Universal Sampling Lower Bounds for Quantum Error Mitigation

Numerous quantum error-mitigation protocols have been proposed, motivated by the critical need to suppress noise effects on intermediate-scale quantum devices. Yet, their general potential and limitations remain elusive. In particular, to understand the ultimate feasibility of quantum error mitigation, it is crucial to characterize the fundamental sampling cost -- how many times an arbitrary mitigation protocol must run a noisy quantum device. Here, we establish universal lower bounds on the sampling cost for quantum error mitigation to achieve the desired accuracy with high probability. Our bounds apply to general mitigation protocols, including the ones involving nonlinear postprocessing and those yet-to-be-discovered. The results imply that the sampling cost required for a wide class of protocols to mitigate errors must grow exponentially with the circuit depth for various noise models, revealing the fundamental obstacles in the scalability of useful noisy near-term quantum devices.

Introduction -As recent technological developments have started to realize controllable small-scale quantum devices, a central problem in quantum information science has been to pin down what can and cannot be accomplished with noisy intermediate-scale quantum (NISQ) devices [1].One of the most relevant issues in understanding the ultimate capability of quantum hardware is to characterize how well noise effects could be circumvented.This is especially so for NISQ devices, as today's quantum devices generally cannot accommodate full quantum error correction that requires scalable quantum architecture.As an alternative to quantum error correction, quantum error mitigation has recently attracted much attention as a potential tool to help NISQ devices realize useful applications [2,3].It is thus of primary interest from practical and foundational viewpoints to understand the ultimate feasibility of quantum error mitigation.
Quantum error mitigation protocols generally involve running available noisy quantum devices many times.The collected data is then post-processed to infer classical information of interest.While this avoids the engineering challenge in error correction, it comes at the price of sampling cost -computational overhead in having to sample a noisy device many times.This sampling cost represents the crucial quantity determining the feasibility of quantum error mitigation.If the required sampling cost becomes too large, then such quantum error mitigation protocol becomes infeasible under a realistic time constraint.Various prominent quantum error mitigation methods face this problem, where sampling cost grows exponentially with circuit size [4][5][6][7][8].The crucial question then is whether there is hope to come up with a new error mitigation strategy that avoids this hurdle or if this is a universal feature shared by all quantum error mitigation protocols.To answer this question, we need a characterization of the sampling cost that is universally required for the general error-mitigation protocols, which has hitherto been unknown.
Here, we provide a solution to this problem.We derive lower bounds for the number of samples fundamentally required for general quantum error mitigations to realize the target performance.We then show that the required samples for a wide class of mitigation protocols to error-mitigate layered circuits under various noise models -including the depolarizing and stochastic Pauli noise -must grow exponentially with the circuit depth to achieve the target performance.This turns the conjecture that quantum error mitigation would generally suffer from the exponential sampling overhead into formal relations, extending the previous results on the exponential resource overhead required for noisy circuits without postprocessing [9][10][11].We accomplish these by employing an information-theoretic approach, which establishes the novel connection between the state distinguishability and operationally motivated error-mitigation performance measures.Our results place the fundamental limitations imposed on the capability of general error-mitigation strategies that include existing protocols [5][6][7][8][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] and the ones yet to be discovered, being analogous to the performance converse bounds established in several other disciplines -such as thermodynamics [32][33][34], quantum communication [35,36], and quantum resource theories [37,38] -that contributed to characterizing the ultimate operational capability allowed in each physical setting.
Our work complements and extends several recent advancements in the field.Ref. [39] introduced a general framework of quantum error mitigation and established lower bounds for the maximum estimator spread, i.e., the range of the outcomes of the estimator, imposed on all error mitigation in the class, which provides a sufficient number of samples to ensure the target accuracy.Those bounds were then employed to show that the maximum spread grows exponentially with the circuit depth to mitigate local depolarizing noise.Ref. [40] showed a related result where for the class of error-mitigation strategies that only involve linear postprocessing, in which the target expectation value can be represented by a linear combination of the actually observed quantities, either the maximum estimator spread or the sample number needs to grow exponentially with the circuit depth to mitigate local depolarizing noise.The severe obstacle induced by noise in showing a quantum advantage for variational quantum algorithms has also recently been studied [41][42][43].Our results lift the observations made in these works to rigorous bounds for the necessary sampling cost required for general error mitigation, including the ones involving nonlinear postprocessing that constitute a large class of protocols [7, 12-17, 19, 21, 22, 24, 25, 27, 29-31, 44].
Framework -Suppose we wish to obtain the expectation value of an observable  ∈ O for an ideal state  ∈ S where O and S are some sets of observables and states.We assume that the ideal quantum state  is produced by a unitary quantum circuit U applied to the initial state  ini ∈ S in as  = U ( ini ) where S in is the set of possible input states.The noise in the circuit, however, prevents us from preparing the state  exactly.We consider quantum error mitigation protocols that aim to estimate the true expectation value under the presence of noise in the following manner [39] (see also Fig. 1).
In the mitigation procedure, one can first modify the circuit, e.g., use a different choice of unitary gates with potential circuit simplification, apply nonadaptive operations (enabling, e.g., dynamical decoupling [45,46] and Pauli twirling [6]), and supply ancillary qubits -the allowed modifications are determined by the capability of the available device.Together with the noise present in the modified circuit, this turns the original unitary U into some quantum channel F , which produces a distorted state  ′ .The distorted state can be represented in terms of the ideal state  by  ′ = E () where we call E F • U † an effective noise channel.The second step consists of collecting  samples {E  ()}  =1 of distorted states represented by a set of effective noise channels E {E  }  =1 and applying a trailing quantum process P  over them.The effective noise channels in E can be different from each other in general, as noisy hardware could have different noise profiles each time, or could purposely change the noise strength [5,47].The trailing process P  then outputs an estimate represented by a random variable Ê  () for the true expectation value Tr( ).The main focus of our study is the sampling number , the total number  of distorted states used in the error mitigation process.
