There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Topological photonics is a rapidly emerging field of research in which geometrical and topological ideas are exploited to design and control the behavior of light. Drawing inspiration from the discovery of the quantum Hall effects and topological insulators in condensed matter, recent advances have shown how to engineer analogous effects also for photons, leading to remarkable phenomena such as the robust unidirectional propagation of light, which hold great promise for applications. Thanks to the flexibility and diversity of photonics systems, this field is also opening up new opportunities to realize exotic topological models and to probe and exploit topological effects in new ways. This article reviews experimental and theoretical developments in topological photonics across a wide range of experimental platforms, including photonic crystals, waveguides, metamaterials, cavities, optomechanics, silicon photonics, and circuit QED. A discussion of how changing the dimensionality and symmetries of photonics systems has allowed for the realization of different topological phases is offered, and progress in understanding the interplay of topology with non-Hermitian effects, such as dissipation, is reviewed. As an exciting perspective, topological photonics can be combined with optical nonlinearities, leading toward new collective phenomena and novel strongly correlated states of light, such as an analog of the fractional quantum Hall effect.

Topological Photonics

Surface plasmons (SPs) are coherent oscillations of conduction electrons on a metal surface excited by electromagnetic radiation at a metal -dielectric interface. The growing field of research on such light -metal interactions is known as ‘plasmonics’. 1-3 This branch of research has attracted much attention due to its potential applications in miniaturized optical devices, sensors, and photonic circuits as well as in medical diagnostics and therapeutics. 4-8

Nanostructured plasmonic sensors.

The calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms. We demonstrate that such algorithms can be applied to problems of chemical interest using modest numbers of quantum bits. Calculations of the water and lithium hydride molecular ground-state energies have been carried out on a quantum computer simulator using a recursive phase-estimation algorithm. The recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4. Mappings of the molecular wave function to the quantum bits are described. An adiabatic method for the preparation of a good approximate ground-state wave function is described and demonstrated for a stretched hydrogen molecule. The number of quantum bits required scales linearly with the number of basis functions, and the number of gates required grows polynomially with the number of quantum bits.

http://science.sciencemag.org/content/sci/309/5741/1704.full.pdf

Simulated quantum computation of molecular energies.

QUANTUM MECHANICS States and Operators Observables and Measurement Dynamics of Quantum Systems MODELING OF QUANTUM CONTROL SYSTEMS: EXAMPLES Quantum Theory of Interaction of Particles and Fields Approximations and Modeling: Molecular Systems Spin Dynamics and Control Mathematical Structure of Quantum Control Systems CONTROLLABILITY Lie Algebras and Lie Groups Controllability Test: The Dynamical Lie Algebra Notions of Controllability for the State Pure State Controllability Equivalent State Controllability Equality of Orbits OBSERVABILITY AND STATE DETERMINATION Quantum State Tomography Observability Observability and Methods for State Reconstruction LIE GROUP DECOMPOSITIONS AND CONTROL Decompositions of SU(2) and Control of Two Level Systems Decomposition in Planar Rotations Cartan Decompositions Levi Decomposition Examples of Application of Decompositions to Control OPTIMAL CONTROL OF QUANTUM SYSTEMS Formulation of the Optimal Control Problem The Necessary Conditions of Optimality Example: Optimal Control of a Two Level Quantum System Time Optimal Control of Quantum Systems Numerical Methods for Optimal Control of Quantum Systems MORE TOOLS FOR QUANTUM CONTROL Selective Population Transfer via Frequency Tuning Time Dependent Perturbation Theory Adiabatic Control STIRAP Lyapunov Control of Quantum Systems ANALYSIS OF QUANTUM EVOLUTIONS: ENTANGLEMENT, ENTANGLEMENT MEASURES, AND DYNAMICS Entanglement of Quantum Systems Dynamics of Entanglement Local Equivalence of States APPLICATIONS OF QUANTUM CONTROL AND DYNAMICS Nuclear Magnetic Resonance Experiments Molecular Systems Control Atomic Systems Control: Implementations of Quantum Information Processing with Ion Traps APPENDIX A: POSITIVE AND COMPLETELY POSITIVE MAPS, QUANTUM OPERATIONS, AND GENERALIZED MEASUREMENT THEORY Positive and Completely Positive Maps Quantum Operations and Operator Sum Representation Generalized Measurement Theory APPENDIX B: LAGRANGIAN AND HAMILTONIAN FORMALISM IN CLASSICAL ELECTRODYNAMICS Lagrangian Mechanics Extension of Lagrangian Mechanics to Systems with Infinite Degrees of Freedom Lagrangian and Hamiltonian Mechanics for a System of Interacting Particles and Field APPENDIX C: CARTAN SEMISIMPLICITY CRITERION AND CALCULATION OF THE LEVI DECOMPOSITION The Adjoint Representation Cartan Semisimplicity Criterion Quotient Lie Algebras Calculation of the Levi Subalgebra in the Levi Decomposition Algorithm for the Levi Decomposition APPENDIX D: PROOF OF THE CONTROLLABILITY TEST OF THEOREM 3.2.1 APPENDIX E: THE BAKER-CAMPBELL-HAUSDORFF FORMULA AND SOME EXPONENTIAL FORMULAS APPENDIX F: PROOF OF THEOREM 6.2.1 REFERENCES INDEX Notes and Exercises appear at the end of every chapter.

