Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the problem on an anonymous network, which is known as a network model with minimal assumptions [Angluin, STOC'80]. The algorithm runs in @math rounds if every party initially has the common knowledge of an upper bound @math on the number of parties. This implies that all solvable problems can be solved in @math rounds on average without error (i.e., with zero-sided error) on the network. As a generalization, a quantum algorithm that works in @math rounds is obtained for the problem of exactly computing any symmetric Boolean function, over @math distributed input bits, which is constant over all the @math bits whose sum is larger than @math for @math . All these algorithms work with the bit complexities bounded by a polynomial in @math .
See the surveys @cite_25 @cite_0 @cite_10 and the references therein for more work on distributed quantum computing.
@cite_0 @cite_10 @cite_25
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We present a brief survey of results where quantum information processing is useful to solve distributed computation tasks. We describe problems that are impossible to solve using classical resources but that become feasible with the help of quantum mechanics. We also give examples where the use of quantum information significantly reduces the need for communication. The main focus of the survey is on communication complexity but we also address other distributed tasks. We investigate a general class of quantum key distribution (QKD) protocols using one-way classical communication. We show that full security can be proven by considering only collective attacks. We derive computable lower and upper bounds on the secret-key rate of those QKD protocols involving only entropies of two-qubit density operators. As an illustration of our results, we determine new bounds for the Bennett-Brassard 1984, the 6-state, and the Bennett 1992 protocols. We show that in all these cases the first classical processing that the legitimate partners should apply consists in adding noise. Algorithms such as quantum factoring1 and quantum search2 illustrate the great theoretical promise of quantum computers; but the practical implementation of such devices will require careful consideration of the minimum resource requirements, together with the development of procedures to overcome inevitable residual imperfections in physical systems3,4,5. Many designs have been proposed, but none allow a large quantum computer to be built in the near future6. Moreover, the known protocols for constructing reliable quantum computers from unreliable components can be complicated, often requiring many operations to produce a desired transformation3,4,5,7,8. Here we show how a single technique—a generalization of quantum teleportation9—reduces resource requirements for quantum computers and unifies known protocols for fault-tolerant quantum computation. We show that single quantum bit (qubit) operations, Bell-basis measurements and certain entangled quantum states such as Greenberger–Horne–Zeilinger (GHZ) states10—all of which are within the reach of current technology—are sufficient to construct a universal quantum computer. We also present systematic constructions for an infinite class of reliable quantum gates that make the design of fault-tolerant quantum computers much more straightforward and methodical. We present a technique for proving the security of quantum-key-distribution (QKD) protocols. It is based on direct information-theoretic arguments and thus also applies if no equivalent entanglement purification scheme can be found. Using this technique, we investigate a general class of QKD protocols with one-way classical post-processing. We show that, in order to analyze the full security of these protocols, it suffices to consider collective attacks. Indeed, we give new lower and upper bounds on the secret-key rate which only involve entropies of two-qubit density operators and which are thus easy to compute. As an illustration of our results, we analyze the Bennett-Brassard 1984, the six-state, and the Bennett 1992 protocols with one-way error correction and privacy amplification. Surprisingly, the performance of these protocols is increased if one of the parties adds noise to the measurement data before the error correction. In particular, this additional noise makes the protocols more robust against noise in the quantum channel. Recent advances in theoretical and experimental quantum computing bring us closer to scalable quantum computing devices. This makes the need for protocols that verify the correct functionality of quantum operations timely and has led to the field of quantum verification. In this paper we address key challenges to make quantum verification protocols applicable to experimental implementations. We prove the robustness of the single server verifiable universal blind quantum computing protocol of Fitzsimons and Kashefi (2012 arXiv:1203.5217) in the most general scenario. This includes the case where the purification of the deviated input state is in the hands of an adversarial server. The proved robustness property allows the composition of this protocol with a device-independent state tomography protocol that we give, which is based on the rigidity of CHSH games as proposed by (2013 Nature 496 456–60). The resulting composite protocol has lower round complexity for the verification of entangled quantum servers with a classical verifier and, as we show, can be made fault tolerant. In this paper, we provide a proof of unconditional security for a semi-quantum key distribution protocol introduced in a previous work. This particular protocol demonstrated the possibility of using X basis states to contribute to the raw key of the two users (as opposed to using only direct measurement results) even though a semi-quantum participant