Showing papers on "Quantum error correction published in 2008"

Quantum Discord and the Power of One Qubit

Abstract: We use quantum discord to characterize the correlations present in the model called deterministic quantum computation with one quantum bit (DQC1), introduced by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)]. The model involves a collection of qubits in the completely mixed state coupled to a single control qubit that has nonzero purity. The initial state, operations, and measurements in the model all point to a natural bipartite split between the control qubit and the mixed ones. Although there is no entanglement between these two parts, we show that the quantum discord across this split is nonzero for typical instances of the DQC1 ciruit. Nonzero values of discord indicate the presence of nonclassical correlations. We propose quantum discord as figure of merit for characterizing the resources present in this computational model.

Quantum random access memory.

Abstract: A random access memory (RAM) uses $n$ bits to randomly address $N={2}^{n}$ distinct memory cells. A quantum random access memory (QRAM) uses $n$ qubits to address any quantum superposition of $N$ memory cells. We present an architecture that exponentially reduces the requirements for a memory call: $O(\mathrm{log} N)$ switches need be thrown instead of the $N$ used in conventional (classical or quantum) RAM designs. This yields a more robust QRAM algorithm, as it in general requires entanglement among exponentially less gates, and leads to an exponential decrease in the power needed for addressing. A quantum optical implementation is presented.

Randomized Benchmarking of Quantum Gates

Abstract: A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized $\ensuremath{\pi}/2$ pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.

Amplification and squeezing of quantum noise with a tunable Josephson metamaterial

Abstract: An array of 488 Josephson junctions that amplifies and squeezes noise beyond conventional quantum limits should prove useful in the study and development of superconducting qubits and other quantum devices.

Using measurement-induced disturbance to characterize correlations as classical or quantum

Abstract: In contrast to the seminal entanglement-separability paradigm widely used in quantum information theory, we introduce a quantum-classical dichotomy in order to classify and quantify statistical correlations in bipartite states. This is based on the idea that while in the classical description of nature measurements can be carried out without disturbance, in the quantum description, generic measurements often disturb the system and the disturbance can be exploited to quantify the quantumness of correlations therein. It turns out that certain separable states still possess correlations of a quantum nature and indicates that quantum correlations are more general than entanglement. The results are illustrated in the Werner states and the isotropic states, and are applied to quantify the quantum advantage of the model of quantum computation proposed by Knill and Laflamme [Phys. Rev. Lett. 81, 5672 (1998)].

Colloquium: Quantum annealing and analog quantum computation

Abstract: The recent success in quantum annealing, i.e., optimization of the cost or energy functions of complex systems utilizing quantum fluctuations is reviewed here. The concept is introduced in successive steps through studying the mapping of such computationally hard problems to classical spin-glass problems, quantum spin-glass problems arising with the introduction of quantum fluctuations, and the annealing behavior of the systems as these fluctuations are reduced slowly to zero. This provides a general framework for realizing analog quantum computation.

Towards fault-tolerant quantum computing with trapped ions

Abstract: Like their classical counterparts, quantum computers can, in theory, cope with imperfections—provided that these are small enough. The regime of fault-tolerant quantum computing has now been reached for a system based on trapped ions, in which a gate operation for entangling qubits has been implemented with a fidelity exceeding 99%. Today, ion traps are among the most promising physical systems for constructing a quantum device harnessing the computing power inherent in the laws of quantum physics1,2. For the implementation of arbitrary operations, a quantum computer requires a universal set of quantum logic gates. As in classical models of computation, quantum error correction techniques3,4 enable rectification of small imperfections in gate operations, thus enabling perfect computation in the presence of noise. For fault-tolerant computation5, it is believed that error thresholds ranging between 10−4 and 10−2 will be required—depending on the noise model and the computational overhead for realizing the quantum gates6,7,8—but so far all experimental implementations have fallen short of these requirements. Here, we report on a Molmer–Sorensen-type gate operation9,10 entangling ions with a fidelity of 99.3(1)%. The gate is carried out on a pair of qubits encoded in two trapped calcium ions using an amplitude-modulated laser beam interacting with both ions at the same time. A robust gate operation, mapping separable states onto maximally entangled states is achieved by adiabatically switching the laser–ion coupling on and off. We analyse the performance of a single gate and concatenations of up to 21 gate operations.

