Characterization of Radiation-Induced Errors in Superconducting Qubits Protected with Various Gap-Engineering Strategies

This study demonstrates that gap-engineering strategies in superconducting qubits can mitigate radiation-induced correlated errors by reducing quasiparticle density at Josephson junctions and accelerating recovery through trapping in the capacitor/ground-plane, thereby offering effective pathways to improve radiation resilience.

The Gist

Imagine you are trying to keep a house of cards perfectly balanced in a room where people are constantly throwing tiny, invisible pebbles at the walls. In the world of quantum computing, these "pebbles" are high-energy particles from space (cosmic rays) or tiny bits of radioactive dust in the materials we build with. When they hit the superconducting chips that power quantum computers, they cause "glitches" that knock the delicate quantum information out of whack.

This paper is like a team of engineers and physicists trying to figure out how to build a house of cards that can survive these pebble storms. They tested three different architectural designs to see which one holds up best.

The Problem: The "Pebble Storm"

Superconducting qubits (the brain cells of a quantum computer) are incredibly sensitive. When a high-energy particle hits the chip, it creates a shower of energetic particles called quasiparticles. Think of these quasiparticles as a swarm of angry bees. If they buzz around the Josephson Junction (the critical switch in the qubit), they can cause the qubit to flip its state randomly, creating errors.

The scary part? These errors often happen in clusters. One pebble doesn't just knock over one card; it knocks over a whole row at once. This is bad news because standard error-correction software (the "spell-check" for quantum computers) struggles to fix errors that happen all at the same time.

The Solution: "Gap Engineering" (Building Better Walls)

The researchers tried a strategy called Gap Engineering. Imagine the qubit is a room with a specific temperature. The "gap" is like a thermal barrier or a wall that keeps the heat (energy) from moving where it shouldn't.

They built three different types of chips:

1. The Standard Chip (No-JJ-GE): The walls are thin. If a pebble hits, the "angry bees" (quasiparticles) run wild everywhere, causing immediate chaos.
2. The "JJ-Only" Chip: They built a very thick, high wall right at the critical switch (the Josephson Junction). This stops the bees from crossing the threshold easily. It's like putting a bouncer at the door who only lets in calm bees.
3. The "JJ & M1" Chip: This is the super-chip. It has the thick wall at the switch plus a second strategy: a "trap" in the floor (the capacitor/ground plane). If any bees do get past the bouncer, they fall into a pit where they get stuck and can't cause trouble anymore.

The Experiments: Throwing Pebbles and Bees

To test these designs, they didn't just wait for cosmic rays to hit. They set up two different "shooting ranges":

• The Alpha Particle Test (The Heavy Hitters): They used a source of alpha particles (like tiny, heavy cannonballs) to simulate high-energy cosmic rays.
  • Result: Even the best "bouncer" (the thick wall) couldn't stop the biggest cannonballs completely. However, the chip with the trap (JJ & M1) recovered much faster. The bees got stuck in the pit, and the room returned to normal quickly.
• The Electron Test (The Light Hitters): They used a particle accelerator to shoot electrons (lighter, faster pebbles) at the chips.
  • Result: They found that even the "lighter" pebbles could cause errors, but again, the chip with the trap recovered significantly faster. The "bouncer" reduced the initial damage, but the "trap" cleaned up the mess quickly.

The Key Discoveries (In Plain English)

1. The "Bouncer" Helps, But Isn't Enough: Building a strong barrier at the switch (Gap Engineering) stops many errors, but it doesn't stop the biggest, most energetic cosmic rays. These rays are powerful enough to break through the wall.
2. The "Trap" is the Secret Weapon: The most important finding was that adding a quasiparticle trap (a place to catch the angry bees) made the system recover much faster. It's like having a vacuum cleaner that immediately sucks up the bees after they cause a disturbance.
3. The "House" Orientation Matters: They noticed that the direction the chip was built (whether the "trap" was near the capacitor or the ground) changed how fast the bees were caught. This tells them exactly where to put the traps in future designs.
4. The "Temperature" Spike: When a particle hits, the chip gets momentarily "hot" (in terms of energy). The trap helps cool it down faster.

Why Does This Matter?

