QUANTUM COMPUTING: Back Action 2006

Quantum informatics paradigms and tools for QIPC
View Description Hide DescriptionQuantum information processing and communication (QIPC) theory has developed recently very fast and brought a variety of interesting and important results of the great value also for the whole area of quantum physics. One can also say that the field of QIPC has been so far mainly theory driven and the experiments have been mostly done to show that it is indeed possible, and how difficult is to make it, what theory shows as possible. One of the main reasons for such a fast and successful development of the QIPC science is the fact that paradigms, models, concepts, value system, as well as methods and results of the (theoretical) informatics have been intensively used. The goal of this paper is to go behind this successful crusade and applications of the informatics for QIPC and to present, analyse and illustrate the main ideas, concepts, methods and tools that have been involved. All that should help more physics‐oriented researchers in QIPC to understand that in order to explore the quantum world, new paradigms, concepts, models and so on are now available, and they could and should be used, due to the progress in (theoretical) informatics. Our concentration will be not only on what has been achieved, but even more on the main new challenges. In doing that we will concentrate more on the backgrounds, motivations, goals and implications than on the very technical results.

Publicity, Privacy, and Permanence of Information
View Description Hide DescriptionThe quantum principles of superposition and entanglement have led to a recasting of the foundations of information and computation theory, and are especially helpful in understanding the nature of privacy. The most private information, exemplified by a quantum eraser experiment, is best regarded as existing only conditionally and temporarily—after the experiment is over no trace remains. Less private is classically‐secret information—quantum information that has decohered, and thus become recoverable in principle, though not in practice, from portions of the environment. Finally there is public information, which has been replicated so thoroughly throughout the environment as to be infeasible to conceal. The Internet has caused an explosion of public information, with the beneficial side effect of making it harder for despots to rewrite the history of their misdeeds, and it is tempting to hope that all macroscopic information is permanent, making such cover‐ups impossible in principle if not in practice. However, by comparing entropy flows into and out of the Earth with estimates of the planet’s storage capacity, we conclude that most macroscopic information—for example the pattern of sand grains on an ancient beach—is impermanent, in the sense of becoming irrecoverable in principle from the Earth though still recorded in the Universe.

Approximate Randomization of Quantum States With Fewer Bits of Key
View Description Hide DescriptionRandomization of quantum states is the quantum analogue of the classical one‐time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/ε^{2}) Pauli operators to map any d‐dimensional state to a state that is within trace distance ε of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter, and Ambainis and Smith.
Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d‐dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small ε.

Information Processing Using Quantum Probability
View Description Hide DescriptionThis paper presents an information processing paradigm that introduces collective response of multiple agents (computational units) while the level of intelligence associated with the information processing has been increased manifold. It is shown that if the potential field of the Schroedinger wave equation is modulated using a self‐organized learning scheme, then the probability density function associated with the stochastic data is transferred to the probability amplitude function which is the response of the Schroedinger wave equation. This approach illustrates that information processing of data with stochastic behavior can be efficiently done using quantum probability instead of classical probability. The proposed scheme has been demonstrated through two applications: denoising and adaptive control.

Extremality and Entanglement of States in Coupled Quantum Systems
View Description Hide DescriptionBy analysing the properties of the extreme points in the convex set of all states of a coupled bipartite quantum system with preassigned marginal states and using the unitary error basis of Weyl operators we construct examples of completely entangled subspaces and states. For multipartite systems we introduce the notions of strictly nonseparable states and perfect entanglement and present examples exhibiting these properties. A number of unsolved problems and a conjecture are included.

Indecomposable Positive Maps and a New Family of Inseparable PPT States
View Description Hide DescriptionWe present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres‐Horodecki separability criterion. Their inseparability is demonstrated using Choi‐type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.

Cavity mediated long range interaction for fast quantum logic and quantum entanglement
View Description Hide DescriptionWe show how cavity QED can be used to produce multiparticle pure and mixed state entanglement and quantum logic operations. We further study the effect of environmental couplings and show that under certain conditions the dissipative coupling can create entanglement.

