based on the previous sections , and their limitations .the `` actual envelopes '' corresponding to the absolute distance between the iterated values and the invariant curve are plotted together with the estimation of the envelopes ( eq . ) . in examples presented in sections [ sec : small - gamma_m ] and [ sec : large - gamma_m ]the numerical precision is higher than the minimum amplitude reached by the system ( eq . ) .the effect of introducing a stochastic variable in the system , which plays a similar role as performing simulations with low precision  , is shown in the example of section [ sec : effectnoise ] .   in fig .[ fig : sim06 ] , the increase in mouth pressure is stopped at a relatively small value of the parameter . in consequence ,the amplitude of the oscillations is considerably smaller when the increase is interrupted .a jump in the relative amplitude is observed when , in a logarithmic plot ( see fig .[ fig : sim06](b ) ) .this jump arises because is the distance to the invariant curve before and to the fixed point after .  in this example , 6 iterations ( ) are used to reach the linear approximation .moreover , so that the starting amplitude for the constant parameter phase ( ) is deduced from eq . .the envelope is then computed following the method described above ( see fig . [fig : algo ] ) .  fig .[ fig : sim06](b ) also shows that the prediction is slightly in advance relative to the actual envelope .the reason is that is calculated using a severe approximation ( see eq . ) . for small values of approximation is satisfactory .the advantage of using this approximation is that a single curve can be used for any small value of the growth rate .  , , , , .]   in fig .[ fig : sim09 ] , reaches a higher stable value .this results in higher values of amplitude when the parameter stops increasing .  , , , , .]  the envelopes during the growing phase are estimated using the same method as in the previous example . at iteration , since the system is estimated to have an amplitude that is higher than the difference , the new amplitude is the distance to the invariant curve of the growing phase .the remaining envelope is the exponential , as in the example above .  the biggest difficulty in estimating the amplitude of the static phase arises when the amplitude is similar to . in this case, the iterate at can be either very close to the fixed point curve or at twice the distance between the two reference curves , which will imply very different amplitudes for the static phase .  finally , in fig .[ fig : sim09](b ) , the jump in relative amplitude at the beginning of the static phase exists but it is not clearly visible because the amplitude of the oscillation at is large compared with the case shown in the previous example ( see fig .[ fig : sim06](b ) ) .the disagreement between the iterates and the prediction for may appear to suggest that the prediction is not good here , whereas in fact it is the `` actual envelope '' that is incorrect .this is due to an inaccurate determination of the invariant curve .in fact , the number of terms needed for the invariant curve ( eq . ) makes its analytical computation too complicated .this situation is thus different from the numerical precision problem outlined in fig .[ dyn_bif_digramartrecno ] , where the iterates are in some cases very different from those of the ideal system simulated with infinite precision due to the shift in dynamic threshold .the prediction is valid for most of the simulation between and , and it matches the envelope whenever the invariant curve is valid ( in particular above ) .this shows that the envelope and the dynamic threshold can be fairly well predicted , even with an inaccurate approximation .   in the example of fig .[ fig : sim - noise ] , the simulation is performed adding a stochastic variable to with a uniform probability distribution having a standard deviation .this is roughly equivalent to a simulation without noise but with a numerical precision fixed to 4 significant digits ( i.e. )  .   and added noise .invariant curve ( eq . ) and envelope predictions given by eq . .unlimited precision , , , , , . ]  the sequence ( eq . ) is calculated based on this value of , and its square - root plotted as the envelope prediction .the prediction is valid for , where reaches its minimum value .the departure of the oscillations occurs earlier when compared to the case without noise .when the amplitude is roughly that of the noise , because the noise is added at each new iteration to a value smaller that of the non - linear function applied to the previous iteration .because of the reduction in the bifurcation delay , the oscillations are seen to depart much earlier than the cessation of the increase of the parameter .after the end of the exponential increase in amplitude , the envelope increases with the parameter , following the two - state oscillation given by the static bifurcation diagram .  a discontinuity similar tothat observed in figure [ fig : sim06 ] can arise also in noisy conditions .however , because the envelope curve multiplies a bigger value at , the discontinuity is seen only for smaller values of .  