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Topological Features in Ion Trap Holonomic Computation
Abstract
Topological features in quantum computing provide controllability and noise error avoidance in the performance of logical gates. While such resilience is favored in the manipulation of quantum systems, it is very hard to identify topological features in nature. This paper proposes a scheme where holonomic quantum gates have intrinsic topological features. An ion trap is employed where the vibrational modes of the ions are coherently manipulated with lasers in an adiabatic cyclic way producing geometrical holonomic gates. A crucial ingredient of the manipulation procedures is squeezing of the vibrational modes, which effectively suppresses exponentially any undesired fluctuations of the laser amplitudes, thus making the gates resilient to control errors. Comment: 9 pages, 4 figures, REVTEX
... The idea of holonomic quantum computations (HQC) was mainly developed in the work [1]. In holonomic quantum computer quantum gates are implemented as holonomies (parallel transporters along loops) in the space of external control parameters λ µ [1,2,3]. Practical implementations of such computational scheme [2,4] are based on the systems described by some degenerate hamiltonian which depends on the set of external control parameters λ µ [1,2,3]: ...
... In holonomic quantum computer quantum gates are implemented as holonomies (parallel transporters along loops) in the space of external control parameters λ µ [1,2,3]. Practical implementations of such computational scheme [2,4] are based on the systems described by some degenerate hamiltonian which depends on the set of external control parameters λ µ [1,2,3]: ...
... In holonomic quantum computer quantum gates are implemented as holonomies (parallel transporters along loops) in the space of external control parameters λ µ [1,2,3]. Practical implementations of such computational scheme [2,4] are based on the systems described by some degenerate hamiltonian which depends on the set of external control parameters λ µ [1,2,3]: ...
We investigate the influence of random errors in external control parameters on the stability of holonomic quantum computation in the case of arbitrary loops and adiabatic connections. A simple expression is obtained for the case of small random uncorrelated errors. Due to universality of mathematical description our results are valid for any physical system which can be described in terms of holonomies. Theoretical results are confirmed by numerical simulations. Comment: 7 pages, 3 figures
... The difficulties are (i) to find a quantum system in which the lowest energy eigenvalue is degenerate and (ii) to design a control which leaves the ground state degenerate as the loop is traversed. Several theoretical ideas have been proposed in linear optics [11], trapped ions [12,13,14], and Josephson junction qubits [15]. Recently, an experiment following the proposals made in [12,13], where the coding space is not the lowest eigenspace, has been reported [16]. ...
... First, we focus on X 1 and X 2 in Eq. (14). We obtain ...
... Next, let us consider X 1-2 in Eq. (14). The following observations are useful to calculate the matrix elements: ...
We exactly construct one- and two-qubit holonomic quantum gates in terms of isospectral deformations of an Ising model Hamiltonian. A single logical qubit is constructed out of two spin-1/2 particles; the qubit is a dimer. We find that the holonomic gates obtained are discrete but dense in the unitary group. Therefore an approximate gate for a desired one can be constructed with arbitrary accuracy.
... In holonomic quantum computation (HQC) proposed by Zanardi and Rasetti [1,2], in contrast, the holonomy [3] associated with adiabatic change of the parameters along a loop in a control parameter manifold is employed to implement a unitary gate. Experimental schemes to manipulate the non-Abelian holonomy have been proposed [4] and their uses to realize a unitary gate are also proposed [5,6,7]. For efficient achievement of holonomic computation, it is necessary to find a loop as short as possible in the control manifold. ...
... A realistic system has a restricted control manifold M and a control map f : M → G N,k (C). For physical realization of HQC it is required to find an optimal loop for the holonomy in the pullback bundle f * (S N,k (C)) as discussed in [2,4,5,6,7]. The exact solutions we have obtained here are pulled back by f * to be loops in M. Detailed analysis of physical realization is beyond the scope of this Letter and will be published elsewhere. ...
Holonomic quantum computation is analyzed from geometrical viewpoint. We develop an optimization scheme in which an arbitrary unitary gate is implemented with a small circle in a complex projective space. Exact solutions for the Hadamard, CNOT and 2-qubit discrete Fourier transformation gates are explicitly constructed.
... Recently, it has been pointed out [14,15] that non-Abelian holonomy may be used in the construction of universal sets of quantum gates for the purpose to achieve fault tolerant quantum computation. This has triggered further work on holonomy effects for quantum computation [16,17,18,19,20,21,22,23,24] and quantum information [25,26,27,28]. ...
... An important issue for holonomic quantum information processing is its robustness to imperfections, such as decoherence and random unitary perturbations. The effect of imperfections on the quantum gates has been analyzed [17,29,30,31], supporting the alleged robustness of holonomic quantum computation. It remains however to address the appearance of non-Abelian holonomy related to the motion of the quantal states themselves. ...
