⚛️ quantum physics

Analysis of asymmetric errors in NISQ experiments

This paper extends previous statistical analyses of NISQ experiments by demonstrating that, contrary to theoretical predictions regarding amplitude damping, the Google 2019 "quantum supremacy" data exhibits zero-one fractions determined solely by readout error asymmetry rather than gate errors, a finding supported by simulations on IBM and Google platforms.

Original authors: Gil Kalai, Tomer Shoham, Carsten Voelkmann

Published 2026-09-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Gil Kalai, Tomer Shoham, Carsten Voelkmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the race to build machines that can solve problems beyond the reach of today's computers, scientists are working with a fragile new technology called quantum computing. These machines use tiny particles of matter, known as qubits, which can exist in a state of being both zero and one at the same time, allowing them to process information in ways that classical computers cannot. However, these qubits are incredibly sensitive to their environment. Even the slightest disturbance can cause them to make mistakes, a phenomenon known as noise. To understand if a quantum computer is truly working as intended, researchers must carefully separate the signal of the calculation from the static of these errors. A major focus of recent study has been on "readout errors," which happen when the machine misreads the final state of a qubit, confusing a zero for a one or vice versa. Understanding the precise nature of these mistakes is crucial because it tells scientists whether the machine is performing a genuine quantum calculation or simply producing random noise.

A team of researchers recently turned their attention to a specific question about how these errors behave. They wanted to know if the errors were purely random or if they had a hidden bias. In a perfectly balanced machine, a qubit that should be zero might be read as one just as often as a qubit that should be one is read as zero. However, some theoretical work suggested that the internal operations of the computer, specifically the gates that manipulate the qubits, might introduce a one-sided bias. This theory proposed that as the machine runs, it would naturally push the results toward a higher number of zeros, creating an imbalance that wasn't just a simple measurement mistake but a fundamental drift in the data.

To test this idea, the researchers examined data from the famous 2019 quantum supremacy experiment conducted by Google, which used a processor with 53 qubits. They also looked at data from other quantum computers and from computer simulations designed to mimic these machines. Their goal was to see if the fraction of zeros in the final results was higher than what could be explained by the known measurement errors alone. If the theory were correct, they expected to find a significant surplus of zeros in the experimental data that grew as the number of qubits increased.

The analysis revealed a clear picture that contradicted the theoretical prediction. When the researchers looked at the actual data from Google's 2019 experiment, covering a range of circuits with anywhere from 12 to 53 qubits, they found no evidence of this extra bias toward zeros caused by internal gate operations. The slight imbalance they did observe between zeros and ones reflected the asymmetry of the reported readout errors, though the empirical difference between zeros and ones was roughly 10%-20% lower than the difference expected from the reported error rates alone. In other words, the machine was not secretly pushing the results toward zero due to its internal operations; the observed noise was consistent with the known, physical measurement errors, without the additional skew predicted by theory. This finding held true across the entire range of qubit counts tested in the real-world experiment.

The story was different when the team looked at computer simulations. When they ran similar tests on a simulated version of an IBM quantum computer, the results showed a much larger gap between zeros and ones than the physical error rates could explain. This suggested that the simulation models included effects that were not present in the real Google hardware, or that the simulations were capturing a type of error that the real machine had managed to avoid. The researchers also noted that while the real Google data showed a stable balance, the proportion of ones in the measurements fluctuated over time in a way that suggested the gates themselves were unstable during the experiment, rather than the measurement tools being the source of the drift.

Ultimately, the study provides a reassuring check on the behavior of current quantum hardware. It confirms that for the Google 2019 experiment, the complex internal operations of the computer did not introduce the kind of one-sided error that some theories had predicted. The machine's output was not being skewed by a hidden force pushing it toward zeros; the noise reflected the known asymmetric readout errors without additional gate-induced bias. While simulations showed signs of this bias, the real-world data did not, suggesting that the physical devices are behaving more predictably than some theoretical models had anticipated. This clarity helps scientists refine their understanding of how these machines work and ensures that future claims about their power are built on a solid, accurate foundation of how errors actually behave.

Technical Summary: Analysis of Asymmetric Errors in NISQ Experiments

Problem Statement
This paper extends the authors' previous statistical analysis of Noisy Intermediate-Scale Quantum (NISQ) experiments, specifically focusing on the Google 2019 "quantum supremacy" experiment and related simulations. The central problem addressed is the characterization of asymmetric readout errors in quantum circuits. While previous work ([10], [13]) established tools for symmetric noise models, this study investigates a refined 3-parameter noise model where the probability of a '1' being read as a '0' ( $q_{1\to0}$ ) differs from the probability of a '0' being read as a '1' ( $q_{0\to1}$ ).

A specific motivation for this analysis stems from a theoretical paper by Fefferman et al. [4], which posits that amplitude damping gate errors (non-unital noise, primarily $T_1$ decay) should induce a bias toward '0's in the output samples. The authors aim to determine if this theoretical effect—manifesting as a larger fraction of zeroes than predicted by readout errors alone—is observable in empirical data from Google's Sycamore processor, IBM's quantum processors, and various simulators.

Methodology
The authors employ and extend statistical tools developed in their prior work, specifically utilizing Fourier-Walsh expansions adapted for asymmetric noise.

