⚛️ quantum physics

Cost-effective scalable quantum error mitigation for tiled Ansätze

This paper introduces "tiled M0," a cost-effective quantum error mitigation technique that leverages the structure of tiled Ansätze to exponentially reduce noise characterization costs while maintaining high accuracy in molecular ground state energy calculations on near-term quantum devices.

Original authors: Oskar Graulund Lentz Rasmussen, Erik Kjellgren, Peter Reinholdt, Stephan P. A. Sauer, Sonia Coriani, Karl Michael Ziems, Jacob Kongsted

Published 2026-08-10

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CC BY 4.0

Original authors: Oskar Graulund Lentz Rasmussen, Erik Kjellgren, Peter Reinholdt, Stephan P. A. Sauer, Sonia Coriani, Karl Michael Ziems, Jacob Kongsted

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

Imagine you are trying to listen to a faint, beautiful melody played on a violin, but the room is filled with a chaotic, roaring crowd. This is the current state of quantum computing. Scientists have built machines that can, in theory, solve problems impossible for today's supercomputers, like simulating complex molecules to discover new medicines or materials. However, these machines are currently "noisy." The qubits (the tiny units of information) are incredibly fragile, and the slightest disturbance causes them to make mistakes, drowning out the correct answer in a sea of static.

To fix this, scientists use "error mitigation." Think of this not as repairing the violin, but as using a clever audio filter to subtract the crowd's noise from the recording. One popular method involves running a test to map out exactly how the noise behaves—creating a "confusion matrix" that acts like a dictionary of mistakes. If you know exactly how the noise twists the signal, you can mathematically untwist it later. But here's the catch: for a large machine, mapping every possible way the noise can mess things up is like trying to write down every possible sentence in a language. The amount of work needed grows so fast that it becomes impossible for anything but the tiniest machines. This paper tackles that specific bottleneck, asking: "Can we clean up the noise without having to map every single mistake?"

The researchers behind this study, working with a team from Denmark and the UK, have developed a clever shortcut called "tiled M0." Instead of trying to map the noise for the entire quantum computer at once, they realized they could treat the computer like a mosaic made of repeating tiles. In their specific setup, called the "tUPS Ansatz," the quantum circuit is built from identical blocks (tiles) that repeat over and over. The team's big idea was to assume that the noise behaves similarly within each of these small tiles and that the noise in one tile doesn't wildly affect its neighbors.

By making this "locality approximation," they could measure the noise for just one or two small tiles and then copy that information across the whole machine. This is a massive shortcut. Instead of needing to run millions of test circuits to map a large system, they only needed to run a handful. In their tests, they applied this method to calculate the ground state energy (the most stable energy level) of several molecules, including water, benzene, and butadiene, using quantum computers with up to 12 qubits.

The results were promising, but with a twist. In computer simulations where the noise was perfectly predictable, the "tiled M0" method worked almost as well as the heavy, full-scale method, reducing energy errors by a factor of ten or more. It successfully cleaned up the signal for molecules like benzene, which has 12 qubits—a size where the old method would have been practically impossible to run. However, when they ran these same tests on real, physical quantum computers, the results were a bit messier. For smaller molecules like Lithium Hydride and Hydrogen, the method worked great. But for larger, more complex ones like water and benzene, the improvement was less dramatic, only cutting the error by a factor of two to four.

The authors suggest that this gap between the simulation and the real world isn't because their math was wrong, but because real quantum computers are fickle. The noise on these machines drifts and fluctuates over time. Since their method required a long time to run the experiments (up to two hours for benzene), the "noise map" they created at the start might have become outdated by the time they finished measuring the energy. They suspect that if the machine's noise were more stable, their technique would shine even brighter.

Ultimately, this paper doesn't claim to have solved the noise problem forever. Instead, it offers a cost-effective, scalable tool that makes error mitigation possible for larger machines that were previously out of reach. It suggests that by treating the quantum computer as a collection of local neighborhoods rather than one giant, tangled web, we can save a tremendous amount of time and resources. While the method needs further testing to see how it handles the drifting noise of real hardware, it opens a door to running more accurate quantum simulations on the imperfect machines we have today.

Technical Summary: Cost-effective scalable quantum error mitigation for tiled Ansätze

Problem Statement
Quantum error mitigation (QEM) is essential for performing accurate computational tasks on current Noisy Intermediate-Scale Quantum (NISQ) devices. However, existing QEM techniques, such as readout error mitigation (REM) and the recently proposed Ansatz-based gate and readout error mitigation (M0), suffer from an exponential scaling in Quantum Processing Unit (QPU) measurement costs. Specifically, constructing a fully correlated confusion matrix to characterize noise requires preparing and measuring all possible bitstrings ( $2^n$ ), rendering these methods infeasible for systems with more than a few qubits. While techniques like zero-noise extrapolation (ZNE) and probabilistic error cancellation (PEC) exist, they also face exponential sampling overheads relative to circuit depth and error rates. The authors identify a need for a cost-effective QEM method that can scale to larger systems without incurring prohibitive QPU costs for noise characterization.

