Noisy hardware forms one of the main hurdles to the realization of a near-term quantum internet. Distillation protocols allows one to overcome this noise at the cost of an increased overhead. We consider here an experimentally relevant class of distillation protocols, which distill <italic>n</italic> to <italic>k</italic> end-to-end entangled pairs using bilocal Clifford operations, a single round of communication and a possible final local operation depending on the observed measurement outcomes. In the case of permutationally invariant depolarizing noise on the input states, we find a correspondence between these distillation protocols and graph codes. We leverage this correspondence to find provably optimal distillation protocols in this class for several tasks important for the quantum internet. This correspondence allows us to investigate use cases for so-called non-trivial measurement syndromes. Furthermore, we detail a recipe to construct the circuit used for the distillation protocol given a graph code. We use this to find circuits of short depth and small number of two-qubit gates. Additionally, we develop a black-box circuit optimization algorithm, and find that both approaches yield comparable circuits. Finally, we investigate the teleportation of encoded states and find protocols which jointly improve the rate and fidelities with respect to prior art.

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In the search for scalable, fault-tolerant quantum computing, distributed quantum computers are promising candidates. These systems can be realized in large-scale quantum networks or condensed onto a single chip with closely situated nodes. We present a framework for numerical simulations of a memory channel using the distributed toric surface code, where each data qubit of the code is part of a separate node, and the error-detection performance depends on the quality of four-qubit Greenberger-Horne-Zeilinger (GHZ) states generated between the nodes. We quantitatively investigate the effect of memory decoherence and evaluate the advantage of GHZ creation protocols tailored to the level of decoherence. We do this by applying our framework for the particular case of color centers in diamond, employing models developed from experimental characterization of nitrogen-vacancy centers. For diamond color centers, coherence times during entanglement generation are orders of magnitude lower than coherence times of idling qubits. These coherence times represent a limiting factor for applications, but previous surface code simulations did not treat them as such. Introducing limiting coherence times as a prominent noise factor makes it imperative to integrate realistic operation times into simulations and incorporate strategies for operation scheduling. Our model predicts error probability thresholds for gate and measurement reduced by at least a factor of three compared to prior work with more idealized noise models. We also find a threshold of 4 × 10 2 in the ratio between the entanglement generation and the decoherence rates, setting a benchmark for experimental progress.

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We present and analyze an architecture for a European-scale quantum network using satellite links to connect Quantum Cities, which are metropolitan quantum networks with minimal hardware requirements for the end users. Using NetSquid, a quantum network simulation tool based on discrete events, we assess and benchmark the performance of such a network linking distant locations in Europe in terms of quantum key distribution rates, considering realistic parameters for currently available or near-term technology. Our results highlight the key parameters and the limits of current satellite quantum communication links and can be used to assist the design of future missions. We also discuss the possibility of using high-altitude balloons as an alternative to satellites.

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The vision of a global network that enables quantum communications between any point on Earth is known as the quantum internet. One crucial element of this network is the use of quantum repeater chains, which have the potential to overcome transmission losses and implement entanglement or quantum key distribution protocols over extended distances. There are various proposals for quantum repeaters, but they can generally be evaluated based on two main figures of merit: the average time for end-to-end entanglement delivery and the associated average fidelity. However, characterizing these quantities can be difficult due to factors such as feedback loops, decoherence, entanglement generation being a probabilistic process, and the potential failure of subprotocols. In this talk, I will discuss algorithmic and analytical methods for computing these quantities for relevant families of protocols.

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Flag-style fault-tolerance has become a linchpin in the realization of small fault-tolerant quantum-error correction experiments. The flag protocol's utility hinges on low qubit over-head, which is typically much smaller than in other approaches. However, as in most fault-tolerance protocols, the advantages of flag-style error correction come with a tradeoff: fault tolerance can be guaranteed, but such protocols involve high-depth circuits, due to the need for repeated stabilizer measurements. Here, we demonstrate that a dynamic choice of stabilizer measurements, based on past syndromes, and the utilization of elements from the full stabilizer group, leads to flag protocols with lower-depth syndrome-extraction circuits for the [[5], [1], [3]] code, as well as for the Steane code when compared to the standard methods in flag fault tolerance. We methodically prove that our new protocols yield fault-tolerant lookup tables, and demonstrate them with a pseudothreshold simulation, showcasing large improvements for all protocols when compared to previously-established methods. This work opens the dialogue on exploiting the properties of the full stabilizer group for reducing circuit overhead in fault-tolerant quantum-error correction.@en

In this paper, we introduce a novel heuristic approach designed to optimize the performance of Greenberger-Horne- Zeilinger (GHZ) creation and distillation protocols under decoherence. Our methodology converts these protocols into a practical set of instructions, demonstrating, through simulations, the production of higher-quality GHZ states than previously known protocols. This advancement contributes to the field of distributed quantum computing by addressing the need for high-quality entanglement required for operations between different quantum computers.

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Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to n = 5 (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to n = 8 copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.

