⚛️ quantum physics

Standard estimators cannot represent fault-tolerant workloads at measured error rates: evaluated, evidence-based uncertainty for quantum resource estimation

This paper demonstrates that standard quantum resource estimators, which typically report single-point estimates, fail to accurately represent fault-tolerant workloads at experimentally measured error rates by revealing that realistic hardware uncertainties expand resource intervals by orders of magnitude, introduce structural disagreements between cost models, and cause existing tools to break down entirely for critical algorithms like AES-256.

Original authors: Furqan Nasir, Sher Jeel Ahmad

Published 2026-10-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Furqan Nasir, Sher Jeel Ahmad

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

To understand the stakes of this research, one must first grasp the nature of the machine it seeks to predict. Quantum computers are not simply faster versions of the laptops we use today; they are devices that harness the strange rules of the subatomic world to solve problems that would take ordinary machines thousands of years to crack. Among the most famous of these problems is breaking the encryption that protects global banking, government secrets, and private communications. To do this, a quantum computer must be "fault-tolerant," meaning it can correct its own mistakes as it runs. These machines are not yet built, but governments and security agencies are already planning for their arrival. They need to know when a quantum computer will be powerful enough to break current codes so they can switch to new, unbreakable ones in time. This planning relies on estimates of how many physical components, or qubits, such a machine would need. For years, these estimates have been reported as single, precise numbers, as if the future were a fixed destination rather than a landscape of possibilities.

A team of researchers has now challenged this certainty. They built a new way of looking at these estimates, one that treats the unknowns of hardware not as fixed values to be ignored, but as a range of possibilities to be measured. Instead of asking, "How many qubits are needed if everything goes perfectly?" they asked, "How many qubits are needed given what we have actually measured in the lab so far?" The researchers took the actual error rates from five of the largest quantum chips built by major companies and fed these real-world numbers into their models. They found that when you use the messy, imperfect reality of current hardware instead of an idealized, optimistic guess, the picture changes dramatically. The number of components required does not just go up a little; it explodes. The range of possible outcomes becomes so wide that a single number becomes meaningless. For a task like breaking a standard 2048-bit encryption code, the number of required components could be forty times higher or lower than the most common estimate, with the most likely outcome being nearly five times higher than the standard guess.

The study also revealed a hidden disagreement among the tools used to make these predictions. The researchers ran the same set of real-world data through five different calculation methods, including two popular software tools used by the industry. Even though these tools were built by different groups and use different internal logic, they agreed with each other only within a narrow margin of about ten percent. However, when compared to the full spread of results from all five methods, they consistently disagreed by a factor of two. This means that even if you pick the most popular tool, you are likely missing a massive layer of uncertainty simply because the tool itself is built on a specific set of assumptions that might not match reality. The researchers showed that this uncertainty is not a minor detail; it is a structural feature of the field that single-number reports completely hide.

Perhaps the most striking finding is that the standard tools used by the industry often cannot even handle the data from real hardware. When the researchers fed the actual error rates measured in laboratories into these popular software tools, the tools failed to produce an answer for the vast majority of scenarios. For the encryption-breaking task, one tool refused to calculate a result for over eighty percent of the realistic error rates, while another failed for nearly ninety-five percent. In one extreme case, the number of operations required to break a specific type of encryption was so large that it caused the software's counter to overflow, crashing the calculation entirely. This suggests that the current generation of planning tools is calibrated for a perfect world that does not exist. They are designed to work with optimistic assumptions, and when faced with the actual performance of today's machines, they simply break down.

The researchers did not just point out these problems; they built a framework to measure them and tested their own methods to ensure they were reliable. They used a pre-registered plan, meaning they decided exactly how they would analyze the data before they even started, preventing them from accidentally cherry-picking results that looked good. Their analysis confirmed that the wide ranges they found were real and not just a result of bad math. They also checked their predictions against a history of hardware measurements and found that their method correctly captured the uncertainty of the dominant factor: the error rate of the machine's basic operations. The study concludes that for anyone making critical decisions about when to switch to new security systems, relying on a single number is dangerous. The honest answer is not a point on a graph, but a wide interval that acknowledges the gap between what we hope for and what we have actually built. Until the tools can represent the full range of real-world performance, the timeline for the arrival of powerful quantum computers remains far more uncertain than the optimistic headlines suggest.

Technical Summary: Evidence-Based Uncertainty in Fault-Tolerant Quantum Resource Estimation

Problem Statement
Current fault-tolerant quantum resource estimates are almost universally reported as single point values (e.g., a specific number of physical qubits or runtime hours). These estimates rely on uncertain hardware parameters—most critically the physical two-qubit gate error rate—and divergent cost models. The community's planning convention typically fixes the error rate at $10^{-3}$ . However, large-array superconducting devices (e.g., Google Willow, IBM Heron, USTC Zuchongzhi-3) have demonstrated median error rates closer to $4 \times 10^{-3}$ .

The paper argues that single-number estimates fail to capture two critical realities:

Parametric Uncertainty: The gap between the optimistic planning convention ( $10^{-3}$ ) and measured hardware performance ( $\approx 4 \times 10^{-3}$ ) is not propagated into the final resource numbers.
Structural Uncertainty: Different cost models (analytic scaling, lattice-surgery, third-party tools) produce different results for the same inputs, a discrepancy that point estimates obscure.

