Latency-Constrained Encoded Quantum Teleportation with Punctured Codes Mahmoud Saad Abouamer† , Jakob Kaltoft Søndergaard† , Petar Popovski
arXiv:2607.19770v1 [quant-ph] 22 Jul 2026
Department of Electronic Systems, Aalborg University, 9220 Aalborg, Denmark Email: {mahmoudabo, jakobks, petarp}@es.aau.dk
improved fidelity using local operations and classical communication [3]. However, purification introduces additional processing, coordination, and storage overhead, during which stored entanglement decoheres [4], [5]. Consequently, practical latency and storage constraints can limit the fidelity that can be achieved through resource-level optimization alone. This motivates encoded teleportation, where reliability is improved beyond individual EPR pair fidelity by encoding quantum information using a quantum error-correcting code prior to teleportation [6], [7]. A logical qubit is encoded across multiple physical qubits, each teleported using an independent entangled pair, followed by decoding to detect and correct errors introduced by imperfect entanglement. While logicallevel protection can improve reliability beyond individual EPR pair fidelity, it also increases the number of entangled resources required for communication. In particular, an [[n, k]] code requires n entangled pairs to transmit k logical qubits. While longer codes improve error-correction capability, they also increase resource consumption and latency. However, these works treat entanglement quality as independent of code length and do not account for how acquiring larger entanglement resources can alter their fidelity through waitingtime variability and memory decoherence. As illustrated in Fig. 1, the impact of longer codes in quantum networks extends beyond the local encoding and decoding operations and includes the time required to acquire I. I NTRODUCTION entanglement resources. Since encoded teleportation requires multiple entangled pairs to transmit a logical state, the network Quantum networks enable the distribution of quantum inmust first accumulate these resources before the encoded formation across different quantum processors, supporting teleportation can be completed. Such collections of entangled applications such as distributed quantum computation [1], [2]. pairs, referred to as entanglement packets [8], are generated Quantum teleportation serves as a fundamental primitive for probabilistically, making the acquisition time variable. Under such networks, enabling the transfer of quantum states through latency constraints, the network may need to operate at higher shared entanglement and classical communication [2]. The entanglement generation probabilities to meet the required reliability of teleportation is therefore fundamentally tied to timing, which can result in lower-fidelity entangled pairs [8]. the quality of the shared entanglement. In addition, pairs generated earlier must be stored while the To improve teleportation reliability, one direct approach is remaining pairs are acquired, during which they decohere and to improve the fidelity of the shared entangled pairs before degrade in fidelity. they are used for teleportation. This is commonly achieved Consequently, increasing the packet size to support a longer through entanglement purification, where multiple imperfect code reduces the quality of the entanglement used for teleentangled pairs are processed to obtain fewer pairs with portation. In some regimes, this reduction in pair quality can outweigh the error-correction gain of a longer code, This work was supported, in part, by the Danish National Research so longer codes do not necessarily improve reliability. This Foundation (DNRF), through the Center CLASSIQUE, grant nr. 187. † Equal contribution. effect is driven by probabilistic acquisition delays and subAbstract—Quantum teleportation is a key protocol for transmitting quantum information using entanglement and classical communication. Its reliability is constrained by both the availability and fidelity of shared entangled pairs, which are affected by stochastic generation and memory decoherence. In this work, we focus on encoded teleportation, in which quantum information is encoded using a quantum error-correcting code and transmitted as a codeword. We evaluate reliability in terms of logical error probability, considering latency-constrained settings where entanglement is accumulated over time and degrades while in memory. We develop a unified framework that captures the interaction between entanglement availability, decoherence, and coding decisions. Our results show that the benefits of longer codes depend on the availability and fidelity of entangled pairs, as acquiring additional resources introduces delays that can reduce their quality. To address this latency-reliability tradeoff, we leverage code puncturing to enable flexible encoded teleportation, allowing the effective code length to adapt across different latency regimes while preserving a common stabilizer structure. Numerical results show that encoded teleportation can provide substantial reliability gains over uncoded transmission under a common entanglement-acquisition latency constraint, and that selecting appropriate punctured codes improves performance across varying latency budgets. Overall, our results highlight the importance of resource-aware adaptation for reliable quantum networking. Index Terms—Quantum networks, entanglement distribution, quantum error correction, encoded teleportation, code puncturing.