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Quantum networks can enable quantum communication and modular quantum computation. A powerful approach is to use multi-qubit nodes that provide quantum memory and computational power. Nuclear spins associated with defects in diamond are promising qubits for this role. However, dephasing during optical entanglement distribution hinders scaling to larger systems. Here, we show that a ¹³C-spin quantum memory in isotopically engineered diamond is robust to the optical link operation of a nitrogen-vacancy centre. The memory lifetime is improved by two orders-of-magnitude upon the state-of-the-art, surpassing reported times for entanglement distribution. Additionally, we demonstrate that the nuclear-spin state can survive ionisation and recapture of the nitrogen-vacancy electron. Finally, we use simulations to show that combining this memory with previously demonstrated entanglement links and gates can enable key network primitives, such as deterministic non-local two-qubit gates, paving the way for test-bed quantum networks capable of investigating complex algorithms and error correction.

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The ability to distribute high-quality entanglement between remote parties is a necessary primitive for many quantum communication applications. A large range of schemes for realizing the long-distance delivery of remote entanglement has been proposed, for both bipartite and multipartite entanglement. For assessing the viability of these schemes, knowledge of the time at which entanglement is delivered is crucial. Specifically, if the communication task requires multiple remote-entangled quantum states and these states are generated at different times by the scheme, the earlier states will need to wait and thus their quality will decrease while being stored in an (imperfect) memory. For the remote-entanglement delivery schemes which are closest to experimental reach, this time assessment is challenging, as they consist of nondeterministic components such as probabilistic entanglement swaps. For many such protocols even the average time at which entanglement can be distributed is not known exactly, in particular when they consist of feedback loops and forced restarts. In this work, we provide improved analytical bounds on the average and on the quantiles of the completion time of entanglement distribution protocols in the case that all network components have success probabilities lower bounded by a constant. A canonical example of such a protocol is a nested quantum repeater scheme which consists of heralded entanglement generation and entanglement swaps. For this scheme specifically, our results imply that a common approximation to the mean entanglement distribution time, the 3-over-2 formula, is in essence an upper bound to the real time. Our results rely on a novel connection with reliability theory.

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Quantum communication enables the implementation of tasks that are unachievable with classical resources. However, losses on the communication channel preclude the direct long-distance transmission of quantum information in many relevant scenarios. In principle, quantum repeaters allow one to overcome losses. However, realistic hardware parameters make long-distance quantum communication a challenge in practice. For instance, in many protocols an entangled pair is generated that needs to wait in quantum memory until the generation of an additional pair. During this waiting time the first pair decoheres, impacting the quality of the final entanglement produced. At the cost of a lower rate, this effect can be mitigated by imposing a cutoff condition. For instance, a maximum storage time for entanglement after which it is discarded. In this article, we optimize the cutoffs for quantum repeater chains. First, we develop an algorithm for computing the probability distribution of the waiting time and fidelity of entanglement produced by repeater chain protocols which include a cutoff. Then, we use the algorithm to optimize cutoffs in order to maximize the secret-key rate between the end nodes of the repeater chain. In this article, we find that the use of the optimal cutoff extends the parameter regime for which secret key can be generated and, moreover, significantly increases the secret-key rate for a large range of parameters.@en

The rate at which quantum communication tasks can be performed using direct transmission is fundamentally hindered by the channel loss. Quantum repeaters allow one, in principle, to overcome these limitations, but their introduction necessarily adds an additional layer of complexity to the distribution of entanglement. This additional complexity - along with the stochastic nature of processes such as entanglement generation, Bell swaps, and entanglement distillation - makes finding good quantum repeater schemes nontrivial. We develop an algorithm that can efficiently perform a heuristic optimization over a subset of quantum repeater schemes for general repeater platforms. We find a strong improvement in the generation rate in comparison to an optimization over a simpler class of repeater schemes based on BDCZ (Briegel, Dür, Cirac, Zoller) repeater schemes. We use the algorithm to study three different experimental quantum repeater implementations on their ability to distribute entanglement, which we dub information processing implementations, multiplexed elementary pair generation implementations, and combinations of the two. We perform this heuristic optimization of repeater schemes for each of these implementations for a wide range of parameters and different experimental settings. This allows us to make estimates on what are the most critical parameters to improve for entanglement generation, how many repeaters to use, and which implementations perform best in their ability to generate entanglement.

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Quantum networks will allow to implement communication tasks beyond the reach of their classical counterparts. A pressing and necessary issue for the design of quantum network protocols is the quantification of the rates at which these tasks can be performed. Here, we propose a simple recipe that yields efficiently computable lower and upper bounds on the maximum achievable rates. For this we make use of the max-flow min-cut theorem and its generalization to multi-commodity flows to obtain linear programs. We exemplify our recipe deriving the linear programs for bipartite settings, settings where multiple pairs of users obtain entanglement in parallel as well as multipartite settings, covering almost all known situations. We also make use of a generalization of the concept of paths between user pairs in a network to Steiner trees spanning a group of users wishing to establish Greenberger-Horne-Zeilinger states.