Non-Abelian holonomy in dynamical systems may arise in adiabatic transport of energetically degenerate sets of states. We examine such a holonomy structure for mixtures of energetically degenerate quantal states. We demonstrate that this structure has a natural interpretation in terms of the standard Wilczek-Zee holonomy associated with a certain class of Hamiltonians that couple the system to an ancilla. The mixed state holonomy is analysed for holonomic quantum computation using ion traps. Comment: Minor changes, journal reference added
... A possible way to choose the three ground states is that |1⟩ and | ⟩ are two degenerate Zeeman sub-levels addressed by lasers with different polarizations, and |0⟩ is the ground state with slight different energy, so that it can be addressed by a laser with a different frequency. The same structure can be realized in trapped ions with lasers that couple the vibrational mode with two internal energy levels of the atom [138], neutral atoms in a cavity QED [139], or Ising chain [140]. Interestingly, in a semiconductor-based system [141,63], the ground state of the system is used as the excited state | ⟩, while the three states of the light holes are depicted as |0⟩, |1⟩, and | ⟩. ...
Geometric and holonomic quantum computation utilize intrinsic geometric properties of quantum-mechanical state spaces to realize quantum logic gates. Since both geometric phases and quantum holonomies are global quantities depending only on evolution paths of quantum systems, quantum gates based on them possess built-in resilience to certain kinds of errors. This article provides an introduction to the topic as well as gives an overview of the theoretical and experimental progresses for constructing geometric and holonomic quantum gates and how to combine them with other error-resistant techniques.
... In these contexts, non-Abelian holonomies arise in cases where the parameter dependent Hamiltonian is degenerate and where the measured observables have degenerate eigenvalues. The former scenario has attracted considerable attention in the literature [7,8,9,10,11,12,13] and has recently been shown to be of relevance to robust quantum computation [14,15,16,17,18,19,20,21,22]. While the latter approach to non-Abelian holonomies has been discussed in the limit of dense sequences of projection measurements in Ref. [6], a detailed analysis of the genuinely discrete non-Abelian setting, analogous to Pancharatnam's original discussion [2] of the Abelian geometric phase in the context of interference of light waves transmitted by a filtering analyzer, seems still lacking. ...
Abelian and non-Abelian geometric phases, known as quantum holonomies, have attracted considerable attention in the past. Here, we show that it is possible to associate nonequivalent holonomies to discrete sequences of subspaces in a Hilbert space. We consider two such holonomies that arise naturally in interferometer settings. For sequences approximating smooth paths in the base (Grassmann) manifold, these holonomies both approach the standard holonomy. In the one-dimensional case the two types of holonomies are Abelian and coincide with Pancharatnam's geometric phase factor. The theory is illustrated with a model example of projective measurements involving angular momentum coherent states.
... This general scheme when applied to the field of quantum computation takes the name of holonomic or geometrical quantum computation and the unitary operators thus obtained are called holonomic quantum gates. The problem of exact holonomic implementation of quantum gates is of great interest in the field of quantum computation [1,2,3,4,5,6,7,8,9,10]. This is due to the fact that holonomic quantum computation, being geometrical in nature has a degree of stability against a class of errors [2,3,11]. ...
We find exact solutions for a universal set of quantum gates on a scalable candidate for quantum computers, namely an array of two level systems. The gates are constructed by a combination of dynamical and geometrical (non-Abelian) phases. Previously these gates have been constructed mostly on non-scalable systems and by numerical searches among the loops in the manifold of control parameters of the Hamiltonian. Comment: 1 figure, Latex, 8 pages, Accepted for publication in Physical Review A
... This attractive feature has been analyzed from different perspectives in the Abelian case, such as random unitary perturbations [12,13] and decoherence [14] (for similar analyses of non-Abelian geometric quantum computation, see Refs. [15,16,17,18,19]). We demonstrate fault-tolerance with respect to dissipative decay for an Abelian geometric phase shift gate. ...
In presence of dissipation, quantal states may acquire complex-valued phase effects. We suggest a notion of dissipative interferometry that accommodates this complex-valued structure and that may serve as a tool for analyzing the effect of certain kinds of external influences on quantal interference. The concept of mixed-state phase and concomitant gauge invariance is extended to dissipative internal motion. The resulting complex-valued mixed-state interference effects lead to well-known results in the unitary limit and in the case of dissipative motion of pure quantal states. Dissipative interferometry is applied to fault-tolerant geometric quantum computation. Comment: Slight revision, journal reference added
... It is seen that the Taylor expansion for fidelity does not have the linear on δλ term. The cancellation of the first order terms was noticed for the particular implementation of holonomic quantum computer on trapped ions[3]. ...
