aa r X i v : . [ qu a n t - ph ] M a y Ion-trap quantum information processing: experimental status
D. Kielpinski
Centre for Quantum Dynamics, Griffith University, Nathan QLD 4111, Australia
Atomic ions trapped in ultra-high vacuum form an especially well-understood and useful physicalsystem for quantum information processing. They provide excellent shielding of quantum infor-mation from environmental noise, while strong, well-controlled laser interactions readily providequantum logic gates. A number of basic quantum information protocols have been demonstratedwith trapped ions. Much current work aims at the construction of large-scale ion-trap quantumcomputers using complex microfabricated trap arrays. Several groups are also actively pursuingquantum interfacing of trapped ions with photons.
Contents
I. Introduction II. Ion traps for QIP III. Small ion-trap quantum registers
IV. Errors and error handling in ion-trap QIP
V. Toward large-scale ion trap quantum computing
VI. Conclusion Acknowledgments References I. INTRODUCTION
Quantum mechanics offers algorithms for efficient factorization of large numbers [1] and efficient searching of largedatabases [2], two problems that appear insoluble in classical computing. Because much modern cryptography relieson the difficulty of factoring large numbers, a large-scale quantum computer could have a large impact on many areasof technology, Internet commerce being only one example. At the same time, the delicate and demanding nature ofQIP, with every quantum accounted for, requires a more subtle technology than that used to construct a classicalcomputer. The search for physical systems supporting QIP has ranged far and wide, across optics, atomic physics,and condensed-matter physics [3, 4, 5]. Several physical implementations, namely linear optics, trapped ions, andsuperconducting electronic circuits, demonstrate the essential ingredients of QIP, including initialization to a knownquantum state, efficient readout of the quantum state, long qubit coherence time, and universal quantum logic. Inparticular, small quantum computers have already been constructed with trapped ions [6], and a number of basicQIP algorithms [7], quantum memory schemes [8, 9], and communication protocols [10] have been demonstrated. Inthis review, we discuss the experimental status and prospects of ion-trap QIP, referring to the theory of QIP and toother physical QIP implementations only for context. This is not to underestimate the excellent work in these otherareas, but only to keep the review at a manageable length. For a general overview of QIP, the reader is encouragedto consult Nielsen and Chuang’s essential text [3], and for a recent review, [5].Most quantum information processing devices are made up of two-level systems, “qubits,” where each qubitis analogous to a single bit in a classical computer. The quantum state of the device encodes information, andan appropriate unitary evolution of the state of the register can perform a computing task. In our case, a qubitcorresponds to a trapped ion, with the two qubit states being two electronic energy levels of the ion. Laser coolingof several trapped ions causes the ions to form a Coulomb crystal, in which the ions are held in the equilibriumpositions given by the combination of the trapping force and their mutual Coulomb repulsion. The techniquesof optical pumping and electron shelving, well known for decades, provide efficient initialization and readout.Trapped-ion qubits can have extremely long memory times, at least 20 seconds for optical [11] and 10 minutes formicrowave [12] transitions from the ground state. The qubit energy levels are often identical to those used for thelong-lived transitions of atomic clocks, and techniques originally developed for atomic frequency metrology haveproved invaluable in the development of ion-trap QIP.Universal quantum computation requires both single-qubit operations and a nontrivial two-qubit logic gate. Thekey to two-ion logic is the collective motion of the trapped ions, which can be cooled to the quantum ground stateand can store quantum information for hundreds of milliseconds [13, 14]. Laser coupling of the ion spin statesto this shared quantum degree of freedom allows engineering of indirect spin-spin interactions and thus two-qubitlogic gates. These techniques have enabled the demonstration of many basic QIP protocols with trapped ions[7, 10, 15, 16, 17, 18, 19].Imperfect control of the quantum information processor leads to decoherence , the loss of information throughcorrelation with an uncontrolled environment. In ion-trap QIP, decoherence arises mainly from environmentalelectromagnetic noise, from technical noise in the logic laser, and from spontaneous emission from excited electronicslevels. A decoherence-free subspace (DFS) encoding has proved resistant to the main source of environmental noise[8]. The complementary technique of quantum error correction has recently been demonstrated in