Abstract: Superconducting circuits are macroscopic in size but have generic quantum properties such as quantized energy levels, superposition of states, and entanglement, all of which are more commonly associated with atoms. Superconducting quantum bits (qubits) form the key component of these circuits. Their quantum state is manipulated by using electromagnetic pulses to control the magnetic flux, the electric charge or the phase difference across a Josephson junction (a device with nonlinear inductance and no energy dissipation). As such, superconducting qubits are not only of considerable fundamental interest but also might ultimately form the primitive building blocks of quantum computers.

Abstract: The emerging field of circuit quantum electrodynamics could pave the way for the design of practical quantum computers.

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.

Abstract: A single electron spin confined within a semiconductor nanostructure is an ideal qubit for quantum computing, as it is relatively stable against decoherence and is easily manipulated electrically or optically. Full quantum control involving initialization, spin rotation and detection, has been demonstrated previously using electrically controlled radio-frequency pulses, but this method will be too slow for the construction of quantum computing circuits operating at useful clock speeds. Optical manipulation of electron spin allows much faster operations and has the added advantage that it allows for an optical interface. Press et al. now achieve ultrafast optical control of electron spin in a quantum dot and demonstrate, in combination with optical initialization and detection, a single-qubit logic gate operation, involving a sequence of two optical pulses. Such high-speed operation could conceivably lead to quantum computing devices at gigahertz clock speeds. A basic requirement for quantum information processing systems is the ability to completely control the state of a single qubit1,2,3,4,5,6. For qubits based on electron spin, a universal single-qubit gate is realized by a rotation of the spin by any angle about an arbitrary axis. Driven, coherent Rabi oscillations between two spin states can be used to demonstrate control of the rotation angle. Ramsey interference, produced by two coherent spin rotations separated by a variable time delay, demonstrates control over the axis of rotation. Full quantum control of an electron spin in a quantum dot has previously been demonstrated using resonant radio-frequency pulses that require many spin precession periods7,8,9,10. However, optical manipulation of the spin allows quantum control on a picosecond or femtosecond timescale11,12,13,14,15,16,17,18, permitting an arbitrary rotation to be completed within one spin precession period6. Recent work in optical single-spin control has demonstrated the initialization of a spin state in a quantum dot19,20,21,22, as well as the ultrafast manipulation of coherence in a largely unpolarized single-spin state17. Here we demonstrate complete coherent control over an initialized electron spin state in a quantum dot using picosecond optical pulses. First we vary the intensity of a single optical pulse to observe over six Rabi oscillations between the two spin states; then we apply two sequential pulses to observe high-contrast Ramsey interference. Such a two-pulse sequence realizes an arbitrary single-qubit gate completed on a picosecond timescale. Along with the spin initialization and final projective measurement of the spin state, these results demonstrate a complete set of all-optical single-qubit operations.

Abstract: We are interested in the problem of characterizing the correlations that arise when performing local measurements on separate quantum systems. In a previous work (Navascues et al 2007 Phys. Rev. Lett. 98 010401), we introduced an infinite hierarchy of conditions necessarily satisfied by any set of quantum correlations. Each of these conditions could be tested using semidefinite programming. We present here new results concerning this hierarchy. We prove in particular that it is complete, in the sense that any set of correlations satisfying every condition in the hierarchy has a quantum representation in terms of commuting measurements. Although our tests are conceived to rule out non- quantum correlations, and can in principle certify that a set of correlations is quantum only in the asymptotic limit where all tests are satisfied, we show that in some cases it is possible to conclude that a given set of correlations is quantum after performing only a finite number of tests. We provide a criterion to detect when such a situation arises, and we explain how to reconstruct the quantum states and measurement operators reproducing the given correlations. Finally, we present several applications of our approach. We use it in particular to bound the quantum violation of various Bell inequalities.

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.

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.

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.

Abstract: Quantum computers hold the promise to solve certain computational task much more efficiently than classical computers. We review the recent experimental advancements towards a quantum computer with trapped ions. In particular, various implementations of qubits, quantum gates and some key experiments are discussed. Furthermore, we review some implementations of quantum algorithms such as a deterministic teleportation of quantum information and an error correction scheme.

Abstract: The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with $T$ gates whose underlying graph has a treewidth $d$ can be simulated deterministically in $T^{O(1)}\exp[O(d)]$ time, which, in particular, is polynomial in $T$ if $d=O(\log T)$. Among many implications, we show efficient simulations for log-depth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that one-way quantum computation of Raussendorf and Briegel (Phys. Rev. Lett., 86 (2001), pp. 5188-5191), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a small-treewidth graph with a constant maximum degree. (The requirement on the maximum degree was removed in [I. L. Markov and Y. Shi, preprint:quant-ph/0511069].)

Abstract: We analyze the coherent transport of a single photon, which propagates in a one-dimensional coupled-resonator waveguide and is scattered by a controllable two-level system located inside one of the resonators of this waveguide. Our approach, which uses discrete coordinates, unifies low and high energy effective theories for single-photon scattering. We show that the controllable two-level system can behave as a quantum switch for the coherent transport of a single photon. This study may inspire new electro-optical single-photon quantum devices. We also suggest an experimental setup based on superconducting transmission line resonators and qubits.

