Operating a bistable qubit

This paper presents an adaptive FPGA-based "1-bit feedback" protocol that efficiently mitigates dephasing errors in superconducting qubits caused by bistable two-level-system defects, achieving high-fidelity operation with significant error reduction and stabilization of gate fidelities.

The Gist

Imagine you are trying to tune a radio to a specific station to listen to a clear broadcast. Usually, the station stays on one frequency. But in this experiment, the "radio station" (a quantum bit, or qubit) is being haunted by a mischievous ghost called a TLS (Two-Level System).

This ghost doesn't just make static; it physically moves the radio station's frequency back and forth between two distinct spots. Sometimes the station is at 5.10 GHz (let's call this the "High" mode), and a moment later, the ghost jumps, and the station drops to 5.0996 GHz (the "Low" mode).

If you keep your radio tuned to the middle ground, trying to guess where the station is, the signal will be garbled and weak. This is what happens to quantum computers when these ghosts are present: the information gets scrambled (dephasing), and the computer makes mistakes.

The Problem: The "Blind" Operator

Normally, to fix this, a computer operator would have to constantly check the frequency, take many measurements, and slowly adjust the dial. But in the quantum world, time is precious. The "ghost" might jump again before you finish checking. Also, checking too many times disturbs the delicate quantum state.

The Solution: The "1-Bit Feedback" Trick

The researchers developed a clever, lightning-fast trick called "1-bit feedback."

Think of it like a game of "Hot and Cold," but with a twist. Instead of asking, "Is it warmer or colder?" over and over, they set up a special test that tells them exactly which of the two frequencies the station is on with just one single glance.

Here is how the trick works:

1. The Setup: They prepare the qubit in a superposition (a state that is both "on" and "off" at the same time).
2. The Timing: They wait for a very specific, tiny fraction of a second (about 1.33 microseconds). This time is calculated perfectly so that if the qubit is in the "High" mode, it ends up in one position, but if it's in the "Low" mode, it ends up in the exact opposite position.
3. The Single Shot: They take one quick measurement.
   • If the result is "0," they know instantly: "It's the Low mode!" They immediately tune their controls to the Low frequency.
   • If the result is "1," they know: "It's the High mode!" They instantly tune to the High frequency.

This is like having a magic mirror that, with a single flash of light, tells you whether a chameleon is red or blue, allowing you to instantly change your clothes to match it perfectly.

The Results: Smoother Riding

The team tested this on a superconducting quantum processor. Here is what they found:

• Stopping the Wobble: Without their trick, the quantum signal looked like a shaky, beating drum (called "Ramsey beating") because the frequency was jumping back and forth. With the 1-bit feedback, the signal became smooth and steady.
• Fewer Mistakes: They measured how often the quantum computer made errors when performing a simple task (a "gate").
  • Without the trick: The error rate was high and jumped around wildly (fluctuating between good and bad).
  • With the trick: The error rate dropped significantly. They reduced the errors by about 77%.
• Speed: Because they only needed one measurement instead of dozens, their system could react incredibly fast (about 136,000 times a second). This is fast enough to catch the "ghost" before it causes too much damage.

Why This Matters

The paper argues that as we build bigger quantum computers, we might not be able to eliminate every single "ghost" (TLS defect) in the materials. Instead of trying to build a perfect, ghost-free machine, we can build a machine that is smart enough to adapt to the ghosts in real-time.

By using a fast, classical computer (an FPGA) to make these instant, single-shot decisions, they can keep the quantum computer running smoothly even when the hardware is imperfect. It's a way to get high performance from a "noisy" system by listening to it very closely and reacting instantly.

Technical Summary

Technical Summary: Operating a Bistable Qubit via 1-Bit Feedback

Problem Statement
Parasitic two-level system (TLS) defects are a primary source of decoherence in solid-state quantum processors, including superconducting and semiconductor spin qubits. While ensembles of TLSs typically generate $1/f$-like noise, a single strongly coupled TLS can induce discrete, telegraphic shifts in the qubit frequency. This phenomenon renders the qubit "bistable," causing it to stochastically switch between two distinct frequency modes. Such bistability introduces non-Markovian noise with memory effects, leading to significant dephasing and a reduction in gate fidelities. Standard dynamical decoupling techniques are not universally effective against this specific type of noise, and existing online estimation methods often fail to address qubits that switch between discrete values rather than drifting continuously. Without active mitigation, the qubit cannot maintain coherent superposition states or execute high-fidelity gates, as the control frequency becomes mismatched to the instantaneous qubit state.