We quantify the performance of an error-mitigation protocol by how well the protocol can estimate the expectation values for a given set O of observables and a set S of ideal states, which we call target observables and target states respectively.We keep the choices of these sets general, and they can be flexibly chosen depending on one's interest.For instance, if one is interested in error mitigation protocols designed to estimate the Pauli observables (e.g., virtual distillation [15,16,21]), O can be chosen as the set of Pauli operators.As the trailing process includes a measurement depending on the observable, an errormitigation strategy with target observables O is equipped with a family of trailing processes {P  } ∈O .Similarly, our results hold for an arbitrary choice of S, where one can, for instance, choose this as the set of all quantum states, which better describes the < l a t e x i t s h a 1 _ b a s e 6 4 = " j s q S h J p W v t 5 l M 4 / 0 A n m X 1 x 9 C u 0 s = " > A A A J T O O 3 2 N P s d n u J v c i u 5 6 S U u P l p p C S u v / 9 3 H N t p Q r q e G / F i 8 e / E g 4 e P H j 9 5 O p n L P 3 v + 4 u W r q e n X j S j s q e K 3 w c 6 E a u 8 H G g m 7 n C G a T b u c J x p j u 5 w n G m z V z h w 0 I P c o U j 6 W Y j 6 e q 3 w S V j 6 a 6 Y D k f T 3 T Y 9 6 X k Z L z H 9 v u R s X d d u 9 h k n I u I S C / 0 X r v c x q l / U s j l W v a 6 S e h + j T E k 2 z u J e q G R y G N e r S L 8 G k s M Y d 6 J Q N 5 8 e x 0 Q Q 1 1 U y O Z T 1 F 0 b 2 G W G N L + z W q 7 W 3 1 e l v 0 5 W 5 + s W 3 x i R 6 h J 6 g 5 6 i G 3 q E 5 t I q 2 U A s 5 6 C f 6 h X 6 j P y V U e l Z 6 V X o 9 D L 0 y c Z H z E I 1 s p c Y / K P e L q g = = < / l a t e x i t > Tr(A⇢) s 9 e t B n F P v E o 9 / I y V L s 3 0 4 p i 9 w w a P L z P u 3 4 u B e 4 j m t j r q q O p m Y F S h p p s x 7 p i G q l q r b 5 t e a q s 9 9 n K 5 / r V 9 8 a k 8 Y L 4 5 X x 2 r C M 9 8 Z n Y 9 X Y N Z q G b f w 0 f h m / j T + T / 0 q v S m 9 K b 4 e h E 7 e u c p 4 b I 1 t p 9 j 8 u n 4 u y < / l a t e x i t > " < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 B 0 o h x 0 e n s L E y 1 H m s 4 J Z t j h a i h H r k I Y P q V 8 5 D 5 E c j 0 e R X 4 s y 6 / 0 + p B + 5 + e c n v P p t J T 2 r l w u I / X L i + y s 9 n 5 b K E f + m B q k P P G G N m y S / 8 B P / O L S w = = < / l a t e x i t > ÊA (⇢) < l a t e x i t s h a 1 _ b a s e 6 4 = " w s + 0 N 7 x N l v U 4 S T P 1 + 1 B u I q E i g U o = " > A A A J S H i c d d b b b t o w G A D g t D t R d m q 3 y 9 1 k Q 5 V 2 M S F A 3 U m d p h 4 p P V B Y C 5 Q W o 8 o O D k T N S Y 5 h r d y 8 w m 6 3 Z 9 o b 7 C 1 2 N + 1 u D o f F 5 I d I S Y y / / 3 c c 2 y E h v m 0 F P J f 7 t b B 4 5 + 6 9 + w 9 S S + m H j x 4 / e b q 8 8 q w R e H 1 m 0 L r h 2 R 5 r E h x Q 2 3 J p n V v c p k 2 f U e w Q m 5 6 R q + 3 I z w a U B Z b n 1 v i N T 9 s O 7 r q W a R m Y R 1 U o s J z L 5 U w u m x t u O i z k x 4 W M N t 6 q l y u p L O p 4 R t + h L j d s H A S t f M 7 n b Y E Z t w y b h m n U D 6 i P j S v c p S 1 Z d L F D g 7 Y Y d j b U V 2 V N R z c 9 J n e X 6 8 N a N U N g J w h u H C I j H c x 7 Q d K i y l n W 6 n P z Q 1 t Y r t / n 1 D V G F z L 7 t s 4 9 P b p z v W M x a n D 7 R h a w w S z Z V 9 3 o Y Y Y N L s d n 6 i q E 4 S v K p + 5 D R N f j n m c H Y X p V r X f p V 3 7 N 6 T V / M y w N e 5 d O p 5 H 8 Z X i O g 9 2 O Q L I x 2 W T Y K r T F r c j k Q 8 S w 2 7 U p e o n s Y U F k C u F t O J 3 T c c J W v j 3 J j b K i P R E U E M x C g d Y R 6 8 t W 5 C S E 4 m P 2 r T w h Z n V 7 / B a t J x L s Q N 6 X P F G T o 9 E p m 4 h g o 4 h h P k p e z 5 e 3 N + o W o V 3 L F d F v Z l 2 H e i a v I y o j J h W J v E E i b 5 D M G 8 z O + z T p 6 2 i c p D K q + u f / P W X j g O l 8 Z 1 M G R F N i Y F t s J l t 3 t h X d B r q r 6 C 7 Q s q J l o M e K H g O t K 1 o H 2 l C 0 A b S m a A 3 o j q I 7 Q H t P z C y C e / J 2 C h U c j m 3 2 X X v q x l N g r j b 4 2 U 9 k J 7 p b 3 W 8 t p 7 b U M r a V W t r h l a T / u m f d d + p H 6 m f q f + p P 6 O Q h c X x j n P t a l t a f E f T i 5 9 + g = = < / l a t e x i t > , where E = {E  }  =1 is the set of effective noise channels.A trailing quantum process P  is then applied to  distorted states, producing the final estimation of Tr( ) represented by a random variable Ê  ().We quantify the error-mitigation performance in two ways by studying the property of the distribution of Ê  (); the first is the combination of the accuracy  and the success probability 1 − , and the second is the combination of the bias   () Ê  () − Tr( ) and the standard deviation protocols such as probabilistic error cancellation [5,[47][48][49][50][51][52], or as the set of states in a certain subspace, which captures the essence of subspace expansion [12,14,17].This framework includes many error-mitigation protocols proposed so far [5][6][7][8][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31].It is worth noting that our framework includes protocols that involve nonlinear postprocessing of the measurement outcomes.Error-mitigation protocols typically work by (1) making some set of (usually Pauli) measurements for observables {  }  , (2) estimating their expectation values {⟨  ⟩}  for distorted states, and (3) applying a classical postprocessing function  over them.The protocols with linear postprocessing functions, i.e., the ones with the form  (⟨  ⟩  ) =    ⟨  ⟩, are known to admit simpler analysis [39,40], but numerous protocols -including virtual distillation [15,16,21], symmetry verification [13], and subspace expansion [12,14,17] -come with nonlinear postprocessing functions.In our framework, the sampling number  is the total number of samples used, where we consider the output represented by Ê  () as our final guess and thus do not generally assume repeating some procedure many times and take a statistical average.This enables us to have any postprocessing absorbed in the trailing process P  , making our results valid for the protocols with nonlinear postprocessing functions.
We also remark that our framework includes protocols with much more operational power than existing protocols, as we allow the trailing process to apply any coherent interaction over all distorted states.Our results thus provide fundamental limits on the sampling overhead applicable to an arbitrary protocol in this extended class of error-mitigation protocols.
Sampling lower bounds -We now consider the required samples to ensure the target performance.The performance of quantum error mitigation can be defined in multiple ways.
Here, we consider two possible performance quantifiers that are operationally relevant.
Our first performance measure is the combination of the accuracy of the estimate and the success probability.This closely aligns with the operational motivation, where one would like an error mitigation strategy to be able to provide a good estimate for each observable in O and an ideal state in S at a high probability.This can be formalized as a condition where  is the target accuracy and 1− is the success probability (see also Fig. 1).
The problem then is to identify lower bounds on the number  of distorted states needed to achieve this condition as a function of  and .We address this by observing that the trailing process of quantum error mitigation is represented as an application of a quantum channel and thus can never increase the state distinguishability.To formulate our result, let us define the observable-dependent distinguishability with respect to a set O of observables as This quantity can be understood as the resolution in distinguishing two quantum states by using the measurements of the observables in O.We note that when O = {  |0 ≤  ≤ I} [53], the quantity in (2) becomes the trace distance  tr (, ) = 1 2 ∥  − ∥ 1 [54].
We then obtain the following sampling lower bounds applicable to an arbitrary given set E of effective noise channels.(Proof in Appendix A [55].) Theorem 1. Suppose that an error-mitigation strategy achieves (1) with some  ≥ 0 and 0 ≤  ≤ 1/2 with  distorted states characterized by the effective noise channels E = {E  }  =1 .Then, the sample number  is lower bounded as 1 is the (square) fidelity and (∥) Tr( log ) − Tr( log ) is the relative entropy.