due to the random nature of the system , the prediction should not be interpreted as an approximation to the exact envelope , but rather as the envelope followed on average by a series of runs of the simulation . in fact ,in this case , for a series of runs with different noise samples , the actual envelope was seen to shift towards the right or the left by about 4 iterations .the method can in principle be extended to include frequency independent losses , although this may be hard to acheive analitically .the invariant curve can not be calculated directly from the simple expression of , as in appendix [ sec : perturb ] , requiring the use of the much more complicated expressions of and its derivatives .more complex models of clarinets with frequency dependent losses are known to give rise to long attack transients with similar envelope shapes , and the envelope estimation used in the present article may be similar in models with small dispersion in the reflection function .  when the mouth pressure grows linearly over time , the logarithm of the amplitude is proportional to a predetermined curve , which we call .the proportionality factor depends on the inverse of the growth rate , whereas the offset depends on one of these two factors :  * the initial amplitude ( starting distance to the invariant curve ) , when the precision is high enough ( see eq . ) or * the stochastic level when the simulation is imprecise or the system is noisy ( eq . ) .a stop in the linear growth of the mouth pressure may occur while the system is still oscillating with low amplitude . in this case , when the pressure stops increasing , the oscillation resumes exponentially from a higher amplitude , which is given by the distance between the invariant curve and the fixed point at the particular value of the mouth pressure .a discontinuity in the amplitude envelope is observed if before the mouth pressure stops increasing while the amplitude was still at a value lower than the distance between the invariant curve and the fixed point .bifurcation delay has also been observed in a real instrument .so far it has been hard to relate the amplitude envelope to the value of the mouth pressure . in interrupted ramps of the mouth pressure however , the oscillations seem to be triggered close to the inflection point of the blowing pressure  . in this case , an exponential amplitude growth then resumes .this is as expected for low values of , as shown in the example in section [ sec : small - gamma_m ] .the values of at the start of the oscillation depend on the rate of growth of the mouth pressure , an indication that the system is determined by the stochastic fluctuations in the mouth pressure .for a constantly increasing parameter , the _ dynamic oscillation threshold _  gives the approximate value of the mouth pressure parameter for which an audible sound appears , or in other terms , the distance from the invariant curve becomes `` macroscopic '' .when the linear growth of the mouth pressure is suddenly stopped at and then kept constant at a value , two situations must be distinguished :  1 . : a growing exponential envelope starts at with a fixed starting amplitude , which only depends on the value of ( see section [ sec : small - gamma_m ] ) .audible sound occurs at a fixed time interval from the stop in pressure increase ; 2 . : the audible ( `` macroscopic '' ) sound begins at ( see sections [ sec : large - gamma_m ] and [ sec : effectnoise ] ) .  in most practical casesthe latter situation is more common : because of the limited precision or noise , is effectively reduced to values that are much closer to the static threshold .this work shows that the amplitude envelope produced with a regular increase of blowing pressure in a simplified clarinet system can be described reasonably well by the use of a single function that is a characteristic of the system .this function can be used in exact and `` noisy '' cases to describe the envelope beyond the static threshold .  when the pressure increase is interrupted , the exponential envelope corresponding to the transient of a static - parameter case can be matched with the one corresponding to growing pressures . in many practical cases ,when the interruption occurs at sufficiently low values of the blowing pressure , this corresponds to a fixed starting amplitude so that the transient time measured from the interruption is roughly independent of the previous history of the system .these conclusions show some dramatic effects of the stabilisation of the mouth pressure that are due to the discontinuity in derivative .  in summary , a sudden cessation in the increase inmouth pressure can have a large impact in the initial transient of the clarinet if it appears at a low enough value of mouth pressure .a preliminary comparison with a smoother stabilisation profiles suggests that smoother profiles give rise to slower transients .however , because of the simple mathematic expressions used for the profiles , they are not easy to compare to the piecewise linear profiles shown in this article .   