Methodology: Tiled M0
The paper introduces "tiled M0," a cost-effective QEM technique designed specifically for tiled Ansätze, such as the tiled Unitary Product State (tUPS), Quantum Number Preserving (QNP), and hardware-efficient circuits with tiled structures. The method builds upon the M0 technique, which characterizes noise by including the Ansatz (with variational parameters set to zero) in the noise calibration circuits, thereby encoding both gate and readout noise into the confusion matrix.

Tiled M0 improves upon M0 by leveraging the repeating, local structure of tiled Ansätze through three key approximations:

Locality Approximation: Instead of constructing a single, fully correlated confusion matrix for the entire system, tiled M0 assumes that noise is local to the "tiles" (groups of qubits) within the Ansatz. The full-system confusion matrix is approximated as a tensor product of smaller confusion matrices, one for each unique tile in the first layer.
Layer Approximation: The method assumes that identical tiles across different layers of the Ansatz induce the same noise transitions. This allows the characterization of a single layer's tiles to represent the noise for the entire circuit depth, significantly reducing the number of distinct circuits required.
Separability Approximation: To prevent the overcorrection of readout noise (which would occur if multiple tile matrices shared qubits), the method assumes gate noise and readout noise are independent. It separates the total tile confusion matrix ( $A_{ti}$ ) into a readout component ( $A^{REM}_{ti}$ ) and a gate component ( $A^{gate}_{ti}$ ). The readout part is removed via matrix inversion, and the remaining gate noise matrices are combined.

Key Contributions

Exponential Cost Reduction: By approximating the full-system confusion matrix with a set of smaller tile matrices, tiled M0 reduces the QPU cost of noise characterization from exponential ( $2^n$ ) to linear with respect to the number of qubits. For a tUPS Ansatz, the method requires executing only 64 distinct quantum circuits (plus 4 for readout) regardless of system size (for $n \ge 6$ ), compared to the $2^n$ circuits required by standard M0.
Scalability: The technique enables error mitigation for systems with up to 12 qubits (benzene) and deep circuits (up to 299 CZ gates), which would be computationally intractable for standard M0 due to the sheer number of required shots for matrix construction.
Hybrid Noise Characterization: The method successfully integrates gate noise characterization (via the zero-parameterized Ansatz) with readout error mitigation, providing a unified framework for correcting both error types in a scalable manner.

Results
The authors validated tiled M0 using the tUPS Ansatz on molecular ground state energy calculations for LiH, H $_2$ , H $_2$ O, butadiene, and benzene (ranging from 4 to 12 qubits).

Noisy Simulations: In simulations using noise models imported from IBM backends, tiled M0 consistently reduced energy errors by one to two orders of magnitude compared to raw (unmitigated) results. For smaller systems (LiH, H $_2$ ), the performance was comparable to the full M0 method, validating the layer and separability approximations under controlled noise conditions. In some cases, energies were accurate to within a few millihartrees of the exact reference values.
Quantum Experiments: On real IBM quantum hardware (ibm_fez and ibm_marrakesh), tiled M0 demonstrated significant improvements for LiH, H $_2$ , and butadiene, reducing energy errors by approximately one order of magnitude.
Limitations on Larger Systems: For larger systems (H $_2$ O and benzene), the improvement in quantum experiments was more modest (reducing errors by a factor of 2–4) compared to the simulation results. The authors attribute this discrepancy to noise drift and fluctuations during the longer QPU processing times required for these experiments. The noise characterization was performed at the start of the experiment, and subsequent drift likely rendered the confusion matrices less representative of the noise during energy measurement.
Comparison with M0: In 4-qubit experiments where both methods were feasible, M0 generally performed slightly better or comparably to tiled M0, particularly in deeper circuits (3–4 layers). This suggests that the layer approximation may introduce minor inaccuracies in real hardware settings, though the trade-off in cost is substantial.

Significance and Claims
The paper claims that tiled M0 provides a "simple and cost-effective" solution for mitigating errors in tiled Ansätze, offering a noise characterization cost that is exponentially cheaper than fully correlated methods like M0 or REM. The authors emphasize that while the technique relies on assumptions (locality, layer independence, and noise separability) that may not hold perfectly in all hardware environments, the method is highly effective in noisy simulations and shows promise for near-term applications.

The significance of the work lies in its ability to extend the reach of QEM to larger molecular systems (up to 12 qubits in this study) that were previously inaccessible to high-fidelity error mitigation due to resource constraints. The authors conclude that tiled M0 is particularly well-suited for improved quantum hardware where noise levels are lower and less prone to drift, and they position it as a viable strategy for near-term quantum computing applications. They acknowledge that further validation on hardware is necessary to fully understand the impact of noise drift and the validity of the underlying approximations in diverse physical settings.

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