Understanding NP-complete problems is a central topic in computer science (NP stands for nondeterministic polynomial time). This is why adiabatic quantum optimization has attracted so much attention, as it provided a new approach to tackle NP-complete problems using a quantum computer. The efficiency of this approach is limited by small spectral gaps between the ground and excited states of the quantum computer's Hamiltonian. We show that the statistics of the gaps can be analyzed in a novel way, borrowed from the study of quantum disordered systems in statistical mechanics. It turns out that due to a phenomenon similar to Anderson localization, exponentially small gaps appear close to the end of the adiabatic algorithm for large random instances of NP-complete problems. This implies that unfortunately, adiabatic quantum optimization fails: The system gets trapped in one of the numerous local minima.O ne of the central concepts in computational complexity theory is that of NP (nondeterministic polynomial time) completeness (1). A computational problem belongs to the class NP if its solution can be verified in a time at most polynomial in the input size N; i.e., the verification requires not more than cN k computational steps, where c and k are independent of N. An NPcomplete problem satisfies a second criterion: Any other problem in the class NP can be reduced to it in polynomial time. Remarkably, such problems exist, many of them being of a great practical importance. The question of whether NP-complete problems are "easy to solve," or in other words whether they may be solved in polynomial time, is one of the most fundamental open problems in computer science: This is the famous "P¼ ? NP" question (2). It is commonly believed, however, that it is not the case, i.e., that solving such a problem requires a computational time that is exponential in N. Adiabatic Quantum OptimizationThe discovery of an efficient (polynomial time) quantum algorithm for the factorization of large numbers-a problem in NP but not believed to be NP complete-is a milestone in quantum computing (3), as no algorithm is known to solve this problem efficiently on a classical (nonquantum) computer. However, this success was not extended to NP-complete problems. That was why the proposal of Farhi et al. (4) to use adiabatic quantum optimization (AQO) to solve NP-complete problems has attracted much attention since initial numerical simulations suggested such a possibility (5). The basic idea of AQO is as follows: Suppose that the solution of a computational problem P can be encoded in the ground state (GS) of a HamiltonianĤ P . To implement AQO one needs to construct a physical quantum system that is governed by a HamiltonianĤðsÞ ¼ ð1 − sÞĤ 0 þ sĤ P , where s is a tunable parameter, and H 0 is a Hamiltonian with a known and easy-to-prepare ground state. The idea is to start with s ¼ 0, initialize the system in the ground state ofĤð0Þ ¼Ĥ 0 , and increase s with time as s ¼ t∕T. According to the adiabatic theorem (6), slow enough variation of th...

Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as well as unitary evolution. We derive an expression for hitting time using superoperators, and numerically evaluate it for the discrete walk on the hypercube. The values found are compared to other analogues of hitting time suggested in earlier work. The dependence of hitting times on the type of unitary "coin" is examined, and we give an example of an initial state and coin which gives an infinite hitting time for a quantum walk. Such infinite hitting times require destructive interference, and are not observed classically. Finally, we look at distortions of the hypercube, and observe that a loss of symmetry in the hypercube increases the hitting time. Symmetry seems to play an important role in both dramatic speed-ups and slow-downs of quantum walks.