This result tells that if the noise effect brings states close to each other, it incurs an unavoidable sampling cost to error mitigation.The minimization over E chooses the effective noise channel that least reduces the infidelity and the relative entropy respectively.On the other hand, the maximum over the ideal states represents the fact that to mitigate two states  and  that are separated further than 2 in terms of observables in O, the sample number  that achieves the accuracy  and the success probability 1 −  must satisfy the lower bounds with respect to  and .The maximization over such  and  then provides the tightest lower bound.This also reflects the observation that error mitigation accommodating a larger set O of target observables would require a larger number of samples.
We remark that although the set E -which depends on how one modifies the noisy circuit -ultimately depends on a specific error-mitigation strategy in mind, fixing E to a certain form already provides useful insights as we see later in the context of noisy layered circuits.We also stress that the above bounds hold for an arbitrary choice of E, providing the general relation between the error mitigation performance and the information-theoretic quantity.
The bounds in Theorem 1 depend on the accuracy  implicitly through the constraints on  and  in the maximization.For instance, if one sets  = 0, one can find that both bounds diverge, as the choice of  =  would be allowed in the maximization.In Appendix B, we report an alternative bound that has an explicit dependence on the accuracy .
Let us now consider our second performance measure based on the standard deviation and the bias of the estimate.Let  QEM  () be the standard deviation of Ê  () for an observable  ∈ O, which represents the uncertainty of the final estimate of an error mitigation protocol.Since a good error mitigation protocol should come with a small fluctuation in its outcome, the standard deviation of the underlying distribution for the estimate can serve as a performance quantifier.However, the standard deviation itself is not sufficient to characterize the error mitigation performance, as one can easily come up with a useless strategy that always outputs a fixed outcome, which has zero standard deviation.This issue can be addressed by considering the deviation of the expected value of the estimate from the true expectation value called bias, defined as   () Ê  () −Tr( ) for a state  ∈ S and an observable  ∈ O (see also Fig. 1).
To assess the performance of error-mitigation protocols, we consider the worst-case error among possible ideal states and measurements.This motivates us to consider the maximum standard deviation  QEM max max ∈O max ∈S  QEM  () and the maximum bias  max max ∈O max ∈S   ().Then, we obtain the following sampling lower bound in terms of these performance quantifiers.(Proof in Appendix C.) Theorem 2. The sampling cost for an error-mitigation strategy with the maximum standard deviation  QEM max and the maximum bias  max is lower bounded as This result represents the trade-off between the standard deviation, bias, and the required sampling cost.To realize the small standard deviation and bias, error mitigation needs to use many samples; in fact, the lower bound diverges at the limit of  QEM max → 0 whenever there exist states ,  ∈ S such that  O (, ) ≥ 2 max .On the other hand, a larger bias results in a smaller sampling lower bound, indicating a potential to reduce the sampling cost by giving up some bias.
The bounds in Theorems 1, 2 are universally applicable to arbitrary error mitigation protocols in our framework.Therefore, our bounds are not expected to give good estimates for a given specific error-mitigation protocol in general, just as there is a huge gap between the Carnot efficiency and the efficiency of most of the practical heat engines.Nevertheless, it is still insightful to investigate how our bounds are compared to existing mitigation protocols.In Appendix D, we compare the bound in Theorem 1 to the sampling cost for several error-mitigation methods, showing that our bound can provide nontrivial lower bounds with the gap being the factor of 3 to 6.Although this does not guarantee that our bound behaves similarly for other scenarios in general, this ensures that there is a setting in which the bound in Theorem 1 can provide a nearly tight estimate.We further show in Appendix E that the scaling of the lower bound in Theorem 2 with noise strength can be achieved by the probabilistic error cancellation method in a certain scenario.This shows that probabilistic error cancellation serves as an optimal protocol in this specific sense, complementing the recent observation on the optimality of probabilistic error cancellation established for the maximum estimator spread measure [39].
Noisy layered circuits -The above results clarify the close relation between the sampling cost and state distinguishability.As an application of our general bounds, we study the inevitable sample overhead to mitigate noise in the circuits consisting of multiple layers of unitaries.Although we here focus on the local depolarizing noise, our results can be extended to a number of other noise models as we discuss later.
Suppose that an -qubit quantum circuit consists of layers of unitaries, each of which is followed by a local depolarizing noise, i.e., a depolarizing noise of the form D  = (1 − ) id +I/2 where  is a noise strength, applies to each qubit.We aim to estimate ideal expectation values for the target states S and observables O by using  such noisy layered circuits.Although the noise strength can vary for different locations, we suppose that  layers are followed by the local depolarizing noise with noise strength of at least .We call these layers  1 ,  2 , . . .,   and let  ,, denote the noise strength of the local depolarizing noise on the  th qubit after the  th unitary layer   in the  th noisy circuit, where  ≤ ,  ≤ ,  ≤ .This gives the expression of the local depolarizing noise after  th layer in the  th noisy circuit as ⊗  =1 D  ,, , where  ,, ≥  ∀, , .Here, we focus on the error-mitigation protocols that apply an arbitrary trailing process over  distorted states and any unital operations (i.e., operations that preserve the maximally mixed state) before and after   (Fig. 2).This structure ensures that error correction does not come into play here, as the size of input and output spaces of the intermediate unital channels is restricted to  qubits, as well as that unital channels do not serve as good decoders for error correction.
We show that the necessary number of samples required to achieve the target performance grows exponentially with the number of layers in both performance quantifiers introduced above.Theorem 3. Suppose that an error-mitigation strategy described above is applied to an -qubit circuit to mitigate local depolarizing channels with strength at least  that follow  layers of unitaries, and achieves (1) with some  ≥ 0 and 0 ≤  ≤ 1/2.Then, if there exist at least two states ,  ∈ S such that  O (, ) ≥ 2, the required sample number  is lower bounded as ( The proof can be found in Appendix F. This result particularly shows that the required number of samples must grow exponentially with the circuit depth .We remark that the bound always holds under the mild condition, i.e.,  O (, ) ≥ 2 for some ,  ∈ S.This reflects that, to achieve the desired accuracy  satisfying this condition, error mitigation really needs to extract the expectation values about the observables in O and the states in S, prohibiting it from merely making a random guess.
In Appendix G, we obtain a similar exponential growth of the required sample overhead for a fixed target bias and standard deviation.We also obtain in Appendix H alternative bounds that are tighter in the range of small .
With a suitable modification of allowed unitaries and intermediate operations, we extend these results to a wide class of noise models, including stochastic Pauli, global depolarizing, and thermal noise.The case of thermal noise particularly provides an intriguing physical interpretation: the sampling cost  required to mitigate thermal noise after time  is characterized by the loss of free energy  = Ω(1/[ (  ) −  eq ]) where   is the state at time  and  eq is the equilibrium free energy.This in turn shows that the necessary sampling cost grows as  = Ω(  ent  ) where  ent is a constant characterized by the minimum entropy production rate.We provide details on these extensions in Appendix I.
We remark that Theorem 3 (and related results discussed in the Appendices) extends the previous results showing the exponential resource overhead required for noisy circuits without postprocessing [9][10][11].In Appendix J, we provide further clarifications about the differences between the settings considered in the previous works and ours.
