Efficient noise mitigation technique for quantum computing

Quantum computers have enabled solving problems beyond the current machines’ capabilities. However, this requires handling noise arising from unwanted interactions in these systems. Several protocols have been proposed to address efficient and accurate quantum noise profiling and mitigation. In this work, we propose a novel protocol that efficiently estimates the average output of a noisy quantum device to be used for quantum noise mitigation. The multi-qubit system average behavior is approximated as a special form of a Pauli Channel where Clifford gates are used to estimate the average output for circuits of different depths. The characterized Pauli channel error rates, and state preparation and measurement errors are then used to construct the outputs for different depths thereby eliminating the need for large simulations and enabling efficient mitigation. We demonstrate the efficiency of the proposed protocol on four IBM Q 5-qubit quantum devices. Our method demonstrates improved accuracy with efficient noise characterization. We report up to 88% and 69% improvement for the proposed approach compared to the unmitigated, and pure measurement error mitigation approaches, respectively.


Introduction
Building large-scale quantum computers is still a challenging task due to a plethora of engineering obstacles 1 .One prominent challenge is the intrinsic noise.In fact, implementing scalable and reliable quantum computers requires implementing quantum gates with sufficiently low error rates.There has been substantial progress in characterizing noise in a quantum system [2][3][4] and in building error correcting schemes that can detect and correct certain types of errors [5][6][7] .Numerous protocols have been constructed to characterize the noise in quantum devices.Many of these protocols fail in achieving one of the following desirables: scalability to large-scale quantum computers and efficient characterization of the noise.Quantum Process Tomography 8 is a protocol that can give a complete description of the dynamics of a quantum black box, however, it's not scalable to large-scale quantum systems.Randomized Benchmarking (RB) is another protocol that's typically used to estimate the error rate of some set of quantum gates 9,10 .Although RB is a scalable protocol in principle, it can only measure a single error rate that's used to approximate the average gate infidelity thus providing an incomplete description of noise.Various other protocols based on RB protocol are able to characterize the correlations of noise between the different qubits, however, these protocols lack scalability 9,11,12 .Quantum Error Mitigation 13 (QEM) is a recently emerging field that aims to improve the accuracy of near-term quantum computational tasks.Whereas Quantum Error Correction (QEC) 14,15 necessitates additional qubits to encode a quantum state in a multiqubit entangled state, QEM does not demand any additional quantum resources.It is considered an excellent alternative for enhancing the performance of Noisy Intermediate-Scale Quantum (NISQ) computing 16 .QEM protocols include zero-noise Richardson extrapolation of results from a collection of experiments of varying noise 17 , probabilistic error cancellation through sampling from a set of quantum circuits generated from a target circuit mixed with Pauli gates from an optimized distribution 18,19 , and exploiting state-dependent bias through invert-and-measure techniques to map the predicted state to the strongest state 20 .Measurement Error Mitigation (MEM) is another QEM protocol that models the noise in a quantum circuit as a measurement noise matrix   applied to the ideal output of the circuit.The columns of   are the 1 arXiv:2109.05136v1[quant-ph] 10 Sep 2021 probability distributions obtained through preparing and immediately measuring all possible 2  basis input states 21 .Recently, the authors in 22 developed a protocol based on the RB that relies on the concept of a Gibbs Random Field (GRF) to completely and efficiently estimate the error rates of the Pauli Channel and detect correlated errors between the qubits in a quantum computer.Their effort paves the way to enable quantum error correction and/or mitigation schemes.Herein, we refer to their efficient learning protocol as the {EL protocol}.In this paper, we build upon the EL protocol and decompose the average noise of a quantum circuit of specific depth into State Preparation and Measurement (SPAM) error and average gate error.We propose a linear algebraic based protocol and proof to efficiently construct and model the average behavior of noise in a quantum system for any desired circuit depth without having to run a large number of quantum circuits on the quantum computer or simulator.We then rely on this model to mitigate the noisy output of the quantum device.For an n-qubit quantum system, the average behavior of the noise can be well approximated as a special form of a Pauli Channel [23][24][25] .A Pauli channel  acts on a qubit state  to produce where   is an error rate associated with the Pauli operator   .The   's form a probability distribution (    = 1), and are related to the eigenvalues, , of the Pauli Channel defined as Thus, when a state  is subjected to the noisy channel ,   describes the probability of a multiqubit Pauli error   affecting the system, while   describes how faithfully a given multispin Pauli operator is transmitted. and  are related by Walsh-Hadamard transform where While RB only estimates the average value of all   of the Pauli Channel, the EL protocol estimates the individual   .A complete characterization of the Pauli channel requires learning more than the eigenvalues or error rates associated with single-qubit Pauli operators such as  (1)  or  (3)  ; it requires learning all of the noise correlations in the system, that is, also learning the eigenvalues and error rates associated with multiqubit Pauli operators such as  (1)  ⊗ 1 (2) ⊗  (3)  and how they vary compared to the ones obtained under independent local noise.Estimating these correlations is essential for performing optimal QEC and/or QEM.However, these correlations increase exponentially as the number of qubits increases, so having an efficient noise characterization protocol is crucial to direct the error mitigation efforts to capture the critical noise correlations.Our method relies on the error rates vector  of the Pauli-Channel to decompose the average behavior of noise for circuits of depth  into two noise components: a SPAM error matrix denoted by the matrix  and a depth dependent component comprising an average gate error matrix denoted by the matrix .We evaluate our model for the average noise by predicting the average probability distribution for circuits of depth  and computing the distance between this predicted distribution and the empirically obtained one.Finally, we use our proposed decomposition to mitigate noisy outputs of random circuits and compare our mitigation protocol with the MEM protocol 21 .We applied our noise characterization and mitigation protocols on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem 26 .