East University Boulevard, Tucson, AZ Remote quantum entanglement can enable numerous applications including distributed quantum computation, secure communication, and precision sensing. In this paper, we consider how a quantum network-nodes equipped with limited quantum processing capabilities connected via lossy optical links-can distribute high-rate entanglement simultaneously between multiple pairs of users (multiple flows). We develop protocols for such quantum "repeater" nodes, which enable a pair of users to achieve large gains in entanglement rates over using a linear chain of quantum repeaters, by exploiting the diversity of multiple paths in the network. Additionally, we develop repeater protocols that enable multiple user pairs to generate entanglement simultaneously at rates that can far exceed what is possible with repeaters time sharing among assisting individual entanglement flows. Our results suggest that the early-stage development of quantum memories with short coherence times and implementations of probabilistic Bell-state measurements can have a much more profound impact on quantum networks than may be apparent from analyzing linear repeater chains. This framework should spur the development of a general quantum network theory, bringing together quantum memory physics, quantum information theory, and computer network theory.A quantum network can generate, distribute and process quantum information in addition to classical data [1]. The most important function of a quantum network is to generate long distance quantum entanglement, which serves a number of tasks including the generation of multiparty shared secrets whose security relies only on the laws of physics [2,3], distributed quantum computing [4], improved sensing [5,6], blind quantum computing (quantum computing on encrypted data) [7], and secure private-bid auctions [8].Recent experiments have demonstrated entanglement links, viz., long-range entanglement established between quantum memories separated by a few kilometers using a point-to-point optical link [9]. As illustrated in Fig. 1, measurements performed at nodes in a quantum network can be used to glue together small entanglement links into longer-distance clusters. The nodes contain quantum memories that store qubits up to their coherence time, sources that generate photons entangled with the quantum memory to be sent to neighboring nodes, and local quantum processors that can perform multiqubit joint measurements. Entanglement attempts between neighboring nodes are synchronized on a global clock. The quantum routing protocol dictates the measurements to be performed locally at each node in order to obtain the desired entanglement topology. Possible goals of a routing protocol could be to enable high rate entanglement among multiple user-pairs simultaneously, * mpant@mit.edu or to generate multi-partite entanglement (entanglement between three or more parties).The development of network algorithms and protocols for routing and scheduling information flows was critical for the cr...

We present a resource-performance tradeoff of an all-optical quantum repeater that uses photon sources, linear optics, photon detectors and classical feedforward at each repeater node, but no quantum memories. We show that the quantum-secure key rate has the form R(η) = Dη s bits per mode, where η is the end-to-end channel's transmissivity, and the constants D and s are functions of various device inefficiencies and the resource constraint, such as the number of available photon sources at each repeater node. Even with lossy devices, we show that it is possible to attain s < 1, and in turn outperform the maximum key rate attainable without quantum repeaters, R direct (η) = − log 2 (1 − η) ≈ (1/ ln 2)η bits per mode for η 1, beyond a certain total range L, where η ∼ e −αL in optical fiber. We also propose a suite of modifications to a recently-proposed alloptical repeater protocol that ours builds upon, which lower the number of photon sources required to create photonic clusters at the repeaters so as to outperform R direct (η), from ∼ 10 11 to ∼ 10 6 photon sources per repeater node. We show that the optimum separation between repeater nodes is independent of the total range L, and is around 1.5 km for assumptions we make on various device losses.

We solve an open problem by constructing quantum walks that not only detect but also find marked vertices in a graph. In the case when the marked set M consists of a single vertex, the number of steps of the quantum walk is quadratically smaller than the classical hitting time HT(P, M ) of any reversible random walk P on the graph. In the case of multiple marked elements, the number of steps is given in terms of a related quantity HT + (P , M ) which we call extended hitting time.Our approach is new, simpler and more general than previous ones. We introduce a notion of interpolation between the random walk P and the absorbing walk P ′ , whose marked states are absorbing. Then our quantum walk is simply the quantum analogue of this interpolation. Contrary to previous approaches, our results remain valid when the random walk P is not statetransitive. We also provide algorithms in the cases when only approximations or bounds on parameters p M (the probability of picking a marked vertex from the stationary distribution) and HT + (P , M ) are known.

