Abstract: The pressure of fundamental limits on classical computation and the promise of exponential speedups from quantum effects have recently brought quantum circuits (Proc. R. Soc. Lond. A, Math. Phys. Sci., vol. 425, p. 73, 1989) to the attention of the electronic design automation community (Proc. 40th ACM/IEEE Design Automation Conf., 2003), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 68, p. 012318, 2003), (Proc. 41st Design Automation Conf., 2004), (Proc. 39th Design Automation Conf., 2002), (Proc. Design, Automation, and Test Eur., 2004), (Phys. Rev. A, At. Mol. Opt. Phy., vol. 69, p. 062321, 2004), (IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 22, p. 710, 2003). Efficient quantum-logic circuits that perform two tasks are discussed: 1) implementing generic quantum computations, and 2) initializing quantum registers. In contrast to conventional computing, the latter task is nontrivial because the state space of an n-qubit register is not finite and contains exponential superpositions of classical bitstrings. The proposed circuits are asymptotically optimal for respective tasks and improve earlier published results by at least a factor of 2. The circuits for generic quantum computation constructed by the algorithms are the most efficient known today in terms of the number of most expensive gates [quantum controlled-NOTs (CNOTs)]. They are based on an analog of the Shannon decomposition of Boolean functions and a new circuit block, called quantum multiplexor (QMUX), which generalizes several known constructions. A theoretical lower bound implies that the circuits cannot be improved by more than a factor of 2. It is additionally shown how to accommodate the severe architectural limitation of using only nearest neighbor gates, which is representative of current implementation technologies. This increases the number of gates by almost an order of magnitude, but preserves the asymptotic optimality of gate counts

Abstract: This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form ofan efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequenceof gates from a fixed and finite set. The algorithm can be used, for example, to compileShor's algorithm, which uses rotations of π/2k, into an efficient fault-tolerant form usingonly Hadamard, controlled-not, and π/8 gates. The algorithm runs in O(log2.71(1/e))time, and produces as output a sequence of O(log3.97(1/e)) quantum gates which isguaranteed to approximate the desired quantum gate to an accuracy within e > 0. Wealso explain how the algorithm can be generalized to apply to multi-qubit gates and togates from SU(d).

Abstract: We show how ultracold polar molecules, suggested as a new platform for quantum computation, can be manipulated to switch ``on'' and ``off'' their strong dipole-dipole interactions. This can be accomplished through selective excitation of states with considerably different dipole moments. We discuss different schemes for quantum gates using real molecules: CO, LiH, and CaF, as examples of polar molecules which are being experimentally studied at ultracold temperatures. These schemes can be realized in several recently proposed architectures.

Abstract: Reversible hardware computation, that is, performing logic signal transforimations in a way that allows the original input signals to be recovered from the produced outputs, is helpful in diverse areas such as quantum computing, low-power design, nanotechnology, optical information processing, and bioinformatics. We propose a paradigm for performing such reversible computations in a manner that renders a wide class of circuit faults readily detectable at the circuit's outputs. More specifically, we introduce a class of reversible logic gates (consisting of the well-known Fredkin gate and a newly defined Feynman double-gate) for which the parity of the outputs matches that of the inputs. Such parity-preserving reversible gates, when used with an arbitrary synthesis strategy for reversible logic circuits, allow any fault that affects no more than a single logic signal to be detectable at the circuit's primary outputs. We show the applicability of our design strategy by demonstrating how the well-known, and very useful, Toffoli gate can be synthesized from parity-preserving gates and apply the results to the design of a binary full-adder circuit, which is a versatile and widely used element in digital arithmetic processing.

Abstract: This paper proposes an approach to optimally synthesize quantum circuits by symbolic reachability analysis, where the primary inputs and outputs are basis binary and the internal signals can be nonbinary in a multiple-valued domain. The authors present an optimal synthesis method to minimize quantum cost and some speedup methods with nonoptimal quantum cost. The methods here are applicable to small reversible functions. Unlike previous works that use permutative reversible gates, a lower level library that includes nonpermutative quantum gates is used here. The proposed approach obtains the minimum cost quantum circuits for Miller gate, half adder, and full adder, which are better than previous results. This cost is minimum for any circuit using the set of quantum gates in this paper, where the control qubit of 2-qubit gates is always basis binary. In addition, the minimum quantum cost in the same manner for Fredkin, Peres, and Toffoli gates is proven. The method can also find the best conversion from an irreversible function to a reversible circuit as a byproduct of the generality of its formulation, thus synthesizing in principle arbitrary multi-output Boolean functions with quantum gate library. This paper constitutes the first successful experience of applying formal methods and satisfiability to quantum logic synthesis