ENCODED TELEPORTATION
Generate & store EPR pairs
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• Request completes sooner • Less time in memory ⇒ higher fidelity • Lower coding gain, better entanglement
time • Request completes later • More time in memory ⇒ more decoherence • Higher coding gain, lower entanglement
Implication: [[n1 , k]] is preferable in some regimes, while [[n2 , k]] is preferable in others. Support both using punctured variants of a common base code [[nbase , k]].
Figure 1: Latency–reliability tradeoff in encoded quantum teleportation. The dominant latency arises from entanglement acquisition. With n1 < n2 , shorter effective code lengths require smaller entanglement packets, reducing acquisition burden and memory decoherence, but provide lower coding gain. Longer codes can provide stronger error protection, but require larger entanglement packets and may increase waiting times and storage-induced decoherence. Different latency–reliability regimes can therefore favor different code lengths, motivating adaptive puncturing from a common base code.
sequent memory decoherence, which has no direct analogue glement generation, link quality, and latency requirements can in conventional classical communication resources. The result- vary across operating conditions, a fixed code choice may be ing latency-reliability tradeoff, illustrated in the contrasting suboptimal. We investigate these tradeoffs to identify decision regimes of Fig. 1, depends on both the number and quality regions and motivate a latency-aware code-selection strategy. of available entangled pairs. This work focuses on networkTo support this, we adopt puncturing of quantum errorrelated imperfections arising from stochastic entanglement correcting codes [9] as a mechanism for modifying code length generation, storage-induced decoherence, and noisy shared and distance by selectively removing physical qubits. This entangled pairs. makes puncturing a practical mechanism for adaptive encoded The latency-reliability tradeoff affects the optimized choice teleportation driven by resource availability. In particular, of code length depending on the network conditions. As nodes can operate within a common base-code architecture and illustrated conceptually in Fig. 2, evaluating the logical er- dynamically use only the number of entangled pairs required ror probability under different latency constraints can reveal by the chosen punctured code, enabling adaptation to resource distinct decision regions separated by crossover boundaries. availability while reducing the need to switch between unreAlthough a given code length can, in principle, be operated lated encoding and decoding implementations. Since shared under different latency targets by adjusting the entanglement entangled pairs are consumed as network communication regeneration probability, doing so changes the quality of the sources, whereas encoding and decoding are performed locally, generated entangled pairs and therefore the resulting reliability. puncturing can reduce the number of entangled pairs required Thus, different effective code lengths can minimize the logical for a teleportation request while maintaining a common baseerror probability in different operating regimes, leading to code structure. Moreover, coding gains are obtained only at thresholds where the preferred code size changes. Since entan- certain punctured code lengths rather than continuously with
packet size. We refer to these discrete operating points as puncturing tiers, which create non-trivial trade-offs between the latency required to acquire larger entanglement packets and the reliability gains provided by stronger error correction. While [7] considered punctured encoded teleportation under fixed entanglement availability and quality, independent of code length, this work incorporates the entanglementacquisition process into the code-selection problem. Since an [[n, k]] code requires n entangled pairs, increasing the code length can improve error protection but can also increase acquisition time, memory decoherence, and fidelity heterogeneity across the packet. As a result, the best puncturing tier depends on the latency and link-quality regime, providing a networkaware rationale for when different punctured codes should be used. More broadly, this work bridges entanglement generation and its use in scalable quantum network architectures. Here, entanglement generation refers to the successful establishment of shared entanglement between communicating nodes. In the entanglement generation literature (e.g., [8]), the focus is on generating and maintaining entanglement, often without considering how it is ultimately used in applications such as