Consequently, decision-makers regarding post-quantum cryptography migration and quantum advantage claims are relying on estimates that may be both systematically biased and devoid of uncertainty quantification.

Methodology
The authors developed an open, tool-agnostic framework to propagate evidence-based priors through multiple resource estimators. The methodology consists of four main components:

Evidence-Based Prior Library: Instead of assuming fixed values, the framework constructs priors from published experimental measurements. For the dominant parameter (physical two-qubit gate error rate), a lognormal distribution was fitted using weighted maximum likelihood from five large-array superconducting devices and one small-scale fluxonium device. This prior has a median of $4.0 \times 10^{-3}$ , four times the conventional planning value. Other parameters (cycle time, factory throughput, routing overhead, decoder reaction time) are similarly fitted from available data.
Tool-Agnostic Uncertainty Propagation: The framework uses inverse-transform sampling (Monte Carlo, Latin-hypercube, Sobol sequences) to draw parameter vectors from the priors. These vectors are fed into five distinct surface-code cost models:
- Three transparent analytic/tile-counting models (based on Fowler et al., Litinski, and Fowler/Gidney constructions).
- Two third-party estimators: The Azure Quantum Resource Estimator (QRE) and Qualtran (implementing Gidney/Fowler magic-state costs).
- The framework records "censored" draws where a tool fails to compute an estimate (e.g., due to code distance caps or factory error budgets).
Sensitivity Analysis: To attribute output variance to specific inputs, the authors employ Shapley effects (which handle correlated inputs) alongside Sobol indices. This identifies which parameters drive the width of the resource intervals.
Pre-Registered Evaluation: The study utilizes a three-layer protocol registered prior to confirmatory runs:
- Layer 1: Numerical correctness of propagation against closed-form benchmarks.
- Layer 2: Prior-predictive coverage via leave-one-source-out validation.
- Layer 3: Retrospective hindcasting (exploratory) to test if priors fitted from earlier data would have contained later measurements.

Key Results
Applied to canonical workloads (RSA-2048, ECC-256, AES-256, and a Hubbard-model simulation) on a superconducting surface-code architecture, the framework yields three coupled findings:

Parametric Uncertainty is Large: When propagating the evidence-based error rate prior ( $median \approx 4 \times 10^{-3}$ ), the 90% resource interval for physical qubits spans approximately 40 times (ratio of 95th to 5th percentile). The median of this distribution sits roughly 4.6 times higher than the conventional point estimate calculated at $10^{-3}$ . For RSA-2048, the interval ranges from $2.86 \times 10^7$ to $1.18 \times 10^9$ physical qubits.
Structural Disagreement is Real: Across five independently structured cost models, there is a stable factor-of-two spread in median resource estimates for matched inputs. Notably, two third-party tools (Azure QRE and Qualtran), which agree within ~10% of each other, both fall within the band set by the transparent analytic models. This confirms that model-form uncertainty is a structural feature, not an artifact of a single tool.
Standard Estimators Cannot Represent Measured Error Rates: A significant portion of the evidence-based parameter space lies outside the "representable envelope" of standard tools.
- For RSA-2048, the Azure estimator censors 81% of draws, and Qualtran censors 95%.
- For ECC-256, censored fractions are 87% and 99.7%, respectively.
- AES-256 is 100% censored by both tools; its gate count ( $6 \times 10^{43}$ ) also overflows the 64-bit integer counter of the Azure estimator.
- The tools fail because their default configurations (e.g., Azure's code distance cap of 50, Qualtran's fixed factory error budget) are tuned for the $10^{-3}$ convention and cannot accommodate the higher error rates actually demonstrated by hardware.

Significance and Claims
The paper claims its primary contribution is not a new statistical method (the uncertainty machinery is standard) but the evidence-based application of these methods to fault-tolerant resource estimation.

Evaluation over Assertion: The study provides a pre-registered, evaluated framework that produces intervals rather than point estimates. The authors demonstrate that the dominant parameter (error rate) is well-calibrated within its estimand (Layer 2 coverage reaches ~99% for the sensitivity-weighted metric).
Breakdown of Standard Tools: The most consequential finding is that standard estimators "break down precisely where hardware operates." At measured error rates, canonical workloads often fall outside the representable region of current tools, a fact completely hidden by the optimistic $10^{-3}$ convention.
Implications for Decision Making: For stakeholders timing the migration to post-quantum cryptography, the "optimistic convention" is misleading. The true resource cost is not only higher (median 4.6x) but also vastly more uncertain (40x interval). Furthermore, the factor-of-two disagreement between models implies that any single-tool estimate carries a hidden structural uncertainty band.

The authors conclude that resource estimates should be reported as intervals accompanied by the fraction of the parameter space the tool cannot represent, rather than as single numbers. The framework is designed as a "living estimate" that can be refitted as the hardware evidence base grows and error rates improve.

Technical Summary: Evidence-Based Uncertainty in Fault-Tolerant Quantum Resource Estimation

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