Abstract: We propose a scheme to implement quantum gates on any pair of trapped ions immersed in a large linear crystal, using interaction mediated by the transverse phonon modes. Compared with the conventional approaches based on the longitudinal phonon modes, this scheme is much less sensitive to ion heating and thermal motion outside of the Lamb-Dicke limit thanks to the stronger confinement in the transverse direction. The cost for such a gain is only a moderate increase of the laser power to achieve the same gate speed. We also show how to realize arbitrary-speed quantum gates with transverse phonon modes based on simple shaping of the laser pulses.

Abstract: A crucial building block for quantum information processing with trapped ions is a controlled-NOT quantum gate. In this Letter, two different sequences of laser pulses implementing such a gate operation are analyzed using quantum process tomography. Fidelities of up to 92.6(6)% are achieved for single-gate operations and up to 83.4(8)% for two concatenated gate operations. By process tomography we assess the performance of the gates for different experimental realizations and demonstrate the advantage of amplitude-shaped laser pulses over simple square pulses. We also investigate whether the performance of concatenated gates can be inferred from the analysis of the single gates.

Abstract: We present here a new approach to scalable quantum computing—a 'qubus computer'—which realizes qubit measurement and quantum gates through interacting qubits with a quantum communication bus mode. The qubits could be 'static' matter qubits or 'flying' optical qubits, but the scheme we focus on here is particularly suited to matter qubits. There is no requirement for direct interaction between the qubits. Universal two-qubit quantum gates may be effected by schemes which involve measurement of the bus mode, or by schemes where the bus disentangles automatically and no measurement is needed. In effect, the approach integrates together qubit degrees of freedom for computation with quantum continuous variables for communication and interaction.

Abstract: We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, sub-Riemannian, and Finslerian manifolds. These results generalize the results of [Nielsen, Dowling, Gu, and Doherty, Science 311, 1133 (2006)], which showed that the gate complexity can be related to distances on a Riemannian manifold.

Abstract: We propose a two-qubit collisional phase gate that can be implemented with available atom chip technology and present a detailed theoretical analysis of its performance. The gate is based on earlier phase gate schemes, but uses a qubit state pair with an experimentally demonstrated, very long coherence lifetime. Microwave near fields play a key role in our implementation as a means to realize the state-dependent potentials required for conditional dynamics. Quantum control algorithms are used to optimize gate performance. We employ circuit configurations that can be built with current fabrication processes and extensively discuss the impact of technical noise and imperfections that characterize an actual atom chip. We find an overall infidelity compatible with requirements for fault-tolerant quantum computation.

Abstract: We present controllable generation of various kinds of highly nonclassical states of light, including the single photon state and superposition states of mesoscopically distinct components. The high nonclassicality of the generated states is measured by the negativity of the Wigner function, which is largest ever observed to our knowledge. Our scheme is based on photon subtraction from a nearly pure squeezed vacuum, generated from an optical parametric oscillator with a periodically-poled KTiOPO$_4$ crystal as a nonlinear medium. This is an important step to realize basic elements of universal quantum gates, and to serve as a highly nonclassical input probe for spectroscopy and the study of quantum memory.

Abstract: We show that a large number of ions forming a 2D Coulomb crystal provides an almost ideal system for scalable quantum computation and quantum simulation. In particular, the coupling of the internal states to the motion of the ions transverse to the crystal plane allows one to implement two-qubit quantum gates. We analyze in detail the decoherence induced by anharmonic couplings, and show that very high gate fidelities can be achieved with current experimental setups.

Abstract: We propose a scheme to implement arbitrary-speed quantum entangling gates on two trapped ions immersed in a large linear crystal of ions, with minimal control of laser beams. For gate speeds slower than the oscillation frequencies in the trap, a single appropriately detuned laser pulse is sufficient for high-fidelity gates. For gate speeds comparable to or faster than the local ion oscillation frequency, we discover a five-pulse protocol that exploits only the local phonon modes. This points to a method for efficiently scaling the ion trap quantum computer without shuttling ions.