Introduction to Quantum Control and Dynamics

We consider a generic elementary gate sequence which is needed to implement a general quantum gate acting on $n$ qubits---a unitary transformation with ${4}^{n}$ degrees of freedom. For synthesizing the gate sequence, a method based on the so-called cosine-sine matrix decomposition is presented. The result is optimal in the number of elementary one-qubit gates, ${4}^{n}$, and scales more favorably than the previously reported decompositions requiring ${4}^{n}\ensuremath{-}{2}^{n+1}$ controlled NOT gates.

/pdf/quantum-circuits-for-general-multiqubit-gates-14k9s68ku8.pdf

Quantum circuits for general multiqubit gates.

The basis of synchronous manipulation of individual electrons in solid-state devices was laid by the rise of single electronics about two decades ago1,2,3. Ultrasmall structures in a low-temperature environment form an ideal domain for addressing electrons one by one. In the so-called metrological triangle, voltage from the Josephson effect and resistance from the quantum Hall effect would be tested against current via Ohm’s law for a consistency check of the fundamental constants of nature, ℏ and e (ref. 4). Several attempts to create a metrological current source that would comply with the demanding criteria of extreme accuracy, high yield and implementation with not too many control parameters have been reported5,6,7,8,9,10,11. Here, we propose and prove the unexpected concept of a hybrid normal-metal–superconductor turnstile in the form of a one-island single-electron transistor with one gate, which demonstrates robust current plateaux at multiple levels of e f at frequency f.

/pdf/hybrid-single-electron-transistor-as-a-source-of-quantized-bkv34rhbo6.pdf

Hybrid single-electron transistor as a source of quantized electric current

Optimal implementation of quantum gates is crucial for designing a quantum computer. We consider the matrix representation of an arbitrary multiqubit gate. By ordering the basis vectors using the Gray code, we construct the quantum circuit which is optimal in the sense of fully controlled single-qubit gates and yet is equivalent with the multiqubit gate. In the second step of the optimization, superfluous control bits are eliminated, which eventually results in a smaller total number of the elementary gates. In our scheme the number of controlled NOT gates is $O({4}^{n})$ which coincides with the theoretical lower bound.

Efficient Decomposition of Quantum Gates

Uniformly controlled one-qubit gates are quantum gates which can be represented as direct sums of two-dimensional unitary operators acting on a single qubit. We present a quantum gate array which implements any $n$-qubit gate of this type using at most ${2}^{n\ensuremath{-}1}\ensuremath{-}1$ controlled-NOT gates, ${2}^{n\ensuremath{-}1}$ one-qubit gates, and a single diagonal $n$-qubit gate. To illustrate the versatility of these gates we then apply them to the decomposition of a general $n$-qubit gate and a state preparation procedure. Moreover, we study their implementation using only nearest-neighbor gates. We give upper bounds for the one-qubit and controlled-NOT gate counts for all the aforementioned applications. In all four cases, the proposed circuit topologies either improve on or achieve the previously reported upper bounds for the gate counts. Thus, they provide the most efficient method for general gate decompositions currently known.

/pdf/quantum-circuits-with-uniformly-controlled-one-qubit-gates-39ryl2q4ce.pdf

Quantum circuits with uniformly controlled one-qubit gates

We present the first measurements of the Berry phase in a superconducting Cooper pair pump. A fixed amount of Berry phase is accumulated to the quantum-mechanical ground state in each adiabatic pumping cycle, which is determined by measuring the charge passing through the device. The dynamic and geometric phases are identified and measured quantitatively from their different response when pumping in opposite directions. Our observations, in particular, the dependencies of the dynamic and geometric effects on the superconducting phase bias across the pump, agree with the basic theoretical model of coherent Cooper pair pumping.

/pdf/experimental-determination-of-the-berry-phase-in-a-3eh1td5ctb.pdf

Juha J. Vartiainen

Papers

Quantum circuits for general multiqubit gates.

Hybrid single-electron transistor as a source of quantized electric current

Efficient Decomposition of Quantum Gates

Quantum circuits with uniformly controlled one-qubit gates

Experimental determination of the berry phase in a superconducting charge pump.