Demonstrating quantum error mitigation on logical qubits

Domain: physics
Published: 2025-12-24
Editorial outcome: minor_revision

Abstract
A long-standing challenge in quantum computing is developing technologies to overcome the inevitable noise in qubits. To enable meaningful applications in the early stages of fault-tolerant quantum computing, devising methods to suppress post-correction logical failures is becoming increasingly crucial. In this work, we propose and experimentally demonstrate the application of zero-noise extrapolation, a practical quantum error mitigation technique, to error correction circuits on superconducting processors. By amplifying the noise on physical qubits, the circuits yield outcomes that exhibit a predictable dependence on noise strength, following a polynomial function determined by the code distance. This property enables the effective application of polynomial extrapolation to mitigate logical errors. Our experiments demonstrate a universal reduction in logical errors across various quantum circuits, including fault-tolerant circuits of repetition and surface codes. We observe a favorable performance in multi-round error correction circuits, indicating that this method remains effective when the circuit depth increases. These results advance the frontier of quantum error suppression technologies, opening a practical way to achieve reliable quantum computing in the early fault-tolerant era.

Introduction
Suppressing errors is a problem that lies at the center of quantum computing technologies. Quantum error correction (QEC) and mitigation are the two generic methods of error suppression. Error correction promises to reach an arbitrarily high fidelity provided sufficient qubit resources. Recently, experiments have demonstrated a positive gain in the surface code error correction when scaling the code distance up to 71. However, it is still viewed as a long-term goal to achieve negligible infidelity through error correction, which could need millions or even more qubits and pose a challenge to the experimental technologies regarding scalability2. Error mitigation takes different resources, computing time, to suppress errors, being originally designed for the regime that error correction is lacking3.

Considering the earliest applications of quantum computing, it is highly likely that they will be carried out under strict technology constraints, mainly on the qubit number. In this scenario, error correction can only attain a limited fidelity, and a practical approach taking advantage of the two error suppression methods is necessary4–6. Zero-noise extrapolation (ZNE) is one of the most practical error mitigation techniques, universally applicable to quantum algorithms that evaluate expectation values7,8. The central idea behind ZNE is to amplify the noise in a quantum circuit by a controllable factor r, and then extrapolate the results back to r=0, thereby inferring the behavior of a noiseless circuit. In a recent experiment, ZNE demonstrated substantial error suppression on a superconducting system with more than a hundred physical qubits9, making it the only error mitigation techniques successfully applied at this scale to date. Given these promising results, we consider ZNE the leading candidate for mitigating errors in quantum error correction circuits and report the first experimental demonstration of this application. Specifically, we apply ZNE to reduce post-correction logical errors by amplifying noise on physical qubits. We note that one can also amplify logical noise through code distance scaling, which offers limited tunability but avoids the need for randomized circuit compilation6.

The implementation of ZNE on logical qubits faces two primary experimental challenges. First, it requires a quantum processor with high-fidelity gates and sufficient qubits, which is crucial for executing quantum error correction. Recent progress in qubit fabrication and control technologies have made superconducting qubits a promising platform for investigating quantum error correction and mitigation technologies1,9–14. In this work, we utilize two superconducting quantum processors to experimentally assess residual errors and costs in quantum error correction and mitigation, employing both the repetition code and surface code15–18. Second, accurate error mitigation relies on measuring error rates per operation3,19. While measuring logical error rates is feasible for small codes, it becomes increasingly time-consuming for larger codes due to the small logical error rates20,21. Additionally, for high-encoding-rate quantum error correction codes, such as certain quantum low-density parity-check (qLDPC) codes, logical errors may involve multi-qubit correlations, making the measurement impractical22,23 (see Supplementary Note 2). To overcome this issue, we adopt a strategy of amplifying noise on physical qubits instead of logical qubits. By integrating error mitigation techniques with error correction, this study demonstrates a practical pathway to bridge the gap between the noisy intermediate-scale quantum (NISQ) era and the fault-tolerant quantum computing (FTQC) era, advancing the pursuit of practical quantum computing technologies.

Results
The essential assumption made in ZNE is that the expected value of an observable is a function \langle O \rangle(r) of the noise strength r, and this function can be fitted well by simple functions such as polynomials. This assumption holds in NISQ circuits, which has been illustrated in many theoretical works and experiments3,7,8,13,24–26. In this paper, one of the primary goals is to demonstrate that the assumption is also valid in QEC circuits.

QEC circuits incorporate features beyond typical NISQ circuits27, including mid-circuit state preparation and measurement, feedback, and post-selection. By considering a general formalism for evaluating observables in such circuits, while accounting for the effects of decoding algorithms, we find that the expected value of an observable under noise amplification follows the form \langle O \rangle(r) = \langle O \rangle_{ideal} + \sum_{k=1}^{N} a_k r^k, where N is the number of operations in the circuit (see Supplementary Note 1). Here, the coefficient a_k quantifies the deviation caused by k errors occurring within the circuit. In QEC circuits, errors can be reliably detected and corrected as long as their number remains below \lceil d/2 \rceil, where d is the effective code distance. Motivated by this observation and the interpretation of a_k, we adopt a fitting model of the form \langle O \rangle(r) = \langle O \rangle_{em} + \sum_{k=\lceil d/2 \rceil}^{\lceil d/2 \rceil + K - 1} a_k r^k, which includes K terms of the form r^k, starting from the \lceil d/2 \rceil term (Fig. 1). This model notably differs from conventional ZNE applied to NISQ circuits, where extrapolation begins with the linear r term.

This fitting strategy aligns with numerical results of logical error rates in parity-check circuits23,28, and the underlying reasoning is broadly applicable to a wide range of QEC circuits. This generality extends beyond parity-check circuits to encompass full fault-tolerant quantum computing circuits, including those incorporating magic state distillation, non-Clifford operations, and arbitrary decoding algorithms.

We carry out experimental demonstrations on two superconducting quantum processors to certify that ZNE can be naturally integrated into the FTQC circuits. These demonstrations include a parity check experiment with feedback error correction, a repetition code experiment, and a surface code experiment. The processors used here, each of which contains a lattice of tens of frequency-tunable qubits featuring adjustable nearest-neighbor couplings, have performance similar to those dedicated to quantum error correction experiments in the literature1,13,14. The qubits selected for the experiments are highly coherent, with the median T_1 time being above 100 \mu s (see parameters in Supplementary Table 1), and the controlled \pi-phase (CZ) gates between nearest-neighbor qubits have a median fidelity around 0.995. Since FTQC also relies heavily on measurement quality, we can differentiate between the |0\rangle and |1\rangle states of each qubit, achieving median readout fidelities of 0.995 and 0.991 for Processor I (with Purcell filters, 0.5-\mu s measurement time) and Processor II (without Purcell filters, 2.5-\mu s measurement time), respectively. This is accomplished by pumping the |1\rangle state of each qubit to its next-higher level. Furthermore, even without the pumping technique which is a prerequisite for feedback operations in FTQC circuits, we can still achieve a median readout fidelity of 0.991 on Processor I within 0.5 \mu s, as demonstrated in our repetition code experiment by allowing repetitive measurements on the syndrome qubits up to M=4 rounds (see next).

