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We pose a generalized Boson Sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of Bosons. We describe a quantum optical processor that can solve this problem efficiently based on Gaussian input states, a linear optical network and non-adaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide the first empirical evidence that quantum computers can indeed outperform classical computers and could lead to applications.

The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn's Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d1, ..., dn) and (c1, ..., cn) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form., Comment: 18 pages, 1 figure, material added, references updated, except from figure identical with version to appear in Commun. Math. Phys

We present a scheme for linear optical quantum computation (LOQC) which is highly robust to imperfect single photon sources and inefficient detectors. In particular we show that if the product of the detector efficiency with the source efficiency is greater than 2/3, then efficient LOQC is possible. This threshold is many orders of magnitude more relaxed than those which could be inferred by application of standard results in fault tolerance. The result is achieved within the cluster state paradigm for quantum computation., Comment: New version contains an Added Appendix

We apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of non-deterministic quantum gates. This is most relevant in the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of conditional dynamics; we furthermore show that except for a single layer of gates during the preparation, all subsequent operations can be shifted to the final adapted single qubit measurements. Remarkably, cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available., Comment: 5 pages, 4 figures, discussion of strategies to deal with further imperfections extended, references updated

This is a collection of statements gathered on the occasion of the Quantum Physics of Nature meeting in Vienna., Comment: 3 pages, Quantum Physics of Nature (QUPON) Conference, Vienna, Austria, May 22nd-26th, 2005; v4: more contributions

Standard quantum computation is based on sequences of unitary quantum logic gates which process qubits. The one-way quantum computer proposed by Raussendorf and Briegel is entirely different. It has changed our understanding of the requirements for quantum computation and more generally how we think about quantum physics. This new model requires qubits to be initialized in a highly-entangled cluster state. From this point, the quantum computation proceeds by a sequence of single-qubit measurements with classical feedforward of their outcomes. Because of the essential role of measurement a one-way quantum computer is irreversible. In the one-way quantum computer the order and choices of measurements determine the algorithm computed. We have experimentally realized four-qubit cluster states encoded into the polarization state of four photons. We fully characterize the quantum state by implementing the first experimental four-qubit quantum state tomography. Using this cluster state we demonstrate the feasibility of one-way quantum computing through a universal set of one- and two-qubit operations. Finally, our implementation of Grover's search algorithm demonstrates that one-way quantum computation is ideally suited for such tasks., Comment: 36 pages, 6 figures, 2 tables

It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security., Comment: 18 pages, 10 figures, numerous typos corrected

Although it is impossible for a bit commitment protocol to be both arbitrarily concealing and arbitrarily binding, it is possible for it to be both partially concealing and partially binding. This means that Bob cannot, prior to the beginning of the unveiling phase, find out everything about the bit committed, and Alice cannot, through actions taken after the end of the commitment phase, unveil whatever bit she desires. We determine upper bounds on the degrees of concealment and bindingness that can be achieved simultaneously in any bit commitment protocol, although it is unknown whether these can be saturated. We do, however, determine the maxima of these quantities in a restricted class of bit commitment protocols, namely those wherein all the systems that play a role in the commitment phase are supplied by Alice. We show that these maxima can be achieved using a protocol that requires Alice to prepare a pair of systems in an entangled state, submit one of the pair to Bob at the commitment phase, and the other at the unveiling phase. Finally, we determine the form of the trade-off that exists between the degree of concealment and the degree of bindingness given various assumptions about the purity and dimensionality of the states used in the protocol., Comment: 10 pages, 1 figure. Typos corrected

An algorithm for quantum computing Hamiltonian cycles of simple, cubic, bipartite graphs is discussed. It is shown that it is possible to evolve a quantum computer into an entanglement of states which map onto the set of all possible paths originating from a chosen vertex, and furthermore to subsequently project out all states not corresponding to Hamiltonian cycles., Comment: 4 pages, LaTex, 2 figures