Entanglement scaling in classical and quantum harmonic oscillator lattices
View Description Hide DescriptionWe consider entanglement properties of ground and thermal states of harmonic lattice systems. A theorem connecting entanglement between a region and the rest of the lattice with the surface area of the boundary between the two regions is presented for systems in arbitrary spatial dimensions. The behavior of the block entanglement in the field limit is analysed and a logarithmic divergence is recovered.

Spin Decoherence in Quantum Dots
View Description Hide DescriptionThe decoherence of a spin‐1/2 particle (qubit) interacting with a nuclear spin bath in quantum dots has been studied using an isotropic Heisenberg model. An effective magnetic field (a feature of unitary evolution) and an effective temperature (a non‐unitary feature) describe the effective dynamics of the qubit. The decoherence time scale is calculated as a function of the nuclear bath spin distribution and the nuclear polarizations in various spin channels. The averaged auto correlation function of the qubit spin, averaged over all possible initial states of the qubit, depends on the nuclear spin distribution, but is insensitive to the nuclear polarizations.

Entanglement guides quantum computation
View Description Hide DescriptionOrigin of the speed‐up in quantum algorithms is not yet fully understood. But there are indications that entanglement does play an important role in quantum computation. Algorithms that do not involve entanglement can be efficiently simulated on a classical computer. Here I make a simple observation. I show that infinitesimal change in multi‐particle pure product state always gives rise to an entangled state. During quantum computation even though at each time instant the state is not entangled, it does pass through entangled states at infinitesimal time steps. This applies to multi‐particle pure, pseudo‐pure and general mixed states as well. This suggests that even though unitary operators do not produce any entanglement, quantum entanglement guides the process of quantum computation.

Distinguishability and cloning of entangled states by LOCC
View Description Hide DescriptionThere exist sets of multipartite orthogonal states which can not be distinguished by local operation and classical communication (LOCC). The simplest example is the set of four Bell states. The problem of characterizing a set in terms of distinguishability is a hard problem in general. For maximally entangled states, some results are known. Local cloning of entangled states by supplying necessary entanglement in the blank state is another idea that was introduced very recently. Interestingly there are sets of orthogonal entangled states which can be distinguished locally but can not be copied by local operation (even if necessary entanglement is supplied in the blank state) contrary to the fact that happens in the case of global operation.

Wigner distributions for qudits
View Description Hide DescriptionTwo new approaches to the problem of setting up Wigner distributions for finite level quantum systems are proposed. Both arise by looking at the structure of the familiar Wigner distribution for Cartesian quantum mechanics from different perspectives. The two approaches have one common feature—each involves a ‘square root’ operation though of very different kinds.

Excitation and Entanglement Transfer Versus Spectral Gap
View Description Hide DescriptionCondensed matter systems have recently been proposed as quantum channels for short distance communication. One characteristic of these systems is whether they feature a spectral gap between their ground and excited states. In this paper we study the transfer of excitations and entanglement between two ancillas, which are weakly coupled to a one dimensional chain at different sites. We find that the transfer is almost perfect but slow for large spectral gaps and fast but rather inefficient for small gaps.

Multipartite entanglement configurations: Combinatorial offshoots into (hyper)graph theory and their ramifications
View Description Hide DescriptionComputations in a distributed environment comprising a network of spatially separated nodes may require the exchange of classical and quantum information. The amount of classical communication may be reduced in such computations by using multipartite entanglement. Following the combinatorial approach developed in [25, 27], we study entanglement configurations over a set of nodes, where each entanglement configuration is a collection of multipartite entanglement (CAT or GHZ) states shared within different combinations of subsets of nodes. The main problem is to determine whether LOCC transformations can generate an entanglement configuration B from another entanglement configuration A, written as B < _{LOCC} A. We characterize the resulting partial order introduced on unitarily equivalent classes of entanglement configurations due to LOCC transformations. This study includes the communication complexity of generating higher cardinality multipartite CAT states from smaller sized CAT state configurations. We also study classes of incomparable entanglement configurations where no pair (A, B) of configurations satisfies A < _{LOCC} B. This leads us to investigate certain combinatorial properties of hypergraphs and hypertrees following initial results in [25, 27]. We study the unique reconstruction of vertex labelled r‐uniform hypertrees on n vertices, where r ⩽ n is a constant, and each hyperedge has the same number r, of vertices. We conclude by discussing several problems and open questions in the context of entanglement configurations.