principle [9], butrepeated error correction, which offers the prospect of unlimited coherence time, is still only a dream. Operation ofa large-scale quantum computer will probably require both quantum error correction and DFS encoding.No physical quantum information processor has yet demonstrated computational power anywhere near that ofpresent-day desktop computers. A viable QIP implementation must therefore be scalable . In other words, thepresent-day techniques for QIP in a given physical system must remain useful for much larger versions of thesame system without consuming a disproportionate amount of physical or computational resources. The mostwidely considered roadmap for large-scale ion-trap QIP builds on the successes of small ion-trap QIP devices byproposing a modular architecture of interconnected quantum registers [13, 20]. This roadmap is now being realisedby investigators around the world and steps along the way have already proven crucial for demonstrations of severalbasic QIP protocols.The growth of interest in QIP since the discovery of Shor’s factoring algorithm has fostered work on QIPimplementations in a wide variety of atomic, optical, and condensed-matter systems (see [5] for a recent review).There is broad consensus on the requirements for QIP: efficient initialisation and detection of qubit states, longqubit coherence times, and a set of one- and two-qubit gates that enable access to the entire Hilbert space of thequantum register. Among QIP implementations, only trapped ions, superconducting circuits, and single-photonlinear optics currently meet these criteria. Trapped ions make nearly ideal small quantum registers, but significanttechnical work is still required for large-scale quantum computing. It is easier to see how to fabricate a large numberof superconducting qubits, but decoherence times are still very short. Single photons again have long coherencetime, but no deterministic two-qubit gate has yet been demonstrated, so long chains of gates cannot be reliablyimplemented. While other physical systems may soon achieve the criteria for QIP, it is encouraging that large-scaleQIP already appears feasible (if challenging) in these three cases. II. ION TRAPS FOR QIP
Almost all experiments in ion-trap QIP so far have used radio-frequency (RF) traps to confine ions under ultra-highvacuum. In these traps, one applies large RF voltages to an electrode structure made of conducting material in orderto create a quadrupole electric field with a minimum in free space. For RF voltages of a few hundred volts and anappropriate frequency Ω RF in the MHz range, the RF field induces a deep (few eV) ponderomotive potential thatconfines ions harmonically at the field minimum with an oscillator frequency ν ≪ Ω RF [21]. In the QIP context, theresidual ion motion at the RF drive frequency, or “micromotion,” can generally be neglected once the effects of strayelectric fields have been cancelled. The low pressure ( ∼ − bar) in the vacuum chamber means that collisionsof residual gas atoms with the ions are infrequent, on the order of one every 100 seconds. The ions are then wellisolated from environmental perturbations, enabling the precise quantum state control needed for QIP experiments.One popular method for producing ions uses a resistively heated oven to generate an atomic beam of the speciesto be ionised. The atomic beam passes through the trapping region and intersects a beam of low-energy ( <
100 eV)electrons emitted from a resistively heated thoriated tungsten filament. Electron-impact ionisation produces ions inthe trapping region, and the strong, pseudo-conservative trapping force is sufficient for confining the hot ion sample,which can have a temperature of several thousand kelvin. Some experiments have replaced the oven-generated atomicbeam by laser ablation of a solid target containing the species to be ionised [22, 23]. In many cases one can replaceelectron-impact ionisation by photoionisation [24], which offers the advantage of much higher ionisation efficiencyand isotope selectivity. The latter property is particularly useful for species like Yb + [25], for which several isotopeshave high natural abundance.Linear RF traps are preferred for QIP experiments because they nearly eliminate micromotion along one motionalaxis. A simple electrode structure for a linear trap [26, 27] is shown in Fig. 1. Essentially the trap is a quadrupolemass filter plugged at the ends with static potentials. To operate the trap, one applies RF voltage to the rods 1 and 3of Fig. 1, while rods 2 and 4 are held at RF ground. The induced ponderomotive potential confines the ions to the RFnodal line, which lies along the ˆ z axis. The electrodes on the trap axis are held at a positive DC voltage relative tothe rod electrodes, pushing the (positive) ions toward the center of the trap. For motional amplitudes characteristicof laser-cooled ions, the resulting trap potential is harmonic in all three directions, with trap