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Quantum communication enables the implementation of tasks that are unachievable with classical resources. However, losses on the communication channel preclude the direct long-distance transmission of quantum information in many relevant scenarios. In principle quantum repeaters allow one to overcome losses. However, realistic hardware parameters make long-distance quantum communication a challenge in practice. For instance, in many protocols an entangled pair is generated that needs to wait in quantum memory until the generation of an additional pair. During this waiting time the first pair decoheres, impacting the quality of the final entanglement produced. At the cost of a lower rate, this effect can be mitigated by imposing a cut-off condition. For instance, a maximum storage time for entanglement after which it is discarded. In this work, we optimize the cut-offs for quantum repeater chains. First, we develop an algorithm for computing the probability distribution of the waiting time and fidelity of entanglement produced by repeater chain protocols which include a cut-off. Then, we use the algorithm to optimize cut-offs in order to maximize secret-key rate between the end nodes of the repeater chain. We find that the use of the optimal cut-off extends the parameter regime for which secret key can be generated and moreover significantly increases the secret-key rate for a large range of parameters.

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We consider the asymptotic key rates achieved in the simplest quantum key distribution protocols, namely, the BB84 and the six-state protocols when nonuniform noise is present in the system. We first observe that higher qubit error rates do not necessarily imply lower key rates. Second, we consider protocols with advantage distillation and show that it can be advantageous to use the basis with higher quantum bit error rate for the key generation. We then discuss the relation between advantage distillation and entanglement distillation protocols. We show that applying advantage distillation to a string of bits formed by the outcomes of measurements in the basis with a higher quantum bit error rate is closely connected to the two-to-one entanglement distillation protocol of Deutsch-Ekert-Jozsa-Macchiavello-Popescu-Sanpera [Phys. Rev. Lett. 77, 2818 (1996)PRLTAO0031-900710.1103/PhysRevLett.77.2818]. Finally, we discuss the implications of these results for implementations of quantum key distribution.

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Quantum communication enables a host of applications that cannot be achieved by classical communication means, with provably secure communication as one of the prime examples. The distance that quantum communication schemes can cover via direct communication is fundamentally limited by losses on the communication channel. By means of quantum repeaters, the reach of these schemes can be extended and chains of quantum repeaters could in principle cover arbitrarily long distances. In this work, we provide two efficient algorithms for determining the generation time and fidelity of the first generated entangled pair between the end nodes of a quantum repeater chain. The runtime of the algorithms increases polynomially with the number of segments of the chain, which improves upon the exponential runtime of existing algorithms. Our first algorithm is probabilistic and can analyze refined versions of repeater chain protocols which include intermediate entanglement distillation. Our second algorithm computes the waiting time distribution up to a pre-specified truncation time, has faster runtime than the first one and is moreover exact up to machine precision. Using our proof-of-principle implementation, we are able to analyze repeater chains of thousands of segments for some parameter regimes. The algorithms thus serve as useful tools for the analysis of large quantum repeater chain protocols and topologies of the future quantum internet.

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We provide two algorithms for computing the probability distribution of waiting time and fidelity in quantum repeater chains constructed from probabilistic components. Their polynomial runtimes improve upon existing algorithms’ exponential scaling.@en

Quantum channels enable the implementation of communication tasks inaccessible to their classical counterparts. The most famous example is the distribution of secret key. However, in the absence of quantum repeaters, the rate at which these tasks can be performed is dictated by the losses in the quantum channel. In practice, channel losses have limited the reach of quantum protocols to short distances. Quantum repeaters have the potential to significantly increase the rates and reach beyond the limits of direct transmission. However, no experimental implementation has overcome the direct transmission threshold. Here, we propose three quantum repeater schemes and assess their ability to generate secret key when implemented on a setup using nitrogen-vacancy (NV) centers in diamond with near-term experimental parameters. We find that one of these schemes - the so-called single-photon scheme, requiring no quantum storage - has the ability to surpass the capacity - the highest secret-key rate achievable with direct transmission - by a factor of 7 for a distance of approximately 9.2 km with near-term parameters, establishing it as a prime candidate for the first experimental realization of a quantum repeater.

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Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system.

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Emerging quantum technologies will impact industry and society at large, but how and to what extent we do not know yet. That is why a TU Delft Quantum Vision team set about exploring this impact, with a special focus on quantum internet as the first technology that is likely to become reality. @en

The goal of entanglement distillation is to turn a large number of weakly entangled states into a smaller number of highly entangled ones. Practical entanglement distillation schemes offer a trade-off between the fidelity to the target state and the probability of successful distillation. Exploiting such trade-offs is of interest in the design of quantum repeater protocols. Here, we present a number of methods to assess and optimize entanglement distillation schemes. We start by giving a numerical method to compute upper bounds on the maximum achievable fidelity for a desired probability of success. We show that this method performs well for many known examples by comparing it to well-known distillation protocols. This allows us to show optimality for many well-known distillation protocols for specific states of interest. As an example, we analytically prove optimality of the distillation protocol utilized within the Extreme Photon Loss entanglement generation scheme, even in the asymptotic limit. We proceed to present a numerical method that can improve an existing distillation scheme for a given input state, and we present an example for which this method finds an optimal distillation protocol. An implementation of our numerical methods is available as a Julia package.

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