Showing papers on "Quantum error correction published in 2004"
TL;DR: This work has shown that conventional bounds to the precision of measurements such as the shot noise limit or the standard quantum limit are not as fundamental as the Heisenberg limits and can be beaten using quantum strategies that employ “quantum tricks” such as squeezing and entanglement.
Abstract: Quantum mechanics, through the Heisenberg uncertainty principle, imposes limits on the precision of measurement. Conventional measurement techniques typically fail to reach these limits. Conventional bounds to the precision of measurements such as the shot noise limit or the standard quantum limit are not as fundamental as the Heisenberg limits and can be beaten using quantum strategies that employ “quantum tricks” such as squeezing and entanglement.
2,421 citations
Book•
15 Oct 2004
TL;DR: In this article, the Stochastic Schroedinger Equation (SSE) was applied to apply the Master Equation to the Quantum Langevin Equations (MEE).
Abstract: From the contents: A Historical Introduction.- Quantum Statistics.- Quantum Langevin Equations.- Phase Space Methods.- Quantum Markov Processes.- Applying the Master Equation.- Amplifiers and Measurement.- Photon Counting.- Interaction of Light with Atoms.- Squeezing.- The Stochastic Schroedinger Equation.- Cascaded Quantum Systems.- Supplement.- Bibliography.- Author Index.- Index.
1,316 citations
TL;DR: It is shown that 2log3N is the maximal perfect communication distance for hypercube geometries if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
Abstract: We propose a class of qubit networks that admit the perfect state transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to N-qubit spin networks of identical qubit couplings, we show that 2log3N is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain.
1,014 citations
TL;DR: Un unconditional teleportation of massive particle qubits using atomic (9Be+) ions confined in a segmented ion trap is reported, which achieves an average fidelity of 78 per cent, which exceeds the fidelity of any protocol that does not use entanglement.
Abstract: Quantum teleportation1 provides a means to transport quantum information efficiently from one location to another, without the physical transfer of the associated quantum-information carrier. This is achieved by using the non-local correlations of previously distributed, entangled quantum bits (qubits). Teleportation is expected to play an integral role in quantum communication2 and quantum computation3. Previous experimental demonstrations have been implemented with optical systems that used both discrete and continuous variables4,5,6,7,8,9, and with liquid-state nuclear magnetic resonance10. Here we report unconditional teleportation5 of massive particle qubits using atomic (9Be+) ions confined in a segmented ion trap, which aids individual qubit addressing. We achieve an average fidelity of 78 per cent, which exceeds the fidelity of any protocol that does not use entanglement11. This demonstration is also important because it incorporates most of the techniques necessary for scalable quantum information processing in an ion-trap system12,13.
912 citations
TL;DR: Here a protocol for a high-fidelity transfer of an independently prepared quantum state of light onto an atomic quantum state based on atomic ensembles is proposed and experimentally demonstrated.
Abstract: The information carrier of today's communications, a weak pulse of light, is an intrinsically quantum object. As a consequence, complete information about the pulse cannot be perfectly recorded in a classical memory, even in principle. In the field of quantum information, this has led to the long-standing challenge of how to achieve a high-fidelity transfer of an independently prepared quantum state of light onto an atomic quantum state1,2,3,4. Here we propose and experimentally demonstrate a protocol for such a quantum memory based on atomic ensembles. Recording of an externally provided quantum state of light onto the atomic quantum memory is achieved with 70 per cent fidelity, significantly higher than the limit for classical recording. Quantum storage of light is achieved in three steps: first, interaction of the input pulse and an entangling field with spin-polarized caesium atoms; second, subsequent measurement of the transmitted light; and third, feedback onto the atoms using a radio-frequency magnetic pulse conditioned on the measurement result. The density of recorded states is 33 per cent higher than the best classical recording of light onto atoms, with a quantum memory lifetime of up to 4 milliseconds.
729 citations
TL;DR: This work demonstrates entanglement between a superconducting flux qubit and asuperconducting quantum interference device (SQUID), which provides the measurement system for detecting the quantum states and an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic oscillator.
