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A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors MUQING ZHENG, Lehigh University, USA and Paciic Northwest National Laboratory, USA ANG LI, Paciic Northwest National Laboratory, USA TAMÁS TERLAKY, Lehigh University, USA XIU YANG, Lehigh University, USA Various noise models have been developed in quantum computing study to describe the propagation and efect of the noise which is caused by imperfect implementation of hardware. Identifying parameters such as gate and readout error rates are critical to these models. We use a Bayesian inference approach to identify posterior distributions of these parameters, such that they can be characterized more elaborately. By characterising the device errors in this way, we can further improve the accuracy of quantum error mitigation. Experiments conducted on IBM’s quantum computing devices suggest that our approach provides better error mitigation performance than existing techniques used by the vendor. Also, our approach outperforms the standard Bayesian inference method in some scenarios. CCS Concepts: · Hardware → Quantum error correction and fault tolerance; · Mathematics of computing → Bayesian computation. Additional Key Words and Phrases: error mitigation, gate error, measurement error, Bayesian statistics 1 INTRODUCTION While quantum computing (QC) displays an exciting potential in reducing the time complexity of various problems, the noise from environment and hardware may still undermine the advantages of QC algorithms 23]. [ One of the solutions to this problem is quantum error correction (1QEC) , 9, 13[, 16, 18, 25], which utilizes redundancy to protect the information of a single “logic qubitž from errors. Two representative examples are surface code and color code due to their scalability and high error thresholds 18, 25]. An [ alternative approach to QEC is bosonic codes. In this coding scheme, the single-qubit information is encoded into a higher-dimensional system, like a harmonic oscillator. One advantage of Bosonic codes is that it provides an access to larger Hilbert space with less overhead than traditional QEC codes [9, 13, 16]. However, as described in 23[], in the “noisy intermediate-scale quantum (NISQ)ž era, the small- or medium- sized but noisy quantum computers cannot aford the cost of QEC codes because they impose a heavy overhead cost in number of qubits and number of gates. As a result, quantum error mitigation (QEM) techniques have become attractive, e.g.,4[, 7, 8, 10ś 12, 17, 19, 30, 31], since their cost is much lower than the QEC codes in terms of the circuit depth and the number of qubits. One important area in the error mitigation study is to ilter the measurement errors (or readout errors). These errors are usually modeled by multiplying a stochastic matrix with a probability vector, as such to depict the inluence of the noise on the output of QC algorithms. More precisely, the probability vector represents the desired noiseless output of a QC algorithm, the stochastic matrix describes how the noise afects this output, and the resulting vector consists of the probabilities of observing Authors’ addresses: Muqing Zheng, Lehigh University, Bethlehem, Pennsylvania, USA and Paciic Northwest National Laboratory, Richland, Washington, USA; Ang Li, Paciic Northwest National Laboratory, Richland, Washington, USA; Tamás Terlaky, Lehigh University, Bethlehem, Pennsylvania, USA; Xiu Yang, Lehigh University, 200 West Packer Avenue, Bethlehem, Pennsylvania, USA, 18015, xiy518@lehigh.edu. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor, or ailiate of the United States government. As such, the United States government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for government purposes only. © 2022 Association for Computing Machinery. 2643-6817/2022/9-ART $15.00 https://doi.org/10.1145/3563397 ACM Trans. Quantum Comput. 2 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang each possible state on the quantum device. Here, the stochastic matrix can be constructed from conditional probabilities if only classical errors are considered, or from results of tomography if non-classical errors are not signiicant 4, 7[, 11, 19]. Similarly, the study29 in ] sho [ ws the possibility to simulate bit-lip gate error in some quantum circuits in a classical manner. The goal of QEM from the algorithmic perspective is to recover the noise-free information using data from repeated experiments, which is usually achieved via statistical methods. In the existing error models, the pa- rameters, e.g., error rate of measurement or gates, are usually considered as deterministic values (possibly with conidence interval), and the goal is to ilter the error in estimating the expectation of an operator. Instead, by considering error mitigation as a stochastic inverse problem, we adopt a new Bayesian algorithm 6] to from [ construct the distributions of model parameters and use corresponding backward error models to ilter errors from the outcomes of a quantum device. Note that our framework does not rely on the speciic knowledge of the problems that quantum circuits want to answer, like 12in ], or[hardware calibration, such as 3, 26 [ ]. We aim to estimate the parameters more comprehensively for selected error models as an inverse problem while error mitigation is achieved by using the error model in a backward direction. The paper is organized as following. In Section 2, we provide the measurement error model based on independent classical measurement error and expand the gate error model in 29] to [ multiple-error scenario. In Section 3, we introduce the use of Bayesian algorithm 6]in to infer [ the distributions of parameters of measurement error and gate error models. Then, we demonstrate the creation of our error ilter on IBM’s quantum device ibmqx2 (Yorktown) and apply our ilter together with other existing error mitigation methods on measurement outcome from state tomography, an example of Grover’s search15[], an instance of Quantum Approximate Optimization Algorithm (QAOA) [15], and a 200-NOT-gate circuit in Section 4. The code is available in [32]. 