Quantum Computation and Information Seminar 

I give an overview of the Weyl-Wigner formulation for quantum non-relativistic spinless particles. The main tool of the quantization procedure is the Weyl correspondence rule. Weyl pseudodifferential operators associated with various symbol classes are introduced. The advantages of Weyl quantization over other quantization prescriptions is discussed. The Wigner distribution is basically the symbol of a positive trace-class operator. I study the main properties of Wigner functions and show how they relate to classical Liouville probability measures.

Boyer, Kenigsberg, and Mor [Phys.Rev.Lett.99, 140501(2007)] proposed a new semi-quantum key distribution with four states where a key can be securely distributed between Alice who can perform any quantum operation and Bob who works classically. We have proposed new protocols for semi-quantum key distribution with less than four states and their security has been analyzed in detail. Recently, this idea of "semi-quantumness" has been incorporated also to quantum secret sharing [Phys.Rev. A 82, 022303 (2010)], and a semi-quantum secret sharing protocol has been presented, where a quantum Alice, can share a secret key with two classical participants and they can collaborate to recover the secret, but none can do that alone. In the protocol, a three-particle maximally entangled state plays a crucial role. However, multipartite entangled states are generally difficult to prepare in experiments. Therefore, we also present a new protocol for semi-quantum secret sharing where a quantum participant can share a secret key with two classical participants without using any entanglement. The protocol is also showed to be secure against eavesdropping.

Variational free energy, as used in generative model inversion, is reviewed. It is pointed out to be just a relative entropy (Kullback-Leibler divergence), if non-normalized states are allowed. This allows to formulate model inversion in the framework of information geometry.

In this work we implement the Deutsch-Jozsa algorithm with one qubit, removing its redundancy, which reduced the register size and simplified the evaluation function. For this, we used pairs of photons generated by parametric down conversion, where one is used as triger and the other goes through a double slit leaving it in a state of superposition of paths, i.e. a qubit. Using a spatial modulator and detecting photons in coincidence, we demonstrate the Deutsch-Jozsa algorithm with photons, and using only one qubit.

Traditional tree search algorithms supply a blueprint for modeling problem solving behaviour. A diverse spectrum of problems can be formulated in terms of tree search. Quantum computation, namely Grover’s algorithm, has aroused a great deal of interest since it allows for a quadratic speedup to be obtained in search procedures. In this work we consider the impact of incorporating classical search concepts alongside Grover’s algorithm into a hybrid quantum search system. Some of the crucial points examined include: (1) the reverberations of contemplating the use of non-constant branching factors; (2) determining the consequences of incorporating an heuristic perspective into a quantum tree search model. Joint work with Andreas Wichert.

We discuss the main features of a phase space noncommutative model of quantum mechanics and its application for quantum cosmology and black holes in the context of the Kantowski-Sachs minisuperspace geometry.

The endeavor to extend genuine quantum effects to ever larger systems, and to elucidate whether or not such quantum effects play a non-trivial role in driven, complex molecular systems poses a great technological and, moreover, conceptual challenge. Entanglement is a quantum effect that is required to demonstrate the genuine features of quantum physics beyond the wave-particle duality, namely to violate a Bell-inequality, and thereby proof correlations stronger than explainable by classical physics. In order to understand and efficiently assess entanglement in dynamical, complex systems under realistic conditions, we develop a unified picture of the dynamics of entanglement in general open quantum systems. A detailed algebraic analysis reveals evolution equations of entanglement, which show that it is possible to benchmark the entanglement dynamics with a single test state. A topological perspective for large quantum systems that employs results of high dimensional geometry yields effective, statistical results, which unveil a typical behavior of entanglement evolution. Both approaches thereby simplify the understanding of entanglement in a dynamical system, and stimulate the investigation of the role that entanglement plays in driven, complex systems far from thermal equilibrium.

We report recent experimental results obtained with the generation, transmission and detection of single and entangled photon pairs.

One of the most promising physical properties for implementing quantum technology is light polarization. However, since light polarization is fragile, it is crucial to use quantum error correction in order to make quantum information over optical networks feasible. This paper performs a statistical analysis of a noiseless subsystem technique to correct errors on quantum information sent through light polarization. We discuss the performance of the noiseless subsystem scheme in a noisy channel using a two-dimensional random walk to represent the channel variation. Finally, we propose an expression to measure the efficiency of the analyzed setup using the degree of depolarization of light. Joint work with P. Mateus.

The main problem of many practical random number generators is that they produce non-uniform, i.e. biased, output. Moreover, the actual probability distribution may be not fixed and can be (in a limited way) controlled by an adversary. The main goal of randomness extractors is to postprocess the output of an extractor in such a way that the extractor output is (almost) uniformly distributed. A dual siuation is when the adversary does not control the probability distribution of the random number generator, but can learn some information (fixed number of bits) about the bit sequence output by the generator. It is easy to show that such situation is equivalent to modification of the probability distribution and extractors are able to annihilate adversary's knowledge, i.e. to produce output adversary has (almost) no information about. This is also tightly related to the problem of privacy amplification, where two communicating participants want to eliminate adversary's (limited) knowledge of a commonly shared bit string using public discussion the adversary can eavesdrop.

Quantum properties of many-body states such as macroscopic superposition and large multipartite entanglement are essential resources for quantum information processing. In this talk, we study when and how macroscopic superposition appears in quantum many-body systems, such as a magnon system, quantum computers, and a quantum memory, and show that the emergence of such macroscopic superposition also causes large multipartite entanglement and low gate fidelity in the one-way quantum computation.