Abstract: Characterization of quantum dynamics is a fundamental problem in quantum physics and quantum-information science. Several methods are known which achieve this goal, namely standard quantum-process tomography (SQPT), ancilla-assisted process tomography, and the recently proposed scheme of direct characterization of quantum dynamics (DCQD). Here, we review these schemes and analyze them with respect to some of the physical resources they require. Although a reliable figure-of-merit for process characterization is not yet available, our analysis can provide a benchmark which is necessary for choosing the scheme that is the most appropriate in a given situation, with given resources. As a result, we conclude that for quantum systems where two-body interactions are not naturally available, SQPT is the most efficient scheme. However, for quantum systems with controllable two-body interactions, the DCQD scheme is more efficient than other known quantum-process tomography schemes in terms of the total number of required elementary quantum operations.

Abstract: The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can be applied only to small systems. By contrast, we demonstrate that quantum computers could exactly simulate chemical reactions in polynomial time. Our algorithm uses the split-operator approach and explicitly simulates all electron-nuclear and interelectronic interactions in quadratic time. Surprisingly, this treatment is not only more accurate than the Born–Oppenheimer approximation but faster and more efficient as well, for all reactions with more than about four atoms. This is the case even though the entire electronic wave function is propagated on a grid with appropriately short time steps. Although the preparation and measurement of arbitrary states on a quantum computer is inefficient, here we demonstrate how to prepare states of chemical interest efficiently. We also show how to efficiently obtain chemically relevant observables, such as state-to-state transition probabilities and thermal reaction rates. Quantum computers using these techniques could outperform current classical computers with 100 qubits.

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.

Abstract: We present a protocol which allows a client to have a server carry out a quantum computation for her such that the client's inputs, outputs and computation remain perfectly private, and where she does not require any quantum computational power or memory. The client only needs to be able to prepare single qubits randomly chosen from a finite set and send them to the server, who has the balance of the required quantum computational resources. Our protocol is interactive: after the initial preparation of quantum states, the client and server use two-way classical communication which enables the client to drive the computation, giving single-qubit measurement instructions to the server, depending on previous measurement outcomes. Our protocol works for inputs and outputs that are either classical or quantum. We give an authentication protocol that allows the client to detect an interfering server; our scheme can also be made fault-tolerant. We also generalize our result to the setting of a purely classical client who communicates classically with two non-communicating entangled servers, in order to perform a blind quantum computation. By incorporating the authentication protocol, we show that any problem in BQP has an entangled two-prover interactive proof with a purely classical verifier. Our protocol is the first universal scheme which detects a cheating server, as well as the first protocol which does not require any quantum computation whatsoever on the client's side. The novelty of our approach is in using the unique features of measurement-based quantum computing which allows us to clearly distinguish between the quantum and classical aspects of a quantum computation.

Abstract: We show that the NP-hard quadratic unconstrained binary optimization (QUBO) problem on a graph G can be solved using an adiabatic quantum computer that implements an Ising spin-1/2 Hamiltonian, by reduction through minor-embedding of G in the quantum hardware graph U. There are two components to this reduction: embedding and parameter setting. The embedding problem is to find a minor-embedding G emb of a graph G in U, which is a subgraph of U such that G can be obtained from G emb by contracting edges. The parameter setting problem is to determine the corresponding parameters, qubit biases and coupler strengths, of the embedded Ising Hamiltonian. In this paper, we focus on the parameter setting problem. As an example, we demonstrate the embedded Ising Hamiltonian for solving the maximum independent set (MIS) problem via adiabatic quantum computation (AQC) using an Ising spin-1/2 system. We close by discussing several related algorithmic problems that need to be investigated in order to facilitate the design of adiabatic algorithms and AQC architectures.

Abstract: This chapter contains sections titled: Adiabatic Processes Adiabatic Quantum Computing Exercises

Abstract: Quantum annealing is a generic name of quantum algorithms that use quantum-mechanical fluctuations to search for the solution of an optimization problem. It shares the basic idea with quantum adiabatic evolution studied actively in quantum computation. The present paper reviews the mathematical and theoretical foundations of quantum annealing. In particular, theorems are presented for convergence conditions of quantum annealing to the target optimal state after an infinite-time evolution following the Schrodinger or stochastic (Monte Carlo) dynamics. It is proved that the same asymptotic behavior of the control parameter guarantees convergence for both the Schrodinger dynamics and the stochastic dynamics in spite of the essential difference of these two types of dynamics. Also described are the prescriptions to reduce errors in the final approximate solution obtained after a long but finite dynamical evolution of quantum annealing. It is shown there that we can reduce errors significantly by an ingenious choice of annealing schedule (time dependence of the control parameter) without compromising computational complexity qualitatively. A review is given on the derivation of the convergence condition for classical simulated annealing from the view point of quantum adiabaticity using a classical-quantum mapping.

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.