2 ERROR MODELS The goals of our error modeling include estimating the inluence of bit-lip gate errors and measurement errors in the outputs of a quantum circuit without accessing any quantum device and recovering the error-free (or error-mitigated) output. Throughout this paper, we assume no state-preparation error and only focus on pure state measurements. The three error rates that we care about are as follows: (1) ϵ = the chance of having a bit-lip error in a gate; (2) ϵ = the chance of having a measurement error when measur|e0⟩; m0 (3) ϵ = the chance of having a measurement error when measur|e1⟩. m1 It is reasonable to consider ϵ , 0.5 and ϵ + ϵ , 1 in the current quantum computer3[, 4]. This assumption д m0 m1 is one of the necessary conditions for the existence of the error-mitigation solutions in our following models. 2.1 Measurement Error As is demonstrated in 7],[ classical measurement error is applicable in the device we conduct experiments on, i.e.,ibmqx2. We build measurement error model using conditional probabilities. Consider a single-qubit state α |0⟩ + β |1⟩, its distribution of the noisy measurement outcomes are 2 2 Pr(Measure 0 w/ noise ) = |α| · (1 − ϵ ) + |β| · ϵ m0 m1 2 2 Pr(Measure 1 w/ noise ) = |α| · ϵ + |β| · (1 − ϵ ), m0 m1 which is equivalent to " # ! ! 1 − ϵ ϵ Pr(Measure 0 w/o noise ) Pr(Measure 0 w/ noise ) m0 m1 = , (1) ϵ 1 − ϵ Pr(Measure 1 w/o noise ) Pr(Measure 1 w/ noise ) m0 m1 ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 3 Fig. 1. Single bit-flip error model.U is a noisy-free gate andX represents a bit-flip error with probabilityϵ . ϵ д where “w/ž stands for “withž and “w/ož stands for “without.ž Denoting ϵ and ϵ for qubit i as ϵ and ϵ , m0 m1 m0,i m1,i respectively, we can extend the matrix form in Eq. (1) to n-qubit an case Ar = r˜, (2) where " # 1 − ϵ ϵ m0,i m1,i A := , ϵ 1 − ϵ m0,i m1,i i=1 Pr(Measure 0...00 w/o noise ) * + .Pr(Measure 0...01 w/o noise )/ . / r := . / , . . / . . / Pr(Measure 1...11 w/o noise ) , - Pr(Measure 0...00 w/ noise ) * + .Pr(Measure 0...01 w/ noise )/ . / r˜ := . / , . . / . / Pr(Measure 1...11 w/ noise ) , - A ∈ [0, 1], r ∈ [0, 1], r˜ ∈ [0, 1] by introducing the independence of measurement errors across qubits . We aim to ij i i identify r, but, in practice, we only hav r˜ which e is the probability vector characterizing the observed results from repeated measurements. Note thatA is a nonnegative left stochastic matrix (i.e., each column sums to 1), so if r ≥ 0 and its entries sums tor˜1,≥ 0 and its entries also sums to 1. Ifϵ and ϵ for all i = 1, ..., n are known, the most straightforward denoising method derived from (2) Eq. m0,i m1,i −1 isr := A r˜. As ϵ + ϵ , 1 for all i = 1, ..., n, each individual 2-by-2 matrix has non-zero determinant. Thus m0,i m1,i −1 ∗ A has non-zero determinant andA exists. However, it is not guaranteed that r is a valid probability vector. An alternative is to ind a constrained approximation r := arg min ∥Ar − r˜∥ . (3) 2 n r =1,∀i ∈{1, . . .,2 } r ≥0 i i i=1 2.2 Bit-flip Gate Error For the gate error, we focus on the single bit-lip error in this work, and we adopt the error model proposed in29 [ ]. Of note, there is no direct proof29 in ] to [ validate this model. In this section, we complete the proof and also extend it to a multiple-error case. We irst consider the case when there is only one gate and qubits could have bit-lip errors (all in the sameϵrate ) after this gate, as shown in Figure 1. ACM Trans. Quantum Comput. 4 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang 2.2.1 Single Bit-flip Error.Let p : {0, 1} → [0, 1], where n is the number of qubits, be the Boolean function that represents the noise-free probability distribution of the outcome of a QC algorithm x ∈ {0, 1}and denote the basis used in a QC algorithm. The Fourier expansion of this Boolean function is s .x p(x ) = pˆ(s)(−1) , (4) s ∈{0,1} where pˆ(s) is the Fourier coeicientpof and s.x = s · x [22, p.22]. These Fourier coeicients can be i i i=1 computed from s .x p(s) = p(x )(−1) . x ∈{0,1} Let y be the erroneous version of x induced by the bit-lip error. In other woryds, is a functionxof that adds bit-lip error into the measurement outcomes. The mathematical expression y isof x with probability − ϵ1 i д y = fori = 1, ..., n. ¬x with probability ϵ i д Deinep : {0, 1} → [0, 1] to be the expected distribution function of measurement outcomes under the noise model. Then Eq. (4) implies s .y ˜ ˆ p(x ) = E [p(y)] = p(s)E [(−1) ]. x x s ∈{0,1} It is clear that n s .y s ·y i i E [(−1) ] = E (−1) x y i=1 s ·y i i = E [(−1) ] (5) i=1 f g s ·x s ·¬x i i i i = 1 − ϵ (−1) + ϵ (−1) . д д i=1 Sincex and s are binary bits, there are four possible cases: i i s ·x s ·¬x i i i i • s = 0, x = 0, then 1 − ϵ (−1) + ϵ (−1) = 1; i i д д s ·x s ·¬x i i i i • s = 0, x = 1, then 1 − ϵ (−1) + ϵ (−1) = 1; i i д д s ·x s ·¬x i i i i • s = 1, x = 0, then 1 − ϵ (−1) + ϵ (−1) = 1 − 2ϵ ; i i д д д s ·x s ·¬x i i i i • s = 1, x = 1, then 1 − ϵ (−1) + ϵ (−1) = (1 − 2ϵ ) · (−1). i i д д д To summarize, s ·x s ·¬x s s ·x i i i i i i i 1 − ϵ (−1) + ϵ (−1) = (1 − 2ϵ ) (−1) , д д д for all s ∈ {0, 1} and x ∈ {0, 1}. Consequently, continuing from Eq. (5), i i f g s .x s s ·x |s | s .x i i i E [(−1) ] = (1 − 2ϵ ) (−1) = (1 − 2ϵ ) (−1) , y д д i=1 where |s| = s . Thus, the p with only one bit-lip error is i=1 |s | s .x ˜ ˆ p(x ) = E [p(y)] = (1 − 2ϵ ) p(s)(−1) . x д s ∈{0,1} ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 5 (a) A practical multi-gate bit-flip error model (b) The converted model under commutativity condition Fig. 2. Circuit illustration of a bit-flip noise model and its restricted equivalence under commutativity assumption. 