We present a probabilistic quantum contract signing protocol between two clients that requires no communication with the third trusted party during the commitment (i.e. signature exchange) phase. We discuss its fairness and show that it is possible to design such a protocol for which the probability of a dishonest client to cheat becomes negligible, and scales as N^{-1/2}, where N is the size of the signature, in bits. This way, our protocol over performs the classical probabilistic protocol by Ben-Or et. al., for which the probability to cheat can be as high as 1/4. We discuss the real-life scenario when the measurement errors and qubit state corruption due to noisy channels occur and argue that for real, good enough measurement apparatus and transmission channels, our protocol would still be fair. Our protocol could be implemented by today's technology, as it requires in essence the same type of apparatus as the one needed for BB84 cryptography protocol. Finally, we show that it is possible to generalize our protocol to an arbitrary number of clients.

Quantum channels have a number of capacities that depends fundamentally on the kind of information to be carried, the employed resources and the communication protocol. In this work, we generalize the definition of zero-error capacity of a classical channel to the zero-error capacity of a quantum channel. We propose a new kind of capacity for transmitting classical information through a quantum channel. The quantum zero-error capacity is defined by the maximum amount of classical information per channel use that can be sent over a noisy quantum channel, with the restriction that the probability of error must be equal to zero. The communication protocol used in the definition assumes codewords are built as tensor products of input quantum states, whereas entangled measurements can be performed between several channel outputs. Hence, our communication protocol is similar to the Holevo-Schumacher-Westmoreland protocol. Additionally we introduce some connection between concepts related to quantum channel capacity with concepts found in graph theory.

Derivation of the nonlinear master equation and some simple solutions. Some topics on wavelet local analyses. Derivation of a more general set of quantum uncertainty relations. Presentation of real experiments on the limits of Heisenberg uncertainty relations. Discussion of some quantum mechanical crucial experiments.

A mathematical model of a quantum computer, or q-computer, will be presented, together with related concepts such as q-gates, q-computations and q-algorithms/programs. Emphasis will be given to examples, such as the q-Fourier transform and q-algorithm of Shor to factor integers in polynomial time. The possible physical realizations of the model will be analyzed using an axiomatic version of quantum mechanics. At the end, a few lines for future work will be mentioned.

Proteins are the machines of life since they mediate essentially all the physical and chemical processes that go on in living cells. Many of them function by changing conformation. Here it is proposed that the triggers of protein conformational changes are quantum vibrational excited states (the VES hypothesis). In the Davydov/Scott model it is further assumed that the vibrational excited state behind protein function is the amide I mode of peptide groups. In this talk, the Davydov/Scott model for energy transfer in proteins will be introduced and then applications of the VES hypothesis to several biological processes such as energy transfer in proteins and protein misfolding, will be presented.

Quantum computation and communications offer a new range of possibilities for information processing. However, there is relatively little work on the application of quantum information to computer network issues. In this talk, I will show how the existing quantum communication protocols fit into the classical layered model for computer networks and argue that quantum computer networks can outperform their classical counterparts for some tasks. Two new applications will be presented for the network and the data link layer: a delayed commutation protocol and a quantum multiple access technique. The delayed commutation protocol gives a quantum solution to the processing delay problem in packet switching networks. Hilbert Space Division Multiple Access, HDMA, is proposed as a quantum theory of multiple access.

This talk will discuss the role of quantum topology in quantum computing and quantum information theory. We will begin with a review of basic quantum information theory and the elements of diagrammatic and categorical approaches to quantum topology We then weave correspondences between these fields, discussing teleportation, topological gate construction, topological quantum computing and quantum knots. In particular, we shall discuss a construction of the Fibonacci model for topological quantum computing that is based on the properties of the Jones polynomial, and we shall present quantum algorithms for computing the Jones polynomial, colored Jones polynomials and the Witten-Reshetikhin-Turaev invariant.

Quantum mechanics was developed in the first quarter of the 20th century mainly due to the work of Niels Bohr and his school. As is well known it is a linear theory, based onto the Fourier ontology. It is a very good theory to treat statistically large ensembles. Yet, when applied to one of few quantum entities this approach leads to strange paradoxes and, in last instance, to the indeterminism and consequently to the rejection of the existence of the objective reality. Now, at the beginning of this new century some experiments challenge the general validity of the old linear indeterministic quantum mechanics. These facts lead to the development of the nonlinear quantum physics based in the early works of de Broglie and Einstein. This more general theory, rejects Fourier ontology, and uses instead the local analysis by wavelets. In this way, causality is recovered and the old paradoxes are solved in a very clear and easy way. Furthermore, this nonlinear local and causal approach not only explains the new experiments but also opens a whole new physical realm.

In order to understand computation in a quantum context, it might be useful to introduce as many concepts as possible from the classical computation theory to the quantum case. One of these basic concepts concerns the functioning of finite automata. In this talk I present a model of Quantum Automaton (QA), the quantum transducer, working with qubits and address the problem of minimizing its dimension and the cardinality of its state set. The quantum memory of the QA is formed by a finite number of two-level quantum particles where the information is encoded in the form of qubits. These quantum particles are prepared in an initial quantum sate that is transformed by a finite set of quantum gates (unitary operators), in accordance to a classical program (input string). Finally the QA is measured. The outputs of the QA are a set of probability measures. The quantum states of the QA are represented by density operators. Based on its linearity I use the partial trace operator jointly with the properties of quotient spaces to derive the necessary and sufficient conditions to reduce its dimension and to minimize the cardinality of its state set. These conditions depend uniquely in the structure of the unitary operators and are independent of the initial state of the QA. Based on these conditions I develop an iterative algorithm to find the minimal QA. It is also shown that the minimization of the number of qubits is possible whether the QA is finite or not. The states of the minimal QA are described by reduced density operators, obtained by applying the partial trace operator to the quantum states of the original QA. An immediate conclusion of the minimization procedure here developed is that it is also possible to minimize the number of qubits in a quantum circuit.