teleportation. Conversely, encoded-teleportation studies often abstract away the stochastic acquisition and storage of entanglement resources. We investigate this interface by evaluating how stochastic entanglement availability affects logical teleportation reliability and how punctured codes can adapt to resource constraints. Such adaptiveness is particularly relevant for early-generation quantum access networks, where users exhibit diverse quality-of-service requirements and hardware capabilities vary significantly. In these settings, a common base encoding, shared across a network domain such as a wireless quantum cell [10], can be adaptively punctured to accommodate individual users. This provides a potential architectural approach for balancing performance, complexity, and resource efficiency. The main contributions of this work are as follows: We develop a unified framework for latency-constrained encoded teleportation that jointly models stochastic entanglement generation, memory decoherence, and punctured quantum error-correcting codes. • We investigate the latency-reliability tradeoff by comparing uncoded teleportation and multiple encoded teleportation schemes under a common average entanglementacquisition latency constraint, highlighting how latency, entanglement fidelity, and logical error probability are coupled under resource constraints. • Through numerical simulations, we investigate latencyand link-dependent decision regions in which different puncturing tiers minimize the logical error probability, demonstrating the potential of adaptive puncturing-based code selection to outperform fixed code-selection strategies. •
PL
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Figure 2: Decision regions induced by crossover behavior of the logical error probability PL under latency constraint L. Each curve corresponds to a code length n, and the preferred length n⋆ (L) minimizes PL at that latency. Larger n requires a larger entanglement packet, changing the acquisitiontime/fidelity tradeoff. The crossover points partition the latency axis into regions where different code lengths are preferred.
II. S YSTEM M ODEL In this section, we introduce a unified model for latencyconstrained encoded teleportation based on stochastic entanglement generation. The model captures the interaction between entanglement generation, storage-induced decoherence, and quantum error correction, enabling a cross-layer analysis of latency-reliability trade-offs in quantum networks. Throughout this work, entanglement generation refers to the successful generation and distribution of an EPR pair shared between the communicating nodes. As illustrated in Fig. 1, encoded teleportation proceeds through a sequence of stages including entanglement generation, encoding, teleportation, and decoding. Since encoding and decoding are local operations and classical communication during teleportation can be performed in parallel, we focus on the latency associated with entanglement generation, which is typically the dominant contribution in many physical implementations. In particular, the time required to generate an entanglement packet of n pairs determines the overall latency. Since the packet size n governs both the acquisition time and the strength of error correction, this induces a trade-off between resource availability, fidelity, and logical reliability. In particular, increasing n improves error-correction capability but requires longer waiting times to accumulate entanglement, leading to additional decoherence and reduced fidelity. The goal is to select the code length n that minimizes the logical error probability under a latency constraint. To accommodate varying operating conditions, we consider puncturing of a base code, allowing the effective code length to be adapted while preserving a common stabilizer structure. This provides a structured way to realize different latency-reliability tradeoffs
without switching among unrelated code families, as detailed in the following subsections. A. Encoded Teleportation
The resulting logical error probability is PL = 1 − Psucc,X Psucc,Z .
(6)