2.2.2 Extension to Multiple Bit-flip Errors.The extension is only applicable on gatescommute that withX gate up to a global phase factor. This commutativity condition allows us to move occurred bit-lip errors to the end of the circuit, like the change from Figure 2a to 2b, U wher , ...e,U are still noisy-free unitary gates. The model is 1 m constructed by repeatedly apply the previous proof procedure, instead of considering the cancellation of errors, since our interest is on individual gates but not on the accumulated one. The expected distribution function p˜ of circuit U · · ·U |ϕ⟩ with up tom layers of bit-lip errors can be m 1 recursively deined by (1) (1) p˜ (x ) := E [p(y )] (j ) (j−1) (j−1) p (x ) := E [p (y )] forj = 2, . . . ,m (m) (m−1) (m−1) ˜ ˜ p(x ) = p (x ) = E [p (y )] where p is the error-free output distribution, (j−1) x with probability − ϵ1 y with probability − ϵ1 i д д (1) (j ) y = , and y = , (j−1) i i ¬x with probability ϵ ¬y with probability ϵ i д д fori = 1, ..., n and j = 2, ...,m (to avoid the confusion, the superscriptsp˜on are just indices). Because the expectations are all ovxernot s, we can repeat the process in Section 2.2.1 m times. Each repetition provides a |s | (1 − 2ϵ ) term in the multiplication: X Y X * + |s | s .x |s |·m s .x ˜ ˆ p(x ) = . (1 − 2ϵ ) /p(s)(−1) = (1 − 2ϵ ) pˆ(s)(−1) . (6) д д n n j=1 s ∈{0,1} s ∈{0,1} , - Eq. (6) is also straightforward to compatible with the case when each layer of bit-lip errors have a diferent error rate by indexing ϵ withj X Y * + |s | s .x ˜ . / ˆ p(x ) = (1 − 2ϵ ) p(s)(−1) . д, j n s ∈{0,1} j=1 , - 2.2.3 Bit-flip Error Filter.Let j be the binary representation of a non-negative integer j. Givenϵ and p(x ) for b д allx ∈ {0, 1} , it is possible to recover the noise-free outcomes of a QC algorithm. The irst step is to solve for ˆ ˜ p(s). With knownϵ , p(x ), and x, a linear system derived from Eq. (6) can be built as following: Gρˆ = ρ˜ (7) ACM Trans. Quantum Comput. 6 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang where |(j−1) |m (i−1) .(j−1) n n b b b G := (1 − 2ϵ ) (−1) fori ∈ {1, ..., 2 } and j ∈ {1, ..., 2 } ij д pˆ(0...00) p˜(0...00) * + * + .pˆ(0...01)/ .p˜(0...01)/ . / . / ρˆ := . / , ρ˜ := . / , . . . / . / . . . . / . . / ˆ ˜ p(1...11) p(1...11) , - , - f g n n n n 1 1 2 ×2 2 G ∈ [−1, 1] , ρˆ ∈ − , , and ρ˜ ∈ [0, 1] . Using the algorithm to be introduced in Section 3, we can n n 2 2 estimate the value of ϵ to construct matrixG. Using a suicient number of measurements, we can compute vector ρ˜. Thus, by solving Eq. (7) and substituting the result into(4) Eq. , we can then re-construct the noise-free distribution function p(x ) for all x ∈ {0, 1} . The following lemma implies that the solution (7) of always Eq. exists. Lemma 2.1. G is full-rank for alln ≥ 1. Proof. We decompose G as (n) (n) G = G ◦ G , 1 2 n n n n (n) (n) 2 ×2 2 ×2 where ◦ is element-wise multiplication, G ∈ [0, 1] , G ∈ {−1, 1} , and 1 2 (n) |(j−1) | m n n (G ) := (1 − 2ϵ ) fori ∈ {1, ..., 2 } and j ∈ {1, ..., 2 } ij д (n) (i−1) (j−1) n n b b (G ) := (−1) fori ∈ {1, ..., 2 } and j ∈ {1, ..., 2 } ij (n) We start withG when n = 1. It is easy to examine that " # 1 1 (1) G = 1 −1 (n) (n) s .x is full-rank. Recall that each entrG y ofis(−1) for binary numbers s and x, and each row ofG shares the 2 2 (2) same x while each column shares the same s. Following the little-endian convention, the 16 entries G canin be divided into four divisions equally based on their position " # s = 0, x = 0 s = 0, x = 1 2 2 2 2 s = 1, x = 0 s = 1, x = 1 2 2 2 2 Namely, (1) (1) (1) (1) 0·0 0·1 (−1) G (−1) G G G (2) 2 2 2 2 G = = . (1) (1) (1) (1) 1·0 1·1 (−1) G (−1) G G −G 2 2 2 2 Similarly, we can have (n−1) (n−1) G G (n) 2 2 G = (8) 2 (n−1) (n−1) G −G 2 2 n n (n) (n−1) (n) (1) 2 ×2 SinceG ∈ {−1, 1} , if G is full-rank, the structure in(8) Eq.implies G is full-rank. G As is also 2 2 2 2 (n) full-rank, by induction, G is full-rank fornall ≥ 1. (n) |(j−1) | m Note that the jth column ofG is thejth column ofG multiplied(1by− 2ϵ ) and 1 − 2ϵ , 0. As all д д (n) columns ofG are linearly independent, their non-zero multiples are linearly independent, too. Namely, all columns ofG are linearly independent,Gsois full-rank fornall ≥ 1. □ ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 7 However, similar to the problem in Section 2.1, solving (7) cannot Eq. guarantee a meaningful ρ˜, that is, a ρ˜ ∈ [0, 1] . Nevertheless, we can consider a optimization problem instead. ρˆ := arg min∥Gρˆ − ρ˜∥ s.t. n n 2 2 X X (i−1) .(j−1) b b ρ (−1) = 1 (9) i=1 j=1 (i−1) .(j−1) n b b ρˆ (−1) ≥ 0 for all j ∈ {1, ..., 2 } i=1 As an example, whenn = 1, Eq. (4) yields ˆ ˆ p(0) = p(0) + p(1) (10) p(1) = pˆ(0) − pˆ(1), so pˆ(0) is always as p(0) + p(1) = 1. Thus, when n = 1, Eq. (9) can be simpliied as ρˆ := arg min ∥Gρˆ − ρ˜∥ (11) 1 1 1 ρˆ = ,− ≤ρˆ ≤ 1 2 2 2 2 3 ESTIMATING DISTRIBUTIONS OF NOISE PARAMETERS The bit-lip gate error model Eq. (6) and the measurement error model Eq.(2) together provide us forward models to propagate noise in QC algorithms. Based on these forward models and measurement results from a QC device, we can iltering out measurement errors and, in some scenarios, bit-lip gate errors, as such to recover noise-free information. Here, a critical step is to identify modelϵparameters , ϵ and ϵ using repeated measurements д m0 m1 of a testing circuit. The Bayesian approach is suited to solving this inverse problem. In this work, we will use both the standard Bayesian inference and a novel Bayesian approach calle consistent d Bayesian[6] to infer these parameters. 