We report our recent results in development of the secure fiber-optics communication system based upon quantum key
distribution (QKD). Emphasize is made on the limitation imposed by the state-of-the-art components crucial for the
system performance. We discuss the problem of the interferometer design and highlight the possible security loopholes
known. Together with single photon counting performance it places the main restriction on the distance range and the
secure key rate of the QKD system based upon the weak coherent pulses. Finally we describe the result of the first test
of the system using single photons produced by non-degenerate parametric down-conversion as a source.
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We demonstrate quantum-noise protected data encryption over a 200km-long inline optically-amplified fiber line at 650Mbps rate using off-the-shelf components. In contrast to our previous implementation, this demonstration uses phase-encoded coherent states, resulting in a polarization independent system that is compatible with the existing WDM infrastructure. Security calculations are presented for individual attacks on both the encrypted data as well as the secret key. This demonstration paves the way for widespread deployment of quantum cryptography in WDM networks.
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We describe a technique of parameter estimation and control in a phase-encoded quantum key distribution that uses continuous control of receiver-interferometer differential path length to maintain alignment with the transmitter. In this fiber-based system, a small number of training frames are sent over the quantum channel allowing the receiver to compensate for drift in the transmitter and receiver interferometers due to slow changes in temperature. The minimum mean-square error estimation method used to infer the state of the system incorporates the prior knowledge of the fiber dynamics recursively. The optimal linear-quadratic regulator feedback design is described and combined with the estimator to obtain the stochastic linear regulator of the path-length error.
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The paper proposes algorithmic and environmental modifications to the extant reconciliation algorithms within the BB84 protocol so as to speed up reconciliation and privacy amplification. These algorithms have been known to be a performance bottleneck 1 and can process data at rates that are six times slower than the quantum channel they serve2. As improvements in single-photon sources and detectors are expected to improve the quantum channel throughput by two or three orders of magnitude, it becomes imperative to improve the performance of the classical software. We developed a Cascade-like algorithm that relies on a symmetric formulation of the problem, error estimation through the segmentation process, outright elimination of segments with many errors, Forward Error Correction, recognition of the distinct data subpopulations that emerge as the algorithm runs, ability to operate on massive amounts of data (of the order of 1 Mbit), and a few other minor improvements. The data from the experimental algorithm we developed show that by operating on massive arrays of data we can improve software performance by better than three orders of magnitude while retaining nearly as many bits (typically more than 90%) as the algorithms that were designed for optimal bit retention.
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The security of quantum key distribution against undetected eavesdropping depends on the key-sharing parties (Alice and Bob) making a probabilistic estimate of the ignorance of a maximally adept eavesdropper (Eve) concerning sifted, error-free bits from which Alice and Bob distill a key. For individual attacks on the BB84 protocol, we show how to generalize the defense function and the defense frontier of Slutsky et al. to take advantage of Cachin’s analysis of Renyi entropy of arbitrary order R, here called R-entropy. For a special case of an attack uniform over all bits, an optimum defense frontier is displayed. Evidence is discussed for the conjecture that this defense frontier in terms of R-entropy holds good not just for uniform attacks but for all individual attacks on BB84.
We also show how the entropy estimate fits in to the full suite of key-distillation protocols in a QKD system, in particular how it relates to privacy amplification. After privacy amplification, Eve will have, with high probability, no information about the remaining bits. By choosing the optimal Rényi order R, we can distill secure bits in the presence of a significantly higher error rate.
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The standard four-state (BB84) protocol of key distribution in quantum cryptography, Slutsky, Rao, Sun, and Fainman performed an eavesdropping probe optimization, which an average yields the most information to the eavesdropper for a given error rate caused by the probe. The most general possible probe consistent with unitary was considered, in which each individual transmitted bit is made to interact with the probe so that the carrier and the probe are left in an endangered state, and measurement by the probe, made subsequent to measurement by the legitimate receiver, yield information about the carrier state. The probe optimization is based on maximizing the Renyi information gain by the probe on corrected data for a given error rate induced by the probe in a legitimate receiver. The results were obtained for the standard protocol with an angle of 45 degrees between the signal bases.