this work is part of the research project sdns - aimv `` systmes dynamiques non - stationnaires - application aux instruments vent '' ( anr-09-rpdoc-022 - 01 ) financed by the _ agence nationale de la recherche_. the authors thank prof .joe wolfe for useful suggestions and proof reading .this appendix presents a perturbation method to calculate the invariant curve , using only expressions of function ( see eq . ) . this has the advantage of producing much simpler expressions than using function ( eq . ) .higher order terms are needed only when determining from a simulation . for all other purposes used in this article ,the first order term is usually sufficient .the invariant curve satisfies the following equation :   the perturbation to order consists in expressing as a series of terms depending on powers of ( the perturbation ) :    both and are developed in a power series , around  around the first term the right - hand side of eq .is then , to 2nd order :   by equating expression on the left - hand side and , it is possible to isolate terms on each power of , and extract expressions for each of the functions .the first term is nothing but the definition of the fixed point :   each of the higher order terms is obtained from lower - order ones : all functions and derivatives of the functions are taken at .as expected , all the derivatives of are taken at the fixed point , and this remains true for higher - order terms too .for four different values of . ]  derivatives of function , as given by taillard are hard to calculate , as the expressions are complex . however , the derivatives of are related to those of function . can be defined as a parametric curve ( with parameter ) as the locus of points : the derivative of the curve is : all higher order derivatives can be calculated iterativelly : for instance the second derivative is :   in general these formulas are not of much use because they are functions of instead of .however , it can be proved that the fixed point of corresponds to ( the line corresponds to the axis ) , so that :    and etc .   from eqs  and , the first perturbation terms can be written :    the first perturbation terms ( for 1 to 4 ) are represented graphically in fig .[ fig : icterms ] .according to the shape of , the second - order taylor expansion of around the static oscillation threshold is :    through eq . ,by definition , we have . since is the integral of a known function , at the static threshold , the expression of the first and the second derivatives of are :    eq .can be calculated explicitly from the expression of , given by eq . .the resulting expression estimated in .after calculation we obtain , yielding : with a quadratic approximation close to .using eq . , eq .is developed ,                j. kergomard , j. p. dalmont , j. gilbert , p. guillemain : period doubling on cylindrical reed instruments .proceeding of the joint congress cfa / daga 04 , 22nd-24th march 2004 , strasbourg , france , socit franaise dacoustique - deutsche gesellschaft fr akustik , 113114 .s. farner , c. vergez , j. kergomard , a. lize : contribution to harmonic balance calculations of self - sustained periodic oscillations with focus on single - reed instruments . j. acoust* 119 * ( 2006 ) 1794 .b. bergeot , c. vergez , a. almeida , b. gazengel : prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure : influence of noise .nonlinear dyn .* 74 * ( 2013 ) 591605 .b. bergeot , a. almeida , c. vergez : effect of the shape of mouth pressure variation on dynamic oscillation threshold of a clarinet model .proceedings of the international symposium in music acoustics , 2014 , pp . 535540 .

Abstract: this article uses a basic model of a reed instrument , known as the lossless raman model , to determine analytically the envelope of the sound produced by the clarinet when the mouth pressure is increased gradually to start a note from silence . using results from dynamic bifurcation theory , a prediction of the amplitude of the sound as a function of time is given based on a few parameters quantifying the time evolution of mouth pressure . as in previous uses of this model , the predictions are expected to be qualitatively consistent with simulations using the raman model , and observations of real instruments . model simulations for slowly variable parameters require very high precisions of computation . similarly , any real system , even if close to the model would be affected by noise . in order to describe the influence of noise , a modified model is developed that includes a stochastic variation of the parameters . both ideal and stochastic models are shown to attain a minimal amplitude at the static oscillation threshold . beyond this point , the amplitude of the oscillations increases exponentially , although some time is required before the oscillations can be observed at the `` dynamic oscillation threshold '' . the effect of a sudden interruption of the growth of the mouth pressure is also studied , showing that it usually triggers a faster growth of the oscillations .