🔬 physics

Experimental Observation of Error-Resilience Phase Transition on Superconducting Quantum Processor

This paper presents the first experimental observation of an error-resilience phase transition on IBM's 26-qubit Heron processor, utilizing advanced error mitigation and randomized measurement protocols to detect a critical noise angle that, while lower than theoretical predictions due to circuit depth limitations, confirms the necessity of error mitigation for identifying such transitions on current quantum hardware.

Original authors: Emanuel Dallas, Mario Morford-Oberst, Eric Weine

Published 2026-10-30

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Emanuel Dallas, Mario Morford-Oberst, Eric Weine

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 world of quantum computing, information is incredibly fragile. Unlike the bits in a standard computer, which stay put until you change them, quantum bits are easily disturbed by the slightest whisper of heat or electromagnetic interference from their surroundings. This disturbance, known as noise, usually destroys the delicate quantum state, causing the computer to fail. For years, scientists have sought ways to protect this information, often by spreading it out across many particles so that if one part gets corrupted, the whole message isn't lost. This idea is the foundation of quantum error correction. However, a surprising theoretical discovery suggested that the relationship between noise and protection is not a simple line. Instead, it proposed a sudden shift, or phase transition, where a system could suddenly switch from being able to recover its information to being completely unable to do so, depending on how strong the noise becomes.

A team of researchers has now taken this theoretical idea out of the realm of equations and into the real world. Working with a superconducting quantum processor at IBM, they successfully observed this transition for the first time. They built a specific experiment where they scrambled information across a group of qubits, introduced a controlled amount of noise, and then tried to unscramble it to see if the original message survived. By testing systems ranging from a few qubits up to twenty-six, they watched for the exact moment where the noise became too strong for the system to handle. Their findings confirm that such a tipping point exists in physical hardware, but they also revealed that the real-world version of this transition happens at a slightly different point than the idealized theory predicted.

The experiment was conducted on a device called the ibm pittsburgh Heron r3, which contains a grid of superconducting qubits. The researchers set up a process that mimics a memory protocol. First, they prepared a set of qubits in a known state. Then, they applied a complex sequence of operations to scramble this information, spreading it out so that it was no longer localized to any single qubit. This scrambling is designed to make the information robust against local disturbances. After scrambling, they introduced a layer of identical noise to every qubit. This noise was a specific type of coherent error, meaning it was a consistent, predictable rotation of the quantum state rather than random static. Finally, they attempted to reverse the scrambling process to retrieve the original information.

To determine if the information survived, the team measured three different properties of the final state. The first was a measure of how closely the final state matched the original starting point, essentially asking, "Did we get our message back?" The second property measured how entangled the qubits had become, which indicates how thoroughly the information had been mixed. The third property measured a form of quantum complexity known as non-stabilizerness, which reflects how difficult the state would be to simulate on a classical computer. By varying the strength of the noise, they could watch these three properties change as the system moved from a state where information was safe to a state where it was lost.

The results showed a clear transition. As the noise angle increased, the ability to recover the original information dropped sharply, while the entanglement and complexity of the system rose. This behavior matched the predictions of the theoretical models, which described a switch between an "error-protecting" phase and an "error-vulnerable" phase. However, the exact point where this switch occurred was not exactly where the theory for a perfect, ideal system had predicted. The theoretical model suggested the transition should happen at a specific noise angle of approximately 1.14. In the experiment, the transition was observed at an angle of about 1.01.

To understand why the real experiment differed from the perfect theory, the researchers ran detailed computer simulations using the exact same circuit designs they had built on the hardware, but without any physical noise. When they analyzed these noiseless simulations, they found a transition point of about 1.05. This result was crucial because it showed that the difference between the experiment and the theory was not caused by the messy, unpredictable noise of the actual hardware. Instead, the discrepancy came from the fact that the circuits they built were not deep enough to be truly random in the way the theory assumed. The ideal theory relies on circuits that are infinitely complex and perfectly random, but the physical device had to use simpler, shallower circuits that could actually be run. These shallower circuits scrambled the information less effectively, making the system more vulnerable to noise and causing the transition to happen earlier than expected.

A major part of this success relied on advanced techniques to clean up the data. Quantum computers are notoriously noisy, and without correction, the signal of the phase transition would have been completely buried. The team used two specific methods to fix this. One method, called zero-noise extrapolation, involved running the circuits multiple times with the noise artificially increased, and then mathematically working backward to estimate what the result would have been with no noise at all. The other method, known as matrix-free measurement error mitigation, corrected for errors that happened when the computer read out the final state. When the researchers tried to analyze the raw data without these corrections, the statistical model they used could not find any clear transition. The data was too scattered, and the model could not pinpoint a critical point. This proved that these error-correction techniques were not just helpful extras; they were essential for seeing the phenomenon at all.

The study also highlighted the challenges of measuring these complex quantum properties. To determine the entanglement and complexity, the team had to use a technique called randomized measurements. This involved running the same circuit many times with different random settings and then using statistical methods to piece together the answer. Because the number of possible outcomes grows so fast with the number of qubits, they had to be very careful about how they collected their data. They found that for larger systems, they needed to run the experiment many more times to get a clear signal. Even with these efforts, the measurements of the most complex property were noisier than the others, but the overall pattern remained consistent across all three measurements.

The researchers concluded that while their experimental setup did not perfectly match the ideal mathematical model, it successfully captured the essence of the phase transition. The fact that the transition appeared at a slightly different angle than the theory predicted is not a failure, but a valuable insight. It shows that the depth and randomness of the circuit are critical factors in how well a quantum system can protect information. As quantum computers grow larger and more complex, understanding these limits will be vital. This work demonstrates that even on current, imperfect machines, scientists can observe fundamental shifts in how information behaves, provided they have the right tools to filter out the noise. The ability to see this transition in a real device suggests that the principles of quantum error protection are robust enough to survive the imperfections of today's technology, offering a promising path forward for building more reliable quantum computers in the future.