frequencies up to 50MHz for light ions such as Be + [28]. Because the RF electric fields at the electrode surfaces are so high, electricalbreakdown limits any attempt to radically increase the ion motional frequency [29].In traditional ion-trap experiments, the trap electrodes are fabricated from bulk metal and the electrode assemblyis clamped into a custom jig, usually of ceramic. Since the most widely accepted path to large-scale QIP with iontraps requires the fabrication of large arrays of interconnected traps [13, 20], recent experiments have moved towardmicrofabricated electrodes and quasi-planar (multilayer or single-layer) geometries. The first microfabricated trapsused laser-machined substrates, but traps based on gallium arsenide [30] and silicon [31] have now been fabricated byMEMS techniques. The design and fabrication of trap arrays has been greatly simplified by the introduction of the“surface” trap geometry [32]. In this design all electrodes lie in a plane, and a trap array of great complexity can befabricated on an insulating substrate by a single photolithographic step. Surface traps using a printed-circuit-boardsubstrate [33] allow rapid prototyping of designs for trap arrays. Silicon-based surface traps offer the possibility ofvery large scale integration of such a trap array with CMOS control circuitry [34]. A warning to novices: the intense experimental competition in this field has led some investigators to make exaggerated claims on thebasis of their results, and these claims have sometimes appeared in reputable journals. I do not believe that this paper cites any workof that kind. ionsRF RFGND GNDDCDC
FIG. 1: Electrode structure of a linear RF trap. High-voltage RF applied to the rod electrodes provides ponderomotiveconfinement of ions to the trap axis, while the endcaps are held at DC potential for axial confinement. GND: electrical ground.The electrodes marked RF have equal RF voltages at all times. The DC electrodes are held at RF ground.FIG. 2: A crystal of eight Yb + ions in the trap at Griffith University, Brisbane, Australia. The ions are cooled by a 369.5 nmdiode laser [38] and are viewed by fluorescence imaging on a CCD camera. When multiple ions are present in the trap, one must consider the Coulomb repulsion between ions as well asthe ions’ interaction with the harmonic trapping potential. If the ions are sufficiently cold, the classical equilibriumpositions of the ions are given by minimizing the potential energy. For sufficiently weak axial confinement, theequilibrium positions all lie on the trap axis x = y = 0, so that the ions line up in a string, as shown in Fig. 2.Minimizing the combined trap and Coulomb potential gives the equilibrium positions along the axis [35, 36 ? ]. Inthe case of a two-ion string, the ion-ion distance is given by ( e / (2 πǫ mν z )) / , where m is the ion mass and ν z isthe axial oscillation frequency of a single ion. In current experiments the separation between ions is on the orderof 3 to 30 µ m, with smaller separation for lower mass and for higher trap frequency. Hence it is experimentallydifficult to maintain the positioning of a focused laser beam to the tolerance required for individual laser addressingof the trapped ions. While most current ion-trap QIP experiments use readout and quantum logic schemes that aredesigned to avoid the requirement of individual laser addressing, the Innsbruck group has had remarkable success inion QIP using individual addressing ([37], and see below).After laser cooling, the residual motion of the ions is very small, so we can linearize the total potential about theequilibrium positions. The resulting harmonic oscillations constitute normal modes of the ion crystal. Solving therelevant eigenvalue equation gives the normal mode frequencies and eigenvectors (for numerical values see [36]). Forany number of ions, the lowest-frequency mode is always the center-of-mass (COM) mode, in which the ion stringmoves as a unit, with no relative motion between the ions. The Coulomb interaction then has no effect on thedynamics of the COM mode, so the COM frequency is just equal to the single-ion trap frequency ν z . Because of thesymmetry of the ion string about z = 0, the ions’ relative amplitudes of motion in a given mode are either symmetricor antisymmetric about the center of the string. flfl 〉 ›› 〉 FIG. 3: Atomic energy levels for a trapped-ion qubit. The qubit transition between |↓i and |↑i is shown in green. To detectthe qubit state, one uses a laser to repeatedly drive the transition shown in orange, causing fluorescence from the |↓i state butnot from the |↑i state. The detection laser also cools the ions to the crystalline state. One reinitialises the qubit by driving atransition from |↑i to an auxiliary energy level that decays into |↓i . The auxiliary level can be the upper state of the detectiontransition or some other excited state. The purple arrow schematically indicates this reinitialisation process.