Methodology
The authors propose and experimentally demonstrate an adaptive "1-bit feedback" protocol executed on a commercial field-programmable gate array (FPGA) controller (Quantum Machines OPX1000). The core of the method is a TLS-syndrome circuit designed to discriminate between the two qubit modes using a single-shot measurement.

The protocol operates as follows:

1. Syndrome Preparation: A superposition state is prepared using a $-\pi/2$ rotation around the $X'$-axis.
2. Optimal Evolution: The state evolves for an optimal time $\tau_{opt} \approx 1/(2\Delta_{TLS})$, where $\Delta_{TLS}$ is the frequency splitting between the two modes. This duration is chosen such that the state in the low-frequency (L) mode becomes exactly antiparallel to the state in the high-frequency (H) mode on the Bloch sphere equator.
3. Single-Shot Readout: A second $-\pi/2$ pulse converts the accumulated phase into population, followed by a standard dispersive readout. The binary outcome ($0$ or $1$) exclusively identifies the current qubit mode.
4. Real-Time Feedback: The controller updates the qubit control frequency ($f_c$) to match the identified mode immediately.

This approach treats the problem as hypothesis testing between two discrete Hamiltonians rather than continuous parameter tracking. The system was tested on a transmon qubit with an anharmonicity of $\approx -300$ MHz and a TLS-induced splitting of $\Delta_{TLS} \approx 374$ kHz. The switching rate of the TLS was observed to be on the order of tens of seconds, significantly slower than the estimation cycle time.

Key Results
The protocol was validated through two primary experimental demonstrations:

• Suppression of Ramsey Beating: In standard Ramsey experiments, the bistable nature of the qubit causes a characteristic beating pattern in the oscillation fringes due to the averaging of the two modes. The 1-bit feedback protocol successfully suppressed this beating, restoring clear, high-visibility oscillations by ensuring the control frequency matched the active mode during the superposition time.
• Gate Fidelity Improvement: Randomized benchmarking (RB) was performed over a 14-minute period, interleaving sequences with and without feedback. The results showed a significant reduction in gate infidelity:
  • Without feedback: Average infidelity was $(1.4 \pm 1.1) \times 10^{-3}$, exhibiting telegraphic fluctuations.
  • With feedback: Average infidelity dropped to $(0.65 \pm 0.18) \times 10^{-3}$.
  • This represents a 77% reduction in error for the largest observed errors.
  • The estimated bandwidth of the method is approximately 136 kHz, limited by readout and reset times, which is more than an order of magnitude higher than previous non-adaptive or multi-shot estimation protocols.

The measured infidelities with feedback remained slightly above the theoretical decoherence limit ($\approx 5 \times 10^{-4}$), attributed to intrinsic SPAM (State Preparation and Measurement) errors and decoherence, but the protocol successfully mitigated the dominant error source: the frequency mismatch caused by TLS switching.

Significance and Claims
The paper claims that this work provides a simple yet fundamentally efficient strategy for mitigating dephasing errors induced by strongly coupled TLS defects. By operating at the information limit set by the qubit's intrinsic entropy, the protocol demonstrates that a single bit of information from a single-shot measurement is sufficient to stabilize a bistable qubit.

The authors suggest that as device fabrication improves and the density of TLS defects decreases, future quantum processors may suffer from only a few remaining, discrete instabilities rather than pervasive noise. In such a scenario, efficient active mitigation of these discrete instabilities becomes a critical resource for operating large qubit arrays. The approach is particularly relevant for ancilla qubits in fault-tolerant quantum error correction, where the absence of temporal correlations in noise is a fundamental assumption for syndrome extraction. The protocol offers a control-based, real-time calibration tool that complements improvements in materials and device design, potentially enabling the full utilization of near-perfect quantum processing units.