Conclusions -We established sampling lower bounds imposed on the general quantum error-mitigation protocols.Our results formalize the idea that the reduction in the state distinguishability caused by noise and error-mitigation processes leads to the unavoidable computational overhead in quantum error mitigation.We then showed that error-mitigation protocols with certain intermediate operations and an arbitrary trailing process require the number of samples that grows exponentially with the circuit depth to mitigate various types of noise.We presented these bounds with respect to multiple performance quantifiers -accuracy and success probability, as well as the standard deviation and bias -each of which has its own operational relevance.
Our bounds provide fundamental limitations that universally apply to general mitigation protocols, clarifying the underlying

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In the case of  ≪ 1 and  ≪ 1, the lower bound approximately becomes log 1  4 /(4 E,O ).This bound has the advantage of having explicit dependence on the accuracy and separating the contraction coefficient of the effective noise channel, making explicit the role of the reduction in the state distinguishability.
Proof.Let  E,O,S be the generalized contraction coefficient (c.f., Refs.[58][59][60]) for E defined by Proof.Let   (, ) √︁ 1 −  (, ) be the purified distance [61].For an observable , let   () √︁ Tr[(  − Tr[ ]) 2 ] be the standard deviation of the probability distribution for measuring the observable  for state .Then, as an improvement of a relation reported in Ref. [62], it was shown in Ref. [63] that arbitrary states ,  and an observable  satisfy Using the data-processing inequality and applying (C1) to the error-mitigated classical states, we get for arbitrary ,  ∈ S that where  O is the observable-dependent distinguishability defined in (2).
The bound (C6) can be turned into a lower bound on .To see this, note that (C6) ensures that for arbitrary ,  ∈ S, we have which leads to Noting that where we used the multiplicativity of the fidelity for tensor-product states.This gives for ,  ∈ S such that  O (, ) − 2 max ≥ 0. Taking the inverse and logarithm on both sides, we get we reach

𝑀-qubit GHZ states GHZ
We also assume that .

(E1)
Note that where we used .

(E4)
This gives Focusing on the case when  = 1 2 −  with  ≪ 1, the scaling of the lower bound in (E1) with respect to the noise strength is given by log We now show that this scaling can be achieved by probabilistic error cancellation.The probabilistic error cancellation method is the strategy to mitigate the effect of noise channel E by simulating the action of the inverse noise channel E −1 .The simulation can be accomplished by considering a linear decomposition of the inverse channel E −1 =    B  where   is a (possibly negative) real number and B  is a physical operation that can be implemented on the given device.One then applies the operation B  with probability |  |/ with   |  |, makes a measurement, and multiplies sgn(  ) to the outcome, where sgn(  ) is the sign of   that takes +1 if   ≥ 0 and −1 if   < 0. This constructs a distribution whose expected value coincides with the ideal expectation value with the standard deviation scaled by the factor of .Therefore, by taking the average over  samples from this distribution, one can construct an estimate of the ideal expectation value, which obeys the distribution with the standard deviation ∼ / √ .In general, one can also consider optimizing  over the choices of implementable operations {B  }  .Such optimal cost for mitigating the local dephasing noise is characterized as [39] This implies that the sample number  that achieves some fixed standard deviation becomes  ∝ 2 −  with  ≪ 1, the sample scales as  ∼ 1/(2) 2 , realizing the same scaling in (E6).

Appendix F: Proof of Theorem 3
Proof.Let D  () (1 − )  + I/2 be the single-qubit depolarizing noise.Let also U 1 , . . ., U  be the unitary channels followed by a local depolarizing channel with the noise strength at least  and suppose the  th layer U  in the  th noisy circuit is followed by the local depolarizing channel ⊗  =1 D  ,, .By assumption, we have  ,, ≥  ∀, , .Also, let Λ , and Ξ , be the unital channels applied before and after the  th layer   in the  th noisy layered circuit.
We first note that every D  ,, can be written as D  ,, − • D  .Since the second local depolarizing channel ⊗  =1 D  ,, − is unital, it can be absorbed in the following unital channel Ξ , , allowing us to focus on D ⊗   as the noise after each layer.
Then, the effective noise channel for the  th layered circuit can be written as With a little abuse of notation, we write this as where we let the product sign refer to the concatenation of quantum channels.
Let  in and  in be some input states and Then, the unitary invariance and the triangle inequality of the trace distance imply that where we defined where in the second line we used the additivity of the relative entropy ( We now recall the result in Ref. [59], showing that for arbitrary -qubit state .This implies that for arbitrary state -qubit state , noise strength , unitary U, and unital channels Ξ, Λ, Theorem 3 in the main text shows the exponential sampling overhead for noisy layered circuits with a fixed target accuracy and success probability.Here, we present a similar exponential growth of the required sample overhead for a fixed target bias and standard deviation. Theorem S.2.Suppose that an error-mitigation strategy described above is applied to an -qubit circuit to mitigate local depolarizing channels with strength at least  that follow  layers of unitaries.Then, if the estimator of the error mitigation has the maximum standard deviation  QEM max and maximum bias  max , the required number  of samples is lower bounded as Proof.We first note that   (, ) ≤ √︁ (∥) for arbitrary states ,  [64].Also, the purified distance   satisfies the triangle inequality [61].Therefore, we can repeat the argument in (F2) and (F4) with the purified distance to get Let us investigate alternative bounds that also diverge with the vanishing failure probability  → 0. We first note the inequality [64] (∥) ≥ log  (, ) −1 (H1) that hold for arbitrary states  and .Together with Theorem 1, our goal is to upper bound (E  () ∥E  ()) for the effective noise channel defined in (F1).
To do this, we employ the continuity bounds of the relative entropy [65] where  is the dimension of the Hilbert space that  and  act on.Note that (H2) has the square dependence on the trace distance, while (H3) has the log dependence on the minimum eigenvalue.probability of obtaining 0 (and 1) cannot be away from 1/2 by a constant amount.This implies that to realize the "useful" computation -which should not be simulable by a random guess -the noisy quantum circuit should be restricted to the one with logarithmic depth.The authors showed this result by employing the fact -which was also shown in the same manuscript -that the relative entropy (∥I/2  ) =  − () decreases exponentially with circuit depth.They combined this with the fact that the computational-basis measurement (followed by partial trace) does not increase the relative entropy, showing that the entropy of the probability distribution of the measurement outcome of the first qubit can only be deviated from the maximal value by the amount scaling as 2 − .Therefore, to ensure the probability of obtaining 0 or 1 to be away from 1/2 by a constant amount, one needs to have  = O (log ) to allow the entropy to be away from the maximal value with a large  limit.
Although the above consideration contains important insights, and our results -as well as many other follow-up works after this such as Refs.[40][41][42][43]59] -benefit from their inspirations, the argument there is not directly applicable to the setting involving quantum error mitigation.As explained in the main text, the idea of quantum error mitigation is to use quantum and classical resources in hybrid, in which postprocessing computation after the noisy circuit is essential.In our framework, this part is included in the trailing process (P  in Fig. 1 in the main text) -indeed, our model allows an arbitrary quantum operation as the trailing process, which also includes classical postprocessing computation.Importantly, the trailing process is not unital in general, and therefore, entropy can decrease in the postprocessing step, preventing the results in Ref. [10] from being directly carried over.