Realization of quantum walks with negligible decoherence in waveguide lattices.

Abstract: Quantum random walks are the quantum counterpart of classical random walks, and were recently studied in the context of quantum computation. Physical implementations of quantum walks have only been made in very small scale systems severely limited by decoherence. Here we show that the propagation of photons in waveguide lattices, which have been studied extensively in recent years, are essentially an implementation of quantum walks. Since waveguide lattices are easily constructed at large scales and display negligible decoherence, they can serve as an ideal and versatile experimental playground for the study of quantum walks and quantum algorithms. We experimentally observe quantum walks in large systems (similar to 100 sites) and confirm quantum walks effects which were studied theoretically, including ballistic propagation, disorder, and boundary related effects.

Experimental demonstration of a BDCZ quantum repeater node.

Abstract: At distances beyond about 100 km, quantum communication fails due to photon losses in the transmission channel. To overcome this problem, Briegel, Dur, Cirac and Zoller (BDCZ) introduced the concept of quantum repeaters, combining entanglement swapping and quantum memory to efficiently extend the achievable distances. Their implementation has proved challenging due to the difficulty of integrating a quantum memory. Zhen-ShengYuan et al. realize a building block of the BDCZ quantum repeater, demonstrating entanglement swapping with storage and retrieval of light from atomic quantum memories. Quantum communication is a method that offers efficient and secure ways for the exchange of information in a network. Large-scale quantum communication1,2,3,4 (of the order of 100 km) has been achieved; however, serious problems occur beyond this distance scale, mainly due to inevitable photon loss in the transmission channel. Quantum communication eventually fails5 when the probability of a dark count in the photon detectors becomes comparable to the probability that a photon is correctly detected. To overcome this problem, Briegel, Dur, Cirac and Zoller (BDCZ) introduced the concept of quantum repeaters6, combining entanglement swapping7 and quantum memory to efficiently extend the achievable distances. Although entanglement swapping has been experimentally demonstrated8, the implementation of BDCZ quantum repeaters has proved challenging owing to the difficulty of integrating a quantum memory. Here we realize entanglement swapping with storage and retrieval of light, a building block of the BDCZ quantum repeater. We follow a scheme9,10 that incorporates the strategy of BDCZ with atomic quantum memories11. Two atomic ensembles, each originally entangled with a single emitted photon, are projected into an entangled state by performing a joint Bell state measurement on the two single photons after they have passed through a 300-m fibre-based communication channel. The entanglement is stored in the atomic ensembles and later verified by converting the atomic excitations into photons. Our method is intrinsically phase insensitive and establishes the essential element needed to realize quantum repeaters with stationary atomic qubits as quantum memories and flying photonic qubits as quantum messengers.

$H^{\infty}$ Control of Linear Quantum Stochastic Systems

Abstract: The purpose of this paper is to formulate and solve a H infin controller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology. The paper includes results on the class of such systems for which the quantum commutation relations are preserved (such a requirement must be satisfied in a physical quantum system). A quantum version of standard (classical) dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived. This enables a quantum version of the two Riccati solution to the H infin control problem to be presented. This result leads to controllers which may be realized using purely quantum, purely classical or a mixture of quantum and classical elements. This issue of physical realizability of the controller is examined in detail, and necessary and sufficient conditions are given. Our results are constructive in the sense that we provide explicit formulas for the Hamiltonian function and coupling operator corresponding to the controller.