III. SMALL ION-TRAP QUANTUM REGISTERS
Current experiments in trapped-ion QIP rely on laser excitation of electronic transitions for all QIP operations.Fig. 3 shows a typical set of atomic energy levels for an ion that is being used as a qubit. There are two relevanttransitions out of the electronic ground state |↓i . One of these is a strong optical transition whose upper state | e i decays rapidly by spontaneous emission. This transition is used for initialisation and detection of the qubit. Theother is a coherent optical or RF transition whose upper state |↑i is extremely long-lived. The states |↓i , |↑i formthe logical basis, corresponding to the role of 0 and 1 in a classical computer. In analogy to the spin-1/2 system, oneoften refers to the logical degree of freedom as the “spin”. Coherent rotations of this spin correspond to single-qubitlogic gates. Of course, all real ions have much more complex level structures than shown in Fig. 3, and so there aremany variations and complications of this basic scheme depending on the exact level structure of the ion being used,but in almost all cases, the overall goal of these schemes is to reproduce the level structure of Fig. 3. A. Readout and initialisation of ion qubits
Quantum computation requires the preparation of the computational register in a well-defined input state at thebeginning of the computation and the efficient readout of the state of the register at the end of the computation.These are accomplished with the assistance of the strongly-allowed |↓i → | e i transition of Fig. 3, referred to as thereadout transition. By choosing the correct ion species with an appropriate energy level structure, one arrangesthat the readout laser only induces fluorescence from |↓i , while the laser is far off resonance for |↑i . Then a highfluorescence rate indicates |↓i , and a low one, |↑i . This technique for high-efficiency internal state discrimination hasbeen used in studies of trapped ions for decades [39, 40, 41]. The fluorescence of the |↓i state distinguishes it fromthe |↑i state with >
98% efficiency per ion in each repetition of the experiment. The fluorescence has a total powerof picowatts per ion and is emitted nearly isotropically, so current experiments in ion QIP use complex multi-elementobjective lenses to achieve large numerical aperture (NA) and high light collection efficiency. For current opticalimaging systems, with f-number ∼ f /
1, the readout time can be as short as a few hundred microseconds [43, 44].A recent experiment reports readout error rates as low as 1 . × − [45]. Optical pumping on the same transition Recently another efficient detection method has been demonstrated when the energy levels are not appropriate for the electron shelvingtechnique [42].