3.1 Computational Framework The Bayesian inference considers model parameters conditioned donasdata the posterior distribution π (λ|d), which is proportional to the product of the prior distribution parameters π (λ) and the likelihoπo(dd|λ), i.e. π (λ|d) ∝ π (λ)π (d|λ). It infers the posterior distribution using the stochastic d = Qmap (λ) + ε, where Q is the quantity of interest (QoI) and ε is an assumed error model. In our caseλ, represents model parametersϵ , ϵ and д m0 ϵ , d is the measured data collected from the device,πand (d|λ) characterize the diference between forward m1 model output and the data. Unlike the standard Bayesian inference, the consistent Bayesian directly inverts the observed stochasticity of the data, described as a probability measure or density, using the deterministic Q (λ). This map approach also prior begins with a prior distribution, denote π d (as λ), on the model parameters, which is then updated to construct post a posterior distribution π (λ). But its posterior distribution takes a diferent form: obs π (Q (λ)) post prior D π (λ) = π (λ) , (12) Λ Λ Q (prior ) π (Q (λ)) where λ ∈ Λ and D is the space of the observed data. Each terms in Eq. (12) are explained as follows: ACM Trans. Quantum Comput. 8 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang Fig. 3. A testing circuit example Q (prior ) • π denotes the push-forward of the prior through the model and represents a forward propagation of uncertainty. It represents how the prior knowledge of likelihoods of parameter values deines a likelihood of model outputs. obs • π is the observed probability density of the QoI. It describes the likelihood that the output of the model corresponds to the observed data. 3.2 Implementation Details We take a noisy one-qubit gate U (its noise-free version is denote Ud) by as an example. Suppose we use this gate to build a testing circuit as shown in Figure 3. We set the QoI in our case to be the probability of measuring 0 from the testing circuit. Assume that the measurement operator in the testing circuit is associated with measurement errors ϵ and ϵ . Let λ := (ϵ , ϵ , ϵ ) be the tuple of noise parameters that we want to infer. Note that if m0 m1 д m0 m1 U is a gate like Hadamard gate, the bit-lip gate error in theory will not afect the measurement outcome for testing circuit, which means we only need toϵinfer and ϵ in this case. In terms of measurement error m0 m1 rates estimation, we provide a choice of testing circuit in Section 4.1 consisting of a singlentesting circuit for qubits, which dramatically reduces the number of testing circuits compared with the fully correlated setting. Let Λ := (0, 1) × (0, 1) × (0, 1) denote the space of noise parameters andD := [0, 1] denote the space of QoI. Finally, we use Q : Λ → D to denote a general function combining(6)Eq. and Eq. (2) that compute the probability of measuring|0⟩ when testing circuit has bit-lip gate error and measurement error. The overall algorithm consists of two parts. In the irstLpart, number of QoI’s, denoted byq (j = 1, ..., L) are generated fromL number of prior λ’s, denote as λ forj = 1, ..., L, with function Q. Then, the distribution Q (prior ) π is estimated by Gaussian kernel density (KDE) qusing . Next, in the second part, prior λ ’s are either j j rejected or accepted based on Eq. (12), and those accepted priorλ ’s are the posterior noise parameters that we obs are looking for. The distribution π is the observed probability of measuring |0⟩, i.e., Gaussian KDE of data. The algorithm is summarized in Algorithm 1, which is an implementation of Algorithms 1 and 2 in [6]. In this work, the prior λ are randomly generated from some relatively lat normal distributions due to the little knowledge of its actual characterization. Thus, foriQubit , suppose we have estimated gate and measurement 0 0 0 error rates (ϵ , ϵ , ϵ ) from past experience and their variances (σ , σ , σ ) that make curves lat, ϵ ϵ ϵ д,i m0,i m1,i д,i m0,i m1,i the prior distributions are 0 2 ϵ ∼ N (ϵ , σ ), m0,i m0,i ϵ m0,i 0 2 ϵ ∼ N (ϵ , σ ), m1,i m1,i ϵ m1,i 0 2 ϵ ∼ N (ϵ , σ ). д,i д,i ϵ д,i In this setting, the acceptance rates of all experiments in Section 4 range from 10% to 35%. This is high enough to select suicient number of posterior parameters in this study. To demonstrate and compare the diference between the results of consistent and standard Bayesian algorithms, we also use the same priors and observation datasets to infer noise parameters via the standard Bayesian. For a single Qubit i, let(x ,y ) forj = 1, ..., J represent J number of data pairs, wherxe is the theoretical probability j j j ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 9 Algorithm 1 Consistent Bayesian inference for error model parameters. obs Given a set of prior λ (j = 1, ..., L), Gaussian KDEπ of the observed QoI (i.e., data), model function Q (i.e., combination of Eq. (6), Eq. (2), testing circuit and its input state); for j = 1 to L do Use Q (λ ) to compute q ; j j end for Q (prior ) Generate Gaussian KDEπ fromq ’s; obs π (Q (λ)) Estimateµ := max ; λ∈Λ Q (prior ) π (Q (λ)) for k = 1 to L do Generate a random number ζ ∈ [0, 1] from a uniform distribution; obs π (Q (λ )) 1 D Compute ratioη := · ; Q (prior ) π (Q (λ )) if η > ζ then k k Accept λ ; else Reject λ ; end if end for output Accepted noise parameterλ ’s. of measuring|0⟩ and y is the observed probability of measuring |0⟩. As discussed in Eq. (1) and Eq. (10), we have m m y = ((0.5 + (1 − 2ϵ ) (x − 0.5))(1 − ϵ ) + (0.5 − (1 − 2ϵ ) (x − 0.5))ϵ + ε (13) j д,i j m0,i д,i j m1,i j where m is the number of repetitions of the gate in the testingm cir =cuit 1 in(Figure 3) and ε ∼ N (0, σ ) represents noise in general with standard deviation σ ≥ 0. We use Cauchy(0, 1) as the prior distribution σ .of ε ε Eq. (13) yields the following likelihood function f (y⃗|x⃗, ϵ , ϵ , ϵ , σ ) = f (y |x , ϵ , ϵ , ϵ , σ ), m0,i m1,i д,i ε j j j m0,i m1,i д,i ε j=1 where each f is the probability density function (PDF) m m 2 N (((0.5 + (1 − 2ϵ ) (x − 0.5))(1 − ϵ ) + (0.5 − (1 − 2ϵ ) (x − 0.5))ϵ , σ ). д,i j m0,i д,i j m1,i In this work, we useRStan package inR [24, 27] to implement the standard Bayesian inference. which is summarized in Algorithm 2. 