Abstract: In this paper we study universality for quantum gates acting on qudits. Qudits are states in a Hilbert space of dimension d where d can be any integer ≥ 2. We determine which 2-qudit gates V have the properties: (i) the collection of all 1-qudit gates together with V produces all n-qudit gates up to arbitrary precision, or (ii) the collection of all 1-qudit gates together with V produces all n-qudit gates exactly. We show that (i) and (ii) are equivalent conditions on V , and they hold if and only if V is not a primitive gate. Here we say V is primitive if it transforms any decomposable tensor into a decomposable tensor. We discuss some applications and also relations with work of other authors.

Abstract: We discuss the possibility of realizing quantum computation on the basis of a cluster of single interacting nuclear spins in solids. This idea seems to be feasible because of the combination of two techniques—Single Molecule Spectroscopy and Optically Detected Electron Nuclear Double Resonance. Compared to the well-known bulk Nuclear Magnetic Resonance (NMR), the proposed method of quantum computation has the advantage that quantum computation is performed with pure spin states and the quantum processor is more easily scalable. At the same time, the advantages of NMR quantum computation are kept: long coherence time and easy construction of quantum gates. As a specific system to implement the above idea, we discuss the 13C-nuclear spins in the nearest vicinity of a single nitrogen-vacancy (NV) defect center in diamond, which can be optically detected using the technique of scanning confocal microscopy. Owing to the hyperfine coupling of the ground state electron paramagnetic spin S=1 of the center to 13C nuclear spins in a diamond lattice, the states of nuclear spins in the vicinity of the defect-center can be addressed individually. Preliminary consideration shows that it should be possible to address up to 12 individual 13C nuclear spins. The dephasing time of the nuclear spin states at low temperatures allows realization up to 105 gates.

Abstract: Quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision epsilon is only polynomial in log 1/epsilon. Here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/epsilon, a result which matches the lower bound from counting volume up to constant factor.

Abstract: In this paper, we analyze fault tolerance properties of the Majority Gate, as the main logic gate for implementation with Quantum dots Cellular Automata (QCA), in terms of fabrication defect. Our results demonstrate the poor fault tolerance properties of the conventional design of Majority Gate and thus the difficulty in its practical application. We propose a new approach to the design of QCA-based Majority Gate by considering two-dimensional arrays of QCA cells rather than a single cell for the design of such a gate. We analyze fault tolerance properties of such Block Majority Gates in terms of inputs misalignment and irregularity and defect (missing cells) in assembly of the array. We present simulation results based on semiconductor implementation of QCA with an intermediate dimensional dot of about 5 nm in size as opposed to magnetic dots of greater than 100 nm or molecular dots of 2–5A. Our results clearly demonstrate the superior fault tolerance properties of the Block Majority Gate and its greater potential for a practical realization. We also show the possibility of designing fault tolerant QCA circuits by using Block Majority Gates.

Abstract: We demonstrate storage and manipulation of one qubit encoded into a decoherence-free subspace (DFS) of two nuclear spins using liquid state nuclear magnetic resonance (NMR) techniques. The DFS is spanned by states that are unaffected by arbitrary collective phase noise. Encoding and decoding procedures reversibly map an arbitrary qubit state from a single data spin to the DFS and back. The implementation demonstrates the robustness of the DFS memory against engineered dephasing with arbitrary strength as well as a substantial increase in the amount of quantum information retained, relative to an un-encoded qubit, under both engineered and natural noise processes. In addition, a universal set of logical manipulations over the encoded qubit is also realized. Although intrinsic limitations prevent maintaining full noise tolerance during quantum gates, we show how the use of dynamical control methods at the encoded level can ensure that computation is protected with finite distance. We demonstrate noise-tolerant control over a DFS qubit in the presence of engineered phase noise significantly stronger than observed from natural noise sources.

Abstract: In this tutorial we review physical implementation of quantum computing using a system of cold trapped ions. We discuss systematically all the aspects for making the implementation possible. Firstly, we go through the loading and confining of atomic ions in the linear Paul trap, then we describe the collective vibrational motion of trapped ions. Further, we discuss interactions of the ions with a laser beam. We treat the interactions in the travelling-wave and standing-wave configuration for dipole and quadrupole transitions. We review different types of laser cooling techniques associated with trapped ions. We address Doppler cooling, sideband cooling in and beyond the Lamb-Dicke limit, sympathetic cooling and laser cooling using electromagnetically induced transparency. After that we discuss the problem of state detection using the electron shelving method. Then quantum gates are described. We introduce single-qubit rotations, two-qubit controlled-NOT and multi-qubit controlled-NOT gates. We also comment on more advanced multi-qubit logic gates. We describe how quantum logic networks may be used for the synthesis of arbitrary pure quantum states. Finally, we discuss the speed of quantum gates and we also give some numerical estimations for them. A discussion of dynamics on off-resonant transitions associated with a qualitative estimation of the weak coupling regime and of the Lamb-Dicke regime is included in Appendix.