Abstract: In the emerging field of quantum computation and quantum information, superconducting devices are promising candidates for the implementation of solid-state quantum bits (qubits). Single-qubit operations, direct coupling between two qubits and the realization of a quantum gate have been reported. However, complex manipulation of entangled states-such as the coupling of a two-level system to a quantum harmonic oscillator, as demonstrated in ion/atom-trap experiments and cavity quantum electrodynamics-has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux qubit (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic oscillator. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi oscillations of the coupled system.
635 citations
TL;DR: An approach to optical quantum computation in which a deterministic entangling quantum gate may be performed using a few hundred coherently interacting optical elements using the abstract cluster-state model of quantum computation.
Abstract: We propose an approach to optical quantum computation in which a deterministic entangling quantum gate may be performed using, on average, a few hundred coherently interacting optical elements (beam splitters, phase shifters, single photon sources, and photodetectors with feedforward). This scheme combines ideas from the optical quantum computing proposal of Knill, Laflamme, and Milburn [Nature (London) 409, 46 (2001)], and the abstract cluster-state model of quantum computation proposed by Raussendorf and Briegel [Phys. Rev. Lett. 86, 5188 (2001)].
494 citations
TL;DR: In principle, the approach enables a quantum state to be maintained by means of repeated error correction, an important step towards scalable fault-tolerant quantum computation using trapped ions.
Abstract: Scalable quantum computation1 and communication require error control to protect quantum information against unavoidable noise. Quantum error correction2,3 protects information stored in two-level quantum systems (qubits) by rectifying errors with operations conditioned on the measurement outcomes. Error-correction protocols have been implemented in nuclear magnetic resonance experiments4,5,6, but the inherent limitations of this technique7 prevent its application to quantum information processing. Here we experimentally demonstrate quantum error correction using three beryllium atomic-ion qubits confined to a linear, multi-zone trap. An encoded one-qubit state is protected against spin-flip errors by means of a three-qubit quantum error-correcting code. A primary ion qubit is prepared in an initial state, which is then encoded into an entangled state of three physical qubits (the primary and two ancilla qubits). Errors are induced simultaneously in all qubits at various rates. The encoded state is decoded back to the primary ion one-qubit state, making error information available on the ancilla ions, which are separated from the primary ion and measured. Finally, the primary qubit state is corrected on the basis of the ancillae measurement outcome. We verify error correction by comparing the corrected final state to the uncorrected state and to the initial state. In principle, the approach enables a quantum state to be maintained by means of repeated error correction, an important step towards scalable fault-tolerant quantum computation using trapped ions.
485 citations
TL;DR: In this paper, a configuration of devices that includes a source of some unknown bipartite quantum state that is claimed to be the Bell state Φ+ and two spatially separated but otherwise unknown measurement apparatus one on each side, that are each claimed to execute an orthogonal measurement at an angle θ ∈ {-π/8, 0, π/8} that is chosen by the user.
Abstract: We study, in the context of quantum information and quantum communication, a configuration of devices that includes (1) a source of some unknown bipartite quantum state that is claimed to be the Bell state Φ+ and (2) two spatially separated but otherwise unknown measurement apparatus one on each side, that are each claimed to execute an orthogonal measurement at an angle θ ∈ {-π/8, 0, π/8} that is chosen by the user. We show that, if the nine distinct probability distributions that are generated by the self checking configuration, one for each pair of angles, are consistent with the specifications, the source and the two measurement apparatus are guaranteed to be identical to the claimed specifications up to a local change of basis on each side. We discuss the connection with quantum cryptography.
358 citations
TL;DR: In this paper, an optimal two-qubit gate was constructed with at most 3 controlled-NOT (CNOT) gates and 15 elementary one qubit gates, assuming that the desired two qubit gate corresponds to a purely real unitary transformation.
Abstract: In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, $y$-rotation, $z$-rotation, and phase gates.
341 citations
TL;DR: In this paper, the role of unitary braiding operators in quantum computing is explored and it is shown that a single specific solution of the Yang-Baxterequation is universal for quantum computing in the presence of local unitary transformations.
