Latency-Constrained Encoded Quantum Teleportation with Punctured Codes

Technical Summary: Latency-Constrained Encoded Quantum Teleportation with Punctured Codes

Problem Statement
Quantum teleportation is a fundamental protocol for distributing quantum information, but its reliability is constrained by the stochastic nature of entanglement generation and the decoherence of entangled pairs stored in quantum memory. While encoded teleportation—using quantum error-correcting codes (QECC) to transmit logical qubits across multiple physical qubits—can theoretically improve reliability beyond the fidelity of individual entangled pairs (EPR pairs), it introduces a critical trade-off. Longer codes require larger "entanglement packets" (more EPR pairs), which increases the time required to accumulate resources. This increased acquisition time leads to greater memory decoherence, degrading the fidelity of the stored pairs. Consequently, the benefits of stronger error correction may be negated by the reduced quality of the entanglement resources, particularly under strict latency constraints. Existing works often treat entanglement quality as independent of code length, failing to account for how resource acquisition delays alter fidelity.

Methodology
The authors develop a unified framework that jointly models three interacting components: stochastic entanglement generation, memory-induced decoherence, and punctured quantum error-correcting codes.

1. System Model:

   • Entanglement Generation: Modeled as a discrete-time stochastic process where EPR pairs are generated with probability $p$. The initial fidelity $F_0$ is inversely coupled to the generation rate; higher generation rates (required for tighter latency) result in lower initial fidelity.
   • Decoherence: Stored qubits undergo exponential fidelity decay over time ($T$ is the coherence time). Since entanglement packets are formed probabilistically, pairs generated earlier accumulate "age" and suffer more decoherence than those generated later, leading to heterogeneous fidelities within a single packet.
   • Latency Constraint: A target average latency $L$ dictates the required generation probability $p$. The framework solves for $p$ such that the expected waiting time to form a packet of size $n$ meets $L$.

2. Encoded Teleportation:

   • The system employs CSS codes to encode a logical qubit across $n$ physical qubits, each teleported via an independent EPR pair.
   • Due to heterogeneous fidelities, the resulting Pauli error probabilities vary across qubits. The authors model this using a Poisson-binomial distribution to calculate the logical error probability ($P_L$).
   • Adaptive Puncturing: To navigate the latency-reliability trade-off, the authors utilize code puncturing. Starting from a common base code (a length-17 CSS code), they define discrete "puncturing tiers" (e.g., $n=17, 13, 8$) and an uncoded case ($n=1$). This allows the effective code length to adapt to resource availability while maintaining a common stabilizer structure.

3. Evaluation:

   • The authors perform numerical simulations to evaluate $P_L$ under varying latency constraints ($L$) and link qualities (detection probability $p_d$).
   • They analyze both symmetric and asymmetric noise regimes (where phase-flip errors dominate).
   • The goal is to identify "decision regions" where specific puncturing tiers minimize $P_L$ for a given operating condition.

Key Contributions

• Unified Framework: The paper presents a cross-layer model that integrates entanglement generation statistics, memory decoherence, and coding decisions, explicitly capturing the coupling between latency, resource availability, and entanglement fidelity.
• Latency-Reliability Trade-off Analysis: The work demonstrates that increasing code length does not monotonically improve reliability. Instead, there are distinct regimes where shorter codes (or uncoded transmission) outperform longer codes because the fidelity loss from extended waiting times outweighs the coding gain.
• Adaptive Puncturing Strategy: The authors propose using punctured variants of a base code as a practical mechanism for adaptive encoded teleportation. This allows network nodes to dynamically select the optimal code length based on current latency budgets and link conditions without switching between unrelated encoding implementations.
• Decision Regions: Through simulation, the paper maps out specific operating regimes (defined by latency $L$ and detection probability $p_d$) where different puncturing tiers ($n=1, 8, 13, 17$) are optimal.

Results
Numerical results indicate that:

• Under Tight Latency Constraints: Encoded teleportation significantly outperforms uncoded transmission. When latency is low, the generation rate must be high, resulting in low-fidelity pairs. In this regime, the error-correction capability of codes (e.g., $n=8$ or $n=17$) compensates for the poor entanglement quality.
• Under Loose Latency Constraints: As the allowed latency increases, the generation rate can be lowered, yielding higher-fidelity pairs. In this regime, uncoded transmission often becomes optimal because the individual pairs are sufficiently reliable, and the overhead of transmitting multiple physical qubits (which introduces more opportunities for error) outweighs the coding gain.
• Asymmetric Noise: In regimes where phase-flip errors dominate ($\eta = 10$), the asymmetrically punctured $n=13$ code consistently achieves the lowest logical error probability across most latency and link-quality ranges.
• Link Quality Dependence: For poor link conditions (low detection probability), shorter codes or uncoded transmission are preferred. As link quality improves, longer codes become favorable.

Significance
The paper argues that reliable quantum networking requires resource-aware adaptation. It highlights that the quantity and quality of entanglement are inextricably linked; acquiring more resources (longer codes) inherently degrades their quality due to time-dependent decoherence. Therefore, a fixed coding strategy is suboptimal. The proposed use of punctured codes offers a structured, low-overhead method to adapt to these dynamic constraints, enabling quantum networks to balance performance, complexity, and resource efficiency. This approach is particularly relevant for early-generation quantum access networks where users have diverse quality-of-service requirements and hardware capabilities.