Real-time adaptive tracking of fluctuating relaxation rates in superconducting qubits

This paper presents an FPGA-powered real-time adaptive protocol that overcomes traditional temporal resolution limits to track rapid fluctuations in superconducting qubit relaxation rates, revealing millisecond-scale switching events and fast two-level system dynamics that redefine calibration timescales and deepen the understanding of environmental decoherence.

The Gist

The Big Picture: The "Fickle" Quantum Computer

Imagine you are trying to bake the perfect cake (a quantum calculation) in a kitchen where the oven temperature is wildly unpredictable. Sometimes the oven is perfect at 350°F, but a split second later, it might spike to 500°F or drop to 200°F.

In the world of superconducting quantum computers, the "oven temperature" is called the relaxation rate (or $T_1$). It's a measure of how long a qubit (the computer's basic unit of information) can hold onto its data before it "relaxes" and forgets everything.

The Problem:
For years, scientists measured this "oven temperature" using a slow, clumsy method. They would check the temperature, wait a few seconds, check again, and wait a few more seconds. It was like trying to track a hummingbird's flight path by taking a photo once every minute. You'd miss all the fast, erratic movements. Because of this, scientists thought the temperature only changed slowly over minutes or hours.

The Breakthrough:
This paper introduces a new, super-fast way to track these changes. The researchers built a "smart controller" (powered by a chip called an FPGA) that acts like a high-speed camera. Instead of checking the temperature once a minute, it checks it thousands of times a second.

The Core Innovation: The "Smart Wait"

The researchers didn't just measure faster; they measured smarter.

• The Old Way (Non-Adaptive): Imagine a doctor checking your pulse. They decide to check your pulse for exactly 10 seconds, no matter what. If your heart rate is slow, 10 seconds is great. If your heart rate is racing, 10 seconds is too long and wastes time. They just stick to the plan, even if the plan isn't working well.
• The New Way (Adaptive Bayesian): Imagine a doctor who is a genius detective. They check your pulse for 2 seconds. If the pulse is slow, they say, "Okay, I need to wait a bit longer to be sure." If the pulse is racing, they say, "Got it, I need to check again immediately."
  • The controller uses a mathematical method called Bayesian estimation. After every single measurement, it updates its "guess" of what the temperature is.
  • Based on that guess, it instantly decides: "How long should I wait for the next measurement to get the most information?"
  • This allows them to track changes in milliseconds (thousandths of a second) instead of minutes.

The Surprise: The "Telegraph" Switches

When they turned on their high-speed camera, they saw something shocking.

Previously, scientists thought the relaxation rate was like a drifting cloud—it moved slowly and smoothly.
The new data shows it's more like a light switch that flickers on and off.

• The Discovery: The relaxation time (how long the qubit lasts) can suddenly jump from being very short (bad) to very long (good) and back again in the blink of an eye (tens of milliseconds).
• The Culprit: They believe this is caused by tiny defects in the material called Two-Level Systems (TLS). Think of these as microscopic "gremlins" or "switches" inside the metal of the computer. These gremlins are flipping back and forth incredibly fast (up to 10 times a second), messing with the qubit's stability.

Why This Matters: From "Scheduled Maintenance" to "Real-Time Driving"

The Old Strategy:
Because scientists thought the computer's performance only changed slowly, they treated quantum computers like old cars. You'd drive them for a few hours, then pull over to a garage, run a long diagnostic test (taking minutes or hours), and then recalibrate the engine.

The New Strategy:
Now that we know the engine can sputter and recover in milliseconds, we need a different approach.

• Real-Time Adjustment: The computer can now "feel" when a qubit is about to fail. It can pause the calculation for that specific qubit and switch to a different, healthier qubit instantly.
• Better Screening: When building new quantum chips, manufacturers can now test them in seconds instead of hours, quickly spotting the "bad apples" that have these fast-fluctuating gremlins.

The Analogy Summary

• The Qubit: A fragile soap bubble.
• The Relaxation Rate ($T_1$): How long the bubble lasts before popping.
• The Old Method: Checking the bubble every minute. You miss the moment it gets wobbly.
• The New Method: A high-speed camera that watches the bubble frame-by-frame.
• The Gremlins (TLS): Tiny air currents that make the bubble wobble unpredictably.
• The Result: We can now see the wobbles happening in real-time, allowing us to save the bubble before it pops, or switch to a new bubble instantly.

The Bottom Line

This paper redefines how we understand and control quantum computers. By using a "smart," adaptive controller, the team proved that quantum computers are far more dynamic and chaotic than we thought. They are not just slowly drifting; they are rapidly flickering. But now that we have the tools to see these flickers, we can build better, more reliable quantum computers that adapt to the chaos in real-time.

Technical Summary

1. Problem Statement

The fidelity of operations in solid-state quantum processors is fundamentally limited by environmental decoherence, specifically the relaxation rate ($\Gamma_1$) of superconducting qubits. A major challenge in characterizing these systems is that $\Gamma_1$ fluctuates unpredictably over time due to interactions with environmental defects, primarily Two-Level Systems (TLS).