Abstract: Molecular magnets are a type of molecule containing multiple magnetic ions whose spins are tightly coupled to give a single 'collective' spin. The quantum mechanical properties of this collective spin are attracting attention as a possible basis for the functional unit of a quantum computer. Until now it has not been clear whether the quantum spin states of these molecular entities are sufficiently long-lived to permit useful computation, but new work suggests that they are. Bertaina et al. observed pronounced quantum oscillations between the spin states of one such molecular magnet, consistent with long-lived quantum coherence, in this system at least. They also outline a scheme by which, through careful material design, these properties could be harnessed in a practical context. Molecular magnets are a class of molecule containing multiple magnetic ions whose spins are tightly coupled to give a single 'collective' spin. But it has remained an open question whether the quantum spin states of these molecular entities are sufficiently long-lived to permit useful computation. Pronounced quantum oscillations between the spin states of one such molecular magnet have been observed, indicating that quantum coherence is long-lived. The term ‘molecular magnet’ generally refers to a molecular entity containing several magnetic ions whose coupled spins generate a collective spin, S (ref. 1). Such complex multi-spin systems provide attractive targets for the study of quantum effects at the mesoscopic scale. In these molecules, the large energy barriers between collective spin states can be crossed by thermal activation or quantum tunnelling, depending on the temperature or an applied magnetic field2,3,4. There is the hope that these mesoscopic spin states can be harnessed for the realization of quantum bits—‘qubits’, the basic building blocks of a quantum computer—based on molecular magnets5,6,7,8. But strong decoherence9 must be overcome if the envisaged applications are to become practical. Here we report the observation and analysis of Rabi oscillations (quantum oscillations resulting from the coherent absorption and emission of photons driven by an electromagnetic wave10) of a molecular magnet in a hybrid system, in which discrete and well-separated magnetic clusters are embedded in a self-organized non-magnetic environment. Each cluster contains 15 antiferromagnetically coupled S = 1/2 spins, leading to an S = 1/2 collective ground state11,12,13. When this system is placed into a resonant cavity, the microwave field induces oscillatory transitions between the ground and excited collective spin states, indicative of long-lived quantum coherence. The present observation of quantum oscillations suggests that low-dimension self-organized qubit networks having coherence times of the order of 100 μs (at liquid helium temperatures) are a realistic prospect.

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.

Abstract: The key approaches for machine learning, particularly learning in unknown probabilistic environments, are new representations and computation mechanisms. In this paper, a novel quantum reinforcement learning (QRL) method is proposed by combining quantum theory and reinforcement learning (RL). Inspired by the state superposition principle and quantum parallelism, a framework of a value-updating algorithm is introduced. The state (action) in traditional RL is identified as the eigen state (eigen action) in QRL. The state (action) set can be represented with a quantum superposition state, and the eigen state (eigen action) can be obtained by randomly observing the simulated quantum state according to the collapse postulate of quantum measurement. The probability of the eigen action is determined by the probability amplitude, which is updated in parallel according to rewards. Some related characteristics of QRL such as convergence, optimality, and balancing between exploration and exploitation are also analyzed, which shows that this approach makes a good tradeoff between exploration and exploitation using the probability amplitude and can speedup learning through the quantum parallelism. To evaluate the performance and practicability of QRL, several simulated experiments are given, and the results demonstrate the effectiveness and superiority of the QRL algorithm for some complex problems. This paper is also an effective exploration on the application of quantum computation to artificial intelligence.

Abstract: We present a complete scheme for quantum information processing using the unique features of alkaline-earth-metal atoms. We show how two completely independent lattices can be formed for the 1S0 and 3P0 states, with one used as a storage lattice for qubits encoded on the nuclear spin, and the other as a transport lattice to move qubits and perform gate operations. We discuss how the 3P2 level can be used for addressing of individual qubits, and how collisional losses from metastable states can be used to perform gates via a lossy blockade mechanism.

Abstract: The multidisciplinary field of quantum computing strives to exploit some of the uncanny aspects of quantum mechanics to expand our computational horizons. Quantum Computing for Computer Scientists takes readers on a tour of this fascinating area of cutting-edge research. Written in an accessible yet rigorous fashion, this book employs ideas and techniques familiar to every student of computer science. The reader is not expected to have any advanced mathematics or physics background. After presenting the necessary prerequisites, the material is organized to look at different aspects of quantum computing from the specific standpoint of computer science. There are chapters on computer architecture, algorithms, programming languages, theoretical computer science, cryptography, information theory, and hardware. The text has step-by-step examples, more than two hundred exercises with solutions, and programming drills that bring the ideas of quantum computing alive for today's computer science students and researchers.

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.

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

Abstract: An implementation of weak values is investigated in solid-state qubits. We demonstrate that a weak value can be nonclassical if and only if a Leggett-Garg inequality can also be violated. Generalized weak values are described in which post-selection occurs on a range of weak measurement results. Imposing classical weak values permits the derivation of Leggett-Garg inequalities for bounded operators. Our analysis is presented in terms of kicked quantum nondemolition measurements on a quantum double-dot charge qubit.

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.