Abstract: The promise of tremendous computational power, coupled with the development of robust error-correcting schemes, has fuelled extensive efforts to build a quantum computer. The requirements for realizing such a device are confounding: scalable quantum bits (two-level quantum systems, or qubits) that can be well isolated from the environment, but also initialized, measured and made to undergo controllable interactions to implement a universal set of quantum logic gates. The usual set consists of single qubit rotations and a controlled-NOT (CNOT) gate, which flips the state of a target qubit conditional on the control qubit being in the state 1. Here we report an unambiguous experimental demonstration and comprehensive characterization of quantum CNOT operation in an optical system. We produce all four entangled Bell states as a function of only the input qubits' logical values, for a single operating condition of the gate. The gate is probabilistic (the qubits are destroyed upon failure), but with the addition of linear optical quantum non-demolition measurements, it is equivalent to the CNOT gate required for scalable all-optical quantum computation.

Abstract: We report coherent optical control of a biexciton (two electron-hole pairs), confined in a single quantum dot, that shows coherent oscillations similar to the excited-state Rabi flopping in an isolated atom The pulse control of the biexciton dynamics, combined with previously demonstrated control of the single-exciton Rabi rotation, serves as the physical basis for a two-bit conditional quantum logic gate The truth table of the gate shows the features of an all-optical quantum gate with interacting yet distinguishable excitons as qubits Evaluation of the fidelity yields a value of 07 for the gate operation Such experimental capability is essential to a scheme for scalable quantum computation by means of the optical control of spin qubits in dots

Abstract: Entanglement lies at the heart of quantum mechanics, and in recent years has been identified as an essential resource for quantum information processing and computation The experimentally challenging production of highly entangled multi-particle states is therefore important for investigating both fundamental physics and practical applications Here we report the creation of highly entangled states of neutral atoms trapped in the periodic potential of an optical lattice Controlled collisions between individual neighbouring atoms are used to realize an array of quantum gates, with massively parallel operation We observe a coherent entangling-disentangling evolution in the many-body system, depending on the phase shift acquired during the collision between neighbouring atoms Such dynamics are indicative of highly entangled many-body states; moreover, these are formed in a single operational step, independent of the size of the system

Abstract: Quantum computers have the potential to perform certain computational tasks more efficiently than their classical counterparts. The Cirac–Zoller proposal1 for a scalable quantum computer is based on a string of trapped ions whose electronic states represent the quantum bits of information (or qubits). In this scheme, quantum logical gates involving any subset of ions are realized by coupling the ions through their collective quantized motion. The main experimental step towards realizing the scheme is to implement the controlled-NOT (CNOT) gate operation between two individual ions. The CNOT quantum logical gate corresponds to the XOR gate operation of classical logic that flips the state of a target bit conditioned on the state of a control bit. Here we implement a CNOT quantum gate according to the Cirac–Zoller proposal1. In our experiment, two 40Ca+ ions are held in a linear Paul trap and are individually addressed using focused laser beams2; the qubits3 are represented by superpositions of two long-lived electronic states. Our work relies on recently developed precise control of atomic phases4 and the application of composite pulse sequences adapted from nuclear magnetic resonance techniques5,6.

Abstract: A practical quantum computer, if built, would consist of a set of coupled two-level quantum systems (qubits). Among the variety of qubits implemented, solid-state qubits are of particular interest because of their potential suitability for integrated devices. A variety of qubits based on Josephson junctions have been implemented; these exploit the coherence of Cooper-pair tunnelling in the superconducting state. Despite apparent progress in the implementation of individual solid-state qubits, there have been no experimental reports of multiple qubit gates--a basic requirement for building a real quantum computer. Here we demonstrate a Josephson circuit consisting of two coupled charge qubits. Using a pulse technique, we coherently mix quantum states and observe quantum oscillations, the spectrum of which reflects interaction between the qubits. Our results demonstrate the feasibility of coupling multiple solid-state qubits, and indicate the existence of entangled two-qubit states.

Abstract: We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n + 3 qubits and 0(n3lg(n)) elementary quantum gates in a depth of 0(n3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored.

Abstract: We study the low energy states of finite spin chains with isotropic (Heisenberg) and anisotropic (XY and Ising-like) antiferromagnetic exchange interaction with uniform and nonuniform coupling constants. We show that for an odd number of sites a spin cluster qubit can be defined in terms of the ground state doublet. This qubit is remarkably insensitive to the placement and coupling anisotropy of spins within the cluster. One- and two-qubit quantum gates can be generated by magnetic fields and intercluster exchange, and leakage during quantum gate operation is small. Spin cluster qubits inherit the long decoherence times and short gate operation times of single spins. Control of single spins is hence not necessary for the realization of universal quantum gates.