We study the stability of holonomic quantum computations with respect to errors in assignment of control parameters. The general expression for fidelity is obtaned. In the small errors limit the simple formulae for the fidelity decrease rate is derived.
... However, if we are to use solid state devices for quantum computation, we need to manipulate individual quantum systems, like electrons in quantum dots, that demand a much higher degree of control accuracy than currently available. There has been a number of recent proposals to address the issue of controlability by using geometrical and topological effects [4,5]. The key advantage of these methods is that the resulting geometrical and topological gates do not depend on the overall time of the evolution, nor on small deformations in the control parameters. ...
We present an idealized model involving interacting quantum dots that can support both
the dynamical and geometrical forms of quantum computation. We show that by
employing a structure similar to the one used in the Aharonov–Bohm effect we can
construct a topological two-qubit phase-gate that is to a large degree independent
of the exact values of the control parameters and therefore resilient to control
errors. The main components of the set-up are realizable with present technology.
In this paper, a discussion about quantum holonomies around a possible bridge between two graphene sheets has been made. That bridge is widely known as a graphene wormhole, and some of its characteristics are also showed up here. As well as their build as a zigzag junction between a baggy nanotube and the graphene lattices. And how the localized electronic states could be mimicked by gauge fields in a low-energy regime. Further, the possibility to build holonomies handle by an effective flux from topological defects in junctions of that bridge has been discussed.
In this paper, we discuss a new way to get a quantum holonomy around topological defects in [Formula: see text] fullerenes. For this, we use a Kaluza–Klein extra dimension approach. Furthermore, we discuss how an extra dimension could promote the formation of new freedom degrees which would open a discussion about a possible qubits computation.
Almost everything that happens in classical mechanics also shows up in quantum mechanics when we know where to look for it. A phenomenon in classical mechanics involves topological changes in action-angle loops as a result of passage around a “monodromy circuit.” This phenomenon is known by the short name “Hamiltonian monodromy” (or, more ponderously, “nontrivial monodromy of action and angle variables in integrable Hamiltonian systems”). In this paper, we show a corresponding change in quantum wave functions: These wave functions change their topological structure in the same way that the corresponding classical action-angle loops change.
We propose a new formulation allowing to realize the holonomic quantum computation with neutral particles with a permanent magnetic dipole moments interacting with an external electric field in the presence of a topological defect. We show that both the interaction of the electric field with the magnetic dipole moment and the presence of topological defect generate independent contributions to the geometric quantum phases which can be used to describe any arbitrary rotation on the magnetic dipole moment without using the adiabatic approximation.
We have studied the performance of a geometric phase gate with a
quantized driving field numerically, and developed an analytical
approximation that yields some preliminary insight on the way the qubit
becomes entangled with the driving field.
We show a new proposal for implementing one-qubit quantum gates in a solid associated with the presence of topological defects. We discuss a new way of obtaining quantum holonomies for a spin-half particle, and the implementation of a set of one-qubit quantum gates based on the topological phases provided by the presence of a defect in a crystalline solid.
We discuss holonomic quantum computation based on the scalar Aharonov–Bohm effect for a neutral particle. We show that the interaction between the magnetic dipole moment and external fields yields a non-abelian quantum phase allowing us to make any arbitrary rotation on a one-qubit. Moreover, we show that the interaction between the magnetic dipole moment and a magnetic field in the presence of a topological defect yields an analogue effect of the scalar Aharonov–Bohm effect for a neutral particle, and a new way of building one-qubit quantum gates.
We study quantum decoherence of single-qubit and two-qubit Aharonov-Anandan
(AA) geometric phase gates realized in a multistep scheme. Each AA gate is also
compared with the dynamical phase gate performing the same unitary
transformation within the same time period and coupled with the same
environment, which is modeled as harmonic oscillators. It is found that the
fidelities and the entanglement protection of the AA phase gates are enhanced
by the states being superpositions of different eigenstates of the
environmental coupling, and the noncommutativity between the qubit interaction
and the environmental coupling.
We investigate the decoherence effect of a bosonic bath on the Berry phase of
a spin-1/2 in a time-dependent magnetic field, without making the Markovian
approximation. A two-cycle process resulting in a pure Berry phase is
considered. The low-frequency quantum noise significantly affects the Berry
phase. In the adiabatic limit, the high-frequency quantum noise only has a
small effect. The result is also valid in some more general situations.