The logical reliability above depends on the error probabilities induced by the shared entangled pairs. In practice, these probabilities are determined by how entanglement is generated and stored over time. As entanglement generation is stochastic and dominates the overall latency, the time required to accumulate an entanglement packet directly impacts both the availability and quality of the pairs used for teleportation. This establishes a trade-off between latency and entanglement fidelity. As illustrated in Fig. 2, this trade-off gives rise to − − + + Φ ρ = pI Φ Φ + pZ Φ distinct operating regimes in which different coding strategies + + − − + pX Ψ Ψ + pY Ψ (1) are optimal, depending on the latency constraint. We analyze Ψ , this behavior by modeling entanglement generation and memwhere pI + pZ + pX + pY = 1. 2) Logical Reliability under CSS Codes: To mitigate these ory decoherence in the following subsections. Furthermore, errors, we employ encoded teleportation using CSS codes. as the required packet size n directly affects this trade-off, Instead of transmitting a single qubit, the state is encoded adaptive code selection via puncturing tiers (see Sec. III-B) into an [[n, k, dZ , dX ]] CSS code, requiring an entanglement becomes essential. packet of n EPR pairs, one for each physical qubit. Due to B. Entanglement Generation Rate and Fidelity the stochastic generation process (see Sec. II-B) and storageEntanglement generation is modelled as a discrete-time induced decoherence (see Sec. II-C), these pairs generally have stochastic process, where in each time slot an attempt is non-identical fidelities, resulting in heterogeneous Pauli error made to generate and distribute an EPR pair between the probabilities (pI,i ,pZ,i , pX,i , pY,i ) across qubits. communicating nodes. The duration of a slot depends on the CSS codes are well-suited to Pauli noise as they decouple implementation, with experimental demonstrations typically X and Z error correction. Each branch can therefore be reporting entanglement-generation rates on the order of 1020 treated as a binary symmetric channel with effective error kHz [13], [14]. probabilities [11] The entanglement generation and storage process is illusqX,i = pX,i + pY,i , qZ,i = pZ,i + pY,i , (2) trated in Fig. 3, where successful generation attempts occur over time and are stored until a complete entanglement packet corresponding to bit-flip and phase-flip errors, respectively. has been successfully distributed. This model is adopted from [11] to evaluate logical reliability We model entanglement generation as a sequence of indeacross different puncturing tiers. pendent Bernoulli trials, where each attempt succeeds with Furthermore, to capture realistic noise processes, we conprobability p. When the success probability of a single attempt sider both symmetric and asymmetric Pauli channels. The is very small, the sender may perform a batch of M attempts latter is defined by the ratio within a single time slot, in which case p corresponds to the pZ,i η= , (3) probability that at least one of the attempts succeeds. pX,i The success probability is inherently related to the fidelity allowing phase-flip errors to occur more frequently than bit- of the generated pair; protocols that aim for higher fidelity typically achieve lower generation rate. Following [8], we flip errors [12]. Let tZ = ⌊(dZ − 1)/2⌋ and tX = ⌊(dX − 1)/2⌋ denote model this trade-off by relating the initial fidelity F0 of a the correction radii of the code. Due to heterogeneous error successfully generated pair to the generation probability as probabilities across qubits, errors are independent but not 1 − (1 − p)1/M F0 (p) = 1 − , (7) identically distributed, and the number of errors in each 3pd branch therefore follows a Poisson-binomial distribution. The probabilities of successful decoding of X and Z errors are where M denotes the batch size and pd is the photon detection probability, which captures physical link characteristics such given by as photon collection efficiency and detector geometry that tX X X Y Y qX,i (4) can vary with the spatial configuration of the link and across Psucc,X = 1 − qX,i , users [15]. j=0 S⊆{1,...,n} i∈S i∈S / The fidelity F0 (p) characterizes the fidelity of entangled |S|=j pairs at the time of generation. However, these pairs are stored tZ X X Y Y Psucc,Z = (5) in quantum memory while additional entanglement is accuqZ,i 1 − qZ,i . mulated. During this time, qubits are subject to decoherence, j=0 S⊆{1,...,n} i∈S i∈S / |S|=j causing their fidelities to degrade as illustrated in Fig. 3. In the 1) Imperfect Teleportation: Teleportation using ideal Bell pairs and perfect local operations faithfully transmits an arbitrary quantum state. In realistic settings, however, the shared entangled resource is imperfect, and the protocol can be interpreted as a Pauli channel whose error probabilities {pI , pZ , pX , pY } are determined by the Bell-state decomposition of the resource [6]:
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III. L ATENCY-C ONSTRAINED E NCODED T ELEPORTATION AND A DAPTIVE C ODING In this section, we investigate how latency constraints shape entanglement availability and fidelity, and how these in turn impact the performance of encoded teleportation. In particular, the time required to generate an entanglement packet couples resource availability with decoherence, creating a trade-off between latency and logical reliability. This trade-off directly influences the choice of code length n, motivating adaptive coding strategies such as puncturing, which are presented in this section. A. Latency Requirement and Waiting Time