We analyze an entanglement-based quantum key distribution (QKD) architecture that uses a linear chain of quantum repeaters employing photon-pair sources, spectral-multiplexing, linear-optic Bell-state measurements, multi-mode quantum memories and classical-only error correction. Assuming perfect sources, we find an exact expression for the secret-key rate, and an analytical description of how errors propagate through the repeater chain, as a function of various loss and noise parameters of the devices. We show via an explicit analytical calculation, which separately addresses the effects of the principle non-idealities, that this scheme achieves a secret key rate that surpasses the TGW bound-a recently-found fundamental limit to the rate-vs.-loss scaling achievable by any QKD protocol over a direct optical link-thereby providing one of the first rigorous proofs of the efficacy of a repeater protocol. We explicitly calculate the end-to-end shared noisy quantum state generated by the repeater chain, which could be useful for analyzing the performance of other non-QKD quantum protocols that require establishing long-distance entanglement. We evaluate that shared state's fidelity and the achievable entanglement distillation rate, as a function of the number of repeater nodes, total range, and various loss and noise parameters of the system. We extend our theoretical analysis to encompass sources with non-zero two-pair-emission probability, using an efficient exact numerical evaluation of the quantum state propagation and measurements. We expect our results to spur formal rate-loss analysis of other repeater protocols, and also to provide useful abstractions to seed analyses of quantum networks of complex topologies.Shared entanglement underlies many quantum information protocols such as quantum key distribution (QKD) [1], teleportation [2] and dense coding [3], and is a fundamental information resource that can boost reliable classical and quantum communication rates over noisy quantum channels [4,5]. Optical photons are arguably the only candidate for distributing entanglement across long distances. They however are susceptible to loss and noise in the channel, which is the bane of practical realizations of long-distance quantum communication. The maximum entanglement-generation rate over a lossy optical channel with no classical-communication assistance is zero when the total loss exceeds 3 dB [6]. With two-way classical-communication assistance, the rates achievable for entanglement generation, as well as those for reliable quantum communication and secretkey generation (i.e., QKD) over a lossy optical channel must decay linearly with the channel's transmittance (i.e., exponentially with optical fiber length), regardless of the specific protocol used, for loss exceeding ∼ 5 dB [7], while the rate plunges to zero at a maximum loss threshold that is determined by the excess noise in the channel and detectors. In order to generate entanglement over long distances at high rates, intermediate nodes equipped with quant...

Nonlinear differential equations model diverse phenomena but are notoriously difficult to solve. While there has been extensive previous work on efficient quantum algorithms for linear differential equations, the linearity of quantum mechanics has limited analogous progress for the nonlinear case. Despite this obstacle, we develop a quantum algorithm for dissipative quadratic n-dimensional ordinary differential equations. Assuming R<1, where R is a parameter characterizing the ratio of the nonlinearity and forcing to the linear dissipation, this algorithm has complexity T2q poly(log⁡T,log⁡n,log⁡1/ϵ)/ϵ, where T is the evolution time, ϵ is the allowed error, and q measures decay of the solution. This is an exponential improvement over the best previous quantum algorithms, whose complexity is exponential in T. While exponential decay precludes efficiency, driven equations can avoid this issue despite the presence of dissipation. Our algorithm uses the method of Carleman linearization, for which we give a convergence theorem. This method maps a system of nonlinear differential equations to an infinite-dimensional system of linear differential equations, which we discretize, truncate, and solve using the forward Euler method and the quantum linear system algorithm. We also provide a lower bound on the worst-case complexity of quantum algorithms for general quadratic differential equations, showing that the problem is intractable for R≥2. Finally, we discuss potential applications, showing that the R<1 condition can be satisfied in realistic epidemiological models and giving numerical evidence that the method may describe a model of fluid dynamics even for larger values of R.

Hitting times are the average time it takes a walk to reach a given final vertex from a given starting vertex. The hitting time for a classical random walk on a connected graph will always be finite. We show that, by contrast, quantum walks can have infinite hitting times for some initial states. We seek criteria to determine if a given walk on a graph will have infinite hitting times, and find a sufficient condition, which for discrete time quantum walks is that the degeneracy of the evolution operator be greater than the degree of the graph. The set of initial states which give an infinite hitting time form a subspace. The phenomenon of infinite hitting times is in general a consequence of the symmetry of the graph and its automorphism group. Using the irreducible representations of the automorphism group, we derive conditions such that quantum walks defined on this graph must have infinite hitting times for some initial states. In the case of the discrete walk, if this condition is satisfied the walk will have infinite hitting times for any choice of a coin operator, and we give a class of graphs with infinite hitting times for any choice of coin. Hitting times are not very well-defined for continuous time quantum walks, but we show that the idea of infinite hitting-time walks naturally extends to the continuous time case as well.