Abstract: We study the physical implementation of a qutrit quantum computer in the context of trapped ions Qutrits are defined in terms of electronic levels of trapped ions We concentrate our attention on a universal two-qutrit gate, which corresponds to a controlled-NOT gate between qutrits Using this gate and a general gate of an individual qutrit, any gate can be decomposed into a sequence of these gates In particular, we show how this works for performing the quantum Fourier transform for n qutrits

Abstract: Recently Shi proved that Toffoli and Hadamard are universal for quantum computation This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a 'classical' set of gates quantum universal In this note we give a few lines proof of this fact relying on Kitaev's universal set of gates, and discuss the meaning of the result

Abstract: We define several quantitative measures of the robustness of a quantum gate against noise. Exact analytic expressions for the robustness against depolarizing noise are obtained for all bipartite unitary quantum gates, and it is found that the controlled-NOT gate is the most robust two-qubit quantum gate, in the sense that it is the quantum gate which can tolerate the most depolarizing noise and still generate entanglement. Our results enable us to place several analytic upper bounds on the value of the threshold for quantum computation, with the best bound in the most pessimistic error model being p(th)less than or equal to0.5.

Abstract: We report the realization of an elementary quantum processor based on a linear crystal of trapped ions. Each ion serves as a quantum bit (qubit) to store the quantum information in long lived electronic states. We present the realization of single-qubit and of universal two-qubit logic gates. The two-qubit operation relies on the coupling of the ions through their collective quantized motion. A detailed description of the setup and the methods is included.

Abstract: We show that a wide range of spin clusters with antiferromagnetic intracluster exchange interaction allows one to define a qubit. For these spin cluster qubits, initialization, quantum gate operation, and readout are possible using the same techniques as for single spins. Quantum gate operation for the spin cluster qubit does not require control over the intracluster exchange interaction. Electric and magnetic fields necessary to effect quantum gates need only be controlled on the length scale of the spin cluster rather than the scale for a single spin. Here, we calculate the energy gap separating the logical qubit states from the next excited state and the matrix elements which determine quantum gate operation times. We discuss spin cluster qubits formed by one- and two-dimensional arrays of s=1/2 spins as well as clusters formed by spins s<1/2. We illustrate the advantages of spin cluster qubits for various suggested implementations of spin qubits and analyze the scaling of decoherence time with spin cluster size.

Abstract: Advances in quantum devices have brought scalable quantum computation closer to reality. We focus on the system-level issues of how quantum devices can be brought together to form a scalable architecture. In particular, we examine promising silicon-based proposals. We discover that communication of quantum data is a critical resource in such proposals. We find that traditional techniques using quantum SWAP gates are exponentially expensive as distances increase and propose quantum teleportation as a means to communicate data over longer distances on a chip. Furthermore, we find that realistic quantum error-correction circuits use a recursive structure that benefits from using teleportation for long-distance communication. We identify a set of important architectural building blocks necessary for constructing scalable communication and computation. Finally, we explore an actual layout scheme for recursive error correction, and demonstrate the exponential growth in communication costs with levels of recursion, and that teleportation limits those costs.

Abstract: The paper discusses the evolutionary computation approach to the problem of optimal synthesis of Quantum and Reversible Logic circuits. Our approach uses standard Genetic Algorithm (GA) and its relative power as compared to previous approaches comes from the encoding and the formulation of the cost and fitness functions for quantum circuits synthesis. We analyze new operators and their role in synthesis and optimization processes. Cost and fitness functions for Reversible Circuit synthesis are introduced as well as local optimizing transformations. It is also shown that our approach can be used alternatively for synthesis of either reversible or quantum circuits without a major change in the algorithm. Results are illustrated on synthesized Margolus, Toffoli, Fredkin and other gates and Entanglement Circuits. This is for the first time that several variants of these gates have been automatically synthesized from quantum primitives.

Abstract: Irreversible computation necessarily results in energy dissipation due to information loss. While small in comparison to the power consumption of today's VLSI circuits, if current trends continue this will be a critical issue in the near future. Reversible circuits offer an alternative that, in principle, allows computation with arbitrarily small energy dissipation. Furthermore, reversible circuits are essential components of quantum logic. We consider the problem of testing these circuits, and in particular generating efficient test sets. The reversibility property significantly simplifies the problem, which is generally hard for the irreversible case. We discuss conditions for a test set to be complete, give a number of practical constructions, and consider test sets for worst-case circuits. In addition, we formulate the problem of finding minimal test sets into an integer linear program (ILP) with binary variables. While this ILP method is infeasible for large circuits, we show that combining it with a circuit decomposition approach yields a practical alternative.