cannot directly manipulate such states. In this work, we provide a complete proof of security by deriving a lower bound of the protocol's key rate in the asymptotic scenario. Using this bound, we are able to find an error threshold value such that for all error rates less than this threshold, it is guaranteed that A and B may distill a secure secret key; for error rates larger than this threshold, A and B should abort. We demonstrate that this error threshold compares favorably to several fully quantum protocols. We also comment on some interesting observations about the behavior of this protocol under certain noise scenarios. In the near future, there will likely be special-purpose quantum computers with 40--50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible. First, we study the hardness of sampling the output distribution of a random quantum circuit, along the lines of a recent proposal by the Quantum AI group at Google. We show that there's a natural average-case hardness assumption, which has nothing to do with sampling, yet implies that no polynomial-time classical algorithm can pass a statistical test that the quantum sampling procedure's outputs do pass. Compared to previous work -- for example, on BosonSampling and IQP -- the central advantage is that we can now talk directly about the observed outputs, rather than about the distribution being sampled. Second, in an attempt to refute our hardness assumption, we give a new algorithm, inspired by Savitch's Theorem, for simulating a general quantum circuit with n qubits and depth d in polynomial space and dO(n) time. We then discuss why this and other known algorithms fail to refute our assumption. Third, resolving an open problem of Aaronson and Arkhipov, we show that any strong quantum supremacy theorem -- of the form "if approximate quantum sampling is classically easy, then the polynomial hierarchy collapses"-- must be non-relativizing. This sharply contrasts with the situation for exact sampling. Fourth, refuting a conjecture by Aaronson and Ambainis, we show that there is a sampling task, namely Fourier Sampling, with a 1 versus linear separation between its quantum and classical query complexities. Fifth, in search of a "happy medium" between black-box and non-black-box arguments, we study quantum supremacy relative to oracles in P poly. Previous work implies that, if one-way functions exist, then quantum supremacy is possible relative to such oracles. We show, conversely, that some computational assumption is needed: if SampBPP = SampBQP and NP ⊆ BPP, then quantum supremacy is impossible relative to oracles with small circuits. We consider the question of locality in distributed computing in the context of quantum information. Specifically, we focus on the round complexity of quantum distributed algorithms, with no bounds imposed on local computational power or on the bit size of messages. Linial's LOCAL model of a distributed system is augmented through two types of quantum extensions: (1) initialization of the system in a quantum entangled state, and or (2) application of quantum communication channels. For both types of extensions, we discuss proof-of-concept examples of distributed problems whose round complexity is in fact reduced through genuinely quantum effects. Nevertheless, we show that even such quantum variants of the LOCAL model have non-trivial limitations, captured by a very simple (purely probabilistic) notion which we call "physical locality" (φ-LOCAL). While this is strictly weaker than the "computational locality" of the classical LOCAL model, it nevertheless leads to a generic view-based analysis technique for constructing lower bounds on round complexity. It turns out that the best currently known lower time bounds for many distributed combinatorial optimization problems, such as Maximal Independent Set, bounds cannot be broken by applying quantum processing, in any conceivable way. Quantum computing combines the framework of quantum mechanics with that of computer science. In this paper we give a short introduction to quantum computing and survey the results in the area of distributed quantum computing and its applications to physics. We show how quantum key distribution (QKD) techniques can be employed within realistic, highly secure communications systems, using the internet architecture for a specific example. We also discuss how certain drawbacks in existing QKD point-to-point links can be mitigated by building QKD networks, where such networks can be composed of trusted relays or untrusted photonic switches. This paper extends the consensus framework, widely studied in the literature on distributed computing and control algorithms, to networks of quantum systems. We define consensus situations on the basis of invariance and symmetry properties, finding four different generalizations of classical consensus states. This new viewpoint can be directly used to study consensus for probability distributions, as these can be seen as a particular case of quantum statistical states: in this light, our analysis is also relevant for classical problems. We then extend the gossip consensus algorithm to the quantum setting and prove it converges to symmetric states while preserving the expectation of permutation-invariant global observables. Applications of the framework and the algorithms to estimation and control problems on quantum networks are discussed. It is well understood that the use of quantum entanglement significantly enhances the computational power of systems. Much of the attention has focused on Bell states and their multipartite generalizations. However, in the multipartite case it is known that there are several inequivalent classes of states, such as those represented by the W-state and the GHZ-state. Our main contribution is a demonstration of the special computational power of these