Parity check experiment with feedback error correction
As the first experimental demonstration, we show that ZNE works on an example circuit with a feedback X control to eliminate the bit-flip error on Q_0 (Fig. 2a), where the nominal data qubits (Q_j for j=0, 2, and 4) are each initialized into a superposition state given by |\psi_j\rangle = \cos(\theta_j/2)|0\rangle - i\sin(\theta_j/2)|1\rangle. The first 4 CNOT gates in the sequence diagram are used to encode the parity of the data qubits onto the syndrome qubits (Q_1 and Q_3), followed by operations for algorithmic purpose, and the next 4 CNOT gates serve to decode and identify the bit-flip type of errors that may occur during the operational stage. In the following experiments, we neglect measurement errors in the error mitigation setup; a detailed discussion is provided in Supplementary Note 5. To implement ZNE, one needs to be able to controllably amplify the errors, which can be achieved using schemes such as pulse stretching8,26, sub-circuit repetition29 or simulated error9. In our experiments, we use the simulated-error approach. Here to simulate depolarizing errors occurring at a probability of r_p with r being the scaling factor, we run 43 types of circuit instances which correspond to all combinations of inserting one operation drawn from the list of {I, X, Y, Z} at the operational stage for each of the 3 data qubit, based on which we construct a weighted list of circuit instances so that the insertion probabilities of X, Y, and Z all equal to r_p/3 for each data qubit (see Supplementary Note 3 for more details). Finally all qubits are measured in the Z basis to yield a 5-bit binary string, \mu_0 \dots \mu_4, for each circuit instance, and the no correction data are directly calculated using \mu_0 from all circuit instances. Here, the values \mu=\pm 1 correspond to the states |0\rangle and |1\rangle, respectively. In the case of error correction, since a bit-flip error due to the insertion of X or Y on the pair of data qubits Q_j and Q_{j+2}, j \in {0, 2}, flips the state of Q_{j+1}, the 5-bit binary strings from all instances are selected only when \mu_3=1 to ensure no error on Q_3, following which \mu_2 is numerically reversed due to the feedback X gate only if \mu_0=-1 to eliminate the bit-flip error on Q_1. The newly processed strings are then used to calculate the data with correction.

For the case of no correction, Fig. 2b shows that the expectation values of Z for Q_0, for different input states parametrized by \theta_0, all gradually approach zero as the noise strength r increases. In comparison, the data using processed bit strings with error correction indicate that the impact of noise is reduced, resulting in a slower decline in \langle Z_0 \rangle. Experimental results closely match numerical simulations of circuits with gate and measurement errors (Fig. 2c). Furthermore, when we use ZNE to extrapolate the values at r=0, the results show excellent agreement with numerical predictions, which demonstrates that ZNE effectively works on the circuit.

In the above-mentioned example circuit, small errors occurring at the CNOT gates and measurement cannot be corrected, so that the post-mitigation residual error at r=0 is comparable to the case without error correction. Next, we resort to fault-tolerant circuits of repetition and surface codes, where the logical error rates can be suppressed to an arbitrarily low level by increasing the code distance once physical error rates are below the threshold. On fault-tolerant circuits, we find that the post-mitigation residual error is much smaller than the case without error correction.

Repetition code experiment
We implement repetition codes with distances d=3, 5, 7 using up to 13 qubits on Processor I, as shown in Fig. 3a. Repetition codes protect the encoded logical qubit from bit-flip errors and have been widely used as a prototype for demonstrating error correction across various quantum platforms, including ion traps30,31, nuclear magnetic resonance32,33, and superconducting qubits1,34. Here, we adopt it for our first demonstration of error mitigation on fault-tolerant circuits.

The circuit of a repetition code, schematically depicted in Fig. 3b, consists of d data qubits and d-1 syndrome qubits arranged alternately in a linear chain. The data qubits are initialized in the state |0\rangle, preparing the logical qubit in the logical state |0_L\rangle. Next, we apply error injections and perform parity-check measurements for M successive rounds. The circuit terminates with an additional round of error injection and transversal measurement on data qubits. The transversal measurement facilitates a final round of parity checks and readout of the logical operator Z_L. The parity checks are designed to detect all possible bit-flip errors, whether injected deliberately or introduced by computational operations such as gates, state preparation, and measurements. Each experimental shot produces a bit string of length M(d-1)+d, recording the measurement outcomes. This bit string is then processed by a minimum-weight perfect matching (MWPM) decoder35,36, which outputs the correction gates. These corrections ensure faithful reconstruction of the logical state, provided that the number of bit-flip errors does not exceed \lceil d/2 \rceil - 1; therefore, the circuit is fault-tolerant with respect to bit-flip errors.

We evaluate the performance of error suppression methods using the expectation value of the logical operator Z_L. In an ideal noise-free circuit, the expectation value is \langle Z_L \rangle_{ideal} = 1; the presence of noise reduces this value. For illustrative purpose, we experimentally inject Pauli errors during the operational stage with the unit probability set at p=3.6%, as done in the previous example. Figure 3c shows the experimental results for circuits with one round of parity-check measurements (M=1): the expectation value decreases gradually with increasing noise strength, while error correction slows the rate of decline. As the code distance d increases, the rate of decline approaches zero asymptotically, indicating a below-threshold error regime.

To further suppress errors beyond error correction, we apply ZNE to the results. Specifically, we select K+1 data points corresponding to noise strengths r_0, r_1, \dots, r_K and perform polynomial extrapolation to obtain the error-mitigated expectation value of the logical operator, denoted as \langle Z_L \rangle_{em}. To evaluate the performance of ZNE, we define two metrics: the bias \delta and the sampling overhead \eta. The bias \delta quantifies the deviation of the error-mitigated value from the ideal expectation and is expressed as \delta = |\langle Z_L \rangle_{em} - \langle Z_L \rangle_{ideal}|. The sampling overhead \eta measures the relative increase in sampling cost required to achieve the same variance for the error-mitigated value as for the uncorrected value, and it is defined as \eta = Var(\langle Z_L \rangle_{em}) / Var(\langle Z_L \rangle_{r_0}). Unless otherwise specified, we take r_0=1 in what follows and select the other K data points from the remaining collected dataset (see, e.g., Fig. 3c). Each combination produces a corresponding bias and sampling overhead.

The \delta-\eta scatter plots in Fig. 3d visualize that ZNE consistently provides more accurate estimations compared to the uncorrected value at r_0=1 (indicated by dashed lines) across all cases. The plots also highlight a trade-off between the bias \delta and sampling overhead \eta: while ZNE reduces bias, it increases the sampling overhead. If the noise-boosted data points at r_1, \dots, r_K are chosen closer to r_0 or if a higher-order extrapolation (larger K) is used, the bias generally decreases, but the sampling overhead tends to grow. Figure 3c, d further highlight the advantage of combining ZNE with error correction. When error correction is applied, the \delta-\eta scatter points progressively migrate toward the lower-left corner as the code distance d increases, indicating simultaneous improvements in precision and efficiency. Notably, for d=7, the residual error \delta is reduced to approximately 1 \times 10^{-4}, with a modest sampling overhead of just 5. We also find that, contrary to the trend suggested by Fig. 3d, the improvement does not diminish with increasing code distance, provided that the noise model is sufficiently accurate (see Supplementary Note 5).

Our experiments show that as the code distance and parity-check rounds increase, the performance of ZNE remains nearly unchanged; notice that multi-round parity checks are necessary in practical fault-tolerant quantum computing. This insight can be demonstrated in two complementary ways. First, when the error rate per parity-check round is fixed, increasing the number of parity-check rounds results in a similar relative bias and sampling overhead for ZNE, despite the growth in unmitigated bias (see Supplementary Fig. 5). Second, for a fixed total error rate in all rounds (see Supplementary Table 3), partitioning the circuit and employing multi-round QEC significantly improves both the unmitigated and mitigated results (Fig. 3e, f). Remarkably, these enhancements are achieved while the sampling overhead remains nearly unchanged.

Surface code experiment
As a final experiment, we extend our investigation to the rotated surface code, a quantum error correction code capable of correcting both bit-flip and phase-flip errors. We implement the distance-3 rotated surface code on our Processor II, as illustrated in Fig. 4a. The encoded logical qubit is defined by two anti-commuting logical operators, X_L = X_{D1} X_{D4} X_{D7} and Z_L = Z_{D3} Z_{D4} Z_{D5}, and the system can be initialized by preparing the logical states using digital quantum circuits that consist of single- and two-qubit gates14,37.