In more recent work, a larger set of optimum probe parameters was found than was known previously, and although they all yield the same maximum Renyi information gain by the probe, alternative options are made available for optimum probe design. In the present work, the corresponding optimized unitary transformation, representing the action of the probe on the signal, are calculated. I have determined three classes of unitary transformations yielding the same maximum information to the probe. The simplest one corresponds to a probe having a two-dimensional Hilbert space of states, and is uniquely determined by the error rate. The second class corresponds to a probe having a four-dimensional Hilbert space of states, and is determined by the error rate and two continuous angle parameters, which are mutually constrained by the error rate. The third class corresponds to a probe having a four-dimensional Hilbert space, and is determined by the error rate and two continuous angle parameters, one of which is constrained by the error rate. This work will be useful in the design of an optimum-entangling probe.
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We present quantum key distribution schemes which are
autocompensating (require no alignment) and symmetric (Alice and
Bob receive photons from a central source) for both polarization
and time-bin qubits. The primary benefit of the symmetric
configuration is that both Alice and Bob may have passive setups
(neither Alice nor Bob is required to make active changes for each
run of the protocol). We show that both the polarization and the
time-bin schemes may be implemented with existing technology. The
new schemes are related to previously described schemes by the
concept of advanced waves.
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Strong light signals are detected reliably on a time scale of a nanosecond; however, known detectors of weak light signals used in quantum key distribution (QKD) are much slower; they involve pulse-shaping arbiters based on flip-flops that take many nanoseconds to produce a stable output. Based on a recently shown logical independence of quantum particles from the devices that they are employed to explain, we make use of quantum mechanics fine-tuned so that particles serve not as rigid foundations but as improvised hypotheses useful in models that describe the recorded behavior of devices. On the experimental side, we augment the arbitrating flip-flop of a detector so that it fans out to a matched pair of auxiliary flip-flops, and show how this imparts to a detector a "self-awareness" of its own teetering, as announced by disagreements between the auxiliary flip-flops. We introduce a quantum model of this arrangement, invoking a pair of probe particles, and show this model corresponds well to an experiment. The matched pair of auxiliary flip-flops not only confirms the model of hesitation in a detector, but serves as an instrument, both conceptual and practical, that gives an added dimension to the characterization of signal sources.
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In quantum key distribution (QKD), Alice selects a sequence of characters from a finite alphabet and transmits the corresponding signals, each described by a quantum state, to Bob. Between Alice and Bob, Eve---with her own receiver and transmitter---can eavesdrop. Eve is assumed to know beforehand all the possible states from among which Alice chooses. The central question is: Can Eve gain significant information about the key without influencing what Bob receives in ways that Alice and Bob can
detect? This formulation implies that many options are available to Eve. It is the purpose here to discuss one of these options, where extensive use is made of the Schrodinger equation with spatial variables. This procedure of Eve---which consists of letting the quantum state of Alice scatter from a quantum memory and then, using the information thus obtained, sending a suitably chosen quantum state to Bob---is discussed in detail. Furthermore, there are ways for Alice to defeat this procedure by an unusual choice of her quantum states. This counter-measure is also presented and discussed.
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Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.
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In recent times the interest for quantum models of brain activity has rapidly grown. The Penrose-Hameroff model assumes that microtubules inside neurons are responsible for quantum computation inside brain. Several experiments seem to indicate that EPR-like correlations are possible at the biological level.
In the past year , a very intensive experimental work about this subject has been done at DiBit Labs in Milan, Italy by our research group.
Our experimental set-up is made by two separated and completely shielded basins where two parts of a common human DNA neuronal culture are monitored by EEG.
Our main experimental result is that, under stimulation of one culture by means of a 630 nm laser beam at 300 ms, the cross-correlation between the two cultures grows up at maximum levels.
