Quantum computation with linear optics

Quantum Computing and Quantum Communications,First NASA International Conference, QCQC’98 Palm Springs, California, USA February 17–20, 1998 Selected Papers, Springer Verlag(ISBN:354065514X)info:eu-repo/semantics/publishe

Information-theoretic aspects of quantum copying

An information-theoretic approach to quantum copying is discussed, relying on the notion of quantum loss, a quantity that reflects the transmission quality in a noisy quantum channel. More specifically, an entropic no-cloning inequality is derived for a Hilbert space of arbitrary dimension, which describes the tradeoff between the losses of the channels leading to the two copies. Then, focusing on quantum bits, a family of Pauli cloning machines is introduced. These machines produce two imperfect copies of a single quantum bit that emerge from two distinct Pauli channels. The balance between the quality of the two copies is shown to result from a genuine complementarity principle. In the special probability p and p′, the domain in (√p, √p′-spacelocatedginsideaeparticular ellipse representing close-to-perfect cloning is forbidden. Finally, the class of symmetric Pauli cloning machines is used to provide an upper py and pz. The capacity is proven to be vanishing if (√px, √py, √pz) lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine.SCOPUS: cp.kQuantum Computing and Quantum Communications,First NASA International Conference, QCQC’98 Palm Springs, California, USA February 17–20, 1998 Selected Papers, Springer Verlag(ISBN:354065514X)info:eu-repo/semantics/publishe

Quantum Complexity for Discrete Logarithms and Related Problems

This paper studies the quantum computational complexity of the discrete logarithm (DL) and related group-theoretic problems in the context of generic algorithms -- that is, algorithms that do not exploit any properties of the group encoding. We establish a generic model of quantum computation for group-theoretic problems, which we call the quantum generic group model. Shor's algorithm for the DL problem and related algorithms can be described in this model. We show the quantum complexity lower bounds and almost matching algorithms of the DL and related problems in this model. More precisely, we prove the following results for a cyclic group $G$ of prime order. - Any generic quantum DL algorithm must make $\Omega(\log |G|)$ depth of group operations. This shows that Shor's algorithm is asymptotically optimal among the generic quantum algorithms, even considering parallel algorithms. - We observe that variations of Shor's algorithm can take advantage of classical computations to reduce the number of quantum group operations. We introduce a model for generic hybrid quantum-classical algorithms and show that these algorithms are almost optimal in this model. Any generic hybrid algorithm for the DL problem with a total number of group operations $Q$ must make $\Omega(\log |G|/\log Q)$ quantum group operations of depth $\Omega(\log\log |G| - \log\log Q)$. - When the quantum memory can only store $t$ group elements and use quantum random access memory of $r$ group elements, any generic hybrid algorithm must make either $\Omega(\sqrt{|G|})$ group operations in total or $\Omega(\log |G|/\log (tr))$ quantum group operations. As a side contribution, we show a multiple DL problem admits a better algorithm than solving each instance one by one, refuting a strong form of the quantum annoying property suggested in the context of password-authenticated key exchange protocol

Standard Model Physics and the Digital Quantum Revolution: Thoughts about the Interface

Advances in isolating, controlling and entangling quantum systems are transforming what was once a curious feature of quantum mechanics into a vehicle for disruptive scientific and technological progress. Pursuing the vision articulated by Feynman, a concerted effort across many areas of research and development is introducing prototypical digital quantum devices into the computing ecosystem available to domain scientists. Through interactions with these early quantum devices, the abstract vision of exploring classically-intractable quantum systems is evolving toward becoming a tangible reality. Beyond catalyzing these technological advances, entanglement is enabling parallel progress as a diagnostic for quantum correlations and as an organizational tool, both guiding improved understanding of quantum many-body systems and quantum field theories defining and emerging from the Standard Model. From the perspective of three domain science theorists, this article compiles thoughts about the interface on entanglement, complexity, and quantum simulation in an effort to contextualize recent NISQ-era progress with the scientific objectives of nuclear and high-energy physics.Comment: 63 pages, 5 figure

Cryptanalyse quantique de primitives symétriques

National audienceEtude du crible de Kuperberg et de l'utilisation d'un oracle probabiliste pour l'algorithme de Grover