We demonstrate the effectiveness of ZNE following the sequence diagram shown in Fig. 4a, with three representative initial states: |0_L\rangle, |+_L\rangle, and |\psi_L\rangle = \cos(\pi/6)|0_L\rangle + \sin(\pi/6)|1_L\rangle (Fig. 4b). In the absence of error injections, the initial states reconstructed from the corrected expectation values of the Z_L and X_L observables locate significantly closer to the surface of the Bloch sphere compared to those reconstructed from the uncorrected values (Fig. 4b). This demonstrates reductions in both bit-flip and phase-flip errors that arise during the initial state preparation and parity measurement processes. We present numerical simulations based on the single-qubit depolarizing model, where the depolarizing rate is calibrated by matching the fidelity of the prepared logical state, |0_L\rangle. As shown in Fig. 4b–c, this noise model agrees well with the experimental data, regardless of the initial states. Detailed results for both uncorrected and corrected expectation values of the Z_L observable, with the initial state set to |\psi_L\rangle, are presented in Fig. 4c; see Supplementary Fig. 7 for more experimental results. To implement ZNE, we insert random Pauli gates on data qubits before and after the parity-check measurements to amplify the noise. In surface codes, quantum computing is driven by the parity-check measurements applied on a lattice deforming with time in protocols such as braiding transformation16 and lattice surgery38. ZNE can be applied to such circuits through the same noise amplification strategy as taken in our experiment. By incorporating ZNE, the residual errors in \langle Z_L \rangle and \langle X_L \rangle are further reduced beyond error correction (see Fig. 4d and Supplementary Fig. 7). We find that the improvement obtained by applying ZNE to a surface code is less pronounced than that for a repetition code. We attribute this difference to imperfections in the preparation of logical states. By employing the stabilizer-measurement approach demonstrated in ref. 1, the error in the prepared initial state can be exponentially suppressed as the code distance increases. Consequently, the efficacy of ZNE could be further enhanced—a direction we leave for future exploration.

Methods
Two quantum processors, named Processor I and Processor II, are used in this work for experiments of repetition code and surface code, respectively. Processor I consists of a 6\times 6 transmon qubit array, from which 13 qubits are utilized in the experiment. Processor II has the same architecture as Processor I, but with a qubit array of 11\times 11. In the distance-3 surface code experiments, 17 qubits are used. The median T_1 and T_{2E} times for the qubits used in Processor I (Processor II) are 124 \mu s and 11 \mu s (128 \mu s and 16 \mu s), respectively. In this work, we adopt two different readout schemes: (1) The conventional measurement for multi-round experiments; (2) using state |2\rangle to reduce measurement errors for one-round experiments. The gate and measurement errors of two processors are visualized in Fig. 5.

ZNE protocol
To implement the Kth-order ZNE, we use K+1 data points from experiments, {(r_0, \langle O \rangle(r_0)), (r_1, \langle O \rangle(r_1)), \dots, (r_K, \langle O \rangle(r_K))}. Here, \langle O \rangle(r) denotes the expected value of the observable when error probabilities are amplified by a factor of r, and r_0=1 corresponds to the case without noise amplification. Using these data points and the fitting formula \langle O \rangle(r) = \langle O \rangle_{em} + \sum_{k=\lceil d/2 \rceil}^{\lceil d/2 \rceil + K - 1} a_k r^k, we can compute \langle O \rangle_{em} as

\langle O \rangle_{em} = \sum_{k=0}^{K} b_k \langle O \rangle(r_k) (3)

where b_k = adj(V)_{0k}, and V is a Vandermonde-like matrix with elements

V_{ij} = \begin{cases} 1, & j=0, \\ r_i^{\lceil d/2 \rceil + j - 1}, & j>0 \end{cases} (4)

We characterize the performance of ZNE using two metrics: the bias \delta = |\langle O \rangle_{em} - \langle O \rangle_{ideal}| characterizes the accuracy, and the sampling overhead \eta characterizes the cost. Next, we focus on the overhead \eta. Suppose N_{tot} is the total number of circuit shots used to acquire the K+1 data points. To reduce the variance in ZNE, we allocate the shots among the data points according to the importance sampling principle. Specifically, the number of shots used to measure \langle O \rangle(r_k) is N_k = \frac{|b_k|}{\sum_{j=0}^{K} |b_j|} N_{tot}. The variance in the measurement of \langle O \rangle(r_k) is then given by \sigma_k^2 = \frac{1 - \langle O \rangle^2(r_k)}{N_k}, and the variance of \langle O \rangle_{em} is thus \sigma_{em}^2 = \sum_{k=0}^{K} |b_k|^2 \sigma_k^2. Here we have assumed that the observable O is a Pauli operator. Without ZNE, if the N_{tot} shots are used to measure \langle O \rangle(r_0), the variance is \sigma_{raw}^2 = \frac{1 - \langle O \rangle^2(r_0)}{N_{tot}}. The sampling overhead reads \eta = \frac{\sigma_{em}^2}{\sigma_{raw}^2}, which is the ratio of the two variances.

Data availability
The data generated in this study have been deposited in the Zenodo database under accession code https://doi.org/10.5281/zenodo.17744500.