Solid-state quantum memory using the 31 P nuclear spin

Abstract: The transfer of information between the entities that do the processing and memory is crucial — and problematic — for quantum computation. In classical systems the information transfer can include a copying step, where errors can be spotted and corrected, but in quantum systems copying is fundamentally precluded. Morton et al. demonstrate a technology that could solve the problem: the coherent storage and readout of information between electron-spin processing elements and memory elements based on a nuclear spin. The system utilizes phosphorus-31 spin donors in a silicon-28 crystal. The nuclear spin acts as a memory element that can faithfully store the full state of the electron spin for more than a second, then transfer it back to the electron spin with about 90% efficiency. The transfer of information between processing entities and memory is crucial for quantum computation; it is challenging because the process must remain coherent at all times to preserve the quantum nature of the information. This paper demonstrates coherent storage and readout of information between electron-spin processing elements and memory elements based on a nuclear spin. The transfer of information between different physical forms—for example processing entities and memory—is a central theme in communication and computation. This is crucial in quantum computation1, where great effort2 must be taken to protect the integrity of a fragile quantum bit (qubit). However, transfer of quantum information is particularly challenging, as the process must remain coherent at all times to preserve the quantum nature of the information3. Here we demonstrate the coherent transfer of a superposition state in an electron-spin ‘processing’ qubit to a nuclear-spin ‘memory’ qubit, using a combination of microwave and radio-frequency pulses applied to 31P donors in an isotopically pure 28Si crystal4,5. The state is left in the nuclear spin on a timescale that is long compared with the electron decoherence time, and is then coherently transferred back to the electron spin, thus demonstrating the 31P nuclear spin as a solid-state quantum memory. The overall store–readout fidelity is about 90 per cent, with the loss attributed to imperfect rotations, and can be improved through the use of composite pulses6. The coherence lifetime of the quantum memory element at 5.5 K exceeds 1 s.

Fault-Tolerant Quantum Computation with Constant Error Rate

Abstract: This paper shows that quantum computation can be made fault-tolerant against errors and inaccuracies when $\eta$, the probability for an error in a qubit or a gate, is smaller than a constant threshold $\eta_c$. This result improves on Shor's result [Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE, Los Alamitos, CA, 1996, pp. 56-65], which shows how to perform fault-tolerant quantum computation when the error rate $\eta$ decays polylogarithmically with the size of the computation, an assumption which is physically unreasonable. The cost of making the quantum circuit fault-tolerant in our construction is polylogarithmic in time and space. Our result holds for a very general local noise model, which includes probabilistic errors, decoherence, amplitude damping, depolarization, and systematic inaccuracies in the gates. Moreover, we allow exponentially decaying correlations between the errors both in space and in time. Fault-tolerant computation can be performed with any universal set of gates. The result also holds for quantum particles with $p>2$ states, namely, $p$-qudits, and is also generalized to one-dimensional quantum computers with only nearest-neighbor interactions. No measurements, or classical operations, are required during the quantum computation. We estimate the threshold of our construction to be $\eta_c\simeq 10^{-6}$, in the best case. By this we show that local noise is in principle not an obstacle for scalable quantum computation. The main ingredient of our proof is the computation on states encoded by a quantum error correcting code (QECC). To this end we introduce a special class of Calderbank-Shor-Steane (CSS) codes, called polynomial codes (the quantum analogue of Reed-Solomon codes). Their nice algebraic structure allows all of the encoded gates to be transversal. We also provide another version of the proof which uses more general CSS codes, but its encoded gates are slightly less elegant. To achieve fault tolerance, we encode the quantum circuit by another circuit by using one of these QECCs. This step is repeated polyloglog many times, each step slightly improving the effective error rate, to achieve the desired reliability. The resulting circuit exhibits a hierarchical structure, and for the analysis of its robustness we borrow terminology from Khalfin and Tsirelson [Found. Phys., 22 (1992), pp. 879-948] and Gacs [Advances in Computing Research: A Research Annual: Randomness and Computation, JAI Press, Greenwich, CT, 1989]. The paper is to a large extent self-contained. In particular, we provide simpler proofs for many of the known results we use, such as the fact that it suffices to correct for bit-flips and phase-flips, the correctness of CSS codes, and the fact that two-qubit gates are universal, together with their extensions to higher-dimensional particles. We also provide full proofs of the universality of the sets of gates we use (the proof of universality was missing in Shor's paper). This paper thus provides a self-contained and complete proof of universal fault-tolerant quantum computation in the presence of local noise.

Fault-Tolerant Quantum Computation.