Abstract: We extend the results of Wu & Lidar, eprint quant-ph/0103039: By mapping qubits to parafermions we study the quantum computational power of a generic class of Hamiltonians. We discuss the immunity to decoherence of encoded qubits, symmetry and quantum statistical properties of qubits treated as identical parafermions.

Abstract: We consider stability of a general quantum algorithm with respect to a fixed but unknown residual interaction between qubits, and show a surprising fact, namely that the average fidelity of quantum computation increases by decreasing average time correlation function of the perturbing operator in sequences of consecutive quantum gates. Our thinking is applied to the quantum Fourier transformation where an alternative 'less regular' quantum algorithm is devised which is qualitatively more robust against static random residual n-qubit interaction.

Abstract: We propose a scheme to perform basic gates of quantum computing and prepare entangled states in a system with cold trapped ions located in a single mode optical cavity. General quantum computing can be made with both motional state of the trapped ion and cavity state being qubits. We can also generate different kinds of entangled states in such a system without state reduction, and can transfer quantum states from the ion in one trap to the ion in another trap. Experimental requirement for achieving our scheme is discussed.

Abstract: We investigate the feasibility of an adiabatic logic scheme for cellular automaton systems recently proposed by Toth and Lent [J. Appl. Phys. 85, 2977 (1999)], based on structured arrays of six-dot cells and on the use of a four-phase trapezoidal clock. With respect to the original quantum cellular automaton (QCA) concept, focused on ground state computation, a clocked scheme would have the advantage of being immune to the presence of metastable states and would allow pipelined operation, at the expense of additional complexity, due to the clock distribution circuitry. Based on realistic cell geometries and material systems, we have evaluated the obtainable switching times of QCA cells, and have determined the region in the parameter space that allows operation at a reasonable clock speed.

Abstract: The commonly used circuit model of quantum computing leaves out the problems of imprecision in the initial state preparation, particle statistics (indistinguishability of particles belonging to the same quantum state), and error correction (current techniques cannot correct all small errors). The initial state in the circuit model computation is obtained by applying potentially imprecise Hadamard gate operations whereas useful quantum computation requires a state with no uncertainty. We review some limitations of the circuit model and speculate on the question if a hierarchy of quantum-type computing models exists.

Abstract: The discovery of Shor's prime factorization and Grover's fast database search algorithm have made quantum computing the most rapidly expanding research field recently. Nanotechnology, in particular silicon-based nanoscale devices, have been proposed as one of the candidates that can be used to implement a quantum computer. In this paper, we have derived a systematic procedure to realize any general m-to-n bit combinational Boolean logic using elementary quantum gates. The quantum circuit layout under the locality constraint is then formulated, together with the gate count evaluation function, to reduce the total number of quantum gates required to implement the circuit.

Abstract: The author discusses some of the ideas of quantum computing and then digs into the notation and terminology. He uses physics language and symbology because it is the language of quantum computing. Bracket notation, qubits, multibit registers, entanglement, quantum gates and quantum full adders are discussed.

Abstract: If a quantum mechanical state including a plurality of two-level systems (x 1 , x 2 , . . . , x 2p+1 ) is expressed by a superposition of orthonormal bases in which each two-level system assumes a basic or an excited state, a quantum gate network is used to perform an operation including a combination of a selective rotation operation and an inversion about average operation D in order to configure a desired partly-entangled quantum mechanical state in which the coefficients of the respective bases are all real numbers.

Abstract: We present a finite set of projective measurements that, together with quantum memory and preparation of the |0> state, suffice for universal quantum computation. This extends work of Nielsen [quant-ph/0108020], who proposed a scheme in which an arbitrary unitary operation on n qubits can be simulated using only projective measurements on at most 2n qubits. All measurements in our set involve two qubits, except two measurements which involve three qubits. Thus we improve by one the upper bound, implied by Nielsen's results, on the maximum number of qubits needed to participate in any single measurement to achieve universal quantum computation. Each of our measurements is two-valued, and each can be expressed mathematically as a Boolean combination of single-qubit measurements.

Abstract: We describe a method for achieving arbitrary 1-qubit gates and controlled-NOT gates within the context of the Single Cooper Pair Box (SCB) approach to quantum computing. Such gates are sufficient to support universal quantum computation. Quantum gate operations are achieved by applying sequences of voltages and magnetic fluxes to single qubits or pairs of qubits. Neither the temporal duration, nor the starting time, of a gate operation is used as a control parameter. As a result, the quantum gates have a constant and known duration, and depend upon standard control parameter sequences regardless of when the gate operation begins. This simplifies the integration of quantum gates into parallel, synchronous, quantum circuits. In addition, we demonstrate the ability to fabricate such gates, and large-scale quantum circuits, using current e-beam lithography technology. These features make the SCB-based scheme a credible contender for practical quantum computer hardware.

