Showing papers in "Quantum Information Processing in 2002"

An Example of the Difference Between Quantum and Classical Random Walks

Abstract: In this note, we discuss a general definition of quantum random walks on graphs and illustrate with a simple graph the possibility of very different behavior between a classical random walk and its quantum analog. In this graph, propagation between a particular pair of nodes is exponentially faster in the quantum case. PACS: 03.67.Hk

Nonlocal Properties of Two-Qubit Gates and Mixed States, and the Optimization of Quantum Computations

Abstract: Entanglement of two parts of a quantum system is a nonlocal property unaffected by local manipulations of these parts. It can be described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of nonlocal properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by single-qubit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete. Similarly, nonlocal, entangling properties of two-qubit unitary gates are invariant under single-qubit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-qubit gates). PACS: 03.67-a; 03.67.Lx

Quantum Random Walks in One Dimension

Abstract: This letter treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The dependence of the mth moment on U and initial qubit state p is clarified. A new type of limit theorems for the quantum walk is given. Furthermore necessary and sufficient conditions for symmetry of distribution for the quantum walk is presented. Our results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk. PACS: 03.67.Lx; 05.40.Fb; 02.50.Cw

Approximate Quantum Error Correction

Abstract: The errors that arise in a quantum channel can be corrected perfectly if and only if the channel does not decrease the coherent information of the input state. We show that, if the loss of coherent information is small, then approximate quantum error correction is possible. PACS: 03.67.H, 03.65.U

Quantum Pattern Recognition

Abstract: I review and expand the model of quantum associative memory that I have recently proposed. In this model binary patterns of n bits are stored in the quantum superposition of the appropriate subset of the computational basis of n qbits. Information can be retrieved by performing an input-dependent rotation of the memory quantum state within this subset and measuring the resulting state. The amplitudes of this rotated memory state are peaked on those stored patterns which are closest in Hamming distance to the input, resulting in a high probability of measuring a memory pattern very similar to it. The accuracy of pattern recall can be tuned by adjusting a parameter playing the role of an effective temperature. This model solves the well-known capacity shortage problem of classical associative memories, providing a large improvement in capacity. PACS: 03.67.-a

Coins, Quantum Measurements, and Turing's Barrier

Abstract: Is there any hope for quantum computing to challenge the Turing barrier, i.e., to solve an undecidable problem, to compute an uncomputable function? According to Feynman's '82 argument, the answer is negative. This paper re-opens the case: we will discuss solutions to a few simple problems which suggest that quantum computing is theoretically capable of computing uncomputable functions. Turing proved that there is no “halting (Turing) machine” capable of distinguishing between halting and non-halting programs (undecidability of the Halting Problem). Halting programs can be recognized by simply running them; the main difficulty is to detect non-halting programs. In this paper a mathematical quantum “device” (with sensitivity e) is constructed to solve the Halting Problem. The “device” works on a randomly chosen test-vector for T units of time. If the “device” produces a click, then the program halts. If it does not produce a click, then either the program does not halt or the test-vector has been chosen from an undistinguishable set of vectors Fe, T. The last case is not dangerous as our main result proves: the Wiener measure of Fe, T constructively tends to zero when T tends to infinity. The “device”, working in time T, appropriately computed, will determine with a pre-established precision whether an arbitrary program halts or not. Building the “halting machine” is mathematically possible. To construct our “device” we use the quadratic form of an iterated map (encoding the whole data in an infinite superposition) acting on randomly chosen vectors viewed as special trajectories of two Markov processes working in two different scales of time. The evolution is described by an unbounded, exponentially growing semigroup; finally a single measurement produces the result. PACS: 03.67.Lx

Entanglement, Quantum Phase Transitions, and Density Matrix Renormalization

Abstract: We investigate the role of entanglement in quantum phase transitions, and show that the success of the density matrix renormalization group (DMRG) in understanding such phase transitions is due to the way it preserves entanglement under renormalization. We provide a reinterpretation of the DMRG in terms of the language and tools of quantum information science which allows us to rederive the DMRG in a physically transparent way. Motivated by our reinterpretation we suggest a modification of the DMRG which manifestly takes account of the entanglement in a quantum system. This modified renormalization scheme is shown, in certain special cases, to preserve more entanglement in a quantum system than traditional numerical renormalization methods. PACS: 03.65.Ud, 73.43.Nq, 05.10.-a