Abstract: Fault tolerance is the study of reliable computation using unreliable components. With a given noise model, can one still reliably compute? For example, one can run many copies of a classical calculation in parallel, periodically using majority gates to catch and correct faults. Von Neumann showed in 1956 that if each gate fails independently with probability p, flipping its output bit 0 $ 1, then such a fault tolerance scheme still allows for arbitrarily reliable computation provided that p is below some constant threshold (whose value depends on the model details) [10]. In a quantum computer, the basic gates are much more vulnerable to noise than classical transistors – after all, depending on the implementation, they are manipulating single electron spins, photon polarizations, and similarly fragile subatomic particles. It might not be possible to engineer systems with noise rates less than 10 , or perhaps 10 , per gate. Additionally, the phenomenon of entanglement makes quantum systems inherently fragile. For example, in Schrödinger’s cat state – an equal superposition between a living cat and a dead cat, often idealized as 1= p 2.j0ni C j1ni/ – an interaction with just one quantum bit (“qubit”) can collapse, or decohere, the entire system. Fault tolerance techniques will therefore be essential for achieving the considerable potential of quantum computers. Practical fault tolerance techniques will need to control high noise rates and do so with low overhead, since qubits are expensive. Quantum systems are continuous, not discrete, so there are many possible noise models. However, the essential features of quantum noise for fault tolerance results can be captured by a simple discrete model similar to the one Von Neumann used. The main difference is that, in addition to bit-flip X errors which swap 0 and 1, there can also be phase-flip Z errors which swap jCi 1=p2.j0i C j1i/ and j i 1=p2.j0i j1i/ (Fig. 1). A noisy gate is modeled as a perfect gate followed by independent introduction of X, Z, or Y (which is both X and Z) errors with respective probabilities pX; pZ; pY. One popular model is independent depolarizing noise .pX D pZ D pY p=3/; a depolarized qubit is completely randomized. Faulty measurements and preparations of single-qubit states must additionally be modeled, and there can be memory noise on resting qubits. It is often assumed that measurement results can

Quantum communication with zero-capacity channels.

Abstract: Communication over a noisy quantum channel introduces errors in the transmission that must be corrected. A fundamental bound on quantum error correction is the quantum capacity, which quantifies the amount of quantum data that can be protected. We show theoretically that two quantum channels, each with a transmission capacity of zero, can have a nonzero capacity when used together. This unveils a rich structure in the theory of quantum communications, implying that the quantum capacity does not completely specify a channel's ability to transmit quantum information.

Efficient quantum key distribution over a collective noise channel

Abstract: We present two efficient quantum key distribution schemes over two different collective-noise channels. The accepted hypothesis of collective noise is that photons travel inside a time window small compared to the variation of noise. Noiseless subspaces are made up of two Bell states and the spatial degree of freedom is introduced to form two nonorthogonal bases. Although these protocols resort to entangled states for encoding the key bit, the receiver is only required to perform single-particle product measurements and there is no basis mismatch. Moreover, the detection is passive as the receiver does not switch his measurements between two conjugate measurement bases to get the key.

Quantum Memory for Squeezed Light

Abstract: Introduction Memory for quantum states of light is a necessary component of quantum optical computers and is also required for the implementation of quantum repeaters [1] that would dramatically increase the range of quantum communication. There exists a variety of approaches to implementing quantum optical memory, for example off-resonant interaction of light with spin polarized atomic ensembles [2] and controlled reversible inhomogeneous broadening [3]. One of the most well-studied techniques is adiabatic transfer between optical quantum states and long-lived atomic superposition using electromagnetically induced transparency (EIT) [4]. This method, proposed in 2000 by Fleischhauer and Lukin [5], has been experimentally demonstrated in 2001 with classical light pulses [6]. In 2005, storage and retrieval of single photons, prepared using the Duan-Lukin-Cirac-Zoller

A Quantum Algorithm for the Hamiltonian NAND Tree

Abstract: We give a quantum algorithm for the binary NAND tree problem in the Hamil- tonian oracle model. The algorithm uses a continuous time quantum walk with a running time proportional to p N. We also show a lower bound of W( p N) for the NAND tree problem in the Hamiltonian oracle model.