Code availability
The codes are available upon request from the corresponding author.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors Zhang et al. study error correction circuits (repetition codes and surface code) on two superconducting quantum processors, insert Pauli errors at various rates, decode the logical observables, and apply noise mitigation techniques to improve estimates of logical observables.\nCombining error decoding of logically encoded qubits with error mitigation techniques has a good chance to build a bridge between noisy intermediate-scale quantum (NISQ) circuits and fully error-corrected circuits.\nThe work therefore addresses a timely topic.\nThe experiments seem to have been carried out on state-of-the-art superconducting quantum processing units featuring high-fidelity gates and readout.\nThe paper is well written and likely resonate with a broader readership.\nGenerally, I therefore recommend publication in Nat. Comm. assuming that the authors seriously address the following comments and concerns in their revised manuscript:\n1) The first experimental demonstration studies a circuit which prepares three copies of the same state |theta> = R(theta)|0>, maps pairwise parity onto two ancillas, applies artificial errors, and maps parity again to detect possible errors. While it is instructive to explain the concept on a simple example, I find the particular choice more confusing then helpful, because the essence is hidden in an unnecessary complication.\nSince all instances are rejected in which an error E_2 gets detected, the entire circuit does nothing but checking if a bit flip has occurred on Q0 or not. This could be detected in a much more straightforward way by using only a second qubit Q1, perform the two CNOT gates and measure mu1. All actions involving qubits Q2, Q3, Q4 are therefore completely irrelevant. As such the example and its discussion could be greatly simplified making the main point clearer.\n2) In all the numerical simulation in which observables are simulated vs. noise parameter r (e.g. dashed lines in Fig. 3e), the curves seems to exhibit a slight oscillatory behavior – at least less smooth than one may naively expect when varying a noise parameter continuously. Can the authors clarify?\n3) The third paragraph on pg. 5starts provides a general discussion and resource and performance estimates when scaling to larger systems. Why is this not part of the general discussion but placed between the repetition code and surface code results?\n4) In Fig. 3d the scatter plots appear for the first time and the authors state that extrapolation is done “for all possible choices of r1, … , rK”. This needs to be explained in the main text more clearly what this exactly means.\n5) The discussion about the surface code results falls short. Streamlining the first example as pointed out above in 1) will free up space for extending the discussion on the surface code.", "reviewer_2": "Summary\nThe manuscript reports an experimental study that integrates quantum error correction (QEC) with a practically used error mitigation strategy, zero-noise extrapolation (ZNE), on superconducting processors. The authors implement ZNE at the physical qubit level by deliberately amplifying noise and then extrapolating expectation values to the zero-noise limit using polynomial fitting. They test this approach on repetition and surface codes up to distance seven, showing improved extraction of expectation values and demonstrating that ZNE remains effective in both multi-round error-correction circuits and surface-code experiments. The authors argue that this hybrid strategy can enhance the performance of early fault-tolerant quantum computers and could be extended to more complex logical operations.\nReview\nThe manuscript is technically strong and makes a novel contribution, as error mitigation has so far been demonstrated mainly on noisy intermediate-scale quantum (NISQ) circuits rather than on logical qubits. The hardware performance (coherence times, CZ fidelities, readout fidelities) is state-of-the-art, enabling meaningful improvements from logical encoding and feedback-based error correction. The results clearly show reduced logical error bias compared to both unmitigated and purely error-corrected cases. Experimental data are consistent with numerical simulations, supporting the validity of the noise modeling.\nThe paper is generally well written, though somewhat dense in places. Clarifying certain definitions and presenting figures with mild modifications would significantly improve accessibility and reader comprehension.\n1. Page 2, left column: It was unclear what the authors meant by ‘K non-constant terms’.\n2. Fig.1: I appreciate this thoughtful presentation trying to convey the main message of the paper. However, it took me quite some time to understand that the blue coin represented a noise-free qubit. It would be helpful to mention that explicitly in the caption.\n3. Fig.2a: The figure is slightly misleading as the measurements were performed simultaneously and the conditional X gate was applied in software. I would suggest to aligning all the measurement symbols vertically and modify it to reflect the NOT operation being applied afterwards.\n4. Fig.2 caption: It was unclear why r=1 and 3 only were used. Please elaborate.\n5. Page 3, left column: While the author mentioned about known schemes for increasing ‘r’, they didn’t mention which scheme was actually used in their experiments.\n6. Page 3, left column: It would be nice to mention the mapping of the measurement outcome to the qubit state, i.e. \\mu = -1 indicates the qubit is in the excited state.\n7. Fig.3b: Are the syndrome qubits reset to |0> after each measurement? If yes, the figure should be modified accordingly.\n8. Fig.3c: It would be nice to mention M=1 in the figure.\n9. Page 5, left column: The sentence ‘.. can reduce the logical error by a factor of about 0.025..’ reads a bit strange. While I understand the intention of the authors, it may be better to use the reciprocal of 0.025.\nRecommendation:\nThe manuscript introduces a novel concept supported by clear experimental demonstrations with potential near-term relevance for quantum computing hardware. I recommend acceptance after the authors address the points raised above.", "reviewer_3": "This manuscript demonstrates a series of experiments whereby the zero-noise extrapolation (ZNE) technique is applied to error-corrected qubits on superconducting quantum processors. To bypass the challenges that arise when applying ZNE to logical errors, the authors apply the technique to physical errors in the logical system; they derive the fitting function analytically, design the experimental protocol, and present thorough analyses. They show reductions of logical errors using the repetition code and surface code across multiple code distances and numbers of parity check repetitions, and specify the sampling overhead for each case. Experimental details, including the protocol and implementation tricks, were discussed in the supplementary material, which would be useful for others to perform similar experiments.\nThe paper is well-written, with good motivations in the introduction, detailed justifications of their experiments, and an in-depth analysis of the resources and performance. To the best of our knowledge, this is the first experimental demonstration of a quantum error mitigation (QEM) technique being applied to an error-corrected system, successfully reducing logical errors. The results of this work may inspire similar approaches in the future, including those that aim for early demonstrations of practical quantum computation -- this work may provide useful guidelines. Therefore, we would like to recommend this manuscript for publication in Nature Communications, given that the following comments are appropriately addressed:\n1. The authors mention in the Discussion that this is the first time that a QEM technique has been applied to an error-corrected system. If this is the case, we suggest stating this in the introduction, and highlighting that the cited IBM ZNE experiment was performed on a few hundred uncorrected qubits.\n2. In the introduction, we recommend explicitly comparing your technique to Wahl et al., who considered ZNE on logical qubits via distance-scaling. Would this prior work be more efficient in terms of the sampling overhead?