Path Integration on a Quantum Computer

Abstract: We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j-k with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an e-approximation to path integrals whose integrands are at least Lipschitz. We prove: • Path integration on a quantum computer is tractable. • Path integration on a quantum computer can be solved roughly e-1 times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. • The number of quantum queries needed to solve path integration is roughly the square root of the number of function values needed on a classical computer using randomization. More precisely, the number of quantum queries is at most 4.46 e-1. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. • The number of qubits is polynomial in e-1. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands. PACS: 03.67.Lx; 31.15Kb; 31.15.-p; 02.70.-c

A Quantum Treatment of Public Goods Economics

Abstract: Quantum generalizations of conventional games broaden the range of available strategies, which can help improve outcomes for the participants. With many players, such quantum games can involve entanglement among many states which is difficult to implement, especially if the states must be communicated over some distance. This paper describes a quantum approach to the economically significant n-player public goods game that requires only two-particle entanglement and is thus much easier to implement than more general quantum mechanisms. In spite of the large temptation to free ride on the efforts of others in the original game, two-particle entanglement is sufficient to give near optimal expected payoff when players use a simple mixed strategy for which no player can benefit by making different choices. This mechanism can also address some heterogeneous preferences among the players. PACS: 03.67-a; 02.50Le; 89.65.Gh

Bang–Bang Operations from a Geometric Perspective

Abstract: Strong, fast pulses, called “bang–bang” controls can be used to eliminate the effects of system-environment interactions. This method for preventing errors in quantum information processors is treated here in a geometric setting which leads to an intuitive perspective. Using this geometric description, we clarify the notion of group symmetrization as an averaging technique, provide a geometric picture for evaluating errors due to imperfect bang–bang controls and give conditions for the compatibility of BB operations with other controlling operations. This will provide additional support for the usefulness of such controls as a means for providing more reliable quantum information processing. PACS: 0.365.Yz, 03.67.Lx, 03.67-a

Quantum Computation with Abelian Anyons

Abstract: A universal quantum computer can be constructed using abelian anyons. Two qubit quantum logic gates such as controlled-NOT operations are performed using topological effects. Single-anyon operations such as hopping from site to site on a lattice suffice to perform all quantum logic operations. Anyonic quantum computation might be realized using quasiparticles of the fractional quantum Hall effect. PACS: 03.65-Lx

A priori Probability That Two Qubits Are Unentangled

Abstract: In a previous study (P. B. Slater, Eur. Phys. J. B. 17, 471 (2000)), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m ≤ 4), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 × 4 density matrices of Tilma, Byrd and Sudarshan, we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits—the total SD volume of which is known to be π8 / 1680 = π8/24 ⋅ 3 ⋅ 5 ⋅ 7 a 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of 0.416302 for the SD volume of separable states. We conjecture that this is but an approximation to π6/2310 = π6 / (2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11) a 0.416186. The ratio of the two volumes, 8/11π22 a .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142π7/12285 = 2 ⋅ 71π7/33 ⋅ 5 ⋅ 7 ⋅ 13 a 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states. PACS: 03.67.-; 03.65.Ud; 02.60.Jh; 02.40.Ky

Experimental Concatenation of Quantum Error Correction with Decoupling

Abstract: We experimentally explore the reduction of decoherence via concatenating quantum error correction (QEC) with decoupling in liquid-state NMR quantum information processing. Decoupling provides an efficient means of suppressing decoherence from noise sources with long correlation times, and then QEC can be used more profitably for the remaining noise sources. PACS: 03.67.Lx, 03.65.Bz

Quantum Computing Using Quadrupolar Spins in Solid State NMR

Abstract: Nuclear magnetic resonance (NMR) is a successful method for experimental implementation of quantum information processing. Most of the successful NMR quantum processors are small molecules in liquid state. In this case each spin half particle represents a qubit. Another approach is the usage of higher spin particles as multi qubit systems. We present the first solid state virtual 2-Qubit system, represented by the spin-3/2 nucleus 23Na in a NaNO3 single crystal. For this system we show how to create the pseudo pure states and we derive a set of propagators and logic gates corresponding to the selective excitation of single quantum transitions. With this set, the preparation of an “entangled” state is experimentally verified by state tomography, adjusted to the spin-3/2 system. PACS: 0.367Lx; 76.60-k

Fidelity Decay Saturation Level for Initial Eigenstates

Abstract: We show that the fidelity decay between an initial eigenstate evolved under a unitary chaotic operator and the same eigenstate evolved under a perturbed operator saturates well before the 1/N limit expected for a generic initial state, where N is the dimension of the Hilbert space. We provide a theoretical argument and numerical evidence that, for perturbations of intermediate strength, the saturation level depends quadratically on the perturbation strength. PACS: 05.45.Mt; 03.67.Lx