Technical Summary: Experimental Observation of Error-Resilience Phase Transition on Superconducting Quantum Processor

Problem Statement
The interplay between unitary dynamics and environmental noise is a central topic in quantum information theory. While random unitaries (scrambling) can protect information by distributing it into non-local degrees of freedom, theoretical work by Turkeshi and Sierant [1, 2] predicts that this protection fails once local coherent noise exceeds a critical strength. This phenomenon manifests as a phase transition between an "error-protecting phase" (EPP), where encoded information is recoverable, and an "error-vulnerable phase" (EVP), where information is lost. The transition is characterized by a critical coherent noise angle, $\alpha_c \approx 1.14$ for a code rate of $r=1/2$ , and is observable through order parameters including logical fidelity, entanglement entropy, and non-stabilizerness (magic).

Despite these theoretical predictions, direct experimental observation on existing quantum hardware has remained elusive. Challenges include the requirement for exponentially deep circuits to approximate Haar-random unitaries, the presence of correlated and site-dependent noise on current devices, and the difficulty of measuring entanglement and non-stabilizerness on Noisy Intermediate-Scale Quantum (NISQ) processors without prohibitive overhead.

Methodology
The authors implemented the encoding-decoding model described in [1, 2] on IBM's ibm_pittsburgh Heron r3 device, utilizing system sizes up to $N=26$ qubits.

Circuit Construction: To approximate Haar-random unitaries within linear depth constraints, the authors constructed brickwork circuits of depth $N$ . These circuits utilize native CZ gates interleaved with random single-qubit gates ( $SX$ and $RZ$ ). While not strictly Haar-random for finite $N$ , numerical simulations indicated that this ensemble converges toward Haar-random behavior as $N$ increases.
Noise Injection: Coherent noise was injected via $RZ(\alpha)$ gates following the encoding unitary.
Error Mitigation: To extract meaningful signals from noisy data, the authors employed two primary mitigation techniques:
1. Zero-Noise Extrapolation (ZNE): Using global folding with scaling factors $\lambda \in \{1, 3, 5\}$ and quadratic Richardson extrapolation to estimate noiseless outcomes.
2. M3 (Matrix-Free Measurement Error Mitigation): Applied to correct for State Preparation and Measurement (SPAM) errors.
Order Parameter Measurement:
- Fidelity: Measured via post-selected bit strings.
- Rényi-2 Entanglement Entropy ( $S_2$ ) and Stabilizer Rényi-2 Entropy ( $M_2$ ): Measured using single-copy randomized measurement (RM) protocols. To reduce variance, the authors implemented Importance Sampling (IS) using an approximate state ansatz derived from the measured fidelity.
- Data Analysis: The authors utilized Gaussian Process (GP) Regression to jointly model the finite-size scaling collapse of all three order parameters. This Bayesian approach allowed for the inference of posterior distributions for the critical angle $\alpha_c$ and critical exponents $\nu$ , without assuming a specific functional form for the scaling collapse.

Key Results

Observation of Transition: The experiment successfully detected a phase transition in the error-resilience of the encoded information. The data for fidelity, entanglement entropy, and non-stabilizerness exhibited the characteristic crossing behavior expected near a critical point.
Critical Angle: The GP regression yielded a critical angle of $\alpha_c \approx 1.01$ . This is lower than the theoretical Haar-random value of $\alpha_c \approx 1.14$ .
Source of Discrepancy: By running the same GP model on noiseless simulations of the same experimental circuit ensemble, the authors found a simulated $\alpha_c \approx 1.05$ . This indicates that the deviation from the theoretical value is not primarily due to device noise, but rather because the finite-depth circuits used in the experiment are not deep enough to be truly Haar-random, resulting in weaker information scrambling and a lower noise threshold.
Necessity of Error Mitigation: When the same GP model was applied to unmitigated (raw) experimental data, it failed to identify a coherent critical behavior. The posterior distributions for $\alpha_c$ were broad and inconsistent, and the critical exponents were highly correlated. This demonstrates that error mitigation is essential for observing the transition on current hardware.
Scaling Behavior: The experimental data showed that as system size $N$ increased, the curves for different $N$ crossed at approximately the same $\alpha$ value, consistent with finite-size scaling theory.

Significance and Claims
The paper claims the first experimental observation of the error-resilience phase transition in a noisy quantum system. The significance of this work lies in:

Validation of Theory: It confirms that the theoretical phase transition between error-protecting and error-vulnerable phases exists in physical hardware, despite the challenges of NISQ-era noise.
Methodological Advancement: It demonstrates a scalable workflow for measuring complex order parameters (entanglement and non-stabilizerness) on superconducting processors by combining randomized measurements, variance reduction (Importance Sampling), and advanced error mitigation (ZNE and M3).
Statistical Rigor: The use of Gaussian Process Regression to jointly analyze multiple order parameters provides a robust statistical framework for identifying critical behavior without forcing specific functional forms, allowing for the extraction of posterior distributions for critical parameters.
Hardware Limitations: The work highlights that while error mitigation is crucial, the depth of current circuits limits the ability to perfectly replicate Haar-random dynamics, leading to shifts in critical parameters that must be accounted for in experimental design.

The authors conclude that while their observed $\alpha_c$ differs from the ideal thermodynamic limit due to finite circuit depth, the suite of techniques employed provides a viable path for investigating information-theoretic phase transitions on NISQ devices.

More like this