We impose a latency requirement L, specifying that an entanglement packet must be successfully generated within Figure 3: Example of the generation process with entanglean average of L time slots, reflecting application-level quality ment packet size n = 3. Generated EPR pairs are stored until of service constraints. For a given packet size n and storage a complete packet is formed. Their fidelity degrades over time window w, this latency constraint determines the required due to decoherence, resulting in heterogeneous fidelities across entanglement generation probability p. the packet. Let E[L(w, n)] denote the expected waiting time until an entanglement packet of size n is successfully generated within a window of size w. In the special case where n = 1, the example shown in Fig. 3, a packet of three entangled pairs is waiting time follows a geometric distribution with success requested, but completing the packet requires five generation probability p, thus E[L(w, 1)] = 1/p. However, for larger n, attempts. Since successful pairs must be stored while waiting obtaining the exact expected waiting time requires solving a for the remaining pairs, earlier successes accumulate age linear system of dimension w−1, which quickly becomes n−1 and gradually decohere. Consequently, when the final pair is computationally intractable [8]. Instead, for n > 1, we generated (Age 0), previously generated pairs may have aged approximate the expected waiting time as (e.g., Ages 2 and 4), leading to heterogeneous fidelities within n 1 n the completed packet. −1 w+ + , E[L(w, n)] ≈ (9) Pe 2p p C. Memory Decoherence We model the degradation as exponential decay such that a qubit stored for t time slots has fidelity 1 −t/T 1 e , (8) F t = + F0 − 2 2
where Pe =
w X w ℓ=n−1
ℓ
pℓ (1 − p)w−ℓ
(10)
is the probability that at least n − 1 additional successes occur within a window of w attempts following the first success. Using this approximation, we determine the generation probability p required to satisfy the latency constraint by solving where T is the coherence time of the quantum memory. E[L(w, n)] = L numerically. Comparison with exact evaluaTo mitigate excessive degradation caused by long storage tions shows that (9) conservatively overestimates the expected times, we impose a finite storage window of size w, such that waiting time, and the approximation becomes increasingly entangled pairs are discarded if not used within w time slots accurate as p → 1. Consequently, satisfying a given latency similarly to [8]. On one hand, this prevents severely decohered constraint requires a slightly higher generation probability p pairs from being used in teleportation to reduce the impact than under the exact model, introducing a conservative bias: of excessively aged pairs on the logical error probability. On latency requirements are more likely to be met, but at the cost the other hand, it constrains the time available to accumulate of reduced initial entanglement fidelity due to the coupling an entanglement packet. Consequently, the finite window size between generation rate and fidelity in (7). introduces an inherent trade-off between resource availability Since the generation probability is physically coupled to the and fidelity. The trade-off is further enhanced by latency quality of the generated entanglement, this in turn determines requirements, which determine how long the communicating the initial fidelity through F0 = F0 (p), establishing a direct nodes can wait to form an entanglement packet. trade-off between latency constraint and entanglement quality. Together, the entanglement generation model, memory de- To illustrate this latency-fidelity coupling, we evaluate the coherence, and encoded teleportation framework define the minimum fidelity of the entangled pairs, corresponding to the error characteristics and reliability of the transmitted quantum oldest stored pair in a packet, used for encoded teleportation as a function of the latency constraint L, as shown in Fig. 4. information under realistic network conditions.
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Figure 4: Minimum fidelity of the entangled pairs as a function of the latency constraint L. The minimum corresponds to the oldest stored pair at the time the packet is completed. Allowing larger latency enables higher-quality entanglement as pairs can be generated with lower success probability and thus higher initial fidelity F0 .