Proposed Protocol Theory
The ideal output probability distribution of an -qubit quantum circuit with depth  is perturbed by the SPAM and the average gate errors.Our aim is to construct a comprehensive linear algebraic model that takes into account both these errors for an arbitrary depth .Matrix algebra can then be employed to mitigate the noise as follows: where   is the characterized noise matrix for circuits of depth ,   and   are the ideal and noisy outputs of a given circuit of depth , respectively.The straight-forward approach would be to construct   from empirical simulations in a similar fashion to the   noise matrix that was characterized in the MEM scheme.The columns of   comprise the emperical average probability distributions for basis input states | ∈ {|0 , |1 , . . ., |2  − 1 }, denoted by q(, | ), where q(, | ) are obtained through sampling a number of depth  circuits to incorporate the average gate and SPAM errors.
Building   , however, through empirical simulations can be expensive especially when the circuit depth is large.Herein, we propose a method for an efficient estimation of   where the individual probability distributions q(, | ) are estimated as follows: where   and   are input-specific matrices that represent the SPAM error matrix and average gate error for input | , respectively.Both   and   are extracted empirically using random circuits from a set of small circuit depths  and then used in mitigating the outputs for circuits with higher depths.We first show the construction of  0 and  0 .
The construction of  0 and  0 proceeds by estimating the error rates vector  associated with the Pauli Channel based on the assumption in Equation 1for the average behavior of the noisy quantum device at hand using the EL protocol.The protocol proceeds by constructing  random identity circuits of depth  ∈  11,22 .Each circuit is constructed by initializing the qubits to the all-zeros state |0 followed by choosing a random sequence  ∈   , the set of all length  sequences of one-qubit Clifford gates applied independently on each qubit, followed by an inverse gate for the chosen sequence to ensure an identity circuit.It then estimates the resulting empirical probability distribution q(, |0 ) by averaging over all the empirical probability distributions q(, , |0 ) for the constructed random identity circuits of depth , that is, q(, |0 ) is a vector with 2  entries each corresponding to the possible observed outcome.A Walsh-Hadamard transform is then applied on each q(, |0 ) to obtain Each parameter Λ  () in () is fitted to the model where   is a constant that absorbs SPAM errors and the vector  of all fitted parameters   is a SPAM-free estimate to the eigenvalues of the Pauli Channel defined in Equation 2. Notice that we can rewrite Equation 9as where  is a diagonal matrix where the diagonal entries are   and   is an element-wise exponentiation of a vector.An inverse Walsh-Hadamard Transform is then applied on  to get the error rate vector  of the Pauli Channel as is then projected onto a probability simplex to ensure    = 1.Introducing the GRF model by the EL protocol allows the scalability of estimating  with the increase in the number of qubits.The GRF model assumes the noise correlations are bounded between a number of neighboring qubits depending on the architecture of the quantum computer at hand.Thus, decreasing the number of noise correlations to be estimated.The final outcome  of the EL protocol represents the SPAM-free probability distribution of the average noise in the quantum computer.Each element   ∈  corresponds to the probability of an error of the form  () on an input state |0 .For example, for a 5-qubit quantum computer,  0 corresponds to the probability of no bit flips on the input state, i.e., error of the form     ,  1 to the error of the form     ,  2 to the error of the form     , etc. . .In order to proceed with the proof for our proposed decomposition of Equation ( 6) for input state |0 , we first state the following lemma (the detailed proof of the lemma can be found Section I in the supplementary): Lemma 1 Let  and  be the respective eigenvalues and error rates of a Pauli Channel with  qubits, then   =    |0 where  is a 2  × 2  matrix such that    =   ⊕  ( ⊕  is the bitwise exclusive-OR operator).
Using Lemma 2 and Equations 8 and 10, q(, |0 ) can be estimated as The transition matrix  =  0 represents the average error per gate while the  =  −1  =  0 matrix represents the SPAM errors for an input state |0 .Notice that the average noise for depth  circuits on an input state |0 behaves as a sequence of  average noise gates  0 followed by SPAM errors  0 .
The construction of   and   for input state | proceeds similar to the procedure of constructing  0 and  0 , however, a permutation of q(, | ) is required before applying a Walsh-Hadamard transform to ensure that each element   (| ) in the input-specific error rate vector (| ) corresponds to the probability of an error of the form  () on an input state | .This permutation is done by applying an input-specific permutation matrix   on q(, | ) ∀ where     = 1 if  ⊕  =  and 0 otherwise.