Abstract: This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix)oftheYang-Baxterequationisauniversalgateforquantumcomputing,in thepresenceoflocalunitarytransformations.Weshowthatthissame Rgeneratesa new non-trivial invariant of braids, knots and links. Other solutions of theYang- Baxter equation are also shown to be universal for quantum computation. The paperdiscussestheseresultsinthecontextofcomparingquantumandtopological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.
TL;DR: In this article, the authors consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates.
Abstract: We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them.
TL;DR: In this paper, real-time feedback performed during a quantum non-demolition measurement of atomic spin-angular momentum allowed the authors to influence the quantum statistics of the measurement outcome.
Abstract: Real-time feedback performed during a quantum nondemolition measurement of atomic spin-angular momentum allowed us to influence the quantum statistics of the measurement outcome. We showed that it is possible to harness measurement backaction as a form of actuation in quantum control, and thus we describe a valuable tool for quantum information science. Our feedback-mediated procedure generates spin-squeezing, for which the reduction in quantum uncertainty and resulting atomic entanglement are not conditioned on the measurement outcome.
TL;DR: These proposals offer practical and realistic alternatives to existing schemes for quantum key distribution over optical fibers without resorting to interferometry or two-way quantum communication, thereby circumventing, respectively, the need for high precision timing and the threat of Trojan horse attacks.
Abstract: We present two polarization-based protocols for quantum key distribution. The protocols encode key bits in noiseless subspaces or subsystems and so can function over a quantum channel subjected to an arbitrary degree of collective noise, as occurs, for instance, due to rotation of polarizations in an optical fiber. These protocols can be implemented using only entangled photon-pair sources, single-photon rotations, and single-photon detectors. Thus, our proposals offer practical and realistic alternatives to existing schemes for quantum key distribution over optical fibers without resorting to interferometry or two-way quantum communication, thereby circumventing, respectively, the need for high precision timing and the threat of Trojan horse attacks.
TL;DR: In this work, the matrix representation of an arbitrary multiqubit gate is considered and the number of controlled NOT gates is O(4(n)) which coincides with the theoretical lower bound.
Abstract: Optimal implementation of quantum gates is crucial for designing a quantum computer. We consider the matrix representation of an arbitrary multiqubit gate. By ordering the basis vectors using the Gray code, we construct the quantum circuit which is optimal in the sense of fully controlled single-qubit gates and yet is equivalent with the multiqubit gate. In the second step of the optimization, superfluous control bits are eliminated, which eventually results in a smaller total number of the elementary gates. In our scheme the number of controlled NOT gates is $O({4}^{n})$ which coincides with the theoretical lower bound.
TL;DR: In this paper, a quantum key agreement protocol for quantum teleportation is presented. But the key bits are determined only by random measurement outcomes and are independent of the states transmitted over the channel.
Abstract: By replacing a classical channel with a quantum one during quantum teleportation, a quantum key agreement protocol is presented. The key bits are determined only by the random measurement outcomes and are independent of the states transmitted over the channel.
TL;DR: It is demonstrated that spontaneous parametric down conversion can be used to generate four-photon states which enable the encoding of one qubit in a decoherence-free subspace and the immunity against noise is verified by quantum state tomography of the encoded qubit.
Abstract: Decoherence-free states protect quantum information from collective noise, the predominant cause of decoherence in current implementations of quantum communication and computation. Here we demonstrate that spontaneous parametric down conversion can be used to generate four-photon states which enable the encoding of one qubit in a decoherence-free subspace. The immunity against noise is verified by quantum state tomography of the encoded qubit. We show that particular states of the encoded qubit can be distinguished by local measurements on the four photons only.
TL;DR: In this paper, it was shown that one-dimensional rings of qubits with fixed (time-independent) interactions, constant around the ring, allow high-fidelity communication of quantum states.
Abstract: It has been recently suggested that the dynamics of a quantum spin system may provide a natural mechanism for transporting quantum information. We show that one-dimensional rings of qubits with fixed (time-independent) interactions, constant around the ring, allow high-fidelity communication of quantum states. We show that the problem of maximizing the fidelity of the quantum communication is related to a classical problem in Fourier wave analysis. By making use of this observation we find that if both communicating parties have access to limited numbers of qubits in the ring (a fraction that vanishes in the limit of large rings) it is possible to make the communication arbitrarily good.