These tiers preserve a single logical qubit (k = 1) while offering different packet sizes and error-correction capabilities. In particular, the [[13, 1, 5, 3]] tier provides stronger protection against phase errors than bit-flip errors, making it well suited for asymmetric noise regimes where phase errors dominate [12]. The final tier corresponds to uncoded teleportation using a single entangled pair. Each tier therefore represents a different trade-off between entanglement packet size, acquisition latency, and errorcorrection capability. While larger codes generally provide stronger error correction for equal fidelities, they require larger entanglement packets and longer acquisition times, which reduce the fidelity of stored pairs due to decoherence. This interplay between stochastic entanglement generation, storage-induced decoherence, and coding choices defines the trade-off studied in this work and motivates the need for adaptive encoded teleportation strategies. C. Offline Optimization and Online Deployment In this section, we present an operational use of the proposed framework, illustrated in Fig. 5. The framework consists of two phases for adaptive encoded teleportation. In the first phase, system parameters, including generation statistics, link quality, and memory coherence times, are evaluated offline across candidate puncturing tiers to analyze the trade-offs between logical error probability, latency, and link quality. For each relevant operating point, such as latency requirement or target reliability, the logical error probability is computed for all candidate codes, from which an optimized coding policy can be extracted. In particular, for a given operating point defined by the latency constraint L, detection probability pd , and noise asymmetry η, the selected puncturing tier is
This trade-off plays a central role in determining which coding strategy is preferred under different latency regimes. As shown in Fig. 4, longer codes require more aggressive entanglement generation rates to satisfy a given latency constraint, which reduces the fidelity of the generated pairs. Consequently, the system must balance using a small number of high-fidelity qubits, corresponding to uncoded or short-code teleportation, against using a larger number of lower-fidelity qubits that provide redundancy through error correction. Determining where the reliability gain from coding outweighs the loss in entanglement quality depends jointly on latency constraints and n⋆ (L, pd , η) = arg min PL (n; L, pd , η), (11) link conditions. This motivates adaptive coding strategies that n∈N select the appropriate coding policy for the current operating where N = {1, 8, 13, 17} denotes the set of candidate puncturregime. ing tiers. The resulting policy can be stored as a precomputed B. Puncturing Tiers policy map. In the second phase, the framework is deployed during To adapt to latency-constrained entanglement generation, we network operation. When a teleportation request arrives, the consider puncturing of quantum error-correcting codes, which associated quality-of-service requirements and operating conallow the code length to be reduced by selectively removing ditions are used to determine the appropriate puncturing tier qubits while retaining a common encoding structure at the from the precomputed policy map. The required entanglement cost of lowering code distance [9]. In the context of encoded packet is then requested. After successful generation, the data teleportation, this enables different entanglement packet sizes qubit is encoded using the selected punctured code, and the to be considered under latency constraints, at the cost of resulting physical qubits are teleported using the entanglement reduced error-correction capability. packet and decoded at the receiver. Puncturing a base code can produce many candidate code The numerical results in Sec. IV focus on the first phase to lengths. However, multiple punctured variants may provide investigate these trade-offs and motivate the resulting policy the same error-correction capability, characterized by identical map. (dZ , dX ) values. Since larger n requires more entangled pairs and incurs higher latency, we retain only the smallest-n code IV. N UMERICAL R ESULTS for each distinct (dZ , dX ) pair. This results in a discrete set of In this section, we evaluate the performance of latencyoperating points, referred to as puncturing tiers. In this work, constrained encoded teleportation, with the goal of quantiwe consider four tiers derived from the same length-17 CSS fying the trade-off between latency and logical reliability, base code following [7]: and comparing different coding strategies under a common [[17, 1, 5, 5]], [[13, 1, 5, 3]], [[8, 1, 3, 3]], uncoded. latency constraint. To enable this comparison, we follow the
Phase 1: Tradeoff Analysis & Design Model Entanglement Generation vs. Latency L
Analyze Stochastic Memory Decoherence & Pauli Noise
Evaluate PL vs. L and PL vs. pd Tradeoffs across Puncturing Tiers n
Latency-Aware Decision Rule n⋆ (L, PL )
Inform Adaptive Code Selection
Phase 2: Deployment Pipeline Incoming Request: Latency Lreq & Target Rel. PLreq
Determine n⋆ , Puncture Base Code, & Encode
Teleport over Entangled Channel
Decode & Correct Errors
Figure 5: System framework for adaptive latency-constrained encoded teleportation. Phase 1 represents the tradeoff analysis stage: we model entanglement generation under latency constraints, analyze the resulting stochastic memory decoherence, and evaluate the PL vs. L and PL vs. pd tradeoffs to establish decision regions. Phase 2 represents the physical deployment pipeline: incoming user requests specify constraints (Lreq , PLreq ), which inform the code selection. The base code is punctured to the optimized length n⋆ , after which the data qubit is encoded, followed by quantum teleportation, decoding, and error correction. In this work, we focus on phase 1 to motivate the need for adaptive coding strategies in the system.