Despite at this level of understanding it is impossible to tell if the origin of this non-locality is a genuine quantum effect, our experimental data seem to strongly suggest that biological systems present non-local properties not explainable by classical models.
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Photon entanglement is an essential ingredient for linear optics quantum computing schemes, quantum cryptographic protocols and fundamental tests of
quantum mechanics. Here we describe a setup that allows for the generation of polarisation-entangled N-photon states on demand. The photons are obtained by mapping the entangled state of N atoms, each of them trapped inside an optical cavity, onto the
free radiation field. The required initial state can be prepared by performing postselective measurements on the collective emission from the cavities through a multiport beamsplitter.
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We conjecture that one of the main obstacles to creating new non-abelian quantum hidden subgroup algorithms is the correct choice of a transversal.
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We investigate the entanglement of spins for two electrons
contributing to the acoustoelectric current driven by a surface
acoustic wave (SAW) in two adjacent narrow channels by calculating
their exchange energy (J). Our calculations are done in the s
wave Heitler-London approach, as well as by taking the difference
between the energies of the singlet and triplet states which
we obtain by solving the Schrodinger equation in the adiabatic
approximation. We also calculate the leakage of the electrons
from the lowest states in which they are prepared to excited
states within the moving quantum dots. The leakage from the ground state
is also calculated when an electron is launched in a channel which is then split into
two adjacent channels by gates.
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For the anisotropic XY model in transverse magnetic field, we analyze the
ground state and its concurrence-free point for generic anisotropy,
and the time evolution of initial Bell states created in a fully polarized
background and on the ground state.
We find that the pairwise entanglement
propagates with a velocity proportional to the reduced interaction
for all the four Bell states.
A transmutation from singlet-like to triplet-like states is observed during the propagation.
Characteristic for the anisotropic models is the instantaneous
creation of pairwise entanglement from a fully polarized state;
furthermore, the propagation of pairwise entanglement is suppressed
in favor of a creation of different types of entanglement.
The "entanglement wave" evolving from a Bell state on the ground state
turns out to be very localized in space-time.
Our findings agree with a recently formulated conjecture on entanglement
sharing; some results are interpreted in terms of this conjecture.
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We discuss the operation of the one-qubit quantum feedback loop,
which may be used for
initialization of a qubit in a solid-state quantum computer.
The continuous monitoring of a quantum state, which makes
the feedback possible, is done by means of a weak continuous measurement
and processing of the obtained information via quantum Bayesian
equations.
The properly designed quantum feedback
loop can keep the desired phase of a single-qubit quantum
coherent oscillations for infinitely long time, even in presence
of a dephasing environment. Various nonidealities reduce the
fidelity of the feedback synchronization. We report our
study of the effects of finite available bandwidth and time delay
on the one-qubit quantum feedback performance, and also discuss
the effect of environment-induced dephasing.
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We compare approaches to evaluation of decoherence at low temperatures in two-state quantum systems weakly coupled to the environment. By analyzing an exactly solvable model, we demonstrate that a non-Markovian approximation scheme yields good quality estimates of the reduced density matrix for time scales appropriate for evaluation of quantum computing designs.
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We present a new quantum data compression scheme, which optimally
compresses any completely ergodic quantum information source with
a Markovian property. The scheme does not assume any apriory
knowledge of the source (other than complete ergodicity and the
Markovian property) and uses so-called "weak" (non-demolition)
measurements to estimate the eigenbasis and entropy rate of the
source. The scheme is fully universal in the sense of being source
independent.
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Recent studies suggest that both the quantum Zeno (increase of the natural lifetime of an unstable quantum state by repeated measurements) and anti-Zeno (decrease of the natural lifetime) effects can be made manifest in the same system by simply changing the dissipative decay rate associated with the environment. Here, we
present a model which incorporates the effect of the environment in an exact manner leading to a confirmation of this expectation.