TL;DR: It is shown that the exponential speedup of quantum algorithms is restored if single-charge measurements are added, which enable the construction of a CNOT (controlled NOT) gate for free fermions, using only beam splitters and spin rotations.
Abstract: It is known that a quantum computer operating on electron-spin qubits with single-electron Hamiltonians and assisted by single-spin measurements can be simulated efficiently on a classical computer. We show that the exponential speedup of quantum algorithms is restored if single-charge measurements are added. These enable the construction of a CNOT (controlled NOT) gate for free fermions, using only beam splitters and spin rotations. The gate is nearly deterministic if the charge detector counts the number of electrons in a mode, and fully deterministic if it only measures the parity of that number.
TL;DR: In this article, the quantum Zeno effect was used to suppress the failure events that would otherwise occur in a linear optics approach to quantum computing, which would allow the implementation of deterministic logic gates without the need for high-efficiency detectors.
Abstract: We show that the quantum Zeno effect can be used to suppress the failure events that would otherwise occur in a linear optics approach to quantum computing. From a practical viewpoint, that would allow the implementation of deterministic logic gates without the need for ancilla photons or high-efficiency detectors. We also show that the photons can behave as if they were fermions instead of bosons in the presence of a strong Zeno effect, which leads to an alternative paradigm for quantum computation.
TL;DR: It is shown that, if the size of the blocks that can be coherently attacked by an eavesdropper is fixed and much smaller than the key size, then the optimal attack for a given signal-to-noise ratio in the transmission line is an individual Gaussian attack.
Abstract: A general study of arbitrary finite-size coherent attacks against continuous-variable quantum cryptographic schemes is presented. It is shown that, if the size of the blocks that can be coherently attacked by an eavesdropper is fixed and much smaller than the key size, then the optimal attack for a given signal-to-noise ratio in the transmission line is an individual Gaussian attack. Consequently, non-Gaussian coherent attacks do not need to be considered in the security analysis of such quantum cryptosystems.
TL;DR: Some of quantum algorithms for search problems are reviewed: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.
Abstract: We review some of quantum algorithms for search problems: Grover's search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks.
17 Oct 2004
TL;DR: The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown as mentioned in this paper, but it is known that it is polynomially equivalent to the standard quantum circuit model.
Abstract: The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.
TL;DR: In this article, the authors present a linear nearest-neighbor architecture for computing integer factorization, which requires 8L4 2-qubit gates arranged in a circuit of depth 32L3.
Abstract: Shor's algorithm, which given appropriate hardware can factorise an integer N in a time polynomial in its binary length L, has arguably spurred the race to build a practical quantum computer. Several different quantum circuits implementing Shor's algorithm have been designed, but each tacitly assumes that arbitrary pairs of qubits within the computer can be interacted. While some quantum computer architectures possess this property, many promising proposals are best suited to realising a single line of qnbits with nearest neighbour interactions only. In light of this, we present a circuit implementing Shor's factorisation algorithm designed for such a linear nearest neighbour architecture. Despite the interaction restrictions, the circuit requires just 2L + 4 qubits and to leading order requires 8L4 2-qubit gates arranged in a circuit of depth 32L3 -- identical to leading order to that possible using an architecture that can interact arbitrary pairs of qubits.
Book•
03 Sep 2004
TL;DR: This article reviews Quantum Computing: A Short Course from Theory to Experiment, which was published in Wiley, Hoboken, NJ, 2004.