4 EXPERIMENTS Because using our bit-lip error model only is not suicient for the analyzing gate errors in a complicate algorithm like Grover’s search or QAOA, the inference for gate errors is performed for a few prototype circuits. For more sophisticated algorithms, we only investigate the measurement error. All experiments are conducted on IBM’s 5-qubit quantum computeribmqx2. We compare both the consistent Bayesian (Algorithm 1) and the standard Bayesian method (Algorithm 2) with the measurement error ilter inCompleteMeasFitter Qiskit [2] and the method in19 [ ] based on quantum detector tomography (QDT) to demonstrate the eiciency of our approaches. ACM Trans. Quantum Comput. 10 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang Algorithm 2 Standard Bayesian inference for error model parameters. Data. Number of repetitions of gates in the testing cir m, the cuit oretical probabilities of measuring |0⟩ x , observed 0 0 0 probabilities of measuring |0⟩ y (j = 1, ..., J), prior mean of noise parameters (ϵ , ϵ , ϵ ), and prior m0,i m1,i д,i variance of noise parameters (σ , σ , σ ). ϵ ϵ ϵ д,i m0,i m1,i Model Parameters. posterior(ϵ , ϵ , ϵ ) ∈ (0, 1) and σ ≥ 0. m0,i m1,i д,i ε Prior Distributions. σ ∼ Cauthy(0, 1); 0 2 ϵ ∼ N (ϵ , σ ); m0,i m0,i m0,i 0 2 ϵ ∼ N (ϵ , σ ); m1,i m1,i ϵ m1,i 0 2 ϵ ∼ N (ϵ , σ ); д,i д,i д,i Likelihood Function. Only measurement errors: ∀j, y ∼ N (x (1 − ϵ ) + (1 − x )ϵ , σ ). j j m0 j m1 Gate and measurement errors: m m 2 ∀j, y ∼ N (((0.5 + (1 − 2ϵ ) (x − 0.5))(1 − ϵ ) + (0.5 − (1 − 2ϵ ) (x − 0.5))ϵ , σ ). j д j m0 д j m1 Stan parameters. Default No-U-Turn Sampler, 10,000 iterations, 2000 warm-up iterations, adapt_delta = 0.99, and other parameters are default. Fig. 4. Testing circuit for measurement error parameter inference 4.1 Measurement Errors Filtering Experiment 4.1.1 Construction of Error Filter.We use the circuit in Figure 4 to infer measurement error parameters in every single qubit, i.e ϵ ., , ϵ fori ∈ {1, 2, 3, 4} on ibmqx2. Here, H is the Hadamard gate for each qubit. m0,i m1,i Theoretically, the observed resultsHof|0⟩ is invariant under bit-lip and phase-lip errors. Consequently, in this case, only the measurement error afects the distribution of measurement outputs, and we do not infer gate error rate ϵ . The testing circuit is executed for×1024 128 times, where the fraction of measuring 0 in each ensemble consisting of 1024 runs provides estimated probability of measuring 0 from the testing circuit. Thus, we have 128 data points in total, i.e L =., 128 in Algorithm 1 Jor= 128 in Algorithm 2. For qubit i, 0 2 the prior(ϵ , ϵ ) ⊆ (0, 1) × (0, 1) are random number from truncated normal distribution N (ϵ , 0.1 ) m0,i m1,i m0,i 0 2 0 0 and N (ϵ , 0.1 ), respectively, wherϵe and ϵ are corresponding values provided by IBM in Qiskit API m1,i m0,i m1,i IBMQbackend.properties() after the daily calibration. Then, we use Algorithm 1 to generate the posterior distributions. We note that in this test, the results by the consistent Bayesian is very close to the standard Bayesian, so we present the former only. ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 11 Table 1. Outcomes from the consistent Bayesian inference Qubit 1 Qubit 2 Qubit 3 Qubit 4 Q (posterior ) obs KL-div(π , π ) 0.001014 0.002243 0.000777 0.001610 D D Post. Mean (1 − ϵ , 1 − ϵ ) (0.9354,0.9009) (0.9537,0.8184) (0.9457,0.8976) (0.8272,0.9492) m0,i m1,i Post. MAP (1 − ϵ , 1 − ϵ ) (0.9797,0.9128) (0.9863,0.8243) (0.9858,0.9180) (0.8426,0.9846) m0,i m1,i Figure 5 displays the joint and marginal distribution of posterior distributions of error model parameters for qubits 1-4 using the consistent Bayesian approach. Using these posterior distributions of error model parameters, we can compute the posterior distribution of the QoI by substituting samples of these distributions in the forward Q (post) modelQ. Figure 6 shows that these posteriors of the QoI, denotedπas , approximate the distribution of the obs observed data π very well. For a more quantitative comparison, we list the posterior mean and the maximum a posterioriprobability (MAP) in Table 1. We can see, in general, ϵ is higher than ϵ , which is consistent with the description m1,i m0,i obs in14 [ ]. Also, Tables 1 presents the KullbackśLeibler (KL) divergence between PDFs of the observedπdataand Q (post) the posterior distribution of the π QoI for each qubit in Figure 6, which illustrates the accuracy of our error model. 0 2 0 2 In this test, our prior distribution N (ϵ , 0.1 ) and N (ϵ , 0.1 ) are quite lat and not informative. This is m0,i m1,i 0 0 because the vendor-providedϵ and ϵ are not always good estimations. This can be veriied by the error m0,i m1,i mitigation results. When we use relation (2) Eq. and Eq. (3) to construct measurement error ilters using the 0 0 vendor-provided(ϵ , ϵ ) and our posteriors, then apply those ilters on the 128 outputs of circuit in Figure 4 m0,i m1,i (i.e., 128 observed probability of measuring |0⟩), we obtain diferent results as shown in Figure 7. The theoretical probability of measuring 0 for circuit in Figur .5, but e 4the ispr0ovided parameters rarely gives this value, and its mean and peak of Gaussian KDE are not even close to.5.0On the other hand, the ilters created by our posteriors can make sure the mean and peak of the denoised probability of measuring |0⟩ are around the ideal value .5.0 The results in Figure 4 indicate that when applying (3) toEq. mitigate the measurement error, one has a larger chance to obtain a denoised QoI close to the ideal value .5 by0using the parameters inferred by our method. More importantly, the result by our method is unbiased as the mean value of the denoised.QoI 5. is 0 This test indicates that we can use the circuit shown in Figure 4 to estimate measurement error in multiple qubits at the same time. It only requires to prepare initial state using ground states, and the total number of gates are linearly dependent on the number of qubits. 