Proposal for erasure conversion in integer fluxonium qubits

1. Problem Statement

Quantum error correction (QEC) is essential for fault-tolerant quantum computing, but its overhead is heavily dependent on the nature of errors. Erasure errors (errors where the location is known) are significantly easier to correct than standard non-heralded errors, leading to higher error thresholds and lower logical error rates.
While erasure qubits have been demonstrated in natural atoms and composite superconducting circuits (e.g., dual-rail transmons, Floquet fluxonium molecules), these often require complex multi-mode architectures. The paper addresses the question: Can erasure conversion be engineered using the bare energy levels of a single-mode superconducting artificial atom (fluxonium) without increasing architectural complexity?

The authors focus on Integer Fluxonium Qubits (IFQs), which are biased at the zero-flux sweet spot. They propose utilizing higher excited states to create computational subspaces that are naturally protected against certain error types, converting dominant relaxation errors into detectable erasures.

2. Methodology

The authors propose and analyze two distinct computational configurations for the IFQ, utilizing the three lowest energy levels: $|g\rangle$ (ground), $|e\rangle$ (first excited), and $|f\rangle$ (second excited).

A. Qubit Configurations

1. The $|e\rangle-|f\rangle$ Qubit:

   • Mechanism: Uses the transition between the first and second excited states.
   • Protection: The transition frequency is low (tens to hundreds of MHz), suppressing dielectric loss. The $|e\rangle \to |g\rangle$ decay is symmetry-forbidden or exponentially suppressed due to weak tunneling, making the primary error channel the decay from $|e\rangle$ to $|g\rangle$.
   • Erasure Conversion: Leakage from $|e\rangle$ to $|g\rangle$ is treated as an erasure event.
   • Control: Operated via flux pulses or microwave drives.

2. The $|g\rangle-|f\rangle$ Qubit:

   • Mechanism: Uses the ground state $|g\rangle$ and the second excited state $|f\rangle$.
   • Protection: Direct transitions between $|g\rangle$ and $|f\rangle$ are forbidden by parity symmetry. The primary error is the decay from $|f\rangle$ to $|e\rangle$.
   • Erasure Conversion: Decay to $|e\rangle$ is detected and converted to an erasure. The high frequency of the $|g\rangle-|e\rangle$ transition prevents thermal re-excitation.
   • Control: Requires a two-photon Raman process (via an intermediate state $|h\rangle$) to drive the $|g\rangle \leftrightarrow |f\rangle$ transition.

B. Erasure Detection Protocol (Dispersive Readout)

To detect leakage without destroying the coherence of the computational subspace, the authors propose a modified dispersive readout:

• Bright-Dark Scheme: The resonator is driven at a frequency dressed by the "bright" (leakage) state.
• Selection Rules: By carefully tuning the resonator frequency ($\omega_r$) and coupling strength, the dispersive shift ($\chi$) for the leakage state is made distinct ($\chi_L \neq \chi_{comp}$), while the shifts for the two computational states are matched ($\chi_a \approx \chi_b$).
• Outcome:
  • Bright Resonator: Indicates the qubit has leaked (Erasure detected).
  • Dark Resonator: Indicates the qubit remains in the computational subspace.
• Dephasing Mitigation: The protocol minimizes the average photon number in the resonator when coupled to computational states to reduce AC-Stark shift-induced dephasing (phase smearing).

C. Two-Qubit Gates

For the $|g\rangle-|f\rangle$ qubit, the authors propose a geometric Controlled-Z (CZ) gate using direct capacitive coupling:

• Challenge: Weak frequency differences between desired and unwanted transitions in the weak coupling regime.
• Solution: Utilization of Selective Darkening. By applying simultaneous drives to both qubits with specific amplitude ratios, unwanted transitions (e.g., $|gf\rangle \to |gh\rangle$) are destructively interfered (darkened), while the desired transition ($|ff\rangle \to |fh\rangle$) is driven.
• Optimization: Numerical pulse shaping (using auto-differentiation) is employed to correct for nonlinear effects and leakage at high drive amplitudes.

3. Key Contributions

1. Single-Mode Erasure Qubits: Demonstrated that a single-mode fluxonium circuit can function as a high-performance erasure qubit by selecting specific computational subspaces ($|e\rangle-|f\rangle$ or $|g\rangle-|f\rangle$), eliminating the need for complex composite qubit architectures.
2. Erasure Conversion via Dispersive Readout: Developed a theoretical framework for converting amplitude damping (relaxation) into heralded erasure errors using a "bright-dark" dispersive readout scheme that minimizes dephasing in the computational subspace.
3. High-Fidelity Gate Design:
   • Proposed Raman X-gates for the $|g\rangle-|f\rangle$ qubit with computational errors $\sim 5 \times 10^{-5}$.
   • Designed a geometric CZ gate for coupled $|g\rangle-|f\rangle$ qubits achieving fidelities $> 0.999$ at gate times of $\sim 150-180$ ns, utilizing selective darkening and pulse optimization.
4. Parameter Optimization: Identified distinct circuit parameter regimes ($E_J, E_C, E_L$) required to maximize coherence for each configuration, noting that a single physical device cannot easily switch between the two modes.

4. Results

• Coherence Times:
  • $|e\rangle-|f\rangle$ Qubit: Achieves high $T_2$ due to low frequency and suppressed charge matrix elements. Simulations show computational subspace errors of $10^{-4}$ and erasure rates of $3 \times 10^{-4}$ for a 40 ns gate.
  • $|g\rangle-|f\rangle$ Qubit: Shows order-of-magnitude improvement in effective coherence time compared to conventional fluxonium. Simulations show computational errors of $5 \times 10^{-5}$ and erasure rates of $4 \times 10^{-4}$ for a 50 ns Raman gate.
• Readout Performance:
  • The proposed bright-dark readout achieves a Signal-to-Noise Ratio (SNR) of $\approx 7$, resulting in readout assignment errors of $\sim 7 \times 10^{-7}$.
  • Dephasing induced by the readout in the computational subspace is negligible ($\sim 10^{-5}$) provided the dispersive shift mismatch ($\Delta \chi$) and AC-Stark shift are small.
• Gate Fidelity:
  • The optimized CZ gate for $|g\rangle-|f\rangle$ qubits achieves a coherent fidelity of 0.99976 at 180 ns.
  • The gate successfully suppresses leakage to unwanted levels (e.g., $|gh\rangle$) through dynamic pulse shaping.

5. Significance

This work provides a hardware-efficient pathway to fault-tolerant quantum computing. By leveraging the intrinsic symmetry and energy level structure of the integer fluxonium:

• It reduces the hardware overhead typically associated with erasure qubits (no need for dual-rail or multi-mode cavities).
• It offers a dominant erasure error structure, which significantly relaxes the requirements for QEC codes, allowing for higher error thresholds.
• It demonstrates that high-fidelity single- and two-qubit gates are compatible with these erasure-protected subspaces.
• The results suggest that the $|g\rangle-|f\rangle$ configuration, in particular, is a superior candidate for next-generation quantum processors due to its robustness against flux noise and its ability to convert the dominant relaxation error into a correctable erasure.

The paper concludes that with proper circuit design and control, integer fluxonium qubits can achieve effective coherence times that make them prime candidates for scalable, fault-tolerant quantum computation.