Abstract: Preface. 1 Introduction and survey. 1.1 Information, computers and quantum mechanics. 1.1.1 Digital information. 1.1.2 Moore's law. 1.1.3 Emergence of quantum behavior. 1.1.4 Energy dissipation in computers. 1.2 Quantum computer basics. 1.2.1 Quantum information. 1.2.2 Quantum communication. 1.2.3 Basics of quantum information processing. 1.2.4 Decoherence. 1.2.5 Implementation. 1.3 History of quantum information processing. 1.3.1 Initial ideas. 1.3.2 Quantum algorithms. 1.3.3 Implementations. 2 Physics of computation. 2.1 Physical laws and information processing. 2.1.1 Hardware representation. 2.1.2 Quantum vs. classical information processing. 2.2 Limitations on computer performance. 2.2.1 Switching energy. 2.2.2 Entropy generation and Maxwell's demon. 2.2.3 Reversible logic. 2.2.4 Reversible gates for universal computers. 2.2.5 Processing speed. 2.2.6 Storage density. 2.3 The ultimate laptop. 2.3.1 Processing speed. 2.3.2 Maximum storage density. 3 Elements of classical computer science. 3.1 Bits of history. 3.2 Boolean algebra and logic gates. 3.2.1 Bits and gates. 3.2.2 2 bit logic gates. 3.2.3 Minimum set of irreversible gates. 3.2.4 Minimum set of reversible gates. 3.2.5 The CNOT gate. 3.2.6 The Toffoli gate. 3.2.7 The Fredkin gate. 3.3 Universal computers. 3.3.1 The Turing machine. 3.3.2 The Church Turing hypothesis. 3.4 Complexity and algorithms. 3.4.1 Complexity classes. 3.4.2 Hard and impossible problems. 4 Quantum mechanics. 4.1 General structure. 4.1.1 Spectral lines and stationary states. 4.1.2 Vectors in Hilbert space. 4.1.3 Operators in Hilbert space. 4.1.4 Dynamics and the Hamiltonian operator. 4.1.5 Measurements. 4.2 Quantum states. 4.2.1 The two dimensional Hilbert space: qubits, spins, and photons. 4.2.2 Hamiltonian and evolution. 4.2.3 Two or more qubits. 4.2.4 Density operator. 4.2.5 Entanglement and mixing. 4.2.6 Quantification of entanglement. 4.2.7 Bloch sphere. 4.2.8 EPR correlations. 4.2.9 Bell's theorem. 4.2.10 Violation of Bell's inequality. 4.2.11 The no cloning theorem. 4.3 Measurement revisited. 4.3.1 Quantum mechanical projection postulate. 4.3.2 The Copenhagen interpretation. 4.3.3 Von Neumann's model. 5 Quantum bits and quantum gates. 5.1 Single qubit gates. 5.1.1 Introduction. 5.1.2 Rotations around coordinate axes. 5.1.3 General rotations. 5.1.4 Composite rotations. 5.2 Two qubit gates. 5.2.1 Controlled gates. 5.2.2 Composite gates. 5.3 Universal sets of gates. 5.3.1 Choice of set. 5.3.2 Unitary operations. 5.3.3 Two qubit operations. 5.3.4 Approximating single qubit gates. 6 Feynman's contribution. 6.1 Simulating physics with computers. 6.1.1 Discrete system representations. 6.1.2 Probabilistic simulations. 6.2 Quantum mechanical computers. 6.2.1 Simple gates. 6.2.2 Adder circuits. 6.2.3 Qubit raising and lowering operators. 6.2.4 Adder Hamiltonian. 7 Errors and decoherence. 7.1 Motivation. 7.1.1 Sources of error. 7.1.2 A counterstrategy. 7.2 Decoherence. 7.2.1 Phenomenology. 7.2.2 Semiclassical description. 7.2.3 Quantum mechanical model. 7.2.4 Entanglement and mixing. 7.3 Error correction. 7.3.1 Basics. 7.3.2 Classical error correction. 7.3.3 Quantum error correction. 7.3.4 Single spin flip error. 7.3.5 Continuous phase errors. 7.3.6 General single qubit errors. 7.3.7 The quantum Zeno effect. 7.3.8 Stabilizer codes. 7.3.9 Fault tolerant computing. 7.4 Avoiding errors. 7.4.1 Basics. 7.4.2 Decoherence free subspaces. 7.4.3 NMR in Liquids. 7.4.4 Scaling considerations. 8 Tasks for quantum computers. 8.1 Quantum versus classical algorithms. 8.1.1 Why Quantum? 8.1.2 Classes of quantum algorithms. 