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Normally, the spontaneous emission of photons due to transitions
between functional levels of a qubit is not an important
decohering factor because of low probability. However, when the
distance between the levels corresponds to frequencies below
100GHz, the wavelength of the photons is in the millimeter
range or longer. If this range includes N identical qubits, the
probability of the photon emission goes up by a factor N due to
Dicke superradiance. This can drastically increase the decoherence
rate. Numerical examples with spin and SQUID based qubits are
given to illustrate this effect.
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A theoretical model is presented for the interaction of a quantum system with an orthogonal polarized entangled photon-pair measurement probe. The theoretical framework is based on solving the generalized Jaynes-Cummings and Shroeder's equations to determine the phase evolution of the interacting system. The measurement-induced decoherence is expressed in terms of the temporal evolution of the relative phases of the superposition states induced by the measurement probe. The method is applied to determine the rate of decoherence of a two-qubit rubidium quantum system. Quantitative results are given to contrast measurement-induced of (i) single photon probe and (ii) orthogonal polarized and entangled probe.
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Weak measurement is a technique whereby the coupling between the measuring device and the observable is sufficiently weak that the uncertainty in a single measurement is large compared with the separation between the eigenvalues of the observable. It is found here that weak measurement can be achieved with a single qubit. An additional feature of using single qubits as weak measurement devices is that the entanglement that results between the measured and measuring qubits can be easily quantified. An analysis is provided of the effect of entanglement using an example similar to the Hardy paradox of double interferometers with positron-electron annihilation.
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We describe experiments with photon pairs to evaluate, correct for, and
avoid sources of error in optical quantum information processing.
It is well known that a simple beamsplitter can
non-deterministicially prepare or select entangled polarization
states. We use quantum process tomography (QPT) to fully
characterize this effect, including loss and decoherence. The QPT
results identify errors and indicate how well they can
be corrected. To evade decoherence in a
noisy quantum channel, we identify decoherence-free subspaces
using experimental channel characterization, without need for a
priori knowledge of the decoherence mechanism or simplifying
assumptions. Working with pairs of polarization-encoded photonic
qubits, we use tomographic and adaptive techniques to identify 2-
and 3-state decoherence-free subspaces for encoding
decoherence-free qubits and qutrits within the noisy channel.
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We describe quantum control for implementing fast universal quantum gates using Josephson junction-based phase qubits. We have determined the effects of quantum tunneling and leakage on the gate fidelity by performing nonperturbative simulations of the Schrodinger equation. Phase qubits are seen to have many attractive features, and an experimental demonstration of a simple phase-qubit quantum gate should be possible in the near future.
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In this paper, we propose a technique to characterise the energy level structure of a superconducting charge qubit. The technique relies on the backreaction of a solid-state qubit on its environment and the incoherent transfer of energy from a high frequency mode to a low frequency mode due to the stochastic transitions of the qubit between energy eigenstates. We consider a coupled system consisting of a model charge qubit and several classical degrees of freedom. The qubit is coupled to three electromagnetic modes: a low frequency bias field, a higher frequency mode (which is used to pump the qubit from the ground state to an excited state), and a lossy reservoir (which represents the cavity that contains the qubit and control fields). The reservoir provides a mechanism to allow the qubit to dissipate energy and to induce spontaneous decays from an excited state into the ground state. We show that these spontaneous decays can have a significant effect on the noise in the classical bias field, and that this noise can be used to characterise the energy level structure of the qubit.
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A realistic implemetation of the macroscopic quantum computer requires that
individual qubits are realized as quantum systems with degenerate ground
states which are well protected from the effects of the environment. I show
how this protection can be achieved in the new class of Josephson arrays
with non-trivial symmetry or topology. For such "topologically protected"
arrays the effect of noise is exponentially small in the array size which
allows one in principle to get extremely small error rates and extremely
long dephasing times in these systems. In this review I present a simple
physical picture of the topological protection and explain why it is related
to the presence of the topological order parameter in these systems. I
formulate a set of general mathematical requirements on a model that ensures
the appearance of the protected degenerate states and show how these
conditions can be satisfied in a simple spin model. Finally I present
Josephson junction array that is described by the mathematical model which
satisfy these conditions and discuss its physical properties and how one can
test these predictions experimentally.
