Digital Predistortion for Flux Control of Tunable Superconducting Qubits

This paper presents a digital predistortion framework using IIR and FIR filters to characterize and compensate for flux-control signal distortions in superconducting qubits, thereby enabling automated rapid calibration and significantly improving gate fidelity on quantum processing units.

The Gist

Imagine you are trying to bake a perfect cake, but your oven has a weird quirk: when you turn the dial to "350 degrees," the temperature doesn't jump there instantly. Instead, it spikes to 400, drops to 300, and wobbles around for a few seconds before finally settling at 350. If you put your cake in during that wobble, it burns or stays raw.

This is exactly the problem scientists face with superconducting qubits (the tiny brains of quantum computers).

The Problem: The "Wobbly" Quantum Switch

To make these quantum computers work, scientists need to flip switches very quickly using magnetic fields (called "flux"). Think of these switches as the volume knob on a radio. You want to turn the knob from "off" to "loud" instantly to play a specific song (a quantum calculation).

However, the path the signal takes to get to the chip is full of obstacles:

1. The Generator: The machine making the signal isn't perfect.
2. The Wires: The cables leading into the freezing cold fridge (cryostat) act like old, stretchy rubber bands.
3. The Chip: The quantum chip itself reacts a bit sluggishly.

When you try to send a clean, straight "step" signal (like turning a light switch on), these imperfections cause the signal to overshoot, undershoot, and wiggle before it settles. In the quantum world, this "wobble" ruins the calculation, causing errors in the final result.

The Solution: The "Pre-Emptive" Fix (Digital Predistortion)

Usually, if a signal is wobbly, you might try to wait for it to settle down before doing your work. But in quantum computing, time is money (or rather, time is quantum coherence). Waiting too long means the quantum information disappears.

Instead, the authors of this paper came up with a clever trick called Digital Predistortion (DPD).

The Analogy: The "Anti-Wobble" Driver
Imagine you are driving a car with a very heavy, sluggish steering wheel. If you want to turn left 90 degrees, you know the car will initially turn too far, then correct back, then wobble.

• Normal Driver: Turns left 90 degrees, waits for the car to stop wobbling, then drives straight. (Too slow!)
• Predistortion Driver: Knows the car wobbles. So, they turn the wheel less than 90 degrees initially, then quickly nudge it right, then left again. By the time the car's natural physics kicks in, the car ends up turning exactly 90 degrees smoothly.

The computer does the same thing. It calculates exactly how the signal will wobble and then pre-distorts the signal it sends. It sends a "weird" looking signal that, after passing through all the messy wires and the chip, comes out looking perfectly straight and clean.

How They Did It: The Two-Stage Filter

The team used a two-step cleaning process, like a high-tech water filter:

1. The IIR Filter (The "Quick Fix"): This is like a fast-acting shock absorber. It handles the immediate, sharp wiggles that happen in the first few nanoseconds (billionths of a second). It gets the signal 99.35% of the way to perfect.
2. The FIR Filter (The "Fine Polish"): This is the detail work. It smooths out the tiny remaining ripples. After this step, the signal is 99.83% perfect.

The Result

They tested this on a real quantum computer (a 20-qubit machine).

• Before: The signal was messy and unreliable.
• After: The signal was so clean that it deviated from the perfect target by less than 0.17%.

Why This Matters

This isn't just about fixing one wire; it's about automation.

• Speed: Instead of engineers manually tweaking knobs for hours to fix these wobbles, this system can automatically measure the distortion and calculate the "pre-distortion" recipe in seconds.
• Scale: As quantum computers grow from 20 qubits to 1,000 or 1 million, we can't have humans fixing every single wire. This method allows the computer to fix itself automatically.

In short: The paper teaches us how to "lie" to the quantum computer's wires just enough so that the truth arrives perfectly on time, allowing us to build bigger, faster, and more reliable quantum computers.

Technical Summary

1. Problem Statement

Flux-tunable superconducting qubits are essential for implementing high-fidelity two-qubit entangling gates (such as the Controlled-Z gate) in quantum algorithms. These gates rely on precise, fast magnetic flux pulses to modulate the qubit's transition frequency. However, the control signals generated by room-temperature Arbitrary Waveform Generators (AWGs) suffer from Dynamic Distortions (DDs) as they travel through the control chain.

These distortions arise from four primary sources:

1. AWG Pulse Generation: Non-idealities in the instrument's output (modeled as an under-damped second-order system).
2. Cryogenic Flux Line: Parasitic components and coaxial cables causing exponential under/overshoot.
3. Bias-Tee Filtering: High-pass filtering effects causing exponential decay (though this was mitigated in the specific experimental setup).
4. Quantum Chip Response: The non-linear voltage-to-flux transfer function of the qubit's SQUID loop.