\n3. FIG. 1 can be made clearer by explaining in the caption the difference between the paired lines and shadows, as well as by using the same colour for the pair.\n4. In the Results section, to improve clarity, we recommend including an overview of the experiments that were run in between the device descriptions and the experiments (or elsewhere), listing the 1) parity check experiment with and without error correction, 2) repetition code experiment, and 3) surface code experiment. Optionally, structuring into titled subsections would allow the readers to more easily navigate through the paper.\n5. Since you inject noise using random Pauli gates, you could also apply probabilistic error cancellation (PEC) instead. Could you comment on the choice of using ZNE? More generally, how much of this work is applicable to other QEM techniques?\n6. In the supplementary material S5, you mention that error bars are too small to visualize in the main text. This should be stated in the corresponding figure captions in the main text, or else readers would not be sure how reliable the data are.\n7. How did authors choose the unit error probabilities? Are they the minimum achievable error rates on the devices?\n8. Please explain what the term “operational stage” means, and mention at this point that the experiments in the main text do not consider errors in stabiliser measurements, but the supplementary materials do.\n9. In FIG. 2, could authors comment on why the extrapolation is performed using only two data points, even though many other data points were taken?\n10. Authors state that “our protocol can reduce the logical error by a factor of about 0.025”, but “reduced by a factor of” means dividing by the number. We think the correct phrasing should be along the lines of “reduced by a factor of 40”.\n11. In FIG. 3, it is not immediately clear that subfigures c and e correspond to the unmitigated case. Moreover, could you explain why the x axes of subfigures c and e have different scales? Is it because subfigure e deals with a “fixed total error rate in all rounds”? It is a bit difficult to compare subfigures in FIG. 3.\n12. “In surface codes, quantum computing are driven by the parity-check measurements” should replace “are” with “is”.\n13. In the caption of FIG. 2b, could authors mention that this is experimental data? Moreover, in the caption of FIG. 4b, it can be made clearer which results are error-corrected and which are not.\n14. Regarding FIG. 3, it is noted that “the performance of ZNE can be further improved by accounting for parity-check measurement errors in the noise amplification”. This is noted towards the end of the following paragraph, but we think it is already worth mentioning here that you observe that the improvement does not diminish with the increase of code distance, as opposed to what FIG. 3d suggests.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "List of changes\nAllchangesaremarkedinredintherevisedmaintext.\n• Theintroductionwasslightlymodifiedtobetteremphasizethenoveltyofourwork.\n• We improved the structure of the Results section by adding a sentence outlining our three\nexperimentsandbyintroducingsubsectiontitles.\n• The discussion on scalability has been moved from the Results section to the Discussion\nsection.\n• Figure1anditscaptionhavebeenrevisedforclarity;thefigurenowusesthesamecolorfor\neachline–shadowpair.\n• Figure2hasbeenupdatedbyverticallyaligningthemeasurementsymbolsandrevisingthe\ncaption.\n• Figure3anditscaptionhavebeenrevisedforclarity.\n• Typosandmisleadingpresentationswerecorrected.\nPoint-by-Point Response\nResponsetoReviewer#1\nThe authors Zhang et al. study error correction circuits (repetition codes and surface code) on\ntwo superconducting quantum processors, insert Pauli errors at various rates, decode the logical\nobservables, and apply noise mitigation techniques to improve estimates of logical observables.\nCombiningerrordecodingoflogicallyencodedqubitswitherrormitigationtechniqueshasagood\nchance to build a bridge between noisy intermediate-scale quantum (NISQ) circuits and fully\nerror-corrected circuits. The work therefore addresses a timely topic. The experiments seem to\nhavebeencarriedoutonstate-of-the-artsuperconductingquantumprocessingunitsfeaturinghigh-\nfidelitygatesandreadout. Thepaperiswellwrittenandlikelyresonatewithabroaderreadership.\nGenerally, I therefore recommend publication in Nat. Comm. assuming that the authors seriously\naddressthefollowingcommentsandconcernsintheirrevisedmanuscript:\nWe are grateful for the reviewer’s favorable evaluation and insightful feedback, and we have ad-\ndressedtheirspecificcommentsindetailinthefollowingresponse.\nComment 1: The first experimental demonstration studies a circuit which prepares three copies of the same state |θ⟩ = R(θ)|0⟩, maps pairwise parity onto two ancillas, applies artificial errors,\nand maps parity again to detect possible errors. While it is instructive to explain the concept on a\nsimple example, I find the particular choice more confusing then helpful, because the essence is\nhiddeninanunnecessarycomplication.\nSince all instances are rejected in which an error E gets detected, the entire circuit does nothing\n2\nbut checking if a bit flip has occurred on Q or not. This could be detected in a much more\n0\nstraightforward way by using only a second qubit Q , perform the two CNOT gates and measure\n1\nµ . All actions involving qubits Q , Q , Q are therefore completely irrelevant. As such the\n1 2 3 4\nexampleanditsdiscussioncouldbegreatlysimplifiedmakingthemainpointclearer.\nWe thank the reviewer for suggesting the two-qubit example circuit. We would like to clarify\nthat ourfive-qubit circuit is deliberately designedto incorporatethe followingkey ingredients: (i)\ncapability of error correction, (ii) feedback operation, (iii) post-selection and (iv) non-stabilizer\nstate. These elements are essential components for universal fault-tolerant quantum computation.\nA two-qubit version would indeed allow for an illustration of the basic idea, but it would lack\neither feedback or post-selection. To include both, at least two detection qubits (Q and Q ) are\n1 3\nrequired.\nComment2: Inallthenumericalsimulationinwhichobservablesaresimulatedvs. noiseparam-\neter r (e.g. dashed lines in Fig. 3e), the curves seems to exhibit a slight oscillatory behavior – at\nleast less smooth than one may naively expect when varying a noise parameter continuously. Can\ntheauthorsclarify?\nThe slight oscillations arise because the data are generated via Monte Carlo simulations: the ob-\nservedzigzagbehaviorisduetostatisticalfluctuations. AsshowninFig.R1ofthisreply,increas-\ningthenumberofsimulatedcircuitinstancesto50timestheoriginalnumbers(seeTableS2inour\nSupplementaryInformation)producesmuchsmoothercurves.\n1.0\n0.9\n0.8\n1 1.5 2 2.5\nr\nZ L\n®\n®\nM = 1\nM = 2\nM = 3\nM = 4 Simulation\nFIG.R1. SimilartoFig.3einthemaintext,butwithmoresimulatedcircuitinstances.\nComment 3: The third paragraph on pg.5 starts provides a general discussion and resource and\nperformance estimates when scaling to larger systems. Why is this not part of the general discus-\nsionbutplacedbetweentherepetitioncodeandsurfacecoderesults?\nSincewepresentedtheexperimentalresultsofscalingthecodedistanceinthepreviousparagraph,\nwe believe it is natural to provide further discussion of this topic using theoretical analysis and\nnumericalsimulationsasafollow-up. Thatsaid,weagreewiththereviewerthatitcanbeincluded\naspartofthegeneraldiscussion,andhaverevisedthemanuscriptaccordingly.\nComment4: InFig. 3dthescatterplotsappearforthefirsttimeandtheauthorsstatethatextrap-\nolation is done “for all possible choices of r , ... , r ”. This needs to be explained in the main\ntextmoreclearlywhatthisexactlymeans.\nWethankthereviewerforpointingthisout. Inthepreviousversion,weclarifiedthisimmediately\nfollowingtheEq.2,stating“Unlessotherwisespecified,wetaker = 1inwhatfollowsanditerate\n0\noverallotherdatapointsforr ,··· ,r .” Werealizedthatthisdescriptionisnotclearordetailed;\ntherefore, we have expanded it to “Unless otherwise specified, we take r = 1 in what follows\n0\nand select the other K data points from the remaining collected dataset (see, e.g., Fig. 3c). Each\ncombinationproducesacorrespondingbiasandsamplingoverhead.”\nComment 5: The discussion about the surface code results falls short. Streamlining the first\nexample as pointed out above in 1) will free up space for extending the discussion on the surface\ncode.