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As interest in quantum computing evolves, consideration must be given to the development of new methods to improve the current design of quantum computers. Such ideas are not only helping the advance towards practical quantum computation applications, but are also providing clearer understanding of quantum computation itself. Eventually, several new exploratory efforts to increase the efficiency beyond the inherent advantage of quantum computational systems to classical systems will materialize. As a part of these exploration efforts, this paper presents a modified version of the qubit, which we refer to as a "Qubit", that allows a smaller number of Qubits than qubits to reach the same result in applications such as Shor’s algorithm for the factorization of large numbers. The current model of the qubit consists of a quantum bit with two states, a zero and a one in a quantum superposition state. The Qubit, which consists of more than two states, is introduced and explained. A mathematical analysis of the Qubit within Hilbert space is given. We present examples of applications of the Qubit to several quantum computing algorithms, including discussion of the advantages and disadvantages that are involved. Finally a physical model to construct such a Qubit is considered.
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This paper proposes the definition of a quantum knot as a linear superposition of classical knots in three dimensional space. The definition is constructed and applications are discussed. Then the paper details extensions and also limitations of the Aravind Hypothesis for comparing quantum measurement with classical topological measurement. We propose a separate, network model for quantum evolution and measurement, where the background space is replaced by an evolving network. In this model there is an analog of the Aravind Hypothesis that promises to directly illuminate relationships between physics, topology and quantum knots.
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Quantum-mechanical phenomena are playing an increasing role in information processing, as transistor sizes approach the nanometer level, and quantum circuits and data encoding methods appear in the securest forms of communication. Simulating such phenomena efficiently is exceedingly difficult because of the vast size of the quantum state space involved. A major complication is caused by errors (noise) due to unwanted interactions between the quantum states and the environment. Consequently, simulating quantum circuits and their associated errors using the density matrix representation is potentially significant in many applications, but is well beyond the computational abilities of most classical simulation techniques in both time and memory resources. The size of a density matrix grows exponentially with the number of qubits simulated, rendering array-based simulation techniques that explicitly store the density matrix intractable. In this work, we propose a new technique aimed at efficiently simulating quantum circuits that are subject to errors. In particular, we describe new graph-based algorithms implemented in the simulator QuIDDPro/D. While previously reported graph-based simulators operate in terms of the state-vector representation, these new algorithms use the density matrix representation. To gauge the improvements offered by QuIDDPro/D, we compare its simulation performance with an optimized array-based simulator called QCSim. Empirical results, generated by both simulators on a set of quantum circuit benchmarks involving error correction, reversible logic, communication, and quantum search, show that the graph-based approach far outperforms the array-based approach.
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In [1] Gingrich and Williams extended the standard quantum circuit model of quantum computation to include gates that perform arbitrary linear, yet non-unitary, transformations of their input state. In this paper we use this theory to construct quantum circuits that perform desired non-unitary state transformations of the sort arising in lattice-based quantum search. Such n-qubit non-unitary gates cannot be achieved deterministically, but they can be achieved probabilistically using a single ancilla. Our approach is to use the n-qubit non-unitary gate to design an (n+1)-qubit Hamiltonian that can be seen to induce a suitable conditional dynamics such that whenever the output value of the ancilla qubit is measured and found to be |O>, then the remaining (unmeasured) n qubits will contain the desired non-unitary transform of the n-qubit input state. The scheme is necessarily probabilistic because we cannot force the measurement on the ancilla qubit to return the value we want. Fortunately, however, failed attempts are found to disturb the n-qubit input state very little. This allows us to re-use the result of successive “failed” attempts until the success condition is finally attained, and we give analytic expressions for the success probability and net fidelity in this case. By using our previous method for designing a quantum circuit for an arbitrary n-qubit gate in conjunction with our new probabilistic non-unitary procedure we are able to compute an explicit quantum circuit sufficient to implement an arbitrary linear, yet non-unitary, transformation of an input pure or mixed n-qubit state, using only a single copy of the input state. This allows us to extend the repertoire of computations that may be performed on a quantum computer, and in particular, gives the first explicit construction of the quantum circuits needed to perform some of the key operations arising in quantum lattice search.