Theory of Single Spin Detection with STM

Abstract: We propose a mechanism for detection of a single spin center on a non-magnetic substrate. In the detection scheme, the STM tunnel current is correlated with the spin orientation. In the presence of magnetic field, the spin precesses and the tunnel current is modulated at the Larmor frequency. The mechanism relies on the effective spin-orbit interaction between the injected unpolarized STM current and the local spin center, which leads to the nodal structure of the spatial signal profile. Based on the proposed mechanism, the strongest spin-related signal can be expected for the systems with large spin-orbit coupling and low carrier concentration. PACS: 74.40.Gk; 72.70.+m; 73.63.Kv; 85.65.+h

On the Generation of Sequential Unitary Gates from Continuous Time Schrödinger Equations Driven by External Fields

Abstract: In most of the proposals for quantum computers, a common feature is that the quantum circuits are expected to be made of cascades of unitary transformations acting on the quantum states. Such unitary gates are normally assumed to belong to a given discrete set of transformations. However, arbitrary superposition of quantum states may be achieved by utilizing a fixed number of transformations, each depending on a parameter. A framework is proposed to dynamically express these parameters directly in terms of the control inputs entering into the continuous time forced Schrouml;dinger equation. PACS: 03.67.Lx; 03.65.Fd; 02.30.Mv; 02.30.Xy

The Real Density Matrix

Abstract: We introduce a nonsymmetric real matrix which contains all the information that the usual Hermitian density matrix does, and which has exactly the same tensor product structure. The properties of this matrix are analyzed in detail in the case of multi-qubit (e.g., spin = 1/2) systems, where the transformation between the real and Hermitian density matrices is given explicitly as an operator sum, and used to convert the essential equations of the density matrix formalism into the real domain. PACS: 03.65.Ca; 03.67-a; 33.25.+k; 02.10.Xm

Qubit Geometry and Conformal Mapping

Abstract: Identifying the Bloch sphere with the Riemann sphere (the extended complex plane), we obtain relations between single qubit unitary operations and Mobius transformations on the extended complex plane. PACS: 03.67.-a, 03.67.Lx, 03.67.Hk

Quantum Algorithms for Highly Structured Search Problems

Abstract: We consider the problem of identifying a base k string given an oracle which returns information about the number of correct components in a query, specifically, the Hamming distance between the query and the solution, modulo r = max{2, 6 − k}. Classically this problem requires Ω(n logrk) queries. For k ∈ {2, 3, 4}, we construct quantum algorithms requiring only a single quantum query. For k > 4, we show that O(√k) quantum queries suffice. In both cases the quantum algorithms are optimal. PACS: 03.67.Lx

The Relative Entropy of Entanglement of Schmidt Correlated States and Distillation

Abstract: We show that a mixed state ρ = ∑mnamn|m〉〈n| can be realized by an ensemble of pure states {pk, |pk〉} where |\phi_k\rangle\ = \sum_m \sqrt{a_mm} e^{i\theta^k_m}|m\rangle. Employing this form, we discuss the relative entropy of entanglement of Schmidt correlated states. Also, we calculate the distillable entanglement of a class of mixed states. PACS: 03.67.-a; 03.65.Bz; 03.65.Ud

Quantum Codes for Simplifying Design and Suppressing Decoherence in Superconducting Phase-Qubits

Abstract: We introduce simple qubit-encodings and logic gates which eliminate the need for certain difficult single-qubit operations in superconducting phase-qubits, while preserving universality. The simplest encoding uses two physical qubits per logical qubit. Two architectures for its implementation are proposed: one employing N physical qubits out of which N/2 are ancillas fixed in the v1〉 state, the other employing N/2+1 physical qubits, one of which is a bus qubit connected to all others. Details of a minimal set of universal encoded logic operations are given, together with recoupling schemes, that require nanosecond pulses. A generalization to codes with higher ratio of number of logical qubits per physical qubits is presented. Compatible decoherence and noise suppression strategies are also discussed. PACS: 03.67.Lx; 85.25.Hv; 03.67.-a; 89.70.+c

Natural Majorization of the Quantum Fourier Transformation in Phase-Estimation Algorithms

Abstract: We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms. Our result relies on the fact that states which are mixed by Hadamard operators at any stage of the computation only differ by a phase. This property is a consequence of the structure of the initial state and of the QFT, based on controlled-phase operators and a single action of a Hadamard gate per qubit. The detail of our proof shows that Hadamard gates sort the probability distribution associated to the quantum state, whereas controlled-phase operators carry all the entanglement but are immaterial to majorization. We also prove that majorization in phase-estimation algorithms follows in a most natural way from unitary evolution, unlike its counterpart in Grover's algorithm. PACS: 03.67.-a, 03.67.Lx