We solve this problem by first reducing the performance of error mitigation to the distinguishability of two noisy quantum states, which eventually leads to Theorems 1 and 2 providing the necessary sampling cost to achieve the target error mitigation performance with respect to entropic measures.It is a priori not obvious how entropic quantity is quantitatively related to the operational performance quantifiers such as accuracy-probability and bias-standard deviation of quantum error mitigation.Theorems 1 and 2 establish the connections between these quantities, which then allow us to focus on analyzing the entropic measures, where we can apply the previous findings, such as the exponential decay of relative entropy from the maximally mixed state under local depolarizing noise, to show the exponential sampling overhead for the general error mitigation.
In addition, our quantitative bounds -which give nearly tight estimates for certain cases -provide concrete sampling lower bounds beyond just the exponential scaling behavior for layered circuits, which could also be used as a benchmark for error mitigation strategies.
Let us also remark on how our results are related to the prior works that investigated the capability of general error mitigation, which involves postprocessing operations.Refs.[39,40] studied the maximum estimator spread, i.e., the range of outcomes of the estimator, imposed on general error mitigation protocols.These works showed that, for error mitigation with linear postprocessing, the maximum estimator spread must grow exponentially with the circuit depth for noisy layered circuits under local depolarizing noise.Although these results hinted that the exponential samples would be required in general error mitigation, they were not conclusive because the maximum estimator spread provides only a sufficient number of samples to ensure a certain accuracy.Our results close this gap by showing that exponential samples are necessary.Ref. [43] studied the general framework introduced in Ref. [39] and obtained a concentration bound, which places an upper bound for the probability of getting an estimate away from the value for the maximally mixed state by a certain amount.Their bound implies that, when the circuit depth is Ω(log ), the probability of achieving a certain constant accuracy will become exponentially small with respect to the accuracy multiplied by the Lipschitz constant of the estimator.However, as the authors pointed out, the Lipschitz constant of the estimator becomes exponentially large for many error mitigation protocols, which severely restricts the applicability of their bound.Also, their bound does not directly provide a sampling lower bound to achieve a certain accuracy.Our results, on the other hand, encompass the standard estimators with exponentially large Lipschitz constants and provide the first explicit sampling lower bounds that apply to general error mitigation protocols.
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⇠FIG. 1 .
FIG. 1. Framework of quantum error mitigation.For an ideal state  ∈ S and an observable  ∈ O of interest, we first prepare  copies of distorted states {E  ()}  =1, where E = {E  }  =1 is the set of effective noise channels.A trailing quantum process P  is then applied to  distorted states, producing the final estimation of Tr( ) represented by a random variable Ê  ().We quantify the error-mitigation performance in two ways by studying the property of the distribution of Ê  (); the first is the combination of the accuracy  and the success probability 1 − , and the second is the combination of the bias   ()Ê  () − Tr( ) and the standard deviation

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a z a X c m / / b 9 R y C < / l a t e x i t > U l a t e x i t s h a 1 _ b a s e 6 4 = " g J z / N P y u G V 5 7 / w I r P u t g x t 4 x I P 0 = " > A A A I W n i c d d V Z T 9 s w H A B w w w 5 K 2 Q H b 3 v a S r Z q 0 p 6 p l b A z x w k 1 h l H a 9 K D Q V s l 2 n R O S o H H c T M v 4 I e 9 0 + 2 6 R 9 m D k 9 F D d / a s m J 5 Z / / i a / E Z O C 5 k S g U / i 4 s P n r 8 5 O l S Z j m 7 8 u z 5 i 5 e r a 6 9 a U T j k l D V p 6 I W 8 T X D E P D d g T e E K j 7 U H n G G f e O y C 3 O 7 H f v G D 8 c g N g 4 a 4 G 7 C u j / u B 6 7 g y 1 6 g z m m Q k / P z F s I x 7 d M z I x D x u 8 T Y e h F K q u T H b C f N P R 9 H P S k r d v q C N V Z 7 8 p 7 m S s q m + O g 7 z H 7 n e 2 N C j K 3 r u 7 V b E z P V 5 1 i d x o b R 8 U 5 1 S g i m C t p b 9 t 8 q J + i 51 j J r f x n f b O 5 2 7 8 R 9 / Z 2 K s C L d L f 1 j T n C H t / y q R Z 8 3 G I U b y u V H o q / q z E e K s W e 3 E 3 3 x 9 8 3 d B / o o a G H Q M u G l o G e G 3 o O t G l o E 2 j L 0 B b Q h q E N o A e G H g A 9 M f Q E 6 K m h p 0 C P D D 0 C e m b o W V q 5 F y / R e B O N t 1 A 6 / N g I P w Y P L x l a A l o x t A J 0 z 9 A 9 o N 8 M / Q a 0 a m g V a M 3 Q G t C 6 o X W g b U P b Q C 8 N v Q R 6 Z e g V U D J d J 0 I e W C Z y k i j c A K S U K J x n U k 4 U b n h y n i j c 8 K S S K