For our numerical results, we use a coherence time of T = 10000 time slots. Since the proposed framework is independent of the physical duration of a time slot, the coherence time should be interpreted relative to the underlying implementation. Relative to the representative time-slot durations discussed in Sec. II-B, this corresponds to a coherence time on the order of one second, consistent with reported coherence times of We then simulate the stochastic entanglement generation long-lived quantum memories [16]. This procedure enables a comparison across coding and storage process described in Sec. II-B and Sec. II-C until schemes, as each operates under the same average latency a successful entanglement packet of the corresponding size is constraint while capturing the coupling between entanglement obtained. Because successful pairs are generated at different generation rate, fidelity, and logical reliability. In particular, time instants, the qubits in a packet experience different longer codes require higher generation probabilities to satisfy storage durations prior to teleportation. This results in heterothe latency constraint, which reduces the fidelity of the gengeneous fidelities across the n qubits, which are computed erated entanglement and can offset the benefits of increased using (8). The resulting fidelities are then mapped to qubiterror-correction capability. wise Pauli error probabilities (pI,i ,pZ,i , pX,i , pY,i ) according to the teleportation channel model in (1). Specifically, for A. Impact of Average Latency Constraint L a qubit with fidelity Fi , we set pI,i = Fi and distribute We evaluate the logical error probability PL as a function the remaining probability mass 1 − Fi across Pauli errors of the latency constraint L for different puncturing tiers, according to the underlying noise model: under symmetric corresponding to different code lengths n. This allows us to noise pX,i = pY,i = pZ,i = (1 − Fi )/3, while under demonstrate how an optimized coding strategy depends on the asymmetric noise pX,i = pY,i and η = pZ,i /pX,i , with all available latency budget, and to identify regimes in which probabilities satisfying pI,i + pZ,i + pX,i + pY,i = 1. encoded teleportation outperforms uncoded transmission, as modeling framework developed in Sec. II-B–Sec. II-C and Sec. III-A. For a given latency constraint L and code length n, we determine the entanglement generation probability p by solving E[L(w, n)] = L using the approximation in (9). This ensures that all schemes satisfy the same latency constraint on average. The resulting value of p determines the initial fidelity of generated entangled pairs through F0 (p) in (7).
Using these heterogeneous error probabilities, we evaluate well as regimes where adaptive coding across puncturing encoded teleportation under CSS codes corresponding to the tiers provides significant performance gains over fixed coding puncturing tiers described in Sec. III-B. For each Monte strategies. In Fig. 6, we show the logical error probability PL as a Carlo realization, we simulate the stochastic entanglementgeneration and storage process to obtain the waiting times and function of the latency constraint L for different coding tiers storage ages of the qubits in the entanglement packet, which n, with n = 1 corresponding to the uncoded case. Increasing determine their heterogeneous fidelities. These fidelities are the allowed latency budget improves entanglement quality, then mapped to qubit-wise Pauli error probabilities, and the reducing logical error probabilities across all transmission corresponding logical error probability PL is computed using strategies. Under tight latency constraints, encoded teleporta(6). The reported results are obtained by averaging PL over tion outperforms uncoded transmission, with distinct decision 105 independent realizations. regions in which the n = 8 punctured code (briefly) and the
LEP vs L: T=10000, w=20, p_d=0.40, M=300, Eta= 1
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Figure 6: Logical error probability PL as a function of allowed latency L for different puncturing tiers n under symmetric errors. LEP vs L: T=10000, w=20, p_d=0.75, M=300, Eta= 10
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Figure 7: Logical error probability PL as a function of latency constraint L for different puncturing tiers n under asymmetric errors (η = 10).