Experiments
In this section, we evaluate the accuracy of the model in Equation 12in predicting the average probability output, q(, |0 ), for identity circuits of higher depths by estimating  0 and (|0 ) using only simulations of lower depths identity circuits.Denote by  (, |0 ) the predicted average probability distribution obtained using Equation 12.We select a training set of depths  = {1, 2, . . .,   } to estimate  0 and  using the EL protocol followed by the construction of the average gate error matrix  0 and SPAM error matrix  0 where  0   =   ⊕  (|0 ) and  0 =  −1  0 .A new testing set of depths  = {  + 1,   + 2, . . ., 100} is then selected where we compute the Jensen-Shannon Divergence () between q( , |0 ) and  ( , |0 ) ∀ ∈  .The  measures the similarity between the two probability distributions 27 .The lower the , the closer the two distributions are.More information about the  can be found in Section II in the supplementary.Figure 8 presents the computed  for different quantum computers while varying   .Figure 5  We rely on  (, | ) to construct and evaluate the mitigation power of   for different depths.We first select a training set of depths  = {1, 20, 40, 60, 80, 100} to estimate   and (| ) for each input state | using the EL protocol followed by the construction of   using (| ) and   =  −1   .We then estimate q(, | ) as  (, | ) for all inputs using Equation 6 in order to construct   using Equation 5. We then choose a new testing set of depths  = {10, 30, 50, 70, 90} so that   is used in mitigating the outputs for circuits of depth  ∈  where for a given identity circuit of depth  with input | and sequence  of gates, the mitigated output q(, , | )  in obtained as q(, , | )  =  −1  q(, , | ) q(, , | )  is projected onto a probability simplex to ensure a probability distribution.The  between q(, , | )  and the ideal output | is computed and then averaged over all input states and all random circuits of depth .We also compare our proposed mitigation protocol using   with the MEM scheme (Figure 6).We report upto 88% improvement in the  value for the proposed approach compared to the unmitigated approach, and upto 69% improvement compared to MEM approach.Note that for the results presented here, we rely on the average SPAM free error rate,   = 1