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The technique of cavity quantum electrodynamics was proposed for coupling solid-state qubits. We study the prospects of employing superconducting stripline resonators as cavities for this purpose. The strong nonlinearity in these resonators originated by the effect of kinetic inductance may provide new tools for manipulating the state of the system and for measuring the outcome. Based on theoretical analysis and experimental results we show that such nonlinearity can be exploited for achieving strong intermodulation gain as well as quantum squeezing. Moreover, we demonstrate the ability to tune the mode frequencies of the resonator, which is needed for implementing these effects for coupling qubits.
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Recent papers by Khaneja, Brockett and Glaser obtained efficient RF pulse trains for two-spin and three-spin NMR systems by finding sub-Riemannian geodesics on a quotient space of SU(4). This paper outlines a method for extending their results via the Griffiths formalism for constrained variational problems.
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Recent research on the topic of quantum computation provides us with some quantum algorithms with higher efficiency and speedup compared to their classical counterparts. In this paper, it is our intent to provide the results of our investigation of several applications of such quantum algorithms - especially the Grover's Search algorithm - in the analysis of Hyperspectral Data. We found many parallels with Grover's method in existing data processing work that make use of classical spectral matching algorithms. Our efforts also included the study of several methods dealing with hyperspectral image analysis work where classical computation methods involving large data sets could be replaced with quantum computation methods. The crux of the problem in computation involving a hyperspectral image data cube is to convert the large amount of data in high dimensional space to real information. Currently, using the classical model, different time consuming methods and steps are necessary to analyze these data including: Animation, Minimum Noise Fraction Transform, Pixel Purity Index algorithm, N-dimensional scatter plot, Identification of Endmember spectra - are such steps. If a quantum model of computation involving hyperspectral image data can be developed and formalized - it is highly likely that information retrieval from hyperspectral image data cubes would be a much easier process and the final information content would be much more meaningful and timely. In this case, dimensionality would not be a curse, but a blessing.
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The prospects for practical quantum computing have improved
significantly over the past few years, and there is an increasing
motivation for developing quantum algorithms to address
problems that are presently impractical to solve using classical
computing. In previous work we have indentified such problems
in the areas of computer graphics applications,
and we have derived quantum-based solutions. In this paper
we examine quantum-based solutions to problems arising in
the area of computational geometry. These types of problems
are important in a variety of scientific, industrial and military
applications such as large scale multi-object simulation,
virtual reality systems, and multi-target tracking. In particular,
we present quantum algorithms for multidimensional searches,
convex hull construction, and collision detection.
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A simulator for quantum information systems cannot be both general,
that is, easily used for every possible system, and efficient.
Therefore, some systems will have aspects which can only be simulated
by cunning modeling. On the other hand, a simulation may conveniently
do extra-systemic processing that would be impractical in a real system.
We illustrate with examples from our quantum computing simulator, QCSim.
We model the [3,1] Hamming code in the
presence of random bit flip or generalized amplitude damping noise,
and calculate the expected result in one simulation run, as opposed
to, say, a Monte Carlo simulation, and keep the original state to
compute the chance of successful transmission, too. We also model the
BB84 protocol with eavesdropping and random choice of basis and
compute the chance of information received faithfully. Finally, we
present our simulation of teleportation as an example of the trade-off
between complexity of the simulation model and complexity of
simulation inputs and as an example of modeling
measurements and classical bits.
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An important result from the mid
nineties shows that any unitary
evolution may be realized as a sequence
of controlled-not and one-qubit gates.
This work surveys especially efficient
circuits in this library, in the special
case of evolutions on two-quantum bits.
In particular, we show that to construct
an arbitrary two-qubit state from
|00>, one CNOT gate suffices.