states in the context of paradigmatic problems from classical distributed computing. Concretely, we show that the W-state is the only pure state that can be used to exactly solve the problem of leader election in anonymous quantum networks. Similarly we show that the GHZ-state is the only one that can be used to solve the problem of distributed consensus when no classical post-processing is considered. These results generalize to a family of W- and GHZ-like states. At the heart of the proofs of these impossibility results lie symmetry arguments. Abstract Quantum cryptography has been shown to be an effective technology for the secure distribution of keys on point-to-point optical links. We show how the existing techniques can be extended to allow multi-user secure key distribution on optical networks. We demonstrate that using network configurations typical of those found in passive optical network architectures any of the current quantum key distribution protocols can be adapted to implement secure key distribution from any user to any other user. An important feature of these adapted protocols is that the broadcaster, or service provider on the network, does not have to be trusted by the two users who wish to establish a key. The performances of Quantum Key Distribution (QKD) systems have notably progressed since the early experimental demonstrations and several recent works indicate that the pace of this progression is very likely to be maintained -if not increased- in the future years. In parallel to this fast progression of QKD techniques, commercial products are also being developed, making QKD deployment for securization of some specific ?real? data networks more and more likely to occur. It is the goal to the European project Secoqc to deploy a secure long-distance network based on quantum cryptography. It implies the conception of a specific architecture able to connect multiple users that may possibly be very far away from each other while QKD links are currently?point-to-point only? and intrinsically limited in distance. We construct a practically implementable classical processing for the Bennett-Brassard 1984 (BB84) protocol and the six-state protocol that fully utilizes the accurate channel estimation method, which is also known as the quantum tomography. Our proposed processing yields at least as high a key rate as the standard processing by Shor and Preskill. We show two examples of quantum channels over which the key rate of our proposed processing is strictly higher than the standard processing. In the second example, the BB84 protocol with our proposed processing yields a positive key rate even though the so-called error rate is higher than the 25 limit. Computational quantum technologies are entering a new phase in which noisy intermediate-scale quantum computers are available, but are still too small to benefit from active error correction. Even with a finite coherence budget to invest in quantum information processing, noisy devices with about 50 qubits are expected to experimentally demonstrate quantum supremacy in the next few years. Defined in terms of artificial tasks, current proposals for quantum supremacy, even if successful, will not help to provide solutions to practical problems. Instead, we believe that future users of quantum computers are interested in actual applications and that noisy quantum devices may still provide value by approximately solving hard combinatorial problems via hybrid classical-quantum algorithms. To lower bound the size of quantum computers with practical utility, we perform realistic simulations of the Quantum Approximate Optimization Algorithm and conclude that quantum speedup will not be attainable, at least for a representative combinatorial problem, until several hundreds of qubits are available. We consider quantum key distribution in the device-independent scenario, i.e., where the legitimate parties do not know (or trust) the exact specification of their apparatus. We show how secure key distribution can be realized against the most general attacks by a quantum adversary under the condition that measurements on different subsystems by the honest parties commute. In this paper, we investigate single-state, semi-quantum key distribution protocols. These are protocols whereby one party is limited to measuring only in the computational basis, while the other, though capable of measuring in both computational and Hadamard bases, is limited to preparing and sending only a single, publicly known qubit state. Such protocols rely necessarily on a two-way quantum communication channel making their security analysis difficult. However, we will show that, for single-state protocols, we need only consider a restricted attack operation by Eve. We will also describe a new single-state protocol that permits "reflections" to carry information and use our results concerning restricted attacks to show its robustness. Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n . But using ideas from computational learning theory, we show that one can do exponentially better in a statistical setting. In particular, to predict the outcomes of most measurements drawn from an arbitrary probability distribution, one needs only a number of sample measurements that grows linearly with n . This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense. The theorem also has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice. We present the optimal collective attack on a quantum key distribution protocol in the "device-independent" security scenario, where no assumptions are made about the way the quantum key distribution devices work or on what quantum system they operate. Our main result is a tight bound on the Holevo information between one of the authorized parties and the eavesdropper, as a function of the amount of violation of a Bell-type inequality.