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arXiv:2302.13791

Repeated Purification versus Concatenated Error Correction in Fault Tolerant Quantum Networks

Michel Barbeau, Joaquín García-Alfaro, Evangelos Kranakis C

Published Feb 27, 2023• 0 citations
arXivHTMLPDF

Abstract

Entanglement distribution is a core mechanism for the future quantum Internet. The quantum world is, however, a faulty environment. Hence, successful entanglement swapping is error-prone. The occurrence of quantum state errors can be mitigated using purification and error correction, which can be repeated in the former case and concatenated in the latter case. Repeated purification merges low-fidelity qubits into higher-quality ones, while concatenated error correction builds upon the redundancy of quantum information. In this article, we study in-depth and compare the two options: repeated purification and concatenated error correction. We consider using repeated purification and concatenated error correction to mitigate the presence of faults that occur during the establishment of Bell pairs between remote network nodes. We compare their performance versus the number of repetitions or concatenations, to reach a certain level of fidelity in quantum networks. We study their resource requirements, namely, their work memory complexity (e.g., number of stored qubits) and operational complexity (e.g., number of operations). Our analysis demonstrates that concatenated error correction, versus repeated purification, requires fewer iterations and has lower operational complexity than repeated purification to reach high fidelity at the expense of increased memory requirements.