n = 17 code respectively achieve the lowest logical error probability. In this regime, the fidelity of a single entangled pair is too low for reliable uncoded transmission, and error correction through encoding reduces logical error probability despite requiring teleportation across multiple physical qubits. However, beyond approximately L = 200, uncoded transmission becomes favorable, as the fidelity of individual entangled pairs becomes sufficiently high that physical error probabilities are already very small. In this regime, encoded teleportation requires transmitting multiple physical qubits through independent teleportation channels, and the additional exposure to physical errors can outweigh the coding gains of larger codes. These decision regions demonstrate that puncturing enables adaptive selection across coding tiers under varying latency constraints. In Fig. 7, we show that under asymmetric errors with η = 10, the role of the asymmetrically punctured n = 13 code becomes pronounced, achieving the lowest logical error probability over most of the latency range. In contrast to the
n=1 n=8 n=13 n=17
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Figure 8: Logical error probability PL as a function of detection probability pd for different puncturing tiers n under symmetric errors.
symmetric case, encoded teleportation remains favorable for nearly all values of L, with uncoded transmission and the n = 17 code only catching up briefly at the largest allowed latencies. These results show that asymmetry can significantly shift the best-performing coding tier and further enhance the benefits of adaptive puncturing. Moreover, under resource constraints, encoded teleportation provides a significant reduction in logical error probability (corresponding to improved logical reliability) across nearly the entire latency regime considered. Taken together, the symmetric and asymmetric results demonstrate that adaptive puncturing can accommodate distinct error environments while maintaining a common stabilizer structure and flexibly selecting coding tiers matched to the operating regime. B. Impact of Detection Probability pd In addition to latency-constrained evaluations, we next examine how physical link quality affects logical reliability by studying the logical error probability PL as a function of the detection probability pd , which enters the initial fidelity model in (7). By evaluating PL as a function of pd , we assess how different coding strategies perform under heterogeneous link conditions while maintaining the same underlying coding framework. In Fig. 8, we show the logical error probability PL as a function of detection probability pd for symmetric errors under a fixed latency constraint L = 20. At very low detection probabilities, corresponding to poor link quality or longer link distances, uncoded transmission achieves the lowest logical error probability, as coding provides limited error-correction benefit while increasing transmission error accumulation across multiple physical qubits. For moderate detection probabilities, approximately 0.2 ≤ pd ≤ 0.4, the n = 8 punctured code achieves the lowest logical error probability, balancing error correction capability with reduced multi-qubit error accumulation relative to larger codes. Beyond approximately pd = 0.5, corresponding to improved link
LEP vs _d: T=10000, w=20, L=40,M=300, Eta= 10 n=1 n=8 n=13 n=17
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0.8
Figure 9: Logical error probability PL as a function of detection probability pd for different puncturing tiers n under asymmetric errors (η = 10).
quality or shorter link distances, the n = 17 code becomes favorable and achieves the lowest logical error probability. These decision regions show that different link conditions require different coding tiers, further motivating adaptive puncturing to accommodate heterogeneous network conditions. In Fig. 9, we show that under asymmetric errors, the asymmetrically punctured n = 13 code again plays a dominant role, achieving the lowest logical error probability over most of the detection probability range. While uncoded transmission performs best briefly at very low detection probabilities, the n = 13 code becomes favorable once link quality improves, providing a significant reduction in logical error probability across most operating regimes. As in the latency-constrained results, these findings show that adaptive puncturing provides a flexible mechanism for selecting coding tiers under both symmetric and asymmetric error environments based on link conditions. V. C ONCLUSIONS AND F UTURE W ORK In this work, we investigated the reliability of quantum teleportation under realistic network conditions where entanglement must be generated stochastically and degrades over time due to memory decoherence. Focusing on encoded teleportation, we developed a unified framework that captures the interaction between entanglement availability, decoherence, and coding decisions, enabling the evaluation of reliability in terms of logical error probability under latency constraints. Our results demonstrate that encoded teleportation can provide substantial reliability gains over uncoded transmission under entanglement resource constraints. However, these gains depend critically on both the availability and fidelity of entangled pairs, as acquiring additional resources introduces delays that can reduce their effective quality. This interplay gives rise to non-uniform coding gains across operating regimes and highlights an interesting latency-reliability tradeoff. Through numerical investigations, we illustrated decision regions that determine optimized effective code length under
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