In this paper, we study the implementation of nonadiabatic geometrical quantum gates with in semiconductor quantum dots. Different quantum information enconding (manipulation) schemes exploiting excitonic degrees of freedom are discussed. By means of the Aharanov-Anandan geometrical phase, one can avoid the limitations of adiabatic schemes relying on adiabatic Berry phase; fast geometrical quantum gates can be, in principle, implemented.
We study the performance of an adiabatic geometric phase gate with a quantized driving field numerically and develop an analytical approximation that shows how the qubit becomes entangled with the driving field. This results in a scaling of the gate error probability versus the energy in the control field that has the same form as that found for dynamic, nonadiabatic gates, but with a prefactor that would typically be several orders of magnitude larger, because of the adiabaticity constraint. In the approximation we have used, which should be valid for sufficiently large control fields, the main source of decoherence (and hence error) is the “which path” information carried by the photons that would be radiated by the driven qubit.
A possibility of holonomic quantum computation based on the defect- mediated properties of graphite cones is discussed. Using a geometric description for the conical graphene, we demonstrate how one can construct the most important one-qubit quantum gates without invoking the adiabatic approximation. The control parameter which defines a particular qubit configuration is directly linked with the number of removed sectors in the graphene layer needed to create a particular conical configuration. Copyright (C) EPLA, 2009
Steering a quantum harmonic oscillator state along cyclic trajectories leads to a path-dependent geometric phase. Here we describe its experimental observation in an electronic harmonic oscillator. We use a superconducting qubit as a nonlinear probe of the phase, which is otherwise unobservable due to the linearity of the oscillator. We show that the geometric phase is, for a variety of cyclic paths, proportional to the area enclosed in the quadrature plane. At the transition to the nonadiabatic regime, we study corrections to the phase and dephasing of the qubit caused by qubit-resonator entanglement. In particular, we identify parameters for which this dephasing mechanism is negligible even in the nonadiabatic regime. The demonstrated controllability makes our system a versatile tool to study geometric phases in open quantum systems and to investigate their potential for quantum information processing.
We consider one possible implementation of Hadamard gate for optical and ion trap holonomic quantum computers. The expression for its fidelity determining the gate stability with respect to the errors in the single-mode squeezing parameter control is analytically derived. We demonstrate by means of this expression the cancellation of the small squeezing control errors up to the fourth order on their magnitude.
We present a scheme to study non-Abelian adiabatic holonomies for open Markovian systems. As an application of our framework, we analyze the robustness of holonomic quantum computation against decoherence. We pinpoint the sources of error that must be corrected to achieve a geometric implementation of quantum computation completely resilient to Markovian decoherence.
Control theoretic aspects of holonomic quantum computation are presented. A generating set of holonomic transformations is constructed from the product bundle arising from the two-qubit system of holonomic quantum computation with squeezed coherent states.
We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We analytically obtain both the purity of the final state and the fidelity for Hadamard gate when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate the purity and the fidelity oscillations depending on the choice of the initial superimposed state. We derive a linear formulae connecting the gate fidelity and the purity of the final state.
We study decoherence induced by stochastic squeezing control errors
considering the particular implementation of Hadamard gate on optical and ion
trap holonomic quantum computers. We find the fidelity for Hadamard gate and
compute the purity of the final state when the control noise is modeled by
Ornstein-Uhlenbeck stochastic process. We demonstrate that in contradiction to
the case of the systematic control errors the stochastic ones lead to
decoherence of the final state. In the small errors limit we derive a simple
formulae connecting the gate fidelity and the purity of the final state.
A quantum dot proposal for the implementation of topological quantum computation is presented. The coupling of the electron charge to an external magnetic field via the Aharonov-Bohm effect, combined with the control dynamics of a double dot, results in a two-qubit control phase gate. The physical mechanisms of the system are analysed in detail and the conditions for performing quantum computation resilient to control errors are outlined and found to be realisable with present technology.
We propose a spin manipulation technique based entirely on electric fields applied to acceptor states in $p$-type semiconductors with spin-orbit coupling. While interesting in its own right, the technique can also be used to implement fault-resilient holonomic quantum computing. We explicitly compute adiabatic transformation matrix (holonomy) of the degenerate states and comment on the feasibility of the scheme as an experimental technique. Comment: 5 pages
Quantum computing in terms of geometric phases, i.e. Berry or Aharonov-Anandan phases, is fault-tolerant to a certain degree. We examine its implementation based on Zeeman coupling with a rotating field and isotropic Heisenberg interaction, which describe NMR and can also be realized in quantum dots and cold atoms. Using a novel physical representation of the qubit basis states, we construct $\pi/8$ and Hadamard gates based on Berry and Aharonov-Anandan phases. For two interacting qubits in a rotating field, we find that it is always impossible to construct a two-qubit gate based on Berry phases, or based on Aharonov-Anandan phases when the gyromagnetic ratios of the two qubits are equal. In implementing a universal set of quantum gates, one may combine geometric $\pi/8$ and Hadamard gates and dynamical $\sqrt{\rm SWAP}$ gate. Comment: published version, 5 pages
A new non-perturbative method of solution of the nonlinear
Heisenberg equations in a finite-dimensional subspace is
illustrated. The method, being a counterpart of the traditional
Schrödinger picture method, is based on a finite operator
expansion into the elementary processes. It provides us with
insight into the nonlinear quantal interaction from a
different point of view. Thus, one can investigate the nonlinear
system in both pictures of quantum mechanics.