Quantum Complexity for Discrete Logarithms and Related Problems

This paper studies the quantum computational complexity of the discrete logarithm and related group-theoretic problems in the context of ``generic algorithms\u27\u27---that is, algorithms that do not exploit any properties of the group encoding. We establish a generic model of quantum computation for group-theoretic problems, which we call the quantum generic group model, as a quantum analog of its classical counterpart. Shor\u27s algorithm for the discrete logarithm problem and related algorithms can be described in this model. We show the quantum complexity lower bounds and (almost) matching algorithms of the discrete logarithm and related problems in this model. More precisely, we prove the following results for a cyclic group $\mathcal G$ of prime order. (1) Any generic quantum discrete logarithm algorithm must make $\Omega(\log |\mathcal G|)$ depth of group operation queries. This shows that Shor\u27s algorithm that makes $O(\log |\mathcal G|)$ group operations is asymptotically optimal among the generic quantum algorithms, even considering parallel algorithms. (2) We observe that some (known) variations of Shor\u27s algorithm can take advantage of classical computations to reduce the number and depth of quantum group operations. We introduce a model for generic hybrid quantum-classical algorithm that captures these variants, and show that these algorithms are almost optimal in this model. Any generic hybrid quantum-classical algorithm for the discrete logarithm problem with a total number of (classical or quantum) group operations $Q$ must make $\Omega(\log |\mathcal G|/\log Q)$ quantum group operations of depth $\Omega(\log\log |\mathcal G| - \log\log Q)$. In particular, if $Q={\rm poly}\log |\mathcal G|$, classical group operations can only save the number of quantum queries by a factor of $O(\log\log |\mathcal G|)$ and the quantum depth remains as $\Omega(\log\log |\mathcal G|)$. (3) When the quantum memory can only store $t$ group elements and use quantum random access memory (qRAM) of $r$ group elements, any generic hybrid quantum-classical algorithm must make either $\Omega(\sqrt{|\mathcal G|})$ group operation queries in total or $\Omega(\log |\mathcal G|/\log (tr))$ quantum group operation queries. In particular, classical queries cannot reduce the number of quantum queries beyond $\Omega(\log |\mathcal G|/\log (tr))$. As a side contribution, we show a multiple discrete logarithm problem admits a better algorithm than solving each instance one by one, refuting a strong form of the quantum annoying property suggested in the context of password-authenticated key exchange protocol

Fases topológicas de la materia y sistemas cuánticos abiertos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Teórica I Física Teórica II (Métodos Matemáticos de la física), leída el 02/11/2016. Tesis formato europeo (compendio de artículos)The growing field of topological orders has been extensively studied both form the communities of condensed matter and quantum simulation. However, very little is known about the fate of topological order in the presence of disturbing effects such as external noise or dissipation. In the first part of this thesis, we start by studying how the edge states of a topological insulator become unstable when interacting with thermal baths. Motivated by these results, we generalise the notion of Chern insulators from the well-known Hamiltonian case to Liouvillian dynamics. We achieve this goal by defining a new topological witness that is still related to the quantum Hall conductivity at finite temperature. The mixed character of edge states is also well captured by our formalism, and explicit models for topological insulators and dissipative channels are considered. Additionally, we find new topological phases that remain quantised at finite temperature. The construction is based on the Uhlmann phase, a geometric quantum phase defined for general density matrices. Using this new tool, we are able to characterise topological insulators and superconductors at finite temperature both in one and two spatial dimensions. From the experimental side, we propose a state-independent protocol to measure the topological Uhlmann phase in the context of quantum simulation. Symmetry-protected topological orders have traditionally emerged from shortrange interactions. It remains very much unknown what the role played by longrange interactions is, within the physics of these topological systems. In the second part of this thesis, we analyse how topological superconducting phases are affected by the inclusion of long-range couplings. Remarkably, we unveil new topological quasi-particles due to long-range interactions, that were absent in short-range models. We also study how topological invariants are modified by the presence of long-range effects. In the appendix section of the thesis, we explore new numerical methods for driven-dissipative phase transitions. We consider quantum systems with a dissipative term driving the system into a non-equilibrium steady state. The inclusion of short-range fluctuations out-of-equilibrium deeply modifies the shape of the phase-diagram, something never observed in equilibrium thermodynamics.Una transición de fase es una transformación entre dos estados de la materia con propiedades físicas diferentes, por ejemplo cuando el agua líquida se convierte en hielo. Tradicionalmente, la física de las transiciones de fase ha sido perfectamente descrita por la teoría de Landau. Esta teoría propone la existencia de un parámetro de orden local que es capaz de distinguir entre dos fases distintas. Además, al atravesar la transición de fase se rompe espontáneamente una simetría del sistema. A partir de los años 80 se empezaron a encontrar un tipo de transiciones de fase que no estaban bien descritas por la teoría de Landau. Estas fases de la materia se denominan órdenes topológicos y constituyen el principal objeto de esta tesis doctoral. Para estas transiciones no existe un parámetro de orden local que pueda distinguir entre fases con propiedades físicas distintas. Por el contrario, vienen caracterizadas por un parámetro de orden global que es capaz de retener la información topológica del sistema. La otra principal diferencia con respecto a las transiciones de orden, descritas por la teoría de Landau, es el papel que juegan las simetrías. En las transiciones de fase topológicas, cuando se cambia de una fase a otra, no se rompe ninguna simetría. De manera adicional, las fases topológicas de la materia vienen caracterizadas por un conjunto de propiedades distintivas: (1) el estado fundamental está separado por un gap del resto de excitaciones y está degenerado, (2) el sistema presenta estados gapless localizados en el borde, (3) las excitaciones son anyones con estadística exótica, etc...Depto. de Física TeóricaFac. de Ciencias FísicasTRUEunpu