8.2 The Deutsch algorithm: Looking at both sides of a coin at the same time. 8.2.1 Functions and their properties. 8.2.2 Example: one qubit functions. 8.2.3 Evaluation. 8.2.4 Many qubits. 8.2.5 Extensions and generalizations. 8.3 The Shor algorithm: It's prime time. 8.3.1 Some number theory. 8.3.2 Factoring strategy. 8.3.3 The core of Shor's algorithm. 8.3.4 The quantum Fourier transform. 8.3.5 Gates for the QFT. 8.4 The Grover algorithm: Looking for a needle in a haystack. 8.4.1 Oracle functions. 8.4.2 The search algorithm. 8.4.3 Geometrical analysis. 8.4.4 Quantum counting. 8.4.5 Phase estimation. 8.5 Quantum simulations. 8.5.1 Potential and limitations. 8.5.2 Motivation. 8.5.3 Simulated evolution. 8.5.4 Implementations. 9 How to build a quantum computer. 9.1 Components. 9.1.1 The network model. 9.1.2 Some existing and proposed implementations. 9.2 Requirements for quantum information processing hardware. 9.2.1 Qubits. 9.2.2 Initialization. 9.2.3 Decoherence time. 9.2.4 Quantum gates. 9.2.5 Readout. 9.3 Converting quantum to classical information. 9.3.1 Principle and strategies. 9.3.2 Example: Deutsch Jozsa algorithm. 9.3.3 Effect of correlations. 9.3.4 Repeated measurements. 9.4 Alternatives to the network model. 9.4.1 Linear optics and measurements. 9.4.2 Quantum cellular automata. 9.4.3 One way quantum computer. 10 Liquid state NMR quantum computer. 10.1 Basics of NMR. 10.1.1 System and interactions. 10.1.2 Radio frequency field. 10.1.3 Rotating frame. 10.1.4 Equation of motion. 10.1.5 Evolution. 10.1.6 NMR signals. 10.1.7 Refocusing. 10.2 NMR as a molecular quantum computer. 10.2.1 Spins as qubits. 10.2.2 Coupled spin systems. 10.2.3 Pseudo / effective pure states. 10.2.4 Single qubit gates. 10.2.5 Two qubit gates. 10.2.6 Readout. 10.2.7 Readout in multi spin systems. 10.2.8 Quantum state tomography. 10.2.9 DiVincenzo's criteria. 10.3 NMR Implementation of Shor's algorithm. 10.3.1 Qubit implementation. 10.3.2 Initialization. 10.3.3 Computation. 10.3.4 Readout. 10.3.5 Decoherence. 11 Ion trap quantum computers. 11.1 Trapping ions. 11.1.1 Ions, traps and light. 11.1.2 Linear traps. 11.2 Interaction with light. 11.2.1 Optical transitions. 11.2.2 Motional effects. 11.2.3 Basics of laser cooling. 11.3 Quantum information processing with trapped ions. 11.3.1 Qubits. 11.3.2 Single qubit gates. 11.3.3 Two qubit gates. 11.3.4 Readout. 11.4 Experimental implementations. 11.4.1 Systems. 11.4.2 Some results. 11.4.3 Problems. 12 Solid state quantum computers. 12.1 Solid state NMR/EPR. 12.1.1 Scaling behavior of NMR quantum information processors. 12.1.2 31P in silicon. 12.1.3 Other proposals. 12.1.4 Single spin readout. 12.2 Superconducting systems. 12.2.1 Charge qubits. 12.2.2 Flux qubits. 12.2.3 Gate operations. 12.2.4 Readout. 12.3 Semiconductor qubits. 12.3.1 Materials. 12.3.2 Excitons in quantum dots. 12.3.3 Electron spin qubits. 13 Quantum communication. 13.1 Quantum only tasks. 13.1.1 Quantum teleportation. 13.1.2 (Super ) Dense coding. 13.1.3 Quantum key distribution. 13.2 Information theory. 13.2.1 A few bits of classical information theory. 13.2.2 A few bits of quantum information theory. Appendix. A. Two spins 1/2: Singlet and triplet states. B. Symbols and abbreviations. Bibliography. Index.
Dissertation•
01 Jan 2004
TL;DR: In this article, the authors explore two approaches to designing quantum algorithms based on continuous-time Hamiltonian dynamics, and demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk, and construct an oracular problem that a quantum walk can solve exponentially faster than any classical algorithm.