Quantum computers are trying to reach a point where they can fix their own errors automatically (called "Quantum Error Correction"). But right now, these "pebble storms" are causing too many simultaneous errors for the software to handle.

This paper says: "Don't just build a stronger wall; build a better cleanup crew."

By combining a strong barrier at the switch with a trap to catch the stray particles, we can make quantum computers much more resilient. This is a crucial step toward building quantum computers that can actually work in the real world, rather than just in a perfectly shielded lab.

In a nutshell: They figured out that to stop cosmic rays from ruining quantum computers, you need a double defense: a strong gate to keep the bad guys out, and a net to catch the ones that slip through, so the computer can bounce back instantly.

Technical Summary

1. Problem Statement

Superconducting qubits are highly susceptible to errors induced by environmental radiation (cosmic rays, terrestrial radioactivity). High-energy particles interacting with the substrate generate phonons, which create non-equilibrium Bogoliubov quasiparticles (QPs). These QPs cause correlated errors (simultaneous errors across multiple qubits) that are difficult to correct with standard Quantum Error Correction (QEC) codes.

While "Josephson Junction (JJ) gap engineering" (creating a superconducting gap difference, $\delta\Delta_{JJ}$, across the junction larger than the qubit energy, $hf_{qb}$) has been shown to mitigate errors from low-energy radiation, a residual error floor (approximately $10^{-10}$ per gate, or one error per hour) persists. The origin of this residual error and the limits of current gap-engineering strategies remain unclear. Specifically, it is unknown if high-energy hadronic cosmic rays or $\alpha$-particles can overcome JJ protection, and whether engineering the gap at the capacitor/ground-plane interface ($\delta\Delta_{M1}$) aids in QP trapping and recovery.

2. Methodology

The authors designed and fabricated three distinct arrays of transmon qubits, each with a unique superconducting gap profile, to test different mitigation strategies:

1. No-JJ-GE (Control): Standard design with $\delta\Delta_{JJ} < hf_{qb}$ and minimal $\delta\Delta_{M1}$.
2. JJ-Only: Engineered with a large $\delta\Delta_{JJ}$ ($\approx 10$ GHz) to block QP relaxation, but minimal $\delta\Delta_{M1}$.
3. JJ&M1: Engineered with a large $\delta\Delta_{JJ}$ ($\approx 7$ GHz) and a significant gap difference at the base metal interface ($\delta\Delta_{M1} \approx 3$ GHz) to act as a QP trap.

The arrays were exposed to two distinct radiation sources:

• $\alpha$-particles: Using a $^{241}\text{Am}$ source (5.5 MeV $\alpha$-particles) to simulate high-energy terrestrial radiation and hadronic cosmic ray impacts.
• Electrons: Using the CLIQUE facility at Johns Hopkins Applied Physics Laboratory (10–20 MeV electrons) to provide tunable, time-tagged, minimum-ionizing radiation with precise energy deposition control (14.5 keV to 1 MeV).

Measurement Techniques:

• Relaxation/Excitation Rates: Used heralded measurement sequences (Sequences A, B, C, and D) to detect transient increases in $T_1$ (relaxation) and excitation rates following particle impacts.
• Frequency Shifts: Employed Ramsey interferometry to measure transient shifts in qubit frequency, serving as a proxy for local QP density near the JJ.
• Modeling: Developed a time-dependent phenomenological model based on Rothwarf-Taylor equations to simulate QP dynamics, including generation, recombination, trapping, and tunneling across the engineered gaps.

3. Key Contributions

• Identification of Residual Error Sources: Demonstrated that even with robust JJ gap engineering ($\delta\Delta_{JJ} > hf_{qb}$), qubits remain sensitive to high-energy particles (MeV scale), supporting the hypothesis that hadronic cosmic rays are the primary source of the residual correlated error floor observed in state-of-the-art QEC experiments.
• Validation of $\delta\Delta_{M1}$ as a Recovery Mechanism: Showed that introducing a gap difference at the capacitor/ground-plane interface ($\delta\Delta_{M1}$) significantly hastens the recovery of qubit transition rates and frequency shifts by trapping QPs in the base metal layer.
• Orientation Dependence: Quantified how the orientation of the JJ (whether the low-gap side contacts the capacitor or the ground plane) affects recovery times, providing insight into QP trapping efficiency in different geometries.
• Comprehensive QP Dynamics Model: Constructed a six-variable differential equation model that qualitatively reproduces the experimental observations of relaxation, excitation, and frequency recovery across different gap profiles.