DOI
10.48550/arxiv.2302.13791

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Entanglement distribution is a core mechanism for the future quantum Internet. The quantum world is, however, a faulty environment. Hence, successful entanglement swapping is error-prone. The occurrence of quantum state errors can be mitigated using purification and error correction, which can be repeated in the former case and concatenated in the latter case. Repeated purification merges low-fidelity qubits into higher-quality ones, while concatenated error correction builds upon the redundancy of quantum information. In this article, we study in-depth and compare the two options: repeated purification and concatenated error correction. We consider using repeated purification and concatenated error correction to mitigate the presence of faults that occur during the establishment of Bell pairs between remote network nodes. We compare their performance versus the number of repetitions or concatenations, to reach a certain level of fidelity in quantum networks. We study their resource requirements, namely, their work memory complexity (e.g., number of stored qubits) and operational complexity (e.g., number of operations). Our analysis demonstrates that concatenated error correction, versus repeated purification, requires fewer iterations and has lower operational complexity than repeated purification to reach high fidelity at the expense of increased memory requirements.
One-way quantum repeaters where loss and operational errors are counteracted by quantum error-correcting codes can ensure fast and reliable qubit transmission in quantum networks. It is crucial that the resource requirements of such repeaters, for example, the number of qubits per repeater node and the complexity of the quantum error-correcting operations are kept to a minimum to allow for near-future implementations. To this end, we propose a one-way quantum repeater that targets both the loss and operational error rates in a communication channel in a resource-efficient manner using code concatenation. Specifically, we consider a tree-cluster code as an inner loss-tolerant code concatenated with an outer 5-qubit code for protection against Pauli errors. Adopting flag-based stabilizer measurements, we show that intercontinental distances of up to 10,000 km can be bridged with a minimized resource overhead by interspersing repeater nodes that each specialize in suppressing either loss or operational errors. Our work demonstrates how tailored error-correcting codes can significantly lower the experimental requirements for long-distance quantum communication.
Entanglement purification with two entangled resource pairs is widely employed in the literature on quantum repeater networks to counteract fidelity degradation introduced by noisy quantum memories and entanglement swapping across multiple hops. Standard purification protocols assume both resource pairs carry identical fidelity. In practice, entanglement generation is stochastic, the two resource pairs are heralded at different times, and so the first pair decoheres in memory while the second is being generated. Thus a fidelity asymmetry is a structural feature of any network operating under realistic memory conditions, leading to the question: when is it beneficial to perform purification? We derive a closed-form fidelity asymmetry tolerance delta(F) that governs whether a purification attempt is beneficial. We determine a universal upper bound delta_max of approximately 0.076 beyond which purification is always counterproductive. Our simulations show that with exponential memory decoherence, purification yields benefits in only approximately 14% of purification attempts on two resource pairs in a two-hop repeater chain. We define three network objectives: fidelity only, time only, and a combination of time and fidelity, to deliver end-to-end entanglement. We show that when the application fidelity requirement is achievable through swapping alone, no-purification is the superior policy, with its advantage increasing with the number of hops. When the fidelity requirement cannot be met with swapping alone and purification is necessary, to be effective, it must be conditioned on delta(F) between resource pairs. We introduce DeltaPurify, a policy that conditions purification decisions on local fidelity information, and show it reduces time-to-serve relative to both naive purification and no-purification across several fidelity thresholds and hops of a repeater chain.
In this paper, we place bounds on when it is impossible to purify a noisy two-qubit state if all the gates used in the purification protocol are subject to adversarial, local, independent noise. It is found that the gate operations must be subject to less than 5.3% error. An existing proof that purification is equivalent to error correction is used to show that this bound can also be applied to concatenated error correcting codes in the presence of noisy gates, and hence gives a limit to the tolerable error rate for a fault-tolerant quantum computer formed by concatenation. This is shown to apply also to the case where error detection and postselection, as proposed by Knill, is used to enhance the threshold. We demonstrate the trade-off between gate and environmentally induced faulty rotations and qubit loss errors.
This work compares the overhead of quantum error correction with concatenated and topological quantum error-correcting codes. To perform a numerical analysis, we use the Quantum Resource Estimator Toolbox (QuRE) that we recently developed. We use QuRE to estimate the number of qubits, quantum gates, and amount of time needed to factor a 1024-bit number on several candidate quantum technologies that differ in their clock speed and reliability. We make several interesting observations. First, topological quantum error correction requires fewer resources when physical gate error rates are high, white concatenated codes have smaller overhead for physical gate error rates below approximately 10E-7. Consequently, we show that different error-correcting codes should be chosen for two of the studied physical quantum technologies - ion traps and superconducting qubits. Second, we observe that the composition of the elementary gate types occurring in a typical logical circuit, a fault-tolerant circuit protected by the surface code, and a fault-tolerant circuit protected by a concatenated code all differ. This also suggests that choosing the most appropriate error correction technique depends on the ability of the future technology to perform specific gates efficiently.
Typically, fault-tolerant operations and code concatenation are reserved for quantum error correction due to their resource overhead. Here, we show that fault tolerant operations have a large impact on the performance of symmetry based error mitigation techniques. We also demonstrate that similar to results in fault tolerant quantum computing, code concatenation in fault-tolerant quantum error mitigation (FTQEM) can exponentially suppress the errors to arbitrary levels. For a family of circuits, we provide analytical error thresholds for FTQEM with the repetition code. These circuits include a set of quantum circuits that can generate all of reversible classical computing. The post-selection rate in FTQEM can also be increased by correcting some of the outcomes. Our threshold results can also be viewed from the perspective of quantifying the number of gate operations we can delay checking the stabilizers in a concatenated code before errors overwhelm the encoding. The benefits of FTQEM are demonstrated with numerical simulations and hardware demonstrations.
We study the use of entanglement purification for quantum communication over long distances. For distances much longer than the coherence length of a corresponding noisy quantum channel, the fidelity of transmission is usually so low that standard purification methods are not applicable. It is possible, however, to divide the channel into shorter segments that are purified separately and then connected by the method of entanglement swapping. This method can be much more efficient than schemes based on quantum error correction, as it makes explicit use of two-way classical communication. An important question is how the noise, introduced by imperfect local operations (that constitute the protocols of purification and the entanglement swapping), accumulates in such a compound channel, and how it can be kept below a certain noise level. To treat this problem, we first study the applicability and the efficiency of entanglement purification protocols in the situation of imperfect local operations. We then present a scheme that allows entanglement purification over arbitrary long channels and tolerates errors on the percent level. It requires a polynomial overhead in time, and an overhead in local resources that grows only logarithmically with the length of the channel.