Abstract: The two-level systems (TLSs) naturally occurring in Josephson junctions constitute a major obstacle for the operation of superconducting phase qubits. Since these TLSs can possess remarkably long decoherence times, we show that such TLSs can themselves be used as qubits, allowing for a well controlled initialization, universal sets of quantum gates, and readout. Thus, a single current-biased Josephson junction can be considered as a multiqubit register. It can be coupled to other junctions to allow the application of quantum gates to an arbitrary pair of qubits in the system. Our results indicate an alternative way to realize superconducting quantum information processing.

Abstract: Quantum algorithms may be described by sequences of unitary transformations called quantum gates and measurements applied to the quantum register of n quantum bits, qubits. A collection of quantum gates is called universal if it can be used to construct any n-qubit gate. In 1995, the universality of the set of one-qubit gates and controlled NOT gate was shown by Barenco et al. using QR decomposition of unitary matrices. Almost ten years later the decomposition was improved to include essentially fewer elementary gates. In addition, the cosine-sine matrix decomposition was applied to efficiently implement decompositions of general quantum gates. In this chapter, we review the different types of general gate decompositions and slightly improve the best known gate count for the controlled NOT gates to (23/48)4^n in the leading order. In physical realizations, the interaction strength between the qubits can decrease strongly as a function of their distance. Therefore, we also discuss decompositions with the restriction to nearest-neighbor interactions in a linear chain of qubits.

Abstract: We present a complete all-optical-processing polarization-based binary-logic system, by which any logic gate or processor can be implemented. Following the new polarization-based logic presented in [Opt. Express 14, 7253 (2006)], we develop a new parallel processing technique that allows for the creation of all-optical-processing gates that produce a unique output either logic 1 or 0 only once in a truth table, and those that do not. This representation allows for the implementation of simple unforced OR, AND, XOR, XNOR, inverter, and more importantly NAND and NOR gates that can be used independently to represent any Boolean expression or function. In addition, the concept of a generalized gate is presented which opens the door for reconfigurable optical processors and programmable optical logic gates. Furthermore, the new design is completely compatible with the old one presented in [Opt. Express 14, 7253 (2006)], and with current semiconductor based devices. The gates can be cascaded, where the information is always on the laser beam. The polarization of the beam, and not its intensity, carries the information. The new methodology allows for the creation of multiple-input-multiple-output processors that implement, by itself, any Boolean function, such as specialized or non-specialized microprocessors. Three all-optical architectures are presented: orthoparallel optical logic architecture for all known and unknown binary gates, singlebranch architecture for only XOR and XNOR gates, and the railroad (RR) architecture for polarization optical processors (POP). All the control inputs are applied simultaneously leading to a single time lag which leads to a very-fast and glitch-immune POP. A simple and easy-to-follow step-by-step algorithm is provided for the POP, and design reduction methodologies are briefly discussed. The algorithm lends itself systematically to software programming and computer-assisted design. As examples, designs of all binary gates, multiple-input gates, and sequential and non-sequential Boolean expressions are presented and discussed. The operation of each design is simply understood by a bullet train traveling at the speed of light on a railroad system preconditioned by the crossover states predetermined by the control inputs. The presented designs allow for optical processing of the information eliminating the need to convert it, back and forth, to an electronic signal for processing purposes. All gates with a truth table, including for example Fredkin, Toffoli, testable reversible logic, and threshold logic gates, can be designed and implemented using the railroad architecture. That includes any future gates not known today. Those designs and the quantum gates are not discussed in this paper.