Language Is Physical

Abstract: Some aspects of the physical nature of language are discussed. In particular, physical models of language must exist that are efficiently implementable. The existence requirement is essential because without physical models no communication or thinking would be possible. Efficient implementability for creating and reading language expressions is discussed and illustrated with a quantum mechanical model. The reason for interest in language is that language expressions can have meaning, either as an informal language or as a formal language associated with mathematical or physical theories. It is noted that any universally applicable physical theory, or coherent theory of physics and mathematics together, includes in its domain physical models of expressions for both the informal language used to discuss the theory and the expressions of the theory itself. It follows that there must be some formulas in the formal theory that express some of their own physical properties. The inclusion of intelligent systems in the domain of the theory means that the theory, e.g., quantum mechanics, must describe, in some sense, its own validation. Maps of language expressions into physical states are discussed. A spin projection example is discussed as are conditions under which such a map is a Godel map. The possibility that language is also mathematical is very briefly discussed. PACS: 03.67−a; 03.65.Ta; 03.67.Lx

How to Steer a Quantum System over a Schrödinger Bridge

Abstract: A new approach to the steering problem for the Schrodinger equation relying on stochastic mechanics and on the theory of Schrodinger bridges is presented. Given the initial and final states ψ0 and ψ1, respectively, the desired quantum evolution is constructed with the aid of a reference quantum evolution. The Nelson process corresponding to the latter evolution is used as reference process in a Schrodinger bridge problem with marginal probability densities vψ0v2 and vψ1v2. This approach is illustrated by working out a simple Gaussian example. PACS: 03.65.-w

How Do Two Observers Pool Their Knowledge About a Quantum System

Abstract: In the theory of classical statistical inference one can derive a simple rule by which two or more observers may combine independently obtained states of knowledge together to form a new state of knowledge, which is the state which would be possessed by someone having the combined information of both observers. Moreover, this combined state of knowledge can be found without reference to the manner in which the respective observers obtained their information. However, we show that in general this is not possible for quantum states of knowledge; in order to combine two quantum states of knowledge to obtain the state resulting from the combined information of both observers, these observers must also possess information about how their respective states of knowledge were obtained. Nevertheless, we emphasize this does not preclude the possibility that a unique, well motivated rule for combining quantum states of knowledge without reference to a measurement history could be found. We examine both the direct quantum analog of the classical problem, and that of quantum state-estimation, which corresponds to a variant in which the observers share a specific kind of prior information. PACS: 03.67.-a, 02.50.-r, 03.65.Bz

On States, Channels, and Purification

Abstract: In this note we introduce purification for a pair (ρ, Φ), where ρ is a quantum state and Φ is a channel, which allows in particular a natural extension of the properties of related information quantities (mutual and coherent informations) to the channels with arbitrary input and output spaces. PACS: 03.67.Hk

Thermodynamic Interpretation of the Quantum Error Correcting Criterion

Abstract: Shannon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannon's 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criterion of necessary and sufficient conditions for the errors to be detectable and correctable. We show that this criterion has a thermodynamical interpretation. PACS: 03.67; 05.30; 63.10

Data Compression Limit for an Information Source of Interacting Qubits

Abstract: A system of interacting qubits can be viewed as a non-iid quantum information source A possible model of such a source is provided by a quantum spin system, in which spin-1/2 particles located at sites of a lattice interact with each other We establish the limit for the compression of information from such a source and show that asymptotically it is given by the von Neumann entropy rate Our result can be viewed as a quantum ana-logue of Shannon's noiseless coding theorem for a class of non-iid quantum informa-tion sources From the probabilistic point of view it is an analog of the Shannon-McMillan-Breiman theorem considered as a cornerstone of modern Information Theory PACS: 0367-a; 0367Lx

Liouville Invariance in Quantum and Classical Mechanics

Abstract: The density-matrix and Heisenberg formulations of quantum mechanics follow—for unitary evolution—directly from the Schrodinger equation. Nevertheless, the symmetries of the corresponding evolution operator, the Liouvillian L = i[⋅, H], need not be limited to those of the Hamiltonian H. This is due to L only involving eigenenergy differences, which can be degenerate even if the energies themselves are not. Remarkably, this possibility has rarely been mentioned in the literature, and never pursued more generally. We consider an example involving mesoscopic Josephson devices, but the analysis only assumes familiarity with basic quantum mechanics. Subsequently, such L-symmetries are shown to occur more widely, in particular also in classical mechanics. The symmetry's relevance to dissipative systems and quantum-information processing is briefly discussed. PACS: 03.65.-w, 03.67.-a, 45.20.Jj, 74.50.+r