F w j U k s U z i S p J w p n k j Q S h Z 8 D u U o U z p U z / V o c / Y N 8 4 H N x j k y H s + n U T I 9 7 n r U P m D 5 C O C v r 2 s q A c S x C L m 2 h f 3 s 6 z 9 G G 1 s Z c d X t K 6 j x H m Z J s n k X 9 Q M n 4 M q 9 X o f5 1 x p c 5 T s N A P 3 5 0 n d M C O 4 6 S 8 S W r D 9 3 p y W r N L 7 T W 8 8 U v + Y 3 v G 7 m d T 5 P j N 4 P e o v f o I y q i T b S D S q i K m o i i P v q F f q M / S / 8 y i 5 n l z M q 4 6 e L C J O Y 1 m k m Z N / 8 B s s A c n Q = = < / l a t e x i t > UL < l a t e x i t s h a 1 _ b a s e 6 4 = " u v e d o H J x v B y J t N 4 M M e T E B s V 8 g B o = " > A A A J s n i c f d b b b t s 2 G A B g 9 b A t 9 k 5 p d 7 k b d U a B X Q S G 7 D l d g g 5 D 2 6 R p j O Z c O 0 l j G Q F J U Q 4 R n U D S b g J G b 9 K n 2 e 3 2 A n u b U Z I V 0 f r d C R B E 8 / t / i v p F S 8 J J w I R 0 n H 8 f P H z 0 + K u v v 1 l p N L / 9 7 v s f f l x 9 8 v R U x F N O 6 J D E Q c z P M R I 0 Y B E d S i Y D e p 5 w i k I c 0 D N 8 v Z X 5 2 Y x y w e J o I G 8 T O g 7 R J G I + I 0 j q r s v V F 8 r N B x n x C R 4 r p 7 2 5 6 f R 6 G 2 t O e 9 3 p d n v r u u H 8 1 t 3 Y 6 K T u n h 7 U Q 5 c q W u u k 6 e V q q 4 y 1 y 1 i 7 j L U 7 b S f f W t Z 8 O 7 p 8 s u K 5 X k y m I Y 0 k C Z A Q o 4 6 T y L F C X D I S 0 L T p T g V N E L l G E z r S z Q i F V I x V P r f U f q 5 7 P N u P u d 4 j a e e 9 Z o Z C o R C 3 I d a R I Z J X o m 5 Z 5 z I b T a W / M V Y s S q a S R q Q 4 k T 8 N b B n b W b V s j 3 F K Z H C r G 4 h w p u d q k y v E E Z G 6 p g t n w R x d U 7 l 4 H d 6 M J W J + J T f F p T S f u 9 t U F 4 D T f T 2 Z w 4 R y J G O u X M l T J b / M A 8 2 D n C P 6 i c R h i C K v 6 F U u D 2 3 d W D i z 0 l H y R t I b u Z a 3 8 u t v N h e S 9 W z 1 n N N R d 6 z u V E v f Y I 6 i S U D d Z 2 6 Q N 1 S r m 9 6 l i z l e m I 4 6 4 z I 3 y 8 r 2 W p D A S M / L f e n y q R 5 F 3 + V U b b b X 9 c H l b H I l 7 9 y X t Y R A 6 M L p A / W l W x z a t Q h e R O T 5 b v 1 8 f 5 S 5 x b y 1 c m r 6 n / e Z f B 6 w 7 h q 6 C 3 T P 0 L 2 6 8 i A r d b G M 7 m / G Q v o 7 I / 0 d G H z f 0 H 2 g R 4 Y e A T 0 0 9 B D o G 0 P f A H 1 v 6 H u g H w z 9 A P T Y 0 G O g p 4 a e A j 0 z 9 A z o u a H n Q D8 a + h H o h a E X Q H G 5 p j F e s q T x d q V w a e G d S u H i w f 1 K 4 c L D + 5 X C + 4 s P K o V / F n x Y K b y / + K T S E 6 i D S u F f C V 9 U C m v l l 9 X w 9 Q N + S T 3 8 H d N h R X z f d B 9 6 3 3 R Y M / / E d H h t o n x 6 C c K X P L 1 E 3 2 A 4 u h g Y D E q T a C k e + J h O W K S y 3 5 z d p H a r Y 7 t U R 5 Q d t b x Z L W 9 W z 5 v d 5 z X / / 3 2 3 X J m X K r 0 3 9 e s w f 3 W r Y 8 I 4 m T K Z f Z e U H x / 2 l x u n 3 X b n R b t 3 3 G u 9 6 s 2 / U F a s n 6 1 f r F + t j v W 7 9 c r a t Y 6 s o U W s z 9 Z f 1 t / W P 4 1 e 4 6 K B G q Q I f f h g n v O T t b A 1 g v 8 A f x a i n w = = < / l a t e x i t > ⇤ n,1< l a t e x i t s h a 1 _ b a s e 6 4 = " 6 F I n i 0 0 v B e v Y 4 a T x M 8 4 U m 5 O m L 7 8 = " > A A A J s n i c f d b b b t s 2 G A B g 9 b A t 9 k 5 p d 7 k b d U a B X Q S G 7 D l d g g 5 D 2 6 R p j O Z c O 0 l j G Q F J U Q 4 R n U D S b g J G b 9 K n 2 e 3 2 A n u b U Z I V 0 f r d C R B E 8 / t / i v p F S 8 J J w I R 0 n H 8 f P H z 0 + K u v v 1 l p N L / 9 7 v s f f l x 9 8 v R U x F N O 6 J D E Q c z P M R I 0 Y B E d S i Y D e p 5 w i k I c 0 D N 8 v Z X 5 2 Y x y w e J o I G 8 T O g 7 R J G I + I 0 j q r s v V F 8 r N B x n x C R 4 r p 7 2 5 6 f R 6 G 2 t O e 9 3 p d n v r u u H 8 1 t 3 Y 6 K T u n h 7 U Q 5 c q W u u m 6 e V q q 4 y 1 y 1 i 7 j L U 7 b S f f W t Z 8 O 7 p 8 s u K 5 X k y m I Y 0 k C Z A Q o 4 6 T y L F C X D I S 0 L T p T g V N E L l G E z r S z Q i F V I x V P r f U f q 5 7 P N u P u d 4 j a e e 9 Z o Z C o R C 3 I d a R I Z J X o m 5 Z 5 z I b T a W / M V Y s S q a S R q Q 4 k T 8 N b B n b W b V s j 3 F K Z H C r G 4 h w p u d q k y v E E Z G 6 p g t n w R x d U 7 l 4 H d 6 M J W J + J T f F p T S f u 9 t U F 4 D T f T 2 Z w 4 R y J G O u X M l T J b / M A 8 2 D n C P 6 i c R h i C K v 6 F U u D 2 3 d W D i z 0 l H y R t I b u Z a 3 8 u t v N h e S 9 W z 1 n N N R d 6 z u V E v f Y I 6 i S U D d Z 2 6 Q N 1 S r m 9 6 l i z l e m I 4 6 4 z I 3 y 8 r 2 W p D A S M / L f e n y q R 5 F 3 + V U b b b X 9 c H l b H I l 7 9 y X t Y R A 6 M L p A / W l W x z a t Q h e R O T 5 b v 1 8 f 5 S 5 x b y 1 c m r 6 n / e Z f B 6 w 7 h q 6 C 3 T P 0 L 2 6 8 i A r d b G M 7 m / G Q v o 7 I / 0 d G H z f 0 H 2 g R 4 Y e A T 0 0 9 B D o G 0 P f A H 1 v 6 H u g H w z 9 A P T Y 0 G O g p 4 a e A j 0 z 9 A z o u a H n Q D 8 a + h H o h a E X Q H G 5 p j F e s q T x d q V w a e G d S u H i w f 1 K 4 c L D + 5 X C + 4 s P K o V / F n x Y K b y / + K T S E 6 i D S u F f C V 9 U C m v l l 9 X w 9 Q N + S T 3 8 H d N h R X z f d B 9 6 3 3 R Y M / / E d H h t o n x 6 C c K X P L 1 E 3 2 A 4 u h g Y D E q T a C k e + J h O W K S y 3 5 z d p H a r Y 7 t U R 5 Q d t b x Z L W 9 W z 5 v d 5 z X / / 3 2 3 X J m X K r 0 3 9 e s w f 3 W r Y 8 I 4 m T K Z f Z e U H x / 2 l x u n 3 X b n R b t 3 3 G u 9 6 s 2 / U F a s n 6 1 f r F + t j v W 7 9 c r a t Y 6 