4. Key Results

A. Response to $\alpha$-particles (High Energy)

• Correlated Errors Persist: Despite $\delta\Delta_{JJ} > hf_{qb}$, both the "JJ-Only" and "JJ&M1" arrays exhibited correlated relaxation and excitation errors upon $\alpha$-particle impact.
• Energy Threshold: The errors were driven by QPs with energy $E_{QP} > \delta\Delta_{JJ} - hf_{qb}$, indicating that high-energy particles deposit enough energy to create "hot" QPs capable of overcoming the gap barrier.
• Recovery Time: The recovery timescale was approximately 100 $\mu$s, an order of magnitude faster than non-engineered qubits, but still significant.
• Implication: The rate of these events matches the expected flux of cosmic rays above 1 MeV, confirming hadronic cosmic rays as a major contributor to the error floor.

B. Response to Electrons (CLIQUE)

• Sensitivity to Low Energy: Even with JJ protection, qubits showed sensitivity to low-energy depositions (14.5–145 keV), though the effect is smaller than for high-energy impacts.
• Impact of $\delta\Delta_{JJ}$: The "JJ-Only" array showed a 3$\times$ lower peak error rate compared to the control, confirming the efficacy of gap engineering in suppressing the initial error magnitude.
• Impact of $\delta\Delta_{M1}$ (Recovery): The "JJ&M1" array (with large $\delta\Delta_{M1}$) recovered significantly faster than the "JJ-Only" array.
  • Relaxation/Excitation: Recovery time constants were reduced by an order of magnitude.
  • Frequency Shift: The QP density near the JJ (inferred from frequency shift) returned to equilibrium much faster in the "JJ&M1" array.
• Mechanism: The large $\delta\Delta_{M1}$ creates an energy barrier that traps QPs in the base metal (M1), preventing them from diffusing back to the JJ and causing further errors.

C. JJ Orientation Effects

• "Slow" vs. "Fast" Orientations: Qubits where the low-gap side of the JJ contacts the isolated capacitor ("Slow") showed different recovery dynamics compared to those contacting the large ground plane ("Fast").
• Trapping Efficiency: The "Fast" orientation generally recovered faster due to the ground plane's ability to trap QPs via features like air bridges. The "JJ&M1" design reduced the difference between orientations, suggesting the engineered trap ($\delta\Delta_{M1}$) dominates the recovery process.

D. Model Validation

• The developed QP dynamics model successfully reconstructed the magnitude and timescales of both relaxation and excitation rates for all arrays.
• The model confirmed that the faster recovery in the "JJ&M1" array is due to QPs being trapped in the M1 layer, effectively removing them from the JJ vicinity.

5. Significance and Future Directions

• Radiation Resilience Strategy: The paper establishes that dual-gap engineering is necessary for optimal radiation resilience:
  1. $\delta\Delta_{JJ}$: Suppresses the severity (magnitude) of the initial error burst by blocking low-energy QPs.
  2. $\delta\Delta_{M1}$: Accelerates the recovery time by trapping high-energy QPs away from the junction.
• Mitigation of Hadronic Cosmic Rays: Since JJ engineering cannot fully block MeV-scale hadronic cosmic rays, the authors suggest that future quantum processors must operate in shallow underground facilities (15–30m depth) to attenuate these particles, alongside material purification to remove $\alpha$-emitting contaminants.
• Material Recommendations: The work highlights a disadvantage of using high-gap base metals with Al-based JJs (which creates a negative $\delta\Delta_{M1}$, turning the JJ into a trap). Instead, it motivates the development of higher-gap JJs (e.g., NbN/AlN) and improved film quality for base metals to maximize the trapping effect.
• Modeling Tool: The provided QP dynamics model offers a framework for future simulations of radiation effects in superconducting circuits, potentially integrable with tools like G4CMP.

In conclusion, this work provides a critical roadmap for improving the radiation hardness of superconducting quantum processors, moving beyond simple gap engineering at the junction to a holistic approach involving substrate interface engineering and environmental shielding.