To simulate an arbitrary two-qubit operator up to relative phases,
two CNOTs suffice. To simulate an
arbitrary two-qubit operator up to
global phase, three CNOTs
suffice. In each case, we construct
an explicit circuit and prove optimality
in the generic case. We also contribute
a procedure to determine the minimal
number of CNOT gates necessary to simulate a given two-qubit operator up
to global phase. We use this procedure
to discuss timing a given Hamiltonian
to simulate the CNOT and to determine
an optimal circuit for the two-qubit Quantum Fourier Transform.
Our constructive proofs amount to
circuit synthesis algorithms and have
been coded in C++.
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We present a distributed implementation of Shor's quantum factoring algorithm on a distributed quantum network model. This model provides a means for small capacity quantum computers to work together in such a way as to simulate a large capacity quantum computer. In this paper, entanglement is used as a
resource for implementing non-local operations between two or more quantum
computers. These non-local operations are used to implement a distributed
factoring circuit with polynomially many gates. This distributed version of Shor's algorithm requires an additional overhead of O((log N)^2)
communication complexity, where N denotes the integer to be factored.
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This paper describes a new approach to global optimization and
control uses geometric methods and modern quantum mathematics.
Polynomial extremal problems (PEP) are considered. PEP
constitute one of the most important subclasses of nonlinear
programming models. Their distinctive feature is that an objective
function and constraints can be expressed by polynomial functions
in one or several variables. A general approach to optimization
based on quantum holonomic computing algorithms and instanton
mechanism. An optimization method based on geometric Lie -
algebraic structures on Grassmann manifolds and related with Lax
type flows is proposed. Making use of the differential geometric
techniques it is shown that associated holonomy groups properly
realizing quantum computation can be effectively found concerning
polynomial problems. Two examples demonstrating calculation
aspects of holonomic quantum computer and maximum clique problems
in very large graphs,
are considered in detail.
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The nonlinear Schrodinger (NLS) equation in a self-defocusing Kerr medium supports dark solitons. Moreover the mean field description of a dilute Bose-Einstein condensate (BEC) is described by the Gross-Pitaevskii equation, which for a highly anisotropic (cigar-shaped) magnetic trap reduces to a one-dimensional (1D) cubic NLS in an external potential. A quantum lattice algorithm is developed for the dark solitons. Simulations are presented for both black (stationary) solitons as well as (moving) dark solitons. Collisions of dark solitons are compared with the exact analytic solutions and coupled dark-bright vector solitons are examined. The quantum algorithm requires 2 qubits per scalar field at each spatial node. The unitary collision operator quantum mechanically entangles the on-site qubits, and this transitory entanglement is spread throughout the lattice by the streaming operators. These algorithms are suitable for a Type-II quantum computers, with wave function collapse induced by quantum measurements required to determine the coupling potentials.
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The work starts with a general idea of how to realize a dynamic programming algorithm as a quantum circuit. This realization is not tied to a specific design model, technology or a class of dynamic algorithms, it shows an approach for such synthesis. As an illustration of the efficiency of this approach, the class of all multiple-output symmetric functions is realized in a dynamic programming algorithm manner as a reversible circuit of Toffoli type elements (NOT, CNOT, and Toffoli gates). The garbage and quantum cost (found based on Barenco et al. paper)
of the presented implementation are calculated and compared to the costs of previously described reversible synthesis methods. The summary of results of this comparison is as follows. The quantum cost of the proposed method is less than the quantum cost of any other reported systematic approach. The garbage is usually lower, except for comparison with the synthesis methods, whose primary goal is synthesis with theoretically minimal garbage. The presented algorithm application to the symmetric function synthesis results in circuits suitable for quantum technology realizations.
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We propose a new simple algorithm for non-destructive estimation
of quantum relative entropy S(ρ||σ), where the density operators ρ and σ are defined on the same
finite-dimensional Hilbert space but are otherwise completely
unspecified (unknown). See text for formula.
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