Noise Model: The study utilizes a 3-parameter noise model:
$R(q_{1\to0}, q_{0\to1})(sP_C + (1-s)/M)$
where $P_C$ is the ideal probability distribution, $s$ is a fidelity-like parameter, $M=2^n$ , and $q_{1\to0}, q_{0\to1}$ represent the asymmetric readout error rates. This generalizes the 2-parameter symmetric model where $q_{1\to0} = q_{0\to1} = q$ .
Estimation Techniques:
- Maximum Likelihood Estimation (MLE): The authors develop MLE estimators for the parameters $(s, q_{1\to0}, q_{0\to1})$ based on sample data.
- Fourier-Walsh Analysis: They extend the Fourier framework from [10] to the $p$ -biased measure space to handle asymmetric errors, allowing for efficient parameter estimation.
- Comparative Analysis: The study compares three methods for estimating error rates:
  1. Statistical inference from the empirical fraction of 0s and 1s in circuit outputs.
  2. Direct readout error rates reported by hardware providers (Google's som_params_by_qubit.csv and readout_raw_data.tar).
  3. MLE derived from the circuit output distributions.
Data Sources:
- Google Sycamore: Empirical data from the 2019 supremacy experiment ( $12 \le n \le 53$ qubits) and subsequent runs ( $n=67, 69$ ).
- IBM Hardware: 5-qubit circuits run on "Nairobi" and "Jakarta" processors.
- Simulators: Google's "QVM Weber" and IBM's "Fake Guadalupe" simulators.
- Temporal Analysis: Examination of the temporal stability of the proportion of measured 1s across bitstring measurement sequences.

Key Results

Absence of Gate-Induced Asymmetry in Google Data:
Contrary to the hypothesis suggested by Fefferman et al. [4], the analysis of Google's 2019 experimental data ( $12 \le n \le 53$ ) shows no evidence of an additional bias toward '0's caused by amplitude damping gate errors.
- The empirical difference between the fraction of 0s and 1s is consistent with, or even slightly lower than, the difference predicted solely by the reported readout errors ( $q_{1\to0} - q_{0\to1}$ ).
- MLE estimates for $q_{1\to0}$ and $q_{0\to1}$ derived from the experimental data are similar to (or slightly lower than) the physical readout errors reported by Google.
- This holds true across the entire range of qubit counts tested.
Discrepancy in Simulators:
- IBM "Fake Guadalupe": In contrast to the Google hardware data, the IBM simulator shows a significant discrepancy. The "effective" readout errors estimated via MLE ( $q_{1\to0} \approx 0.081, q_{0\to1} \approx 0.048$ ) are considerably higher than the physical readout errors ( $q_{1\to0} \approx 0.028, q_{0\to1} \approx 0.007$ ). The asymmetry in the simulator data is larger than what is reported in the physical parameters, suggesting the simulator's noise model may inadvertently incorporate gate-error effects similar to those predicted by Fefferman et al.
- Google "QVM Weber": Results for the Google simulator are inconclusive due to ambiguities in the reported error parameters (differences between "isolated" and "parallel" calibration data).
Temporal Instability:
The study identifies a temporal instability in the proportion of measured 1s during the execution of Google's Sycamore and USTC's Zuchongzhi experiments.
- The proportion of 1s fluctuates significantly over time (shifting by many theoretical standard deviations), exhibiting high positive autocorrelation.
- However, Google's initialization and readout raw data (readout_raw_data.tar) do not show similar temporal variations.
- The authors conclude that this instability likely originates from the applied gates during the circuit cycles rather than from temporal variations in state preparation or readout errors.
Anticoncentration:
The study notes that the Google experimental data exhibits a lack of anticoncentration (high values of the parameter $c$ in $\sum p(x)^2 = c/2^n$ ) that is stronger than expected from the Google noise model. This effect is not attributed to the specific asymmetric readout bias ( $q_{1\to0} > q_{0\to1}$ ) predicted by Fefferman et al., as the data does not show the corresponding bias toward zeroes.

Significance and Claims
The paper claims to provide a rigorous statistical test of the theoretical prediction that non-unital gate errors (amplitude damping) manifest as a specific type of asymmetric readout error in random circuit sampling.

Refutation of Observed Effect: The primary claim is that while Fefferman et al.'s theoretical mechanism is plausible, it is not witnessed in the Google 2019 supremacy data or subsequent runs. The observed asymmetry in the data is fully explained by the reported readout errors, with no additional contribution from gate-induced bias toward '0's.
Methodological Extension: The paper successfully extends Fourier-Walsh analysis and MLE techniques to the 3-parameter asymmetric noise model, providing a framework for future analysis of NISQ data.
Simulation vs. Reality: The paper highlights a divergence between real hardware (Google) and certain simulators (IBM Fake Guadalupe), where the latter appears to exhibit the "effective" error inflation predicted by theory, whereas the former does not.

The authors maintain a modest tone, noting that while the specific effect of gate-induced asymmetry was not found in the Google data, further samples and simulations are required for a complete understanding, particularly given the inconclusive nature of the Google simulator data and the observed temporal instabilities in gate cycles.

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