A significant development in computing has been the discovery that the computational power of quantum computers exceeds that of Turing machines. Central to the experimental realization of quantum information processing is the construction of fault-tolerant quantum logic gates. Their operation requires conditional quantum dynamics, in which one sub-system undergoes a coherent evolution that depends on the quantum state of another sub-system; in particular, the evolving sub-system may acquire a conditional phase shift. Although conventionally dynamic in origin, phase shifts can also be geometric. Conditional geometric (or 'Berry') phases depend only on the geometry of the path executed, and are therefore resilient to certain types of errors; this suggests the possibility of an intrinsically fault-tolerant way of performing quantum gate operations. Nuclear magnetic resonance techniques have already been used to demonstrate both simple quantum information processing and geometric phase shifts. Here we combine these ideas by performing a nuclear magnetic resonance experiment in which a conditional Berry phase is implemented, demonstrating a controlled phase shift gate.
We propose a scheme for measuring the Berry phase in the vibrational degree of freedom of a trapped ion. Starting from the ion in a vibrational coherent state we show how to reverse the sign of the coherent state amplitude by using a purely geometric phase. This can then be detected through the internal degrees of freedom of the ion. Our method can be applied to preparation of entangled states of the ion and the vibrational mode.
We analyse dissipation in quantum computation and its destructive impact on
efficiency of quantum algorithms. Using a general model of decoherence, we
study the time evolution of a quantum register of arbitrary length coupled with
an environment of arbitrary coherence length. We discuss relations between
decoherence and computational complexity and show that the quantum
factorization algorithm must be modified in order to be regarded as efficient
and realistic.
In this paper the idea of holonomic quantum computation is realized within quantum optics. In a non-linear Kerr medium the degenerate states of laser beams are interpreted as qubits. Displacing devices, squeezing devices and interferometers provide the classical control parameter space where the adiabatic loops are performed. This results into logical gates acting on the states of the combined degenerate subspaces of the lasers, producing any one qubit rotations and interactions between any two qubits. Issues such as universality, complexity and scalability are addressed and several steps are taken towards the physical implementation of this model.
In the holonomic approach to quantum computation information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and manipulated by the associated holonomic gates. These are realized in terms of the non-abelian Berry connection and are obtained by driving the control parameters along adiabatic loops. We show how it is possible, for a specific model, to explicitly determine the loops generating any desired logical gate, thus producing a universal set of unitary transformations. In a multi-partite system unitary transformations can be implemented efficiently by sequences of local holonomic gates. Moreover a conceptual scheme for obtaining the required Hamiltonian family, based on frequently repeated pulses, is discussed, together with a possible process whereby the initial state can be prepared and the final one can be measured. Comment: 5 pages, no figures, revtex, minor changes, version accepted by Phys. Rev A (rapid comm.)
Decoherence in quantum computers is formulated within the Semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for error-less quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and moreover, that universal quantum computation is possible within them.
We describe how two vibrational degrees of freedom of a single trapped ion can be coupled through the action of suitably-chosen laser excitation. We concentrate on a two-dimensional ion trap with dissimilar vibrational frequencies in the x- and y-directions of motion, and derive from first principles a variety of quantized two-mode couplings, concentrating on a linear coupling which takes excitations from one mode to another. We demonstrate how this can result in a state rotation, in which it is possible to transfer the motional state of the ion from say the x-direction to the y-direction without prior knowledge of that motional state. Comment: 12 pages, 5 figures, to appear in PRA
In this paper we study a model quantum register $\cal R$ made of $N$ replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for $N$ large enough, in the Hilbert space of $\cal R$ there exists a linear subspace ${\cal C}_N$ which is dynamically decoupled from the environment. The states in ${\cal C}_N$ evolve unitarily and are therefore decoherence-dissipation free. The space ${\cal C}_N$ realizes a noiseless quantum code in which information can be stored, in principle, for arbitrarily long time without being affected by errors.