4.1.2 Application on State Tomography. After obtaining the error model parameters, we can further use this model mitigate the measurement errors in other circuits. We irst apply error ilters to the results of state tomography on circuits that make bell basis fr |00om ⟩ and |000⟩. Qubit 0 and 1 are used for 2-qubit state and Qubit 0 to 2 are used for 3-qubit state in ibmqx2. The idelity between density matrices from (corrected) state tomography result and theoretical quantum state is listed in Table 2. For the 2-qubit state tomography, ilters constructed from posterior means by the consistent Bayesian and the standard Bayesian provides similar idelity as that by the Qiskit ilter. However, for 3-qubit tomography, ilters from both Bayesian methods yield better idelity, and their performance are similar. We note that the Qiskit ilter assumes correlation in the measurements, which requires more model parameters, while our model does not. The idelity in Table 2 indicates that Bayesian methods enable us to use fewer parameters to obtain better results. 4.1.3 Application on Grover’s Search and QAOA. Next, we apply our ilter on Grover’s search and QAOA circuits from [15]. We measure Qubit 1 and 2 in ibmqx2 for Grover’s search circuit. The exact solution of this Grover’s ACM Trans. Quantum Comput. 12 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang (a) ubit 1 (b) ubit 2 (c) ubit 3 (d) ubit 4 Fig. 5. Joint and marginal posterior distributions of measurement error parameters in the testing circuit shown in Figure 4. Here, (1 − ϵ , 1 − ϵ ) are shown for demonstration purpose. m0,i m1,i search example is|11⟩ and the theoretical probability is 1. Thus, in this case, we compare the probability of measuring|11⟩ by running in the real device ibmqx2 and the denoised probabilities from error ilters based on Qiskit methoCompleteMeasFitter d , QDT in19 [ ], mean and MAP of posteriors from standard Bayesian, and mean and MAP of posteriors from the consistent Bayesian. All circuits for the Qiskit ilter and QDT ilters are executed for 8192 shots, and each probability used in both ilters is estimated from 8192 measurement outcomes. ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 13 (a) ubit 1 (b) ubit 2 (c) ubit 3 (d) ubit 4 obs Fig. 6. PDF of the QoI (i.e., the probability of measuring |0⟩) obtained from data (π ) and from evaluating Q using inferred Q (post) measurement error parametersπ( ). Table 2. Fidelity of state tomography results filtered by various error filters Fidelity State Raw Data By Qiskit Method By Cons. Mean By Stand. Mean (|00⟩ + |11⟩) 0.9051 0.9800 0.9781 0.9783 (|01⟩ + |10⟩) 0.9157 0.9803 0.9806 0.9808 (|000⟩ + |111⟩) 0.7389 0.9227 0.9390 0.9391 (|010⟩ + |101⟩) 0.6719 0.8970 0.9203 0.9207 (|100⟩ + |011⟩) 0.7006 0.9121 0.9254 0.9207 √ (|110⟩ + |001⟩) 0.6974 0.8863 0.9443 0.9446 “Qiskit Methodž means toCompleteMeasFitter in Qiskit 2]. “Cons. [ meanž implies the transition matrix is created from posterior mean by Algorithm 1. “Stand. Meanž means the transition matrix is created from posterior mean by standard Bayesian ACM Trans. Quantum Comput. 14 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang (a) ubit 1 (b) ubit 2 (c) ubit 3 (d) ubit 4 Fig. 7. PDFs of the probability of measuring |0⟩ denoised by vendor-provided parameters (priors) and by posteriors In addition, as we do not expect the quantum computer has a stable environment, in order to see the robustness of each method in comparison, after the data for creating error ilters are collected, we run our Grover’s search circuit at several diferent time and then apply the same set of ilters. All results are listed in Table 3. As shown in Table 3, both Bayesian methods yield best performance among all the methods while the ilters constructed from posterior mean are better than the ilters constructed from posterior MAP. In all six time slots, the mean and MAP from the consistent Bayesian provide sightly better denoising efect than those from the standard Bayesian. The QAOA example includes two rounds and parameters for QAOA circuits are set (γ ,as β ) = (0.2π , 0.15π ) 1 1 and (γ , β ) = (0.4π , 0.05π ) [15]. The graph of the QAOA example in15[] is shown in Figure 8, which has 2 2 maximum objective value 3 in Max-Cut problem and 6 bit-string optimal|0010 solution ⟩, |0101⟩, |0110⟩, |1001⟩, |1010⟩, |1101⟩ (ibmqx2 uses little-endien convention, so the rightmost bit is Node 1 and the leftmost bit is Node 4). Because the graph in Figure 8 is a subgraph of the coupling map ibmqx2 of , we map the nodes to qubits exactly. The average size of a cut and the probability of measuring an optimal solution are two quantities to compare in this experiment. Moreover, the results from simulator is also provided as a indicator for the situation without noise. The remaining procedures are the same as those in Grover’s search experiment. The data is reported in Table 4 and 5. Here, “QDTž error ilter from19 [ ] is built under the assumption that measurement operations are ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 15 Table 3. Probability of measuring |11⟩ in Grover’s search example a b Method/Source Hour 0 Hour 2 Hour 4 Hour 8 Hour 12 Hour 16 Raw Data 0.6727 0.6930 0.6724 0.6740 0.6917 0.6841 Qiskit Method 0.7097 0.7335 0.7104 0.7120 0.7323 0.7241 QDT 0.7107 0.7332 0.7087 0.7108 0.7305 0.7224 Stand. Mean 0.9099 0.9324 0.9063 0.9088 0.9290 0.9192 Stand. MAP 0.8378 0.8635 0.8372 0.8392 0.8616 0.8522 Cons. Mean 0.9128 0.9351 0.9088 0.9114 0.9316 0.9219 Cons. MAP 0.8920 0.9158 0.8914 0.8936 0.9128 0.9034 “Qiskit Methodž means toCompleteMeasFitter in Qiskit 2], QD [ T refers to ilter in 19], [ “Stand.ž stands for Standard Bayesian, and “Cons.