Complexity
So far in the estimation of   and   for each input state | using the EL protocol,  random circuits are generated for each depth 1 ≤  ≤   where the EL protocol requires  (2 2 ) for the Walsh-Hadamard transform which can be reduced into  ( 2 ) using fast Walsh-Hadamard transform.Thus, the overall complexity of the construction of   and   for all input states is  (   2 2  ).Furthermore, the GRF model factors the error rates vector into a product of  ∼  () factors, depending on the architecture of the quantum computer, where each factor depends on a subset of adjacent qubits of cardinality  <<  (typically  = 4).Thus, the complexity is reduced further into  (    2 2  ).The construction of   would be based on Equation 6for each input state | where    can be computed efficiently using the Singular Value Decomposition (SVD) of   , thus the construction of   is  (2 3 ).For the MEM scheme, the construction of   requires only generating  circuits with no gates for each input state | , thus the complexity is  (2  ).For mitigation, both protocols are based on matrix inversion, thus the complexity for mitigation is  (2 3 ).

Discussion
The proposed mitigation protocol builds upon the SPAM-free noise characterization protocols for low circuit depths to generate a SPAM-error matrix   and an average gate error matrix   for each input state | .It then constructs a noise mitigation matrix   for arbitrary circuit depths  where the columns of   are the estimated average probability distributions  (, | ) =      .The mitigated output q(, )  of a given circuit of depth  with sequence  of gates is obtained by applying   −1 on the empirical circuit output q(, ).
We evaluated the accuracy of our model in estimating the average probability distributions for high depth circuits and evaluated our mitigation protocol on the IBM Q 5-qubit quantum devices: Belem, Lima, and Manila.For the model accuracy evaluations, for the different   values, we reported on average a test  ( q(, |0 ),  (, |0 )) value around 0.022-0.028for Manilla, 0.03-0.055for Lima, and 0.028-0.048for Belem.For   = 20, the test JSD values varied between 0.005 and 0.05 for Lima computer, 0.01 and 0.06 for Manila computer, and 0.02 and 0.09 for Belem computer.We note that for   = 20 the test spans  ∈ [21 − 100].For higher depths  ∈ [80 − 100], on average   = 80 resulted in better model error than   = 50 and   = 20.Results for IBM Q Athens are presented in the supplementary.
Finally, we report upto 88% JSD improvement for the proposed approach compared to the unmitigated approach with significant mitigation improvement compared to MEM at mid to higher depths.Specifically, for  = 90, we reported 58%, 66% and 85% JSD improvement for the proposed approach compared to the unmitigated on Belem, Manilla, and Lima respectively.This is compared 12%, 17% and 51% respectively for the MEM.On average for all the test depths across the different machines, we report 68.4% JSD improvement for the proposed versus versus 38.2% for the MEM improvement compared to the unmitigated approach.

Methods
In evaluating the accuracy of the model, we run  = 1000 random identity circuits with each submission requesting 1024 shots for each depth  ∈ {1, 2, . . ., 100}.In evaluating the mitigation power of   , we run  = 1000 random identity circuits for depths  ∈ {1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100} for each basis input state | with each circuit requesting 1024 shots.The constructed circuits contain single qubit Clifford gates only.We run the circuits on the following IBM Q 5-qubit quantum computers: Manila, Lima, and Belem.Theoretical derivations and numerical details essential to the study are presented in the Results section.More details can be found in the Supplementary Information.For the configurations and noise profiles of the IBM quantum machines, please go to IBM Quantum Experience at http://www.research.ibm.com/quantum.