We consider pump-probe spectroscopy of a single ion with a highly metastable (probe) clock transition which is monitored by using the quantum jump technique. For a weak clock laser we obtain the well known Autler-Townes splitting. For stronger powers of the clock laser we demonstrate the transition to a new regime. The two regimes are distinguished by the transition of two complex eigenvalues to purely imaginary ones which can be very different in magnitude. The transition is controlled by the power of the clock laser. For pump on resonance we present simple analytical expressions for various linewidths and line positions. Comment: 6 figures. accepted for publication in PRA
We propose a new approach to the implementation of quantum gates in which decoherence during the gate operations is strongly reduced. This is achieved by making use of an environment induced quantum Zeno effect that confines the dynamics effectively to a decoherence-free subspace. Comment: replaced with version accepted for publication in PRL
We describe in detail a general strategy for implementing a conditional
geometric phase between two spins. Combined with single-spin operations, this
simple operation is a universal gate for quantum computation, in that any
unitary transformation can be implemented with arbitrary precision using only
single-spin operations and conditional phase shifts. Thus quantum geometrical
phases can form the basis of any quantum computation. Moreover, as the induced
conditional phase depends only on the geometry of the paths executed by the
spins it is resilient to certain types of errors and offers the potential of a
naturally fault-tolerant way of performing quantum computation.
A geometrical approach to quantum computation is presented, where a non-abelian connection is introduced in order to rewrite the evolution operator of an energy degenerate system as a holonomic unitary. For a simple geometrical model we present an explicit construction of a universal set of gates, represented by holonomies acting on degenerate states.
We present an alternative scheme for the generation of a 2-qubit quantum gate interaction between laser-cooled trapped ions. The scheme is based on the AC Stark shift (lightshift) induced by laser light resonant with the ionic transition frequency. At {\it specific} laser intensities, the shift of the ionic levels allows the resonant excitation of transitions involving the exchange of motional quanta. We compare the performance of this scheme with respect to that of related ion-trap proposals and find that, for an experimental realisation using travelling-wave radiation and working in the Lamb-Dicke regime, an improvement of over an order of magnitude in the gate switching rate is possible. Comment: 12 pages, 4 figures; Main changes: a) Added quantitative gate fidelity estimate; b) Added reference to related work by Steane et al (quant-ph/0003087); other minor changes throughout. To appear in PRA
We propose a new implementation of a universal set of one- and two-qubit gates for quantum computation using the spin states of coupled single-electron quantum dots. Desired operations are effected by the gating of the tunneling barrier between neighboring dots. Several measures of the gate quality are computed within a newly derived spin master equation incorporating decoherence caused by a prototypical magnetic environment. Dot-array experiments which would provide an initial demonstration of the desired non-equilibrium spin dynamics are proposed. Comment: 12 pages, Latex, 2 ps figures. v2: 20 pages (very minor corrections, substantial expansion), submitted to Phys. Rev. A
An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of tw parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors ar caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such proces can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogue of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behav in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such state can be corrected by measurements which determine the error while yielding no information about which codeword or superpositio of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysi which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: th quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneousl many times before the computation is complete.
The subject of quantum computing brings together ideas from classical information theory, computer science, and quantum physics. This review aims to summarize not just quantum computing, but the whole subject of quantum information theory. Information can be identified as the most general thing which must propagate from a cause to an effect. It therefore has a fundamentally important role in the science of physics. However, the mathematical treatment of information, especially information processing, is quite recent, dating from the mid-20th century. This has meant that the full significance of information as a basic concept in physics is only now being discovered. This is especially true in quantum mechanics. The theory of quantum information and computing puts this significance on a firm footing, and has led to some profound and exciting new insights into the natural world. Among these are the use of quantum states to permit the secure transmission of classical information (quantum cryptography), the use of quantum entanglement to permit reliable transmission of quantum states (teleportation), the possibility of preserving quantum coherence in the presence of irreversible noise processes (quantum error correction), and the use of controlled quantum evolution for efficient computation (quantum computation). The common theme of all these insights is the use of quantum entanglement as a computational resource.
It turns out that information theory and quantum mechanics fit together very well. In order to explain their relationship, this review begins with an introduction to classical information theory and computer science, including Shannon's theorem, error correcting codes, Turing machines and computational complexity. The principles of quantum mechanics are then outlined, and the Einstein, Podolsky and Rosen (EPR) experiment described. The EPR-Bell correlations, and quantum entanglement in general, form the essential new ingredient which distinguishes quantum from classical information theory and, arguably, quantum from classical physics.