ž refers to Algorithm 1. MAP and mean represent the error ilters are created from the MAP and mean of posteriors. “Hour Xž means the experiment is conducted X hours after the data for error ilers of all listed methods are collected Fig. 8. The graph of QAOA example in [15] Table 4. Average size of a sampled cut in QAOA example Method/Source Hour 0 Hour 2 Hour 4 Hour 8 Hour 12 Hour 16 Simulator 2.8637 2.8642 2.8651 2.8652 2.8642 2.8626 Raw Data 2.3005 2.3579 2.3197 2.2823 2.3063 2.2871 Qiskit Method 2.3783 2.4623 2.4247 2.3786 2.4063 2.3926 QDT 2.3812 2.4453 2.4016 2.3589 2.3851 2.3612 Stand. Mean 2.4878 2.5483 2.5059 2.4551 2.4860 2.4581 Stand. MAP 2.4080 2.4708 2.4222 2.3800 2.4059 2.3829 Cons. Mean 2.4911 2.5518 2.5089 2.4578 2.4891 2.4612 Cons. MAP 2.4407 2.4996 2.4554 2.4109 2.4382 2.4133 independent between each qubit due to the large amount of testing circuits that correlation assumption requires (6 circuits if assume qubits are correlated in measurement). ACM Trans. Quantum Comput. 16 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang Table 5. Probability of measuring an optimal solution in QAOA example Method/Source Hour 0 Hour 2 Hour 4 Hour 8 Hour 12 Hour 16 Simulator 0.8930 0.8937 0.8941 0.8943 0.8940 0.8940 Raw Data 0.5784 0.6038 0.5895 0.5725 0.5748 0.5740 Qiskit Method 0.5968 0.6456 0.6316 0.6074 0.6140 0.6155 QDT 0.6400 0.6698 0.6525 0.6312 0.6331 0.6325 Stand. Mean 0.6952 0.7239 0.7033 0.6766 0.6787 0.6797 Stand. MAP 0.6444 0.6695 0.6508 0.6305 0.6309 0.6317 Cons. Mean 0.6975 0.7265 0.7058 0.6790 0.6810 0.6822 Cons. MAP 0.6610 0.6860 0.6672 0.6431 0.6439 0.6452 The conclusion from Table 4 and 5 is basically the same as that from Table 3. Namely, Bayesian methods, especially ilters from posterior mean, outperform other methods, and parameters inferred by the consistent Bayesian works slightly better than those by the standard Bayesian in all six time slots. From both Grover’s search and QAOA examples, we can see the accuracy of both Bayesian approaches are better than the existing methods. 4.1.4 Application on Random Cliford Circuits. Finally, we test the measurement-error iltering for random 2-Qubit Cliford circuits with 1, 2, 3, and 4 2-Qubit Cliford operators (i.e., length 1, 2, 3, 4). For each length, 16 random circuits are generated to draw a boxplot and each circuit is run for 8192 shots. The results are shown in Figure 9. While the theoretical output of 2-Qubit Cliford |cir 00⟩ cuit withis probability 1, Figure 9 demonstrate that the ilter constructed from posterior mean estimated by standard Bayesian provides best performance. The consistent Bayesian results in almost the same results as the standard Bayesian method. 4.2 Gate and Measurement Error Filtering Experiment We consider the circuit with 200 NOT gates as shown in Figure 10. We still useibmqx2 machine and run the experiment twice separately on Qubit 1 and Qubit 2. In each trial, the circuit is exe×cute 128dtimes 1024 where readouts from every 1024 runs are used to estimate the QoI, i.e., the probability of measuring |0⟩. Namely, we collect 128 samples of the QoI. Because the aforementioned Qiskit metho CompleteMeasFitter d and QDT are for measurement errors, in this section, we only compare the results from standard Bayesian and the consistent Bayesian with the same 0 2 0 2 0 2 priors and dataset. The priors are truncated normal N (ϵ , 0.1 ), N (ϵ , 0.1 ), and N (ϵ , 0.005 ) with range m0,i m1,i д,i 0 0 0 (0, 1). Of note,ϵ , ϵ , ϵ are vendor-provided values from IBM’s daily calibration, where the prior measure m0,i m1,i д,i 0 0 −2 −1 0 error rates ϵ and ϵ are often at the scale of 10 to 10 , and the prior single-qubit gate errorϵrateis m0,i m1,i д,i −4 −3 usually between 10 and 10 . Therefore, we adjust the standard deviations to match the scale of prior means. Consequently, the prior distributions are relatively lat due to the lack of knowledge on these parameters. 4.2.1 Inference for Noise Parameters.Figure 11 shows the distribution ϵ in of Qubit 1 and 2. Both distributions are right-skewed. Table 6 provides the numerical values for mean and MAP. In Table 6, we can see both methods give similar measurement error parameter ϵ and ϵ on Qubit 1 and 2, but the gate error rateϵ are not m0,i m1,i д always similar. More importantly, as shown in Figure 12, posteriors of the QoI from the consistent Bayesian, i.e., Q (post) obs π matches the distribution of data,πi.e.,quite well. On the other hand, the posterior distribution of D D ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 17 (a) Length 1 (b) Length 2 (c) Length 3 (d) Length 4 Fig. 9. Measurement-error filtering for random 2-ubit Cliford circuits. łLengthž represents the number of Cliford operators in the circuit. łProbabilityž means the probability of measuring |00⟩. Fig. 10. Experiment circuit for gate and measurement error mitigation. the QoI generated by posteriors distributions of model parameters from the standard Bayesian can match the empirical mean of the data only while the shape of the PDF is quite diferent. 4.2.2 Error Filtering. Using the posterior means from Table 6, we construct gate and measurement error ilters and apply them on the 128 samples of the QoI. (i.e., probabilities of measuring 0) on Qubit 1 and Qubit 2. The results are displayed in Figure 13. It shows that both Bayesian approaches we use can recover the exact value 1 with high probability. More importantly, the consistent Bayesian outperforms the standard one as it recovers the exact value 1 with larger chance. especially in the test on Qubit 2. The data for error ilters in Section 4.1 and experiment in Section 4.2 were collected within one hour, so it is reasonable to use posteriors in Section 4.2 to denoise the Grover’s search data in Section 4.1. However, comparing the values of measurement error parameters in Table 1 and in Table 6, we can see there are some noticeable ACM Trans. Quantum Comput. 