Abstract: We report the realization of an elementary quantum processor based on a linear crystal of trapped ions. Each ion serves as a quantum bit (qubit) to store the quantum information in long lived electronic states. We present the realization of single-qubit and of universal two-qubit logic gates. The qwo-qubit operation relies on the coupling of the ions through their collective quantized motion. A detailed description of the setup and the methods is included.

Abstract: We present a solid-state implementation of an all-optical spin-based quantum computer. Our proposal for a quantum-computing device is based on the spin degrees of freedom of electrons confined in semiconductor quantum dots, thus benefitting from relatively long coherence times. Combining Pauli blocking effects with properly tailored ultrafast laser pulses, we obtain sub-picosecond spin-dependent switching of the Coulomb interaction, which is the essence of our gating operations. This allows us to realize fast quantum gates which do not translate into fast decoherence times and pave the way for an all-optical spin-based quantum computer.

Abstract: Quantum gaps exist between an origin and a destination that heretofore have prevented reliably utilizing the advantages of quantum computing. To predict the outcome of instructions with precision, the input data, preferably a qubit, is collapsed to a point value (100) within the quantum gap based on a software instruction (102). After collapse the input data is restructured at the destination (104), wherein dynamics of restructuring are governed by a plurality of gap factors as follows: computational self-awareness; computational decision logic; computational processing logic; computational and network protocol and logic exchange; computational and network components, logic and processes; provides the basis for excitability of the Gap junction and its ability to transmit electronic and optical impulses, integrates them properly, and depends on feedback loop logic; computational and network component and system interoperability; and embodiment substrate and network computational physical topology.

Abstract: We introduce quantum hybrid gates that act on qudits of different dimensions. In particular, we develop two representative two-qudit hybrid gates (SUM and SWAP) and many-qudit hybrid Toffoli and Fredkin gates. We apply the hybrid SUM gate to generating entanglement, and find that operator entanglement of the SUM gate is equal to the entanglement generated by it for certain initial states. We also show that the hybrid SUM gate acts as an automorphism on the Pauli group for two qudits of different dimensions under certain conditions. Finally, we describe a physical realization of these hybrid gates for spin systems.

Abstract: Reversible logic has applications in quantum computing, low power CMOS, nanotechnology, optical computing, and DNA computing. The most common reversible gates are the Toffoli gate and the Fredkin gate. Our synthesis algorithm first finds a cascade of Toffoli and Fredkin gates with no backtracking and minimal look-ahead. Next we apply transformations that reduce the size of the circuit. Transformations are accomplished via template matching. The basis for a template is a network with m gates that realizes the identity function. If a sequence in the network to be synthesized matches more than half of a template, then a transformation that reduces the gate count can be applied. In this paper we show that Toffoli and Fredkin gates behave in a similar manner. Therefore, some gates in the templates may not need to be specified-they can match a Toffoli or a Fredkin gate. We formalize this by introducing the box gate. All templates with less than six gates are enumerated and classified. We synthesize all three input, three output reversible functions and compare our results to those obtained previously.

Abstract: Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a new data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, many of these matrices and vectors can be represented in a form that grows polynomially. Using QuIDDs, we implemented a general-purpose quantum computing simulator in C++ called QuIDDPro and tested it on Grover's algorithm. Our QuIDD technique asymptotically outperforms other known simulation techniques.

Abstract: We demonstrate that the unbounded fan-out gate is very powerful. Constant-depth polynomial-size quantum circuits with bounded fan-in and unbounded fan-out over a fixed basis (denoted by QNCf0) can approximate with polynomially small error the following gates: parity, mod[q], And, Or, majority, threshold[t], exact[q], and counting. Classically, we need logarithmic depth even if we can use unbounded fan-in gates. If we allow arbitrary one-qubit gates instead of a fixed basis, then these circuits can also be made exact in log-star depth. Sorting, arithmetical operations, phase estimation, and the quantum Fourier transform can also be approximated in constant depth.

Abstract: We show that quantum feedback control can be used as a quantum error correction process for errors induced by weak continuous measurement. In particular, when the error model is restricted to one, perfectly measured, error channel per physical qubit, quantum feedback can act to perfectly protect a stabilizer codespace. Using the stabilizer formalism we derive an explicit scheme, involving feedback and an additional constant Hamiltonian, to protect art (n-1)-qubit logical state encoded in n physical qubits. This works for both Poisson (jump) and white-noise (diffusion) measurement processes. As an example, we show that detected-spontaneous emission error correction with a driving Hamiltonian can greatly reduce the amount of redundancy required to protect a state from that which has been previously postulated.