If uncorrected, these distortions degrade gate fidelity and repeatability. Traditional compensation methods often involve introducing "settling times" to wait for transients to decay, which drastically increases experiment execution time. The paper addresses the need for a method to actively compensate for these distortions in real-time without sacrificing speed.

2. Methodology

The authors propose a Digital Predistortion (DPD) framework that characterizes the distortion chain and applies an inverse filter to the input signal before it reaches the qubit.

A. Distortion Modeling

The system models distortions using specific mathematical representations:

• AWG: Second-order under-damped system ($H(s)$).
• Cryo Flux Line: Exponential under/overshoot.
• Bias-tee: Exponential decay (high-pass filter).
• Chip Response: A non-linear voltage-to-flux transfer function derived from qubit spectroscopy.

B. Compensation Algorithm

The core compensation mechanism uses a combination of Infinite Impulse Response (IIR) and Finite Impulse Response (FIR) filters applied to the input signal $x[n]$:
$$y[n] = \sum_{i=0}^{M_b} b_i x[n-i] + \sum_{j=1}^{M_a} a_j y[n-j]$$

• Optimization: Filter coefficients ($a_j, b_i$) are calculated using a linear least-squares (LS) algorithm to minimize the Normalized Mean-Squared Error (NMSE) between the corrected output and the ideal target.
• Hardware Efficiency: The number of taps ($M_a, M_b$) is minimized to optimize Field Programmable Gate Array (FPGA) resource usage.

C. Experimental Characterization (The "Cryoscope" Technique)

To determine the necessary correction coefficients, the authors performed two key measurements on a 20-qubit IQM Garnet QPU:

1. 2D Flux Spectroscopy: A 2D experiment varying the Z-line voltage and XY-line frequency to map the qubit frequency ($f_q$) against the applied flux. This data is fitted to the theoretical SQUID equation to extract the voltage-to-flux relationship.
2. Ramsey-Style Experiment: A sequence involving a $\pi/2$ pulse, a variable-duration flux step pulse, and a second $\pi/2$ pulse. By measuring the accumulated phase of the qubit state as a function of the flux pulse duration, the authors extracted the instantaneous frequency detuning. This allowed them to reconstruct the voltage-to-flux transfer function in-situ.

3. Key Contributions

• Integrated DPD Framework: The first demonstration of a DPD system specifically tailored for flux-control lines in superconducting QPUs, combining classical line distortions with quantum chip non-linearities.
• Two-Stage Correction Strategy: The proposal and validation of a hybrid approach using an IIR filter to handle fast transient rise-up effects, followed by an FIR filter to correct residual steady-state errors.
• Automated Calibration: The method enables rapid, automated calibration of flux channels, removing the need for manual tuning or long settling times.
• Hardware-Aware Optimization: Identification of the minimum number of filter taps required for effective compensation, making the solution viable for integration into programmable quantum control hardware (FPGAs).

4. Results

Experiments were conducted using a Keysight Quantum Control System (QCS) which allowed for direct composite DC+BB pulse generation, effectively eliminating bias-tee distortions for the scope of this test.

• Simulation: Simulations showed that with a minimum of 1 feedback tap and 2 feedforward taps, the system could achieve a log(NMSE) < -30 dB for various distortion types.
• Experimental Correction:
  • IIR Correction: Successfully compensated for the initial rise-up transient (occurring within the first 20 ns), reducing deviations from the ideal linear behavior to 0.65%.
  • FIR Correction: When applied after the IIR stage (once the filter memory was full), the combined system reduced deviations to 0.17%.
  • Overall Performance: The final compensated signal showed only sub-percent deviations from the ideal target, effectively linearizing the flux control.

5. Significance

This work is significant for the scalability of superconducting quantum computing for several reasons:

• Gate Fidelity: By eliminating flux pulse distortions, the technique directly improves the fidelity of two-qubit gates, a critical bottleneck for error correction and large-scale algorithms.
• Speed: It eliminates the need for "settling time" delays, allowing for faster gate execution and higher throughput in quantum experiments.
• Scalability: The automated nature of the DPD calibration makes it feasible to scale to QPUs with hundreds or thousands of qubits, where manual calibration of individual flux lines would be impossible.
• Future Path: The authors note that this technique can be extended to tunable couplers and multi-qubit architectures, paving the way for high-fidelity operations in next-generation quantum processors.