Abstract: The Dirac-matrix representation of a general two-qubit system is shown to exhibit quite interesting features. The relativistic symmetries of time reversal T, charge conjugation C, parity P, and their products are reinterpreted here by examining their action on the Bell states. It is shown that only C does not mix the Bell states whereas all others do. The various logic gates of quantum information theory are also expressed in terms of the Dirac matrices. For example, the NOT gate is related to the product of T and P. A two-qubit density matrix is found to be entangled if it is invariant under C.

Abstract: Recently, it is proposed to do quantum computation through the Berry's phase(adiabatic cyclic geometric phase) shift with NMR (Jones et al, Nature, 403, 869(2000)). This geometric quantum gate is hopefully to be fault tolerant to certain types of errors because of the geometric property of the Berry phase. Here we give a scheme to realize the NMR C-NOT gate through Aharonov-Anandan's phase(non-adiabatic cyclic phase) shift on the dynamic phase free evolution loop. In our scheme, the gate is run non-adiabatically, thus it is less affected by the decoherence. And, in the scheme we have chosen the the zero dynamic phase time evolution loop in obtaining the gepmetric phase shift, we need not take any extra operation to cancel the dynamic phase.

Abstract: It was shown by M.A. Nielsen and I.L. Chuang 1997, that it is impossible to build strictly universal programmable quantum gate array, that could perform any unitary operation precisely and it was suggested to use probabilistic gate arrays instead. In present work is shown, that if to use more physical and weak condition of universality (suggested already in earliest work by D.Deutsch 1985) and to talk about simulation with arbitrary, but finite precision, then it is possible to build universal programmable gate array. But now the same no-go theorem by Nielsen and Chuang will have new interesting consequence --- controlling programs for the gate arrays can be considered as pure classical. More detailed design of such deterministic quantum gate arrays universal ``in approximate sense'' is considered in the paper.

Abstract: In this note we make a short review of constructions of n-repeated controlled unitary gates in quantum logic gates.

Abstract: In this chapter we build on the basic model developed in Chapter 1 by extending the notation to handle quantum systems with multiple qubits. With that terminology in place, we can illustrate the ideas which are basic to quantum algorithms and can confirm theoretically that the unitary transformations we need can be implemented as a sequence of operations involving only one or two qubits. Quantum algorithms can be constructed using a small number of quantum gates, and we discuss those gates next. We then use quantum gates to construct an addition subroutine and complete the chapter with a teleportation subroutine, which is the first illustration of the potential importance of entanglement for communication.

Abstract: To build a general-purpose quantum computer, it is crucial for the quantum devices to implement classical boolean logic. A straightforward realization of quantum boolean logic is to use auxiliary qubits as intermediate storage. This inefficient implementation causes a large number of auxiliary qubits to be used. In this paper, we have derived a systematic way of realizing any general m-to-n bit combinational boolean logic using elementary quantum gates. Our approach transforms the m-to-n bit classical mapping into a t-bit unitary quantum operation with minimum number of auxiliary qubits, then a variation of Toffoli gate is used as the basic building block to construct the unitary operation. Finally, each of these building blocks can be decomposed into one-bit rotation and two-bit control-U gates. The efficiency of the network is taken into consideration by formulating it as a constrained set partitioning problem.

Abstract: Summary form only given. The use of optical Raman interactions to excite spin coherences in solids has numerous potential applications, ranging from low-power nonlinear optics to high-temperature spectral hole burning memories to solid-state quantum computing. The interest in Raman excitation lies in the fact that the spin coherences can be efficiently excited and manipulated using optical laser fields yet are weakly coupled to the environment and hence have the long coherence lifetimes needed for optical memories and quantum computing. The interest in nitrogen-vacancy (N-V) color centers in diamond is its large optical oscillator strength. For memory, a large oscillator strength is important for high temperature operation. For quantum computing, the high optical transition rate enables the higher gate speed and the larger number of quantum logic operations that can be performed within the spin coherence lifetime.

Abstract: We present an alternative scheme to exactly implement one-qubit and two-qubit quantum gates with a single trapped cold ion driven by a travelling laser field. The internal degree of freedom of the ion acts as the target qubit and the control qubit is encoded by two Fock states of the external vibration of the ion. The conditions to realize these operations, including the duration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approximation is required. The multiquantum transition between the internal and external degrees of freedom of the ion is considered.

Abstract: We present a proposal for implementing universal quantum gates by means of a selective interaction in trapped ions and Cavity Quantum Electrodynamics (CQED) Selectivity arises from the possibility of tuning to resonance transitions inside a chosen Hilbert subspace, while leaving other transitions dispersive In particular, a quantum phase (QPG) gate might be realized, in trapped ions and CQED, making use of a single pulse of the associated selective interaction

Abstract: We consider a quantum circuit in which shift and rotation operations on qubits are performed by swap gates and controlled swap gates These operations can be useful for quantum computers performing elementary arithmetic operations such as multiplication and a bit-wise comparison of qubits