Quantum Circuit Architecture

Abstract: We present a method for optimizing quantum circuits architecture, based on the notion of a quantum comb, which describes a circuit board where one can insert variable subcircuits. Unexplored quantum processing tasks, such as cloning and storing or retrieving of gates, can be optimized, along with setups for tomography and discrimination or estimation of quantum circuits.

Transforming quantum operations: Quantum supermaps

Abstract: We introduce the concept of quantum supermap, describing the most general transformation that maps an input quantum operation into an output quantum operation. Since quantum operations include as special cases quantum states, effects, and measurements, quantum supermaps describe all possible transformations between elementary quantum objects (quantum systems as well as quantum devices). After giving the axiomatic definition of supermap, we prove a realization theorem, which shows that any supermap can be physically implemented as a simple quantum circuit. Applications to quantum programming, cloning, discrimination, estimation, information-disturbance trade-off, and tomography of channels are outlined.

Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation

Abstract: The model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities. We describe an efficient adiabatic simulation of any given quantum circuit. This implies that the adiabatic computation model and the standard circuit-based quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows one to state the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of Hamiltonians.

Interacting Quantum Observables

Abstract: We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum observables form bialgebra-like structures. We also provide an abstract account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus.

Non-Markovian quantum jumps.

Abstract: Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics.

Continuous-variable quantum cryptography using two-way quantum communication

Abstract: A class of quantum-cryptographic protocols is proposed that involves back-and-forth communication between two parties. The approach is shown to provide enhanced security and should tolerate higher levels of noise and loss than conventional ‘one-way’ protocols.

Architectures for a quantum random access memory

Abstract: A random access memory, or RAM, is a device that, when interrogated, returns the content of a memory location in a memory array. A quantum RAM, or qRAM, allows one to access superpositions of memory sites, which may contain either quantum or classical information. RAMs and qRAMs with $n$-bit addresses can access ${2}^{n}$ memory sites. Any design for a RAM or qRAM then requires $O({2}^{n})$ two-bit logic gates. At first sight this requirement might seem to make large scale quantum versions of such devices impractical, due to the difficulty of constructing and operating coherent devices with large numbers of quantum logic gates. Here we analyze two different RAM architectures (the conventional fanout and the ``bucket brigade'') and propose some proof-of-principle implementations, which show that, in principle, only $O(n)$ two-qubit physical interactions need take place during each qRAM call. That is, although a qRAM needs $O({2}^{n})$ quantum logic gates, only $O(n)$ need to be activated during a memory call. The resulting decrease in resources could give rise to the construction of large qRAMs that could operate without the need for extensive quantum error correction.

One-way quantum computing in the optical frequency comb.

Abstract: One-way quantum computing allows any quantum algorithm to be implemented easily using just measurements. The difficult part is creating the universal resource, a cluster state, on which the measurements are made. We propose a scalable method that uses a single, multimode optical parametric oscillator (OPO). The method is very efficient and generates a continuous-variable cluster state, universal for quantum computation, with quantum information encoded in the quadratures of the optical frequency comb of the OPO.

Quantum criticality as a resource for quantum estimation

Abstract: We address quantum critical systems as a resource in quantum estimation and derive the ultimate quantum limits to the precision of any estimator of the coupling parameters. In particular, if $L$ denotes the size of a system and $\ensuremath{\lambda}$ is the relevant coupling parameters driving a quantum phase transition, we show that a precision improvement of order $1∕L$ may be achieved in the estimation of $\ensuremath{\lambda}$ at the critical point compared to the noncritical case. We show that analog results hold for temperature estimation in classical phase transitions. Results are illustrated by means of a specific example involving a fermion tight-binding model with pair creation (BCS model).

Quantum Circuit Simplification and Level Compaction

Abstract: Quantum circuits are time-dependent diagrams describing the process of quantum computation. Usually, a quantum algorithm must be mapped into a quantum circuit. Optimal synthesis of quantum circuits is intractable, and heuristic methods must be employed. With the use of heuristics, the optimality of circuits is no longer guaranteed. In this paper, we consider a local optimization technique based on templates to simplify and reduce the depth of nonoptimal quantum circuits. We present and analyze templates in the general case and provide particular details for the circuits composed of NOT, CNOT, and controlled-sqrt-of-NOT gates. We apply templates to optimize various common circuits implementing multiple control Toffoli gates and quantum Boolean arithmetic circuits. We also show how templates can be used to compact the number of levels of a quantum circuit. The runtime of our implementation is small, whereas the reduction in the number of quantum gates and number of levels is significant.