\nWeagreewiththereviewerthatthediscussionofthesurfacecodecanbeexpanded. Intherevised\nmanuscript, we have included additional insights into the observed performance gain, as follows:\nWe find that the improvement obtained by applying zero-noise extrapolation (ZNE) to a surface\ncode is less pronounced than that for a repetition code. We attribute this difference to imper-\nfections in the preparation of logical states. By employing the stabilizer-measurement approach\ndemonstrated in Ref. [1], the error in the prepared initial state can be exponentially suppressed as\nthe code distance increases. Consequently, the efficacy of ZNE could be further enhanced — a\ndirectionweleaveforfutureexploration.\nResponsetoReviewer#2\nSummary\nThemanuscriptreportsanexperimentalstudythatintegratesquantumerrorcorrection(QEC)with\na practically used error mitigation strategy, zero-noise extrapolation (ZNE), on superconducting\nprocessors. The authors implement ZNE at the physical qubit level by deliberately amplifying\nnoise and then extrapolating expectation values to the zero-noise limit using polynomial fitting.\nThey test this approach on repetition and surface codes up to distance seven, showing improved\nextractionofexpectationvaluesanddemonstratingthatZNEremainseffectiveinbothmulti-round\nerror-correctioncircuitsandsurface-codeexperiments. Theauthorsarguethatthishybridstrategy\ncan enhance the performance of early fault-tolerant quantum computers and could be extended to\nmorecomplexlogicaloperations.\nReview\nThe manuscript is technically strong and makes a novel contribution, as error mitigation has so\nfar been demonstrated mainly on noisy intermediate-scale quantum (NISQ) circuits rather than\non logical qubits. The hardware performance (coherence times, CZ fidelities, readout fidelities)\nis state-of-the-art, enabling meaningful improvements from logical encoding and feedback-based\nerrorcorrection. Theresultsclearlyshowreducedlogicalerrorbiascomparedtobothunmitigated\nand purely error-corrected cases. Experimental data are consistent with numerical simulations,\nsupporting the validity of the noise modeling. The paper is generally well written, though some-\nwhatdenseinplaces. Clarifyingcertaindefinitionsandpresentingfigureswithmildmodifications\nwouldsignificantlyimproveaccessibilityandreadercomprehension.\nWe are encouraged by the reviewer’s positive assessment of our work. We also thank them for\ntheir constructive suggestions, which have helped us improve the manuscript. Our point-by-point\nresponsesareprovidedbelow.\nComment 1: Page 2, left column: It was unclear what the authors meant by ‘K non-constant terms’.\nWe thank the reviewer for pointing this out. In the revised manuscript, we have clarified the\nwordingto: “K termsoftheformrk,startingfromther⌈d/2⌉ term.”\nComment 2: Fig. 1: I appreciate this thoughtful presentation trying to convey the main message\nof the paper. However, it took me quite some time to understand that the blue coin represented a\nnoise-freequbit. Itwouldbehelpfultomentionthatexplicitlyinthecaption.\nWe thank the reviewer for this helpful suggestion. In the revised manuscript, we have clarified in\nthecaptionbyadding: “Schematicsalongsidethecurvesshowthebasicconcept,wherebluecoins\nrepresent noise-free physical qubits, cylinders on top of the coins denote noise, and green squares\ncontainingmultiplecoinsarelogicalqubits.”\nComment 3: Fig. 2a: The figure is slightly misleading as the measurements were performed\nsimultaneously and the conditional X gate was applied in software. I would suggest to aligning\nall the measurement symbols vertically and modify it to reflect the NOT operation being applied\nafterwards.\nAs suggested, we have updated Fig. 2: all measurement symbols are now vertically aligned, and\nthe conditioned X gate is now depicted with a dashed outline to indicate it is implemented in\nsoftware.\nComment4: Fig. 2caption: Itwasunclearwhyr = 1and3onlywereused. Pleaseelaborate.\nHere, we used two data points (K = 1) for the extrapolation, as this simple setting was sufficient\nto demonstrate the effectiveness of ZNE. We chose r = 1 and r = 3 because this pair gave the\nmostaccurateresultamongalltwo-pointcombinationsofthe experimental data. Wehaverevised\nthecaptionofFig.2accordingly.\nComment5: Page3,leftcolumn: Whiletheauthormentionedaboutknownschemesforincreas-\ning‘r’,theydidn’tmentionwhichschemewasactuallyusedintheirexperiments.\nIn the revised manuscirpt, we have clarified the scheme taken in our experiments: “To implement\nZNE,oneneedstobeabletocontrollablyamplifytheerrors,whichcanbeachievedusingschemes\nsuchaspulsestretching [2,3],subcircuitrepetition[4]orsimulatederror[5]. Inourexperiments,\nweusethesimulated-errorapproach.”\nComment 6: Page 3, left column: It would be nice to mention the mapping of the measurement\noutcometothequbitstate,i.e. µ=-1indicatesthequbitisintheexcitedstate.\nWe have clarified this by adding: “Here, the values µ = ±1 correspond to the states |0⟩ and |1⟩,\nrespectively.”\nComment 7: Fig. 3b: Are the syndrome qubits reset to |0⟩ after each measurement? If yes, the\nfigureshouldbemodifiedaccordingly.\nWe did not reset the syndrome qubits to |0⟩ after each measurement; therefore, the figure is not\nmodified.\nComment8: Fig. 3c: ItwouldbenicetomentionM = 1inthefigure.\nWehaveupdatedFig.3accordingtothissuggestion.\nComment 9: Page 5, left column: The sentence ‘.. can reduce the logical error by a factor of\nabout0.025..’ readsabitstrange. WhileIunderstandtheintentionoftheauthors,itmaybebetter\ntousethereciprocalof0.025.\nWe addressed this byrevising thesentence to“...can reducethe logicalerror bya factorof about\n40...”\nRecommendation:\nThe manuscript introduces a novel concept supported by clear experimental demonstrations with\npotentialnear-termrelevanceforquantumcomputinghardware. Irecommendacceptanceafterthe\nauthorsaddressthepointsraisedabove.\nResponsetoReviewer#3\nThismanuscriptdemonstratesaseriesofexperimentswherebythezero-noiseextrapolation(ZNE)\ntechnique is applied to error-corrected qubits on superconducting quantum processors. To bypass\nthe challenges that arise when applying ZNE to logical errors, the authors apply the technique to\nphysical errors in the logical system; they derive the fitting function analytically, design the ex-\nperimental protocol, and present thorough analyses. They show reductions of logical errors using\nthe repetition code and surface code across multiple code distances and numbers of parity check\nrepetitions, and specify the sampling overhead for each case. Experimental details, including the\nprotocol and implementation tricks, were discussed in the supplementary material, which would\nbeusefulforotherstoperformsimilarexperiments.\nThepaperiswell-written,withgoodmotivationsintheintroduction,detailedjustificationsoftheir\nexperiments,andanin-depthanalysisoftheresourcesandperformance. Tothebestofourknowl-\nedge, this is the first experimental demonstration of a quantum error mitigation (QEM) technique\nbeingappliedtoanerror-correctedsystem,successfullyreducinglogicalerrors. Theresultsofthis\nwork may inspire similar approaches in the future, including those that aim for early demonstra-\ntions of practical quantum computation – this work may provide useful guidelines. Therefore, we\nwould like to recommend this manuscript for publication in Nature Communications, given that\nthefollowingcommentsareappropriatelyaddressed:\nWesincerelythankthereviewerfortheirencouragingremarksandhelpfulsuggestionstoimprove\nourmanuscript. Wehavecarefullyaddressedthecommentsbelow.\nComment 1: The authors mention in the Discussion that this is the first time that a QEM tech-\nnique has been applied to an error-corrected system. If this is the case, we suggest stating this in\nthe introduction, and highlighting that the cited IBM ZNE experiment was performed on a few\nhundreduncorrectedqubits.\nWe thank the reviewer for this helpful suggestion to better highlight the novelty of our work. As\nsuggested, we have updated the introduction by adding “and report the first experimental demon-\nstrationofthisapplication”andbyspecifyingthatIBM’sexperimentwasperformedon“physical\nqubits”.\nComment 2: In the introduction, we recommend explicitly comparing your technique to Wahl et\nal., who considered ZNE on logical qubits via distance-scaling. Would this prior work be more\nefficientintermsofthesamplingoverhead?\nIn the revised introduction, we have included an explicit comparison. For the reviewer’s refer-\nence, we provide more information below comparing the two methods in terms of flexibility and\nsamplingoverhead.