Surface Code with Imperfect Erasure Checks

Technical Summary: Surface Code with Imperfect Erasure Checks

Problem Statement
Recent advancements in quantum error correction (QEC) have focused on "erasure qubits," where dominant physical noise manifests as leakage states that can be detected and converted into erasures. Theoretical studies indicate that QEC protocols utilizing erasure errors possess significantly higher effective error distances and threshold error rates compared to those subject to standard Pauli noise. However, these theoretical advantages often assume perfectly accurate erasure checks performed after every gate operation. In practice, implementing such perfect checks incurs significant time and hardware overhead. For instance, in superconducting dual-rail (DR) qubit architectures, performing a dedicated erasure check after every gate can double the error correction cycle time and introduce new errors during the check itself. Furthermore, practical erasure checks may suffer from imperfections, including:

1. Imperfect Temporal Resolution: The check may detect a leakage event with a delay, meaning a flag raised at time $t$ could correspond to a leakage event at time $t$ or $t-1$.
2. Imperfect Spatial Resolution: The check may indicate that a leakage occurred within a specific two-qubit gate but fail to identify which of the two participating qubits leaked.

This work investigates the consequences of these imperfections on the performance of the surface code, specifically asking whether the advantages of erasure qubits persist when checks are imperfect but overhead-efficient.

Methodology
The authors analyze the surface code under four distinct noise models combining two types of leakage-induced Pauli errors and two types of erasure check imperfections:

• Leakage-Induced Pauli Models:
  • General Pauli Model: A leaked qubit induces a Pauli error ($I, X, Y, Z$) on the interacting qubit with equal probability.
  • Tailored Pauli Model: Motivated by superconducting DR qubits (specifically cavity-based DRs with built-in checks), this model assumes the induced error is restricted. For a CZ gate, the error is drawn from $\{I, Z\}$; for a CX gate, from $\{I, X\}$. This restriction arises from the specific physics of the DR platform.
• Erasure Check Imperfections:
  • Perfect Spatial Resolution: The check identifies the exact leaked qubit.
  • Imperfect Spatial Resolution: The check identifies the gate containing the leak but not the specific qubit, requiring both qubits to be reset.
  • Imperfect Temporal Resolution: Characterized by a parameter $\eta$, representing the probability that a flagged leakage occurred in the current gate rather than the immediately preceding one.

The authors employ Monte Carlo simulations of stochastic circuit-level noise combined with a weighted Union-Find (UF) decoder. The decoder is calibrated to account for the correlated errors introduced by delayed detection (where a single leakage event can propagate errors to multiple qubits over time) and the uncertainty in spatial resolution. They evaluate the threshold error rate ($p_{th}$) and the effective error distance ($d_{eff}$) for various values of $\eta$ and the erasure fraction $R_e$.

Key Results

1. Threshold Performance:

   • Under physically reasonable assumptions (e.g., $\eta \approx 0.986$ and $R_e \approx 0.98$), the threshold for the surface code with imperfect erasure checks remains significantly higher than the threshold for pure Pauli noise ($\sim 1\%$).
   • Specifically, for the tailored Pauli model with imperfect checks, the threshold is approximately 4.23%, which is more than twice the Pauli threshold and only slightly lower than the perfect erasure detection case ($\sim 5.56\%$).
   • Even in the worst-case scenario where temporal resolution is poor ($\eta \to 0$), the threshold remains well above the Pauli limit (ranging from $\sim 2.6\%$ to $\sim 4.9\%$ depending on the model).
   • The difference in threshold between spatially perfect and spatially imperfect checks is negligible for the general Pauli model but remains significant ($>1\%$) for the tailored Pauli model when $\eta \to 1$.

2. Effective Error Distance ($d_{eff}$):

   • General Pauli Model: Imperfect temporal resolution degrades the effective distance. For a code of size $d$, the effective distance to a logical $Z$ error ($Z_L$) drops from $d$ to approximately $(d+1)/2$ as $\eta$ decreases, effectively reverting to the scaling of Pauli noise.
   • Tailored Pauli Model: Crucially, the effective distance remains robust. Even with imperfect temporal and spatial resolution, the effective distance to both $Z_L$ and $X_L$ logical errors remains at $d$. This is because the restricted error set ($\{I, Z\}$ or $\{I, X\}$) prevents a single delayed leakage event from generating the specific correlated errors required to form a minimum-weight logical operator.

3. Hardware Efficiency:

   • The study connects these models to existing superconducting dual-rail architectures. Cavity-based DRs with built-in erasure checks (which are spatially imperfect but temporally efficient) require significantly fewer transmons (approx. 29% of the count required for conventional qubits to achieve the same $d_{eff}$) compared to explicit check schemes.

Significance and Claims
The paper claims that the substantial advantages of erasure qubits—specifically the relaxed system requirements for fault tolerance—are robust against realistic imperfections in erasure checks.

• Overhead Reduction: It demonstrates that one can achieve high-performance surface code memories using "overhead-efficient" erasure checks (e.g., built-in checks with imperfect spatial/temporal resolution) without sacrificing the threshold or effective distance, provided the leakage-induced noise follows the "tailored" structure found in superconducting DR qubits.
• Practical Feasibility: The results suggest that the stringent requirement for perfect, real-time erasure checks is not necessary. Systems can tolerate delayed detection and spatial ambiguity while still maintaining a threshold error rate more than double that of standard Pauli noise.
• Noise Characterization: The work emphasizes the critical importance of experimentally characterizing the detailed noise model (specifically whether leakage induces general or tailored Pauli errors), as this determines whether the code retains its distance advantage under imperfect checks.

The authors conclude that achieving good performance surface code quantum memories with relaxed system requirements is possible, motivating the use of simplified, hardware-efficient erasure check strategies in superconducting quantum processors.