Fault-tolerant linear optical quantum computing with small-amplitude coherent States.

Abstract: Quantum computing using two coherent states as a qubit basis is a proposed alternative architecture with lower overheads but has been questioned as a practical way of performing quantum computing due to the fragility of diagonal states with large coherent amplitudes. We show that using error correction only small amplitudes (alpha>1.2) are required for fault-tolerant quantum computing. We study fault tolerance under the effects of small amplitudes and loss using a Monte Carlo simulation. The first encoding level resources are orders of magnitude lower than the best single photon scheme.

Quantum Computing : From Linear Algebra to Physical Realizations

Abstract: From linear algebra to quantum computing Basics of Vectors and Matrices Vector Spaces Linear Dependence and Independence of Vectors Dual Vector Spaces Basis, Projection Operator, and Completeness Relation Linear Operators and Matrices Eigenvalue Problems Pauli Matrices Spectral Decomposition Singular Value Decomposition (SVD) Tensor Product (Kronecker Product) Framework of Quantum Mechanics Fundamental Postulates Some Examples Multipartite System, Tensor Product, and Entangled State Mixed States and Density Matrices Qubits and Quantum Key Distribution Qubits Quantum Key Distribution (BB84 Protocol) Quantum Gates, Quantum Circuit, and Quantum Computer Introduction Quantum Gates Correspondence with Classical Logic Gates No-Cloning Theorem Dense Coding and Quantum Teleportation Universal Quantum Gates Quantum Parallelism and Entanglement Simple Quantum Algorithms Deutsch Algorithm Deutsch-Jozsa Algorithm and Bernstein-Vazirani Algorithm Simon's Algorithm Quantum Integral Transforms Quantum Integral Transforms Quantum Fourier Transform (QFT) Application of QFT: Period-Finding Implementation of QFT Walsh-Hadamard Transform Selective Phase Rotation Transform Grover's Search Algorithm Searching for a Single File Searching for d Files Shor's Factorization Algorithm The RSA Cryptosystem Factorization Algorithm Quantum Part of Shor's Algorithm Probability Distribution Continued Fractions and Order-Finding Modular Exponential Function Decoherence Open Quantum System Measurements as Quantum Operations Examples Lindblad Equation Quantum Error-Correcting Codes (QECC) Introduction 3-Qubit Bit-Flip Code and Phase-Flip Code Shor's 9-Qubit Code Calderbank-Shor-Steane (CSS) 7-Qubit QECC DiVincenzo-Shor 5-Qubit QECC Physical realizations of quantum computing DiVincenzo Criteria Introduction DiVincenzo Criteria Physical Realizations Beyond DiVincenzo Criteria NMR Quantum Computer Introduction NMR Spectrometer Hamiltonian Implementation of Gates and Algorithms Time-Optimal Control of NMR Quantum Computer Measurements Preparation of Pseudopure State DiVincenzo Criteria Trapped Ions Introduction Electronic States of Ion as Qubit Ions in Paul Trap Ion Qubit Quantum Gates Readout DiVincenzo Criteria Quantum Computing with Neutral Atoms Introduction Trapping Neutral Atoms 1-Qubit Gate Quantum State Engineering of Neutral Atoms Preparation of Entangled Neutral Atoms DiVincenzo Criteria Josephson Junction Qubits Introduction Nanoscale Josephson Junctions and SQUIDs Charge Qubit Flux Qubit Quantronium Current-Biased Qubit Readout Coupled Qubits DiVincenzo Criteria Quantum Computing with Quantum Dots Introduction Mesoscopic Semiconductors Electron Charge Qubit Electron Spin Qubit DiVincenzo Criteria Appendix: Solutions to Selected Exercises Index