Limitations of Traditional Methods:

• Low Temporal Resolution: Conventional protocols for measuring $T_1$ (relaxation time, $T_1 = 1/\Gamma_1$) are non-adaptive. They involve initializing a qubit, waiting for a fixed set of times, and averaging results over many repetitions.
• Averaging Out Dynamics: These methods typically require seconds to minutes to acquire sufficient data, effectively averaging out rapid fluctuations. Previous reports suggested $T_1$ changes occurred on timescales of minutes or hours.
• Inefficiency: Non-adaptive methods do not utilize prior knowledge to optimize subsequent measurements, leading to suboptimal use of experimental time.
• Calibration Lag: Because fluctuations are slow to detect, calibration in Quantum Processing Units (QPUs) is typically performed on hourly cycles, potentially leaving the system operating with sub-optimal parameters for long periods.

2. Methodology

The authors developed a real-time adaptive Bayesian estimation protocol implemented on a Field-Programmable Gate Array (FPGA) to overcome these limitations.

Experimental Setup:

• Device: A 5-qubit superconducting transmon array operated in a dilution refrigerator (<10 mK).
• Controller: A commercial FPGA-based controller (Quantum Machines OPX1000) capable of single-shot readout and real-time processing.
• Qubits: Two fixed-frequency transmon qubits (Q1 and Q2) with average $T_1 \approx 0.17$ ms.

The Adaptive Protocol:

1. Initialization: The qubit is initialized to the excited state $|1\rangle$.
2. Adaptive Waiting Time: Instead of a fixed wait time, the controller calculates an adaptive waiting time $\tau_i$ based on the current probability distribution of $\Gamma_1$. The wait time is set to $\tau_i = c \cdot \hat{T}_1$, where $\hat{T}_1$ is the current estimate and $c$ is a coefficient optimized via binomial statistics.
3. Single-Shot Measurement: The qubit state is measured projectively.
4. Bayesian Update: The controller updates the probability distribution $P(\Gamma_1)$ in real-time (within $\approx 2.2 \, \mu$s) using Bayes' rule.
   • Distribution Choice: The authors model the probability distribution of $\Gamma_1$ using a Gamma distribution (parameterized by shape $k$ and scale $\theta$). This choice is mathematically convenient for FPGA implementation and ensures the distribution remains positive (unlike a Gaussian, which can assign probability to negative rates).
   • Approximation: After each measurement, the posterior distribution is approximated back to a Gamma distribution using the method of moments (matching mean and variance).
5. Iteration: The process repeats for $N$ cycles (e.g., 50 shots) to converge on a precise estimate of $T_1$ within milliseconds.

3. Key Contributions

• Two Orders of Magnitude Speedup: The adaptive protocol achieves a temporal resolution of milliseconds (total estimation time $\approx 11$ ms for 50 shots), compared to the seconds-to-minutes required by traditional non-adaptive methods.
• Real-Time FPGA Implementation: The entire Bayesian update loop (likelihood calculation, posterior update, and next wait time selection) is executed on-chip with microsecond latency, enabling true real-time tracking.
• Discovery of Fast Fluctuations: The method revealed that $T_1$ can switch by nearly an order of magnitude (e.g., from $>500 \, \mu$s to $\approx 100 \, \mu$s) on timescales of tens of milliseconds, rather than the previously reported minutes or hours.
• High-Speed TLS Characterization: The technique resolved TLS switching rates up to 10 Hz, which is four orders of magnitude faster than previously reported rates in transmon qubits.

4. Key Results

• Tracking Performance: The system successfully tracked $T_1$ fluctuations over 72 hours. The estimated $T_1$ showed "telegraphic" switching behavior, indicating discrete jumps between stable states.
• Validation: The adaptive estimates were validated against interleaved non-adaptive measurements. The mean $T_1$ values agreed within uncertainty (e.g., Adaptive: $135.0 \pm 0.9 \, \mu$s vs. Non-adaptive: $136.7 \pm 2.2 \, \mu$s), confirming the accuracy of the fast method.
• Spectral Analysis:
  • Power Spectral Density (PSD) & Allan Deviation: Analysis of the 72-hour trace revealed dominant Lorentzian noise processes (indicative of individual TLSs) alongside $1/f$ noise.
  • Switching Rates: By analyzing Allan deviation peaks, the authors extracted TLS switching rates of 10 Hz and 100 mHz. The 10 Hz rate is particularly significant as it represents a regime previously inaccessible to standard characterization.
• Statistical Correlation: The standard deviation of the Bayesian posterior distribution correlated strongly with the fitted white noise amplitude, validating that the estimator's internal uncertainty metric accurately reflects the true estimation error.

5. Significance and Impact

• Redefining Calibration Timescales: The results suggest that the relevant timescale for qubit calibration in superconducting QPUs should shift from hours to milliseconds. Continuous, real-time monitoring could allow systems to pause operations or reroute qubits when $T_1$ drops below a fidelity threshold.
• Error Mitigation: By identifying "outlier" qubits in real-time, this protocol enables dynamic error mitigation strategies, potentially improving the overall performance of large-scale quantum processors.
• Materials Science: The ability to detect fast TLS switching provides a powerful tool for screening fabrication processes and understanding the microscopic origins of decoherence in superconducting circuits.
• Scalability: Since the protocol uses only single-qubit gates and is highly efficient, it is well-suited for extension to large qubit arrays and integration into future quantum error correction schemes.

In summary, this work bridges the gap between the speed of quantum operations and the speed of environmental noise characterization, providing a critical tool for the stable operation of future fault-tolerant quantum computers.