s o U W s z 9 Z f 1 t / W P 4 1 e 4 6 K B G q Q I f f h g n v O T t b A 1 g v 8 A i C + i o A = = < / l a t e x i t > ⇤ n,2 < l a t e x i t s h a 1 _ b a s e 6 4 = " Q 1 r I L y D K / o T y U 6 m N W 2 v L V A g F q / o = " > A A A J r n i c f d b d b t s 2 F A B g t e u 2 2 P t L t 8 v e a D M K 7 C I w Z N f Z E n Q o 2 i Z N Y 7 T 5 r e 2 4 s Q y D p C m H i P 5 A 0 l 4 C R u / R p 9 n t 9 g p 7 m 1 G S F d E 6 7 g Q I o v m d Q 1 F H t C Q c + 0 x I x / n 3 w c M v H n 3 5 1 d c b t f o 3 3 3 7 3 / Q + b j 3 8 c i G j O C e 2 T y I / 4 E C N B f R b S v m T S p 8 O Y U x R g n 1 7 g 6 7 3 U L x a U C x a F P X k b 0 3 G A Z i H z G E F S d 0 0 2 2 8 r N B h n x G R 4 r p 7 m 7 6 3 Q 6 O 1 t O c 9 t p t z v b u u E 8 a + / s t B J 3 y CY q 3 G o n y W S z U c T Z R Z x d x N m t p p N t D W u 5 n U 4 e b 0 z d a U T m A Q 0 l 8 Z E Q o 5 Y T y 7 F C X D L i 0 6 T u z g W N E b l G M z r S z R A F V I x V N q / E f q p 7 p r Y X c b 2 H 0 s 5 6 z Q y F A i F u A 6 w j A y S v R N X S z n U 2 m k t v Z 6 x Y G M 8 l D U l + I m / u 2 z K y 0 0 r Z U 8 Y p k f 6 t b i D C m Z 6 r T a 4 Q R 0 T q e q 6 c B X N 0 T e X q d U w X L B b L K 7 n J L 6 X + 1 N 2 n u g C cH u n J n M S U I x l x 5 U q e K P l 5 7 m n u Z R z S P 0 k U B C i c 5 r 3 K 5 Y G t G y t n V j p K 3 k h 6 I 7 e y V n b 9 9 f p K s p 6 t n n M y a o / V n W r o m 8 t R O P O p + 7 P r Z w 3 V a C d 3 y W r O N E h G r X G R m 2 a l e y V I Y K T n 5 T 5 3 + V y P o u 9 y o n a b 2 / r g c j a 7 k n f u 8 0 q C L 3 T h 9 I F 6 0 s 0 P z U o E z y O y f L d 6 v j + K 3 H z e W j k 1 / c V 9 J l 8 G r O Y H r 3 R A W i G C f P W q O n q w Z + g e 0 D e G v g F 6 b O g x 0 L 6 h f a A 9 Q 3 t A 9 w 3 d B 9 o 1 t A v 0 w N A D o I e G H g J 9 b + j 7 q n I / L X W + j O 5 v x k r 6 W y P 9 L R j 8 y N A j o K e G n g I 9 M f Q E 6 G t D X w N 9 Z + g 7 o B 8 M / Q D 0 z N A z o A N D B 0 A v D L 0 A O j R 0 C P S j o R + B X h p 6 C R Q X a x r j N U s a 7 5 c K l x Y + K B U u H t w t F S 4 8 f F Q q v L / 4 u F T 4 Z 8 E n p c L 7 i 8 9 L P Y f a K x X + l f B l q b B W X l E N T z / g 1 9 T D O z A d V s T z T P e g d 0 2 H N f P O T Y f X J o q n l y B 8 z d N L d A 2 G o 4 u e w a A 0 s Z b 8 g Y / p j I U q / c 3 Z T W I 3 W r Z L d U T R U c l b V P I W 1 b z F f V 7 9 / 9 9 3 6 5 V N E 6 X 3 u n 4 d Z q 9 u d U Y Y J 3 M m 0 + + S 4 u P D / n x j 0 G 6 2 f m t 2 z j q N l 5 3 l F 8 q G 9 c T 6 x f r V a l m / W y + t Q + v U 6 l v E + m T 9 Z f 1 t / V N z a o P a u D b J Q x 8 + W O b 8 Z K 1 s t a v / A P P n o P g = < / l a t e x i t > ⌅ n,2 < l a t e x i t s h a 1 _ b a s e 6 4 = " C 6 o g + o T M 3 c w G B s C G y x F k Z M f V E p Q = " > A A A J r n i c f d b b b t s 2 G A B g t d 0 h 9 k 5 p d 7 k b b U a B X Q S G 7 D p b g g 5 D 2 6 R p j O Z c 2 3 F j G Q Z J U w 4 R n U D S X g J G 7 7 G n 2 e 3 2 C n u b U Z I V 0 f r d C R B E 8 / t / i v p F S 8 K x z 4 R 0 n H 8 f P X 7 y 2 e d f f L l R q 3 / 1 9 T f f f r f 5 9 N l A R H N O a J 9 E f s S H G A n q s 5 D 2 J Z M + H c a c o g D 7 9 B L f 7 K V + u a B c s C j s y b u Y j g M 0 C 5 n H C J K 6 a 7 L Z V m 4 2 y I j P 8 F g 5 z d 1 d p 9 P Z 2 X K a 2 0 6 7 3 d n W D e d F e 2 e n l b h D N l H h 1 l G S T D Y b R Z x d x N l F n N 1 q O t n W s J b b 2 e T p x t S d R m Q e 0 F A S H w k x a j m x H C v E J S M + T e r u X N A Y k R s 0 o y P d D F F A x V h l 8 0 r s 5 7 p n a n s R 1 3 s o 7 a z X z F A o E O I u w D o y Q P J a V C 3 t X G e j u f R 2 x o q F 8 V z S k O Q n 8 u a + L S M 7 r Z Q 9 Z Z w S 6 d / p B i K c 6 b n a 5 B p x R K S u 5 8 p Z M E c 3 V K 5 e x 3 T B Y r G 8 k t v 8 U u r P 3 X 2 q C 8 D p s Z 7 M a U w 5 k h F X r u S J k p / m n u Z e x i H 9 g 0 R B g M J p 3 q t c H t i 6 s X J m p a P k r a S 3 c i t r Z d d f r 6 8 k 6 9 n q O S e j 9 l j d q 4 a + u R y F M 5 + 6 P 7 p + 1 l C N d n K f r O Z M g 2 T U G h e 5 a V a 6 V 4 I E R n p e 7 k u X z / U o + i 4 n a r e 5 r Q 8 u Z 7 N r e e + + r C T 4 Q h d O H 6 g n 3 f z Q r E T w P C L L d 6 v n + 6 3 I z e e t l V P T f 3 / I 5 M u A 1 f z g t Q 5 I K 0 S Q r 1 5 X R w / 2 D N 0 D + t b Q t 0 B P D D 0 B 2 j e 0 D 7 R n a A / o v q H 7 Q L u G d o E e G H o A 9 N D Q Q 6 B H h h 5 V l f t p q f N l 9 H A z V t L f G e n v w O D H h h 4 D P T P 0 D O i p o a d A 3 x j 6 B u h 7 Q 9 8 D / W D o B 6 D n h p 4 D H R g 6 A H p p 6 C X Q o a F D o B 8 N / Q j 0 y t A r o L h Y 0 x i v W d J 4 v 1 S 4 t P B B q X D x 4 G 6 p c O H h 4 1 L h / c U n p c I / C z 4 t F d 5 f f F H q B d R e q f C v h K 9 K h b X y i m p 4 + g G / p h 7 e g e m w I p 5 n u g e 9 a z q s m X d h O r w 2 U T y 9 B O F r n l 6 i a z A c X f Q M B q W J t e Q P f E x n L F T p b 8 5 u E 7 v R s l 2 q I 4 q O S t 6 i k r e o 5 i 0 e 8 u r / / 7 5 b r 2 y a K L 3 X 9 e s w e 3 W r c 8 I 4 m T O Z f p c U H x / 2 p x u D d r P 1 S 7 N z 3 m m 8 6 i y / U D a s H 6 y f r J + t l v W r 9 c o 6 t M 6 s v k W s P 6 2 / r L + t f 2 p O b V A b 1 y Z 5 6 O N H y 5 z v r Z W t d v 0 f 4 I C h E g = = < / l a t e x i t > ⌅ n, L < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 e i B A A 9 2 H y m W D V b j w x e m Q Z Y h i g P 9 o t 5 7 u G v p L t B X l r 4 C e m j p I d C h p U O g A 0 s H Q H u W 9 o D 2 L e 0 D 3 b N 0 D + i + p f t A 3 1 r 6 t q 4 i y F J d b K P 7 x V g J f 2 2 F v w a d H 1 h 6 A P T Y 0 m O g R 5 Y e A X 1 p 6 U u g b y x 9 A / S d p e + A n l h 6 A v T M 0 j O g 5 5 a e A 7 2 w 9 A L o e 0 v f A 7 2 0 9 B I o K f c 0 I W u 2 N O l V C r c W 2 a s U b h 7 S r x R u P H J Q K V x f c l g p / L O Q o 0 r h + p L T S k + h D i q F f y V y W S n M l V 9 m w z c P + D X 5 8 P d s h x n x f d t 9 6 H 3 b Y c 7 8 U 9 v h 3 G T 5 9 J J U r H l 6 y b 7 F s H c 5 s B i k J j F S P P A J m / F I Z 7 8 F v 0 n d l u c i Z l q U F b W 4 R S 1 u U Y 9 b 3 M c 1 / / 9 9 t 1 7 5 N N X m b J r X Y f 7 q 1 i e U C z r n K v s u K T 8 + 3 A 8 X z r p t b 7 v 9 5 O R J 6 3 l 3 + Y W y 4 X z n / O D 8 6 H j O T 8 5 z Z 9 8 5 d o Y O d f 5 0 / n L + d v 5 p 7 D R w 4 6 r x e 9 H 0 4 Y N l z L f O y t G Q / w H j 9 a U O < / l a t e x i t > D n,1,1 r L 5 q 0 Z 5 J y M x I 2 P d a X f y Y 6 s o e F 1 T e N r x n m 1 v p y j s T T S a 4 T D E E x 1 t e V v d N E 0 n m 6 0 y x C 1 D 3 D L E 9 Z b U c p b H 8 e T R x h R N Y z o P W a R o g K U c e Z 1 E j T U W i t O A p U 0 0 l y z B 9 B r P 2 M g U I x w y O d b 5 E F P 3 s a m Z u n 4 s z B k p N 6 + 1 2 E b 3 i 7 E S / t o K f w 0 6 P 7 D 0 A O i x p c d A j y w 9 A v r S 0 p d A 3 1 j 6 B u g 7 S 9 8 B P b H 0 B O i Z p W d A z y 0 9 B 3 p h 6 Q X Q 9 5 a + B 3 p p 6 S V Q U u 5 p Q t Z s a d K r F G 4 t s l c p 3 D y k X y n c e O S g U r i + 5 L B S + G c h R 5 X C 9 S W n l Z 5 C H V Q K / 0 r k s l K Y K 7 / M h m 8 e 8 G v y 4 e / Z D j P i + 7 b 7 0 P u 2 w 5 z 5 p 7 b D u c n y 6 S W p W P P 0 k n 2 L Y e 9 y Y D F I T W K k e O A T N u O R z n 4 L f p O 6 L c 9 F z L Q o K 2 p x i 1 r c o h 6 3 u I 9 r / v / 7 b r 3 y a a r N 2 T S v w / z V r U 8 o F 3 T O V f Z d U n 5 8 u B 8 u n H X b 3 n b 7 y c m T 1 v P u 8 g t l w / n O + c H 5 0 f G c n 5 z n z r 5 z 7 A w d 6 v z p / O X 8 7 f z T 2 G n g x l X j 9 6 L p w w f L m G + d l a M h / w P t D 6 U P < / l a t e x i t > D n,1,2 < l a t e x i t s h a 1 _ b a s e 6 4 = " O / N w l B / r g u l D c m s R / 9 W y T o j h I N E = " > A A A J u X i c f d b b b t s 2 G A B g t e s h 9 n p I t 8 v e a D U K 7 C I w L j e I o t H 1 f u 9 L G L z c W v l z P R X N 5 F 8 3 m A x P n i j G 6 Q Z o t h X b 6 q 9 B 5 u G b g J 9 Z + g 7 o H u G 7 g H t G 9 o H 2 j O 0 B 7 R j a A d o 1 9 A u 0C 1 D t 4 B u G 7 o N d M f Q n a p y P 0 1 1 v o 1 u F m M h / L 0 R / h 5 0 v m v o L t A D Q w + A 7 h u 6 D / S t o W + B f j D 0 A 9 C P h n 4 E e m j o I d B j Q 4 + B n h h 6 A v T U 0 F O g n w z 9 B P T M 0 D O g p N j T h C z Z 0 q R T K t x a Z K t U u H l I t 1 S 4 8 c h u q X B 9 y V 6 p 8 M 9 C 9 k u F 6 0 u O S j 2 C 2 i s V / p X I W a k w V 2 6 R D V c / 4 J f k w 9 0 y H W b E d U 1 3 o X d N h z l z j 0 y H c x P F 0 0 t Q v u T p J b o G w 9 5 F z 2 C Q m l h L / s A n b O K F K v 3 N v c v E b j g 2 Y r p F U V G J m 1 X i Z t W 4 2 U 1 c / f / f d 8 v V G y d K n 3 X 9 Os x e 3 e q Q e p x O P Z l + l x Q f H / b 3 C 8 f t p r P e f H L 4 p P G 6 P f 9 C W b E e W H 9 a j y 3 H e m a 9 t r a t A 6 t v U e t v 6 x / r X + u / 2 k Y N 1 8 5 r n / O m t 2 / N Y / 6 w F o 6 a + A b Z + K U p < / l a t e x i t > D n, L,1 < l a t e x i t s h a 1 _ b a s e 6 4 = " P Y w h r S W O M w i g o z Z e Z U Q J 2 c i a u z Y = " > A A A J u X i c f d b b c t M 4 G A B g A 3 t o w h 4 K e 8 m N 2 Q w z X H Q y c Q Y K T F k G a C j N 0 D N J W x p l M p I i p 6 Y + j a S E d l Q / z T 7 N 3 u 5 e 8 T b I d l w r / s N 6 x m N F n 3 5 Z + q X Y J r H v C d l q f b 1 1 + 8 4 P P / 7 0 8 0 qt f v e X X 3 / 7 f f X e / W M R T T l l f R r 5 E T 8 l W D D f C 1 l f e t J n p z F n O C A + O y E X m 6 m f z B g X X h T 2 5 F X M h g G e h J 7 r U S x 1 1 W j 1 L 4 W y T g Z 8 Q o a q 1 W x l x 1 p e c N q 6 8 L T l v F h f T 1 D Q G S k 0 w U G A R y p c 2 1 l r J 0 k y W m 0 U I X Y R Y h c h t j O n h j U / D k b 3 V s Z o H N F p w E J J f S z E w G n F c q g w l x 7 1 W V J H U 8 F i T C / w h A 1 0 M c Q B E 0 O V D T G x H + m a s e 1 G X J + h t L N a M 0 L h Q I i r g O i W A Z b n o m p p 5 T I b T K X 7 f K i 8 M J 5 K F t L 8 R u 7 U t 2 V k p 0 m z x x 5 n V P p X u o A p 9 / R Y b X q O O a Z Sp 3 b h L o T j C y Y X 5 z G e e b G Y z + Q y n 0 r 9 E e o w n Q D O d v V g 9 m P G s Y y 4 Q p I n S n 6 f e 5 p 7 G Y f s C 4 3 0 W o T j v F Y h H t i 6 s H B n p V v J S 8 k u 5 V p W y u Z f r y 8 E 6 9 H q M S e D 9 l B d q 4 a T I I 7 D i c / Q Q FIG.2.Each distorted state ( th copy depicted in the figure) is produced by a circuit with  layers  1 , . . .,   followed by a local depolarizing noise with noise strength at least , i.e.,  ,, ≥ , ∀, , .Each layer   can be sandwiched by additional unital operations Λ , and Ξ , .Other layers and depolarizing channels with noise strength smaller than  are absorbed in these operations as they are also unital.