Basic quantum information ideas are next outlined, including qubits and data compression, quantum gates, the `no cloning' property and teleportation. Quantum cryptography is briefly sketched. The universal quantum computer (QC) is described, based on the Church-Turing principle and a network model of computation. Algorithms for such a computer are discussed, especially those for finding the period of a function, and searching a random list. Such algorithms prove that a QC of sufficiently precise construction is not only fundamentally different from any computer which can only manipulate classical information, but can compute a small class of functions with greater efficiency. This implies that some important computational tasks are impossible for any device apart from a QC.
To build a universal QC is well beyond the abilities of current technology. However, the principles of quantum information physics can be tested on smaller devices. The current experimental situation is reviewed, with emphasis on the linear ion trap, high-Q optical cavities, and nuclear magnetic resonance methods. These allow coherent control in a Hilbert space of eight dimensions (three qubits) and should be extendable up to a thousand or more dimensions (10 qubits). Among other things, these systems will allow the feasibility of quantum computing to be assessed. In fact such experiments are so difficult that it seemed likely until recently that a practically useful QC (requiring, say, 1000 qubits) was actually ruled out by considerations of experimental imprecision and the unavoidable coupling between any system and its environment. However, a further fundamental part of quantum information physics provides a solution to this impasse. This is quantum error correction (QEC).
An introduction to QEC is provided. The evolution of the QC is restricted to a carefully chosen subspace of its Hilbert space. Errors are almost certain to cause a departure from this subspace. QEC provides a means to detect and undo such departures without upsetting the quantum computation. This achieves the apparently impossible, since the computation preserves quantum coherence even though during its course all the qubits in the computer will have relaxed spontaneously many times.
The review concludes with an outline of the main features of quantum information physics and avenues for future research.
We seek a quantum‐theoretic expression for the probability that an unstable particle prepared initially in a well defined state ρ will be found to decay sometime during a given interval. It is argued that probabilities like this which pertain to continuous monitoring possess operational meaning. A simple natural approach to this problem leads to the conclusion that an unstable particle which is continuously observed to see whether it decays will never be found to decay!. Since recording the track of an unstable particle (which can be distinguished from its decay products) approximately realizes such continuous observations, the above conclusion seems to pose a paradox which we call Zeno’s paradox in quantum theory. The relation of this result to that of some previous works and its implications and possible resolutions are briefly discussed. The mathematical transcription of the above‐mentioned conclusion is a structure theorem concerning semigroups. Although special cases of this theorem are known, the general formulation and the proof given here are believed to be new. We also note that the known ’’no‐go’’ theorem concerning the semigroup law for the reduced evolution of any physical system (including decaying systems) is subsumed under our theorem as a direct corollary.
We show that the notion of generalized Berry phase i.e., non-abelian holonomy, can be used for enabling quantum computation. The computational space is realized by a n-fold degenerate eigenspace of a family of Hamiltonians parametrized by a manifold . The point of represents classical configuration of control fields and, for multi-partite systems, couplings between subsystem. Adiabatic loops in the control induce non trivial unitary transformations on the computational space. For a generic system it is shown that this mechanism allows for universal quantum computation by composing a generic pair of loops in .
For the quantum estimation problem only a few of the advanced techniques of classical information geometry have been developed and applied. An asymptotic theory for parameter estimation needs geometric structures which are difficult to construct in the framework of quantum information geometry. It is proposed how a phase-space information geometry can be constructed, which would interpolate between the classical and quantum ones. To motivate this proposal a short review is presented of some current results on estimation and reconstruction of quantum states.
A method is developed for calculating the effects of a strong oscillating field on two states of a quantum-mechanical system which are connected by a matrix element of the field. Explicit approximate solutions are obtained for a variety of special cases, and the results of numerical computations are given for others. The effect of an rf field on the $J=2$\rightarrow${}1$ $l$-type doublet microwave absorption lines of OCS has been studied in particular both experimentally and theoretically. Each line was observed to split into two components when the frequency of the rf field was near 12.78 Mc or 38.28 Mc, which are the frequencies separating the $J=1$ and $J=2$ pairs of levels, respectively. By measuring the rf frequency, ${$\nu${}}_{0}$, at which the microwave lines are split into two equally intense components, one may determine the separation between the energy levels. The measured value of ${$\nu${}}_{0}$ depends upon the intensity of the rf field and the form of this dependence has been calculated and found to be in good agreement with the experimental results.
The quantum Zero effect is the inhibition of transitions between quantum states by frequent measurements of the state. The inhibition arises because the measurement causes a collapse (reduction) of the wave function. If the time between measurements is short enough, the wave function usually collapses back to the initial state. We have observed this effect in an rf transition between two 9Be+ ground-state hyperfine levels. The ions were confined in a Penning trap and laser cooled. Short pulses of light, applied at the same time as the rf field, made the measurements. If an ion was in one state, it scattered a few photons; if it was in the other, it scattered no photons. In the latter case the wave-function collapse was due to a null measurement. Good agreement was found with calculations.