Abstract: Quantum mechanical computers can solve certain problems asymptotically faster than any classical computing device. Several fast quantum algorithms are known, but the nature of quantum speedup is not well understood, and inventing new quantum algorithms seems to be difficult. In this thesis, we explore two approaches to designing quantum algorithms based on continuous-time Hamiltonian dynamics.
In quantum computation by adiabatic evolution, the computer is prepared in the known ground state of a simple Hamiltonian, which is slowly modified so that its ground state encodes the solution to a problem. We argue that this approach should be inherently robust against low-temperature thermal noise and certain control errors, and we support this claim using simulations. We then show that any adiabatic algorithm can be implemented in a different way, using only a sequence of measurements of the Hamiltonian. We illustrate how this approach can achieve quadratic speedup for the unstructured search problem.
We also demonstrate two examples of quantum speedup by quantum walk, a quantum mechanical analog of random walk. First, we consider the problem of searching a region of space for a marked item. Whereas a classical algorithm for this problem requires time proportional to the number of items regardless of the geometry, we show that a simple quantum walk algorithm can find the marked item quadratically faster for a lattice of dimension greater than four, and almost quadratically faster for a four-dimensional lattice. We also show that by endowing the walk with spin degrees of freedom, the critical dimension can be lowered to two. Second, we construct an oracular problem that a quantum walk can solve exponentially faster than any classical algorithm. This constitutes the only known example of exponential quantum speedup not based on the quantum Fourier transform.
Finally, we consider bipartite Hamiltonians as a model of quantum channels and study their ability to process information given perfect local control. We show that any interaction can simulate any other at a nonzero rate, and that tensor product Hamiltonians can simulate each other reversibly. We also calculate the optimal asymptotic rate at which certain Hamiltonians can generate entanglement. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)
IBM1
TL;DR: In this paper, it was shown that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy.
Abstract: We present evidence that there exist quantum computations that can be carried out in constant depth, using 2-qubit gates, that cannot be simulated classically with high accuracy. We prove that if one can simulate these circuits classically efficiently then BQP ⊆ AM.
TL;DR: A modified magnetic dipole-dipole interaction is used to correlate the proton spins in a solid sample and observe the decay of the resulting highly correlated states, and the increase of the decoherence rate with the size of the quantum register is measured.
Abstract: Among the most important parameters for the usefulness of quantum computers are the size of the quantum register and the decoherence time for the quantum information. The decoherence time is expected to get shorter with the number of correlated qubits, but experimental data are only available for small numbers of qubits. Solid-state nuclear magnetic resonance allows one to correlate large numbers of qubits (several hundred) and measure their decoherence rates. We use a modified magnetic dipole-dipole interaction to correlate the proton spins in a solid sample and observe the decay of the resulting highly correlated states. By systematically varying the number of correlated spins, we measure the increase of the decoherence rate with the size of the quantum register.
TL;DR: An experimental demonstration of teleportation of the prototypical quantum controlled-NOT (CNOT) gate is reported, assisted with linear optical manipulations, photon entanglement produced from parametric down-conversion, and postselection from the coincidence measurements.
Abstract: Teleportation of quantum gates is a critical step for the implementation of quantum networking and teleportation-based models of quantum computation. We report an experimental demonstration of teleportation of the prototypical quantum controlled-NOT (CNOT) gate. Assisted with linear optical manipulations, photon entanglement produced from parametric down-conversion, and postselection from the coincidence measurements, we teleport the quantum CNOT gate from acting on local qubits to acting on remote qubits. The quality of the quantum gate teleportation is characterized through the method of quantum process tomography, with an average fidelity of 0.84 demonstrated for the teleported gate.
TL;DR: In this article, a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projectionvalued measures used in the original theorem, is presented.
Abstract: We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal projection-valued measures used in the original theorem. The advantage of this method is that it works for two-dimensional quantum systems (qubits) and even for vector spaces over rational fields—settings where the standard theorem fails. Furthermore, unlike the method necessary for proving the original result, the present one is rather elementary. In the case of a qubit, we investigate similar results for frame functions defined upon various restricted classes of POVMs. For the so-called trine measurements, the standard quantum probability rule is again recovered.