Abstract: The notion of quantum instruments is formalized as statistical equivalence classes of all the possible quantum measurements and mathematically characterized as normalized completely positive map valued measures under naturally acceptable axioms. Recently, universally valid uncertainty relations have been established to set a precision limit for any instruments given a disturbance constraint in a form more general than the one originally proposed by Heisenberg. One of them leads to a quantitative generalization of the Wigner–Araki–Yanase theorem on the precision limit of measurements under conservation laws. Applying this, a rigorous lower bound is obtained for the gate error probability of physical implementations of Hadamard gates on a standard qubit of a spin 1/2 system by interactions with control fields or ancilla systems obeying the angular momentum conservation law.

Abstract: The problems related to the management of large quantum registers could be handled in the context of distributed quantum computation: unitary non-local transformations among spatially separated local processors are realized performing local unitary transformations and exchanging classical communication. In this paper, a scheme is proposed for the implementation of universal non-local quantum gates such as a controlled NOT (CNOT) and a controlled quantum phase gate (CQPG). The system chosen for their physical implementation is a cavity-quantum-electrodynamics (CQED) system formed by two spatially separated microwave cavities and two trapped Rydberg atoms. The procedures to follow for the realization of each step necessary to perform a specific non-local operation are described.

Abstract: In this paper initial experiments towards constructing simple quantum gates in a solid state material are presented. Instead of using specially tailored materials, the aim is to select a subset of randomly distributed ions in the material, which have the interaction necessary to control each other and therefore can be used to do quantum logic operations. The experimental results demonstrate that part of an inhomogeneously broadened absorption line can be selected as a qubit and that a subset of ions in the material can control the resonance frequency of other ions. This opens the way for the construction of quantum gates in rare-earth-ion doped crystals.

Abstract: A quantum circuit is generalized to a nonunitary one whose constituents are nonunitary gates operated by quantum measurement. It is shown that a specific type of one-qubit nonunitary gates, the controlled-NOT gate, as well as all one-qubit unitary gates constitute a universal set of gates for the nonunitary quantum circuit, without the necessity of introducing ancilla qubits. A reversing measurement scheme is used to improve the probability of successful nonunitary gate operation. A quantum NAND gate and Abrams-Lloyd's nonlinear gate are analyzed as examples. Our nonunitary circuit can be used to reduce the qubit overhead needed to ensure fault-tolerant quantum computation.

Abstract: Reversible logic has attracted significant attention in recent years. It has applications in low power CMOS, quantum computing, nanotechnology, and optical computing. Traditional gates such as AND, OR, and EXOR are not reversible. In fact NOT is the only reversible gate from the traditional set of gates. Several reversible gates have been proposed. Among them are the controlled NOT (also known as the Feynman gate), the Toffoli gate, and the Fredkin gate. An n-input Toffoli gate has n - 1 control lines which pass through the gate unaltered and a target line on which the value is inverted if all the control lines have value '1'. An n-input Fredkin gate has n - 2 control lines which pass through the gate unaltered and two target lines on which the values are swapped if all the control lines have value '1'. A NOT gate is the special case of a Toffoli gate with no control inputs. Likewise, a SWAP gate is the special case of a Fredkin gate with no control inputs. In this paper, we review a transformation-based synthesis procedure targeted to Toffoli gates and show how it can be extended to allow Fredkin gates. This extension results in circuits with fewer gates. The synthesis of reversible logic differs significantly from traditional irreversible logic synthesis approaches. Fan-outs and loops are not permitted due to the target technology. Outputs from one gate are used as inputs to the next gate. This results in a high degree of interdependence among gates. Our algorithm first finds an initial circuit with no backtracking and minimal look-ahead. We exploit reversibility directly in our synthesis approach. This method always finds a solution. Next we apply a set of template transforms that reduce the size of the circuit. We synthesize all three input, three output reversible functions and compare our results to those obtained previously.

Abstract: The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2 n ×2 n unitary matrices into efficient circuits of ( 1)controlled single-qubit and (n 1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of selfinverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an enumerative algorithm for minimal (n 1)controlled-NOT circuit construction, which we call the Palindromic Optimization Algorithm. Our work dramatically reduces the number of gates generated by the conventional two-level decomposition method for constructing quantum circuits of (n 1)-controlled single-qubit and (n 1)-controlled-NOT gates.