Abstract: PACS 03.67.Lx, 32.80.Pj, 03.75.Lm, 42.50.Pq We review our experiments on quantum information processing with neutral atoms in optical lattices and magnetic microtraps. Atoms in an optical lattice in the Mott insulator regime serve as a large qubit register. A spin-dependent lattice is used to split and delocalize the atomic wave functions in a controlled and coherent way over a defined number of lattice sites. This is used to experimentally demonstrate a massively parallel quantum gate array, which allows the creation of a highly entangled many-body cluster state through coherent collisions between atoms on neighbouring lattice sites. In magnetic microtraps on an atom chip, we demonstrate coherent manipulation of atomic qubit states and measure coherence lifetimes exceeding one second at micron-distance from the chip surface. We show that microwave near-fields on the chip can be used to create state-dependent potentials for the implementation of a quantum controlled phase gate with these robust qubit states. For single atom detection and preparation, we have developed high finesse fiber Fabry-Perot cavities and integrated them on the atom chip. We present an experiment in which we detected a very small number of cold atoms magnetically trapped in the cavity using the atom chip.

Abstract: Optimal control methods for implementing quantum modules with least amount of relaxative loss are devised to give best approximations to unitary gates under relaxation. The potential gain by optimal control using relaxation parameters against time-optimal control is explored and exemplified in numerical and in algebraic terms: it is the method of choice to govern quantum systems within subspaces of weak relaxation whenever the drift Hamiltonian would otherwise drive the system through fast decaying modes. In a standard model system generalising decoherence-free subspaces to more realistic scenarios, openGRAPE-derived controls realise a CNOT with fidelities beyond 95% instead of at most 15% for a standard Trotter expansion. As additional benefit it requires control fields orders of magnitude lower than the bang-bang decouplings in the latter.

Abstract: We propose a potentially practical scheme to implement a three-qubit Toffoli gate by a single resonant interaction in cavity QED. The scheme does not require two-qubit controlled-NOT gates but uses a three-qubit phase gate and two Hadamard gates, where the phase gate can be implemented by only a single resonant interaction of the atoms with the cavity mode. Both the situations with and without cavity decay are considered. We discuss the advantages and the experimental feasibility of our scheme.

Abstract: We analyse the effects of molecular characteristics on the structure of global quantum gates and the complexity of the resulting mechanisms systematically with the goal of rating a molecule's suitability for molecular quantum computing. One decisive property of a molecular vibration is the anharmonicity and in an extension to multimode systems the mode coupling has to be taken into account additionally. In a parametrized two-dimensional model system, we tune these characteristic properties and explore their effects on quantum gates. We find that the interplay of the anharmonicity and the coupling is of prime importance and leads to two basic control mechanisms for all systems. The features of quantum gate laser fields are explained with characteristic transition frequencies, determined by the molecular parameters, and the limits to obtain simple structures are identified.

Abstract: We propose a fast scheme involving atoms fixed in an optical cavity to directly implement the universal controlled-unitary gate. The present technique based on adiabatic passage uses dark states well suited for the controlled-rotation operation. We show that these dark states allow the robust implementation of a gate that is a generalization of the controlled-unitary gate to the case where the control qubit can be selected to be an arbitrary state. This gate has potential applications to the rapid implementation of quantum algorithms such as the projective measurement algorithm. This process is decoherence-free since excited atomic states and cavity modes are not populated during the dynamics.

Abstract: A scheme is proposed to obtain three-qubit gates, such as quantum-phase gate and ${C}^{2}$-NOT gate operations, in a cavity QED system where highly detuned cavity-field modes interact with a four-level system in an inverted-Y configuration. The influence of the Stark shift is also included in such proposed gate operations. Since only the metastable lower levels are involved in the gate operations, the gates are not affected by the atomic decay rates. The potential application of such gates to realize Grover's algorithm is also discussed.

Abstract: Decoherence-free subsystems (DFSs) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem theoretically exist, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFSs. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two physical qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced “strongly modulating pulses” for N...

Abstract: The d-level or qudit one-way quantum computer (d1WQC) is described using the valence bond solid formalism and the generalized Pauli group. This formalism provides a transparent means of deriving measurement patterns for the implementation of quantum gates in the computational model. We introduce a new universal set of qudit gates and use it to give a constructive proof of the universality of d1WQC. We characterize the set of gates that can be performed in one parallel time step in this model.

Abstract: Decoherence-Free Subsystems (DFS) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians theoretically exist that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFS. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two-physical-qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.

Abstract: We employ optimal control theory to design optimized quantum gates for solid-state qubits subject to decoherence. At the example of a gate-controlled semiconductor quantum dot molecule we demonstrate that decoherence due to phonon couplings can be strongly suppressed. Our results suggest a much broader class of quantum control strategies in solids.