A single trapped ion, laser cooled into its quantum ground state of motion, may be used as a very-low-temperature detector of radio-frequency signals applied to the trap end caps. If the signal source is a resonant oscillator of sufficiently high Q, the source may also be placed in its quantum ground state by coupling to the ion. Parametric couplings may be used to cool and detect source modes other than the mode directly coupled to the ion. A theoretical analysis of these cooling and detection processes is presented, and as an example, their application to single trapped electron and proton spectroscopy is examined. Squeezing and low noise detection of one quadrature component of the source oscillation are also discussed. The techniques discussed here may lead to radio-frequency measurements of improved accuracy and sensitivity. Cooling and detection of vibrations of macroscopic oscillators also appear possible.
We propose a scheme for preparing coherent squeezed states of motion in an ion trap based on the multichromatic excitation of a trapped ion by standing- and traveling-wave light fields. The squeezed state is produced when the beat frequency between two standing-wave light fields is equal to twice the trap frequency, and is indicated by a ``dark resonance'' in the fluorescence emitted by the ion.
We report the creation of thermal, Fock, coherent, and squeezed states of motion of a harmonically bound {sup 9}Be{sup +} ion. The last three states are coherently prepared from an ion which has been initially laser cooled to the zero point of motion. The ion is trapped in the regime where the coupling between its motional and internal states, due to applied (classical) radiation, can be described by a Jaynes-Cummings-type interaction. With this coupling, the evolution of the internal atomic state provides a signature of the number state distribution of the motion. {copyright} {ital 1996 The American Physical Society.}
By generalizing a construction of Berry and Simon, it is shown that non-Abelian gauge fields arise in the adiabatic development of simple quantum mechanical systems. Characteristics of the gauge fields are related to energy splittings, which may be observable in real systems. Similar phenomena are found for suitable classical systems.
We give a possible generalization to the example in the paper of Zanardi and Rasetti (quant-ph/9904011). For this generalized one explicit forms of adiabatic connection, curvature and etc. are given. Comment: Latex file, 12 pages
Collective decoherence is possible if the departure between quantum bits is smaller than the effective wave length of the noise field. Collectivity in the decoherence helps us to devise more efficient quantum codes. We present a class of optimal quantum codes for preventing collective amplitude damping to a reservoir at zero temperature. It is shown that two qubits are enough to protect one bit quantum information, and approximately $L+ 1/2 \log_2((\pi L)/2)$ qubits are enough to protect $L$ qubit information when $L$ is large. For preventing collective amplitude damping, these codes are much more efficient than the previously-discovered quantum error correcting or avoiding codes. Comment: 14 pages, Latex
In this paper we generalize the Jaynes--Cummings Hamiltonian by making use of some operators based on Lie algebras su(1,1) and su(2), and study a mathematical structure of Rabi floppings of these models in the strong coupling regime. We show that Rabi frequencies are given by matrix elements of generalized coherent operators (quant--ph/0202081) under the rotating--wave approximation. In the first half we make a general review of coherent operators and generalized coherent ones based on Lie algebras su(1,1) and su(2). In the latter half we carry out a detailed examination of Frasca (quant--ph/0111134) and generalize his method, and moreover present some related problems. We also apply our results to the construction of controlled unitary gates in Quantum Computation. Lastly we make a brief comment on application to Holonomic Quantum Computation.
Explicit forms are given of matrix elements of generalized coherent operators based on Lie algebras su(1,1) and su(2). We also give a kind of factorization formula of the associated Laguerre polynomials.
We present a nonlocal construction of universal gates by means of holonomic (geometric) quantum teleportation. The effect of the errors from imperfect control of the classical parameters, the looping variation of which builds up holonomic gates, is investigated. Additionally, the influence of quantum decoherence on holonomic teleportation used as a computational primitive is studied. Advantages of the holonomic implementation with respect to control errors and dissipation are presented. Comment: 5 pages, 2 figures, REVTEX, title changed, typos corrected
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
The mathematical problem of localizing modular functors to neighborhoods of points is shown to be closely related to the physical problem of engineering a local Hamiltonian for a computationally universal quantum medium. For genus $=0$ surfaces, such a local Hamiltonian is mathematically defined. Braiding defects of this medium implements a representation associated to the Jones polynomial and this representation is known to be universal for quantum computation.
We show that the topological modular functor from Witten-Chern-Simons theory is universal for quantum computation in the sense a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor's state space. A computational model based on Chern-Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation, have topological implications which will be considered elsewhere.
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:
1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.
2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
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