18 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang (a) ubit 1 (b) ubit 2 Fig. 11. Gaussian KDE of ϵ Table 6. Measurement error parameters of the consistent and standard Bayesian inference. Qubit 1 Qubit 2 Consistent Post. Mean(1 − ϵ , 1 − ϵ , ϵ ) (0.9255, 0.8922, 0.004934) (0.9229, 0.8856, 0.003804) m0,i m1,i д Consistent Post. MAP(1 − ϵ , 1 − ϵ , ϵ ) (0.9756, 0.8837, 0.004827) (0.9770, 0.9485, 0.003291) m0,i m1,i д Standard Post. Mean (1 − ϵ , 1 − ϵ , ϵ ) (0.9221, 0.8939, 0.004683) (0.9214, 0.8871, 0.002982) m0,i m1,i д Standard Post. MAP (1 − ϵ , 1 − ϵ , ϵ ) (0.9758, 0.8835, 0.006550) (0.9836, 0.9354, 0.003453) m0,i m1,i д (a) ubit 1 (b) ubit 2 Q (post) Fig. 12. Posterior distribution of the QoI, i.eπ., , generated by the posterior distribution of model parameters from both Bayesian methods diferences. Table 7 provides the results of using parameters in Section 4.2 to ilter out errors in data used in Section 4.1. We can see they are better than Qiskit method and QDT, but worse than values from either Bayesian methods shown in Tabel 3. ACM Trans. Quantum Comput. A Bayesian Approach for Characterizing and Mitigating Gate and Measurement Errors • 19 (a) ubit 1 (b) ubit 2 Fig. 13. Denoised (both gate and measurement) probability of measuring 0. Parameters used are posterior mean from Table 6. Table 7. Probability of measuring |11⟩ in Grover’s search Example denoised by parameters in Section 4.2 Method Hour 0 Hour 2 Hour 4 Hour 8 Hour 12 Hour 16 Stand. Mean 0.8398 0.8680 0.8392 0.8414 0.8656 0.8561 Stand. MAP 0.8116 0.8367 0.8110 0.8131 0.8348 0.8257 Cons. Mean 0.8434 0.8716 0.8428 0.8450 0.8691 0.8595 Cons. MAP 0.7992 0.8240 0.7986 0.8005 0.8221 0.8131 (a) ϵ = 0, ϵ = 0.005 (b) ϵ = 0, ϵ = 0.005 (c) ϵ = 0, ϵ = 0 m1 д m1 д m1 m0 Fig. 14. Sensitivity analysis of forward modelQ with 200 NOT gate. One possible explanation is, with 200 gates, our model is much more sensitive to the gate error than the measurement error. A comparison of the sensitivity is shown in Figure 14, where the results are obtained by using the error models Eqs. (2) and (6). In Figure 14 (a) and (b), we can see that whenϵ is ixed, the QoI changes linearly and slowlyϵ as or ϵ varies. However, as shown in Figure 14 (c) when ϵ and ϵ are ixed, the m0,i m1,i m0,i m1,i QoI changes rapidly as ϵ increases. the estimation of measurement error in Section 4.1 uses circuits that have 0.5 chance to measure either|0⟩ or |1⟩ without noise and this distribution does not change when a Hadamard gate sufers from bit-lip or phase-lip error, so it yields a better performance. ACM Trans. Quantum Comput. 20 • Muqing Zheng, Ang Li, Tamás Terlaky, and Xiu Yang 5 DISCUSSION AND FUTURE WORKS In this work, we extend a bit-lip error model from a single gate case to multiple gate case, and provide theoretical analysis to prove the existence of the error mitigation solution for both cases. In some noise models, such as depolarizing error model, the rate of bit-lip error is associated with the rates of other types 21of , p.errors [ 379]. Thus, the inference of bit-lip error rates could provide a connection to a more general noise model. We propose to use Bayesian approaches to infer parameters in the error models to characterize the propagation of the device noise in QC algorithms more efectively. The experiments in Section 4 demonstrate that our methodology outperforms two existing methods on the same error models over a wide range of time, while the number of testing circuits is linear or constant to the number of qubits. The consistent Bayesian approach is, in general, better than the standard Bayesian. These results indicate that our error models can characterize the device noise quite well, and they help to understand the propagation of such noise in QC algorithms. There are still several limitations in our methodology. One issue that afects the scalability of our method is the exponentially large matrix in the denoising step. The dimension of matrices can be reduced if we can identify the qubits that are independent during the measurement step of an algorithm and ilter their measurement outcome separately. A recent work in20[] also indicates a scheme to reduce the dimension of the transition matrix by limiting the range of bases that are put into consideration. Also, a parallel algorithm 5] proposed in [ can exploit the tensor-product structure of the linear system in the error iltering step to speedup the calculation. On the other hand, because the method of estimating the distribution of model parameters is not limited by the two models we discussed in the paper, a consideration for pairwise-correlated measurement error model discussed in 4][probably be helpful for inferring correlated measurement error rates. As for the gate error model, its applicable gate and error types are limited. A potential extension to the applicable gates is to modify the model to accommodate multi-qubit bit-lip error instead of individual-qubit error. This is because more gates ⊗n ⊗n commute withX than with elements in {X , I} forn ≥ 2. For example,X ⊗ X commute with matrix A ⊗ B −iδ A⊗B ⊗2 ⊗2 −iδ A⊗B and e for(A, B) ∈ {Y , Z } ∪ {I , X } and arbitraryδ, where the forme is generally utilized in quantum simulation [28]. ACKNOWLEDGMENTS This works was partially supported by Defense Advanced Research Projects Agency under Grant No.: W911NF2010022. Xiu Yang was also supported by NSF CAREER DMS-2143915. Muqing Zheng was also supported by U.S. De- partment of Energy, Oice of Science, National Quantum Information Science Research Centers, Quantum Science Center (QSC). Ang Li’s work was supported by the U.S. Department of Energy, Oice of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage QA(C) under contract number DE-SC0012704. The Paciic Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830. REFERENCES [1] Dorit Aharonov and Michael Ben-Or. 2008. 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ACM Transactions on Quantum Computing – Association for Computing Machinery
Published: Feb 24, 2023
Keywords: Error mitigation
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