\nFlexibility. Distance scaling imposes a minimum amplification factor, while our simulated-error\napproachallowsfine-tuning. Forexample,inthesurfacecodethelogicalerrorratecanbeapprox-\nimatedas\n(cid:18) p\n(cid:19)⌈d/2⌉\np = p ,\nL 0\np\nth\nwhere p = 1% is the threshold. Reducing the distance by two yields an amplification factor of\nth\np/p . On current superconducting devices with p ≈ 0.5%, this corresponds to a factor of about\nth\ntwo,whichissuitableforZNE.However,ifpisreducedto0.1%—whichwebelieveisachievable\nin the near future considering recent advancements in gate fidelity and qubit coherence time —\ntheminimumamplificationfactorrisestoten,whichmaypushthecircuitoutoftheregimewhere\nlow-orderZNEiseffective.\nSampling Overhead. The advantage of distance scaling is that it eliminates the need for random-\nized circuit compilation, which can dominate runtime on many current devices since compilation\nmay take much longer than circuit execution. While both approaches require the same number of\ncircuit executions (samples), the simulated-error approach incurs an additional compilation cost.\nThus, when compilation is expensive, distance scaling can be more efficient; otherwise, the costs\narecomparable.\nAccordingly, the revised introduction now states: We note that one can also amplify logical noise\nthrough code-distance scaling, which offers limited tunability but avoids the need for randomized\ncircuitcompilation[Wahletal.].\nComment 3: FIG. 1 can be made clearer by explaining in the caption the difference between the\npairedlinesandshadows,aswellasbyusingthesamecolourforthepair.\nFollowingthesuggestions,wehaveupdatedFig.1inthemaintextasfollows:\n1. Thecaptionhasbeenrevisedtoinclude: “TheexpectationvalueofanobservableOchanges\nwiththenoisestrengthr,asshownbythethreecurves(gold: physicalqubit;skyblue: low-\ndistancelogicalqubit;purple: high-distancelogicalqubit).”\n2. Wenowusethesamecolorforeachline–shadowpair.\nComment 4: In the Results section, to improve clarity, we recommend including an overview of\nthe experiments that were run in between the device descriptions and the experiments (or else-\nwhere),listingthe1)paritycheckexperimentwithandwithouterrorcorrection,2)repetitioncode\nexperiment, and 3) surface code experiment. Optionally, structuring into titled subsections would\nallowthereaderstomoreeasilynavigatethroughthepaper.\nWearegratefulforthereviewer’ssuggestiontoenhancetheclarityofourpaper. Asrecommended,\nwe have added an overview of the three experiments to the Results section and also introduced\nsubsectiontitles.\nComment 5: Since you inject noise using random Pauli gates, you could also apply probabilis-\ntic error cancellation (PEC) instead. Could you comment on the choice of using ZNE? More\ngenerally,howmuchofthisworkisapplicabletootherQEMtechniques?\nWe thank the reviewer for this insightful question. We chose ZNE primarily because it provides\na practical and straightforward method for error mitigation on logical qubits that crucially avoids\nthe need to benchmark the complex logical error model. While other quantum error mitigation\n(QEM)techniques,suchasprobabilisticerrorcancellation(PEC),couldinprinciplebeapplicable\nto logical qubits, their exact implementation remains an area of active research. Our reasoning is\ndetailedbelow.\nTheChallengeofBenchmarkingtheLogicalErrorModel\nTheprimaryobstacleinapplyingmanyadvancedQEMtechniquestologicalqubitsisthedifficulty\nof accurately characterizing the logical error model. This task is practically infeasible for several\nreasons:\nExtremely Low Error Rates. Quantum error correction (QEC) suppresses logical error rates to be\norders of magnitude lower than physical error rates. For example, with the gross code (the code\n[[144,12,12]]fromthebivariatebicyclefamily)[6],aphysicalerrorrateof0.1%leadstoalogical\nerrorrateof2×10−7.\nHigh Dimensionality and Correlations. Logical errors in advanced codes (like quantum low-\ndensity parity-check codes, qLDPC) exhibit complex, multi-qubit correlations. Fully charac-\nterizing the error model for a 12-qubit logical block would require estimating approximately\n412 ≈ 1.7×107 parameters,eachatthelevelof2×10−7/412 ≈ 1.2×10−14.\nZNEonLogicalQubits\nZNE can be implemented by amplifying the underlying physical noise (e.g., by inserting physical\nPauli gates), which indirectly boosts the logical error rate. This approach, which we adopted in\nthiswork,allowsforextrapolationtothezero-noiselimitwithoutrequiringanyexplicitknowledge\nor characterization of the logical error model. It is worth noting that the sampling cost in ZNE,\naside from benchmarking, is determined by the total logical error rate together with the amplified\nlogicalerrorrate. Therefore,insertingphysicalPauligatesandinsertinglogicalPauligatesdonot\ndiffersignificantlyinthisrespect.\nPEConLogicalQubits\nForPEC,however,usingphysicalnoiseinjectionisnotaviableapproach. Thesamplingoverhead\nof PEC scales with the total error rate. If we inject noise at the logical level, this overhead is\ndetermined by the logical error rate. In this setting, QEC and PEC act synergistically: QEC\nsuppresses logical errors, thereby reducing the sampling overhead to a tolerable level, while PEC\neliminates the residual bias. In contrast, if we inject noise at the physical level, the sampling\noverhead is set by the physical error rate, and QEC offers no benefit, since it does not reduce\nphysicalerrors.\nWe note that some of the authors are now pursuing a new theoretical protocol for PEC that cir-\ncumvents the need for benchmarking the logical error model [7], though it is not yet feasible for\ncurrentdevices.\nOtherQEMTechniques\nWealsoexaminedthebroadersetofQEMtechniquesreviewedinRef.[8]. Mostofthesemethods\ncan,atleastinprinciple,beappliedtologicalqubits:\n• ZNE—demonstratedinthiswork;\n• PEC—itsapplicationtologicalqubitswasfirststudiedinRef.[9];\n• Measurementerrormitigation;\n• Symmetryconstraints;\n• Purityconstraints;\n• Subspaceexpansions;\n• Nrepresentability.\nThemainclassthatfacesdifficultiesonlogicalqubitsis\n• Learning-basedmethods,\nincluding learning-based PEC and Clifford regression. For these methods, a model that predicts\nmitigated values from noisy experimental data is trained using data from circuits that can be ef-\nficiently simulated on classical computers. A practical approach is to replace non-Clifford gates\nwith Clifford gates. However, this approach cannot be directly applied to QEC circuits, because\nnon-Cliffordgatesareimplementedviamagicstatesandexhibiterrorratesthatdiffersignificantly\nfromthoseofCliffordgates.\nAlthough many QEM approaches can reduce bias on logical qubits, only ZNE and PEC are sys-\ntematic methods with the potential to approach bias-free quantum computation. For example,\nsymmetry-based methods enforce that the output respects certain symmetries, but symmetry-\npreservingerrorsmaystillpersist.\nIn summary, a central challenge for ZNE and PEC is benchmarking the error model. Our work\nspecifically addresses this by developing and experimentally demonstrating a protocol that avoids\nbenchmarkingthelogicalerrormodel,makingitsuitablefornear-termfault-tolerantexperiments.\nComment 6: In the supplementary material S5, you mention that error bars are too small to\nvisualize in the main text. This should be stated in the corresponding figure captions in the main\ntext,orelsereaderswouldnotbesurehowreliablethedataare.\nWe have clarified in the caption of Fig. 3 by adding: “The standard errors of ⟨Z ⟩ are small, as\nL\nshowninSupplementaryFig.S4.”\nComment7: Howdid authorschoose theunit error probabilities? Are theytheminimum achiev-\nableerrorratesonthedevices?\nWe deliberately set the error probability to p = 3.6% in our experiments to ensure that the com-\nparisons between uncorrected and corrected results are clear and distinct. Although smaller error\nprobabilitiescanbeachievedonourdevices,theymightresultinsubtlerdifferencesthatareharder\ntodetectorvisualize.\nComment 8: Please explain what the term “operational stage” means, and mention at this point\nthat the experiments in the main text do not consider errors in stabiliser measurements, but the\nsupplementarymaterialsdo.