Evolution of entanglement of two‐mode Gaussian states under a dissipative environment
View Description Hide DescriptionThe equivalence of mode squeezing and quantum entanglement for two‐mode Gaussian states under a dissipative environment is discussed. The evolution of two‐mode Gaussian states in a dissipative environment is considered in two different scenarios. In the first case, we squeeze the individual modes, evolve them under a dissipative environment and then entangle them using passive optics. In the second case, we squeeze the modes, entangle them using passive optics and then evolve them under a dissipative environment. The entanglement properties of the final state are compared in these two different cases. It is shown that under certain circumstances, the surviving entanglement is the same in both the cases.

The Quantum Measurement Problem and Physical reality: A Computation Theoretic Perspective
View Description Hide DescriptionIs the universe computable? If yes, is it computationally a polynomial place? In standard quantum mechanics, which permits infinite parallelism and the infinitely precise specification of states, a negative answer to both questions is not ruled out. On the other hand, empirical evidence suggests that NP‐complete problems are intractable in the physical world. Likewise, computational problems known to be algorithmically uncomputable do not seem to be computable by any physical means. We suggest that this close correspondence between the efficiency and power of abstract algorithms on the one hand, and physical computers on the other, finds a natural explanation if the universe is assumed to be algorithmic; that is, that physical reality is the product of discrete sub‐physical information processing equivalent to the actions of a probabilistic Turing machine. This assumption can be reconciled with the observed exponentiality of quantum systems at microscopic scales, and the consequent possibility of implementing Shor’s quantum polynomial time algorithm at that scale, provided the degree of superposition is intrinsically, finitely upper‐bounded. If this bound is associated with the quantum‐classical divide (the Heisenberg cut), a natural resolution to the quantum measurement problem arises. From this viewpoint, macroscopic classicality is an evidence that the universe is in BPP, and both questions raised above receive affirmative answers. A recently proposed computational model of quantum measurement, which relates the Heisenberg cut to the discreteness of Hilbert space, is briefly discussed. A connection to quantum gravity is noted. Our results are compatible with the philosophy that mathematical truths are independent of the laws of physics.

Probing the quantum nature of an apparatus
View Description Hide DescriptionIn this paper I present a foundational application of a qubit in probing the domain of validity of the quantum superposition principle. I show how a qubit can be coupled to a mesoscopic harmonic oscillator to both create and probe a superposition of states which involves distinct coherent states of the oscillator. Many harmonic oscillator systems are being developed as apparatus for a qubit and I argue that simply setting these oscillators to work in an under‐damped regime is enough to implement the scheme I discuss. I present two realizations: one based on a flux qubit coupled to a LCR circuit and the other based on a ion trap qubit coupled to the vibration of N ions.

Circuits for Distributing Quantum Measurement
View Description Hide DescriptionWe provide explicit quantum circuits for the distributed measurement of Bell states in the Hilbert space C^{dn } , where d is qudit dimension. We discuss a method for generalizing this to distributed measurements on any set of orthogonal states distributed among n parties. From the practical viewpoint, we show that such distributed measurements can help lower quantum communication complexity under certain conditions.

A Quantum Computing Approach to Embryonics for Design of Dependable Systems
View Description Hide DescriptionQuantum computers will perform computations at the atomic scale and operate according to the rules of quantum mechanics wherein quantum bits can exist in two or more states at once. Quantum entanglement is a remarkable property which plays a crucial role in quantum computation and can be utilized for applications like quantum teleportation. Another field, embryonics is inspired by the basic processes of molecular biology and by the embryonic development of living beings. Embryonic systems have properties unique to the living world like self‐repair and self‐reconstruction. Self‐repair allows partial reconstruction in case of a minor fault, while self‐replication allows complete reconstruction of the original device in case of a major fault. This paper presents a novel algorithm based on quantum teleportation for development of embryonic systems, called as Quantum Teleportation Embryonic Algorithm (QTEA). An example application of a robust quantum BCD to seven segment decoder is also presented. The results obtained are quite interesting and hold great promise for design of dependable systems through synthesis of quantum computing and embryonic systems.