Number of Photons P r ob a b ilit y (a) (b) FIG. 4: Statistics of detected photons for a single ion prepared in a) the |↑i state and b) the |↓i state. Statistics were collectedover 1000 repetitions of the experiment. Figure courtesy NIST Ion Storage Group. initialises the register with a residual error that is much smaller than the detection error and requires only a fewmicroseconds. These excellent readout and initialisation properties are key ingredients in the success of ion-trapquantum computing.Qubit readout requires us to convert an analog quantity (number of photons measured) to a digital quantity (qubitstate), so a knowledge of the photon statistics is crucial for high readout efficiency. Fig. 4 shows a histogram of thephoton statistics for a single Be + ion prepared in |↑i , and another histogram for an ion prepared in |↓i . While the |↓i histogram is Poissonian, the |↑i distribution has a long, non-Poissonian tail. The tail arises because one generallyprolongs the detection period until off-resonant repumping of the |↑i state to |↓i begins to cause photon scatteringas well [44, 46]. The two histograms are readily distinguished, and by counting each readout of > |↓i and all other readouts as |↑i , one can easily discriminate |↓i from |↑i in a single repetition of the experimentwith 98% efficiency. Recent work has investigated the adaptive readout of an ion qubit over several repetitions ofan experiment, reaching detection errors as low as 10 − , with a requirement of seven repetitions (on average) perreadout [47].In experiments with one ion, one might detect the ion in |↓i on some repetitions and in |↑i on others, for instanceby preparing the ion in a superposition state. Fig. 5 shows the photon statistics for the state |↓i + |↑i . In eachrepetition of this experiment, the ion is projected into either the state |↓i or the state |↑i , in accordance with thequantum measurement postulate. Thus the probability distribution of photon counts is a linear combination ofthat found for |↓i (Fig. 4a) and that found for |↑i (Fig. 4b), a direct verification of “wavefunction collapse” into aneigenstate of the measurement basis.For multiple ions, the simplest readout method is to collect the fluorescence from the entire ion string at once,making no attempt to spatially resolve the ions [6]. In this case one reads out N |↓i , the number of ions in |↓i ,rather than independently reading out each qubit. The photon statistics over many repetitions of the experimentis compared with theoretical distributions that depend on the mean number of photons collected per ion in |↓i , themean number of photons due to background light, and the effect of off-resonant repumping [6, 46]. This method canresolve the relative probability of the outcomes N |↓i for four ions with 98% accuracy over 1000 repetitions and hasbeen used successfully in six-ion experiments [48].Even if the ions are not spatially resolved during detection, individual laser addressing of the ions can provideindividual readout, though at the cost of considerable technical difficulty. Only the Innsbruck experiment routinelyachieves individual addressing [37], and only at the laser wavelength used for quantum logic operations, rather thanthat used for detection. In this technique [16], one performs a SWAP operation between |↓i and an auxiliary Zeemansublevel of the metastable state |↑i on all the ions except the one to be read out. Since all Zeeman sublevels of |↑i are equally dark to the detection laser, fluorescence will be detected from the ion string only if the ion to be read outis in the state |↓i . Successive SWAP operations and detection periods then read out the ions individually. Number of Photons
FIG. 5: Statistics of detected photons for a single ion prepared in the superposition state |↓i + |↑i . Statistics were collectedover 1000 repetitions of the experiment. The solid line is the best fit to the theoretical count distribution; the integral underthe peak near zero photons is approximately equal to that under the peak near 15 photons, indicating nearly equal populationsin the two states. Figure courtesy NIST Ion Storage Group. B. Single-qubit operations
Universal quantum logic requires a set of single-qubit operations that are adequately realized by coherent couplingof the ion internal states [3]. A source of coherent radiation can drive the appropriate coupling if its frequencymatches the energy splitting of the qubit states. Qubits using the hyperfine levels of the ground state (“hyperfinequbits”) are relatively easy to manipulate using commercial sources of microwave radiation, which easily achievefrequency stability on the 10 Hz scale. On the other hand, logic on qubits that use long-lived optical transitions(“optical qubits”) requires an ultrastable laser (less than 1 kHz linewidth) that can be tuned to the qubit transitionwavelength [49]. Such lasers present considerable technical challenges and are rarely found outside laboratoriesdevoted to optical frequency metrology. However, the underlying physics of the atomic interaction with the radiationfield is the same for both hyperfine and optical qubits. The same interaction again underlies nuclear magneticresonance, where the relevant basis states are often the spin states of a proton in a magnetic field, thus the term“spin” for the qubit state.In manipulating an optical qubit, one attempts to achieve the ideal case of highly monochromatic radiation inresonance with an ion transition of very narrow linewidth. In this regime the radiative interaction drives Rabiflopping between the ion internal states (spin states) [13, 50] according to the Hamiltonian H Rabi = 2 ~ Ω h e i ( − δt + ⇀ k · ⇀ x + φ L ) σ + + h.c. i (1)where σ + is