\nTheoperationalstageisintendedforrunningquantumalgorithms;however,inourproof-of-concept\nexperiments we keep it idle. To clarify this, we have revised the caption of Fig. 2 to “...during\nthe operational stage intended for running quantum algorithms, ...” In addition, following the\nReviewer’s suggestion, we have added: “In the following experiments, we neglect measurement\nerrorsintheerrormitigationsetup;adetaileddiscussionisprovidedinSupplementarySectionS6.”\nComment9: InFIG.2,couldauthorscommentonwhytheextrapolationisperformedusingonly\ntwodatapoints,eventhoughmanyotherdatapointsweretaken?\nTwo-point extrapolation (K = 1) is the simplest, whereas using r = 1 and r = 3 gives the most\naccurateextrapolationresultsamongalltwo-pointcombinationsoftheexperimentaldata.\nComment 10: Authors state that “our protocol can reduce the logical error by a factor of about\n0.025”,but“reducedbyafactorof”meansdividingbythenumber. Wethinkthecorrectphrasing\nshouldbealongthelinesof“reducedbyafactorof40”.\nWehavefixeditassuggested.\nComment 11: In FIG. 3, it is not immediately clear that subfigures c and e correspond to the\nunmitigatedcase. Moreover,couldyouexplainwhythexaxesofsubfigurescandehavedifferent\nscales? Isitbecausesubfigureedealswitha“fixedtotalerrorrateinallrounds”? Itisabitdifficult\ntocomparesubfiguresinFIG.3.\nAdded “unmitigated” to the caption of Fig. 3c for clarity. Moreover, it is true that the difference\nbetween the two panels is because panel e deals with a “fixed total error rate in all rounds”. Be-\ncause the logical error rate is exponentially suppressed as the code distance increases, we used\nlarger unit probabilities from Supplementary Table S3 for the curves in panel e (d = 7) to make\nthem clearly distinguishable. Note that, as a result, the data in the two panels are not directly\ncomparable. To clarify this, we also revised the corresponding main text to “Second, for a fixed\ntotalerrorrateinallrounds(seeSupplementaryTableS3),partitioningthe...”\nComment 12: “In surface codes, quantum computing are driven by the parity-check measure-\nments”shouldreplace“are”with“is”.\nThetypohasbeencorrected.\nComment 13: In the caption of FIG. 2b, could authors mention that this is experimental data?\nMoreover, in the caption of FIG. 4b, it can be made clearer which results are error-corrected and\nwhicharenot.\nAs suggested, we have revised the caption of Fig. 2b to “Experimental uncorrected (left) and\ncorrected (right) expectation values ...”, and the caption of Fig. 4b to “...the subsequent plots\nshowthecorrectedresultsunderdifferentnoisyscalingfactors...”\nComment14: RegardingFIG.3,itisnotedthat“theperformanceofZNEcanbefurtherimproved\nby accounting for parity-check measurement errors in the noise amplification”. This is noted\ntowards the end of the following paragraph, but we think it is already worth mentioning here that\nyouobservethattheimprovementdoesnotdiminishwiththeincreaseofcodedistance,asopposed\ntowhatFIG.3dsuggests.\nWe thank the reviewer for pointing out this. As they correctly noted, our intention was to convey\nthat“improvementdoesnotdiminishwiththeincreaseofcodedistance,asopposedtowhatFig.3d\nsuggests.” Wehaverevisedthetexttostatethisexplicitly.\nResponsetoReviewer#4\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriaterecognitionforEarlyCareerResearcherswhoco-reviewmanuscripts.\nWe appreciate the reviewer’s insightful comments and valuable suggestions. Please refer to our\nresponseabove.\nResponsetoReviewer#5\nI co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is\npart of the Nature Communications initiative to facilitate training in peer review and to provide\nappropriaterecognitionforEarlyCareerResearcherswhoco-reviewmanuscripts.\nWe sincerely thank the reviewer for their thoughtful feedback and constructive comments. Please\nrefertoourresponseabove.\n[1] R.Acharya,L.Aghababaie-Beni,I.Aleiner,T.I.Andersen,M.Ansmann,F.Arute,K.Arya,A.Asfaw,\nN. Astrakhantsev, J. Atalaya, R. Babbush, D. Bacon, B. Ballard, J. C. Bardin, J. Bausch, A. Bengts-\nson, A. Bilmes, S. Blackwell, S. Boixo, G. Bortoli, A. Bourassa, J. Bovaird, L. Brill, M. Broughton,\nD.A.Browne,B.Buchea,B.B.Buckley,D.A.Buell,T.Burger,B.Burkett,N.Bushnell,A.Cabrera,\nJ. Campero, H.-S. Chang, Y. Chen, Z. Chen, B. Chiaro, D. Chik, C. Chou, J. Claes, A. Y. Cleland,\nJ.Cogan,R.Collins,P.Conner,W.Courtney,A.L.Crook,B.Curtin,S.Das,A.Davies,L.D.Lorenzo,\nD.M.Debroy, S.Demura, M.Devoret, A.D.Paolo, P.Donohoe, I.Drozdov, A.Dunsworth, C.Earle,\nT. Edlich, A. Eickbusch, A. M. Elbag, M. Elzouka, C. Erickson, L. Faoro, E. Farhi, V. S. Ferreira,\nL. F. Burgos, E. Forati, A. G. Fowler, B. Foxen, S. Ganjam, G. Garcia, R. Gasca, E´lie Genois, W. Gi-\nang,C.Gidney,D.Gilboa,R.Gosula,A.G.Dau,D.Graumann,A.Greene,J.A.Gross,S.Habegger,\nJ. Hall, M. C. Hamilton, M. Hansen, M. P. Harrigan, S. D. Harrington, F. J. H. Heras, S. Heslin,\nP.Heu,O.Higgott,G.Hill,J.Hilton,G.Holland,S.Hong,H.-Y.Huang,A.Huff,W.J.Huggins,L.B.\nIoffe, S. V. Isakov, J. Iveland, E. Jeffrey, Z. Jiang, C. Jones, S. Jordan, C. Joshi, P. Juhas, D. Kafri,\nH. Kang, A. H. Karamlou, K. Kechedzhi, J. Kelly, T. Khaire, T. Khattar, M. Khezri, S. Kim, P. V.\nKlimov,A.R.Klots,B.Kobrin,P.Kohli,A.N.Korotkov,F.Kostritsa,R.Kothari,B.Kozlovskii,J.M.\nKreikebaum,V.D.Kurilovich,N.Lacroix,D.Landhuis,T.Lange-Dei,B.W.Langley,P.Laptev,K.-M.\nLau, L. L. Guevel, J. Ledford, K. Lee, Y. D. Lensky, S. Leon, B. J. Lester, W. Y. Li, Y. Li, A. T. Lill,\nW. Liu, W. P. Livingston, A. Locharla, E. Lucero, D. Lundahl, A. Lunt, S. Madhuk, F. D. Malone,\nA.Maloney,S.Mandra´,L.S.Martin,S.Martin,O.Martin,C.Maxfield,J.R.McClean,M.McEwen,\nS. Meeks, A. Megrant, X. Mi, K. C. Miao, A. Mieszala, R. Molavi, S. Molina, S. Montazeri, A. Mor-\nvan, R. Movassagh, W. Mruczkiewicz, O. Naaman, M. Neeley, C. Neill, A. Nersisyan, H. Neven,\nM. Newman, J. H. Ng, A. Nguyen, M. Nguyen, C.-H. Ni, T. E. O’Brien, W. D. Oliver, A. Opremcak,\nK.Ottosson,A.Petukhov,A.Pizzuto,J.Platt,R.Potter,O.Pritchard,L.P.Pryadko,C.Quintana,G.Ra-\nmachandran, M. J. Reagor, D. M. Rhodes, G. Roberts, E. Rosenberg, E. Rosenfeld, P. Roushan, N. C.\nRubin,N.Saei,D.Sank,K.Sankaragomathi,K.J.Satzinger,H.F.Schurkus,C.Schuster,A.W.Senior,\nM. J. Shearn, A. Shorter, N. Shutty, V. Shvarts, S. Singh, V. Sivak, J. Skruzny, S. Small, V. Smelyan-\nskiy, W. C. Smith, R. D. Somma, S. Springer, G. Sterling, D. Strain, J. Suchard, A. Szasz, A. Sztein,\nD. Thor, A. Torres, M. M. Torunbalci, A. Vaishnav, J. Vargas, S. Vdovichev, G. Vidal, B. Villalonga,\nC. V. Heidweiller, S. Waltman, S. X. Wang, B. Ware, K. Weber, T. White, K. Wong, B. W. K. Woo,\nC. Xing, Z. J. Yao, P. Yeh, B. Ying, J. Yoo, N. Yosri, G. Young, A. Zalcman, Y. Zhang, N. Zhu, and\nN. Zobrist, “Quantum error correction below the surface code threshold,” (2024), arXiv:2408.13687\n[quant-ph].\n[2] K.Temme,S.Bravyi, andJ.M.Gambetta,Phys.Rev.Lett.119,180509(2017).\n[3] A.Kandala,K.Temme,A.D.Co´rcoles,A.Mezzacapo,J.M.Chow, andJ.M.Gambetta,Nature567,\n491(2019).\n[4] A.He,B.Nachman,W.A.deJong, andC.W.Bauer,Phys.Rev.A102,012426(2020).\n[5] Y.Kim,A.Eddins,S.Anand,K.X.Wei,E.vandenBerg,S.Rosenblatt,H.Nayfeh,Y.Wu,M.Zaletel,\nK.Temme, andA.Kandala,Nature618,500(2023).\n[6] S.Bravyi,A.W.Cross,J.M.Gambetta,D.Maslov,P.Rall, andT.J.Yoder,Nature627,778(2024).\n[7] H. Xie, N. Yoshioka, K. Tsubouchi, and Y. Li, “Unbiased quantum error mitigation without reliance\nonanaccurateerrormodel,” (2025),arXiv:2504.12864[quant-ph].\n[8] Z.Cai,R.Babbush,S.C.Benjamin,S.Endo,W.J.Huggins,Y.Li,J.R.McClean, andT.E.O’Brien,\nRev.Mod.Phys.95,045005(2023).\n[9] Y.Suzuki,S.Endo,K.Fujii, andY.Tokunaga,PRXQuantum3,010345(2022)."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Based on the revised manuscript I recommend publication in Nature Communications.", "reviewer_2": "The authors have satisfactorily addressed my previous concerns as well as those raised by the other referees. I now recommend the manuscript for publication in Nature Communications.", "reviewer_3": "The authors have carefully addressed all of our comments. They have included new sentences, corrected typographical errors, and introduced subsections which have significantly improved the readability of the paper. We also appreciate the authors’ detailed clarifications on how the approach compares with distance-scaled ZNE and other QEM techniques.\nWe noticed two minor typos that currently read “noisy scaling factor” in FIG. 4b and the caption of FIG.4, which we think should both be replaced by “noise scaling factor”. We have no more concerns with the paper apart from this minor additional suggestion, and we recommend the paper for publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

Decision letter