Other Quantum Computers
In this section, we show the results of evaluating the accuracy of the model where we include the IBM Q Athens quantum computer.The Athens quantum computer is not included in the main manuscript since it is retired by IBM.Hence, we could not include the results of the mitigation.Similar to Figure 1 in the main manuscript, Figure 1 below presents the computed  for different quantum computers while varying   with the addition of IBM Q Athens that demonstrates  values that range between 0.005 and 0.08. Figure 2    Comparing the performance with   vs   In this section, we evaluate the mitigation power of   when relying on the input specific error rate vectors (| ) to construct   for all inputs compared to relying on the average error rate   = 1 2  2  −1 =0 (| ) to construct   =   .  remains input specific.Similar to Figure 3 in the main manuscript, Figure 3 below compares the average  between the ideal outputs and each of the unmitigated outputs, the mitigated outputs by the MEM protocol, and the mitigated outputs by   using   in addition to the mitigated outputs by   using (| ). Figure 3 shows that the difference between using   and (| ) for the construction of   is negligible for the different quantum computers indicating consistent SPAM-free error rates for all inputs.Average  between the ideal output | and each of the unmitigated outputs q(, , | ), mitigated outputs by the MEM protocol, and mitigated outputs by our proposed noise model using (| ) and using   for each depth  on IBM Q 5-qubit quantum computers.

Average 𝐽𝑆𝐷 for Different Input States
In this section, we elaborate further on Figure 3 in the main manuscript and show the average  between the ideal output | and the mitigated output by our proposed protocol for each depth  and input state | (Figure 4). Figure 5 presents the average and standard deviation of  between the ideal output and mitigated output by our proposed protocol over all depths for each input state | .We notice that, for the different quantum computers, some input states demonstrate lower  values compared to other input states.The Lima quantum computer presents the best average  in the worst input state with a minimum of 0.0023 and a maximum of 0.107 followed by the Belem quantum computer showing a minimum of 0.074 and a maximum of 0.202 and the Manila quantum computer showing a minimum of 0.031 and a maximum of 0.280.Table 1 shows the average reduction in the test  using the proposed protocol over the unmitigated data is about 58.33%, 61.04%, and 85.82% for the Belem, Manila, and Lima computers, respectively, compared to a reduction of 16.67%, 27.12%, and 72.77% for the Belem, Manila, and Lima computers, respectively, using the MEM protocol.We report upto 69% improvement and on average 40% improvement compared to the MEM protocol.

2 𝑛 2 − 1 𝑖𝑛=0Figure 3 .
Figure 3. Average  between the ideal output | and each of the unmitigated output q(, , | ), mitigated output by the MEM protocol, and mitigated output by our proposed noise model for each depth  on IBM Q 5-qubit quantum computers.
presents the average and standard deviation for the test  values for the different quantum computers including IBM Q Athens that demonstrates an average test  between 0.02 and 0.04 for the different   values.

Figure 7 .
Figure 7. Average  between the ideal output | and the mitigated outputs by our proposed noise model for each depth  and each input state | on IBM Q 5-qubit quantum computers.

Figure 8 .
Figure 8.Average and standard deviation of  over all depths  between the ideal output | and the mitigated output by our proposed noise model for each input state | on IBM Q 5-qubit quantum computers.

Table 1 .
Average test  between the ideal output | and each of the unmitigated output, mitigated by the proposed protocol output, and mitigated by the MEM protocol output for the different quantum computers.m Unmitigated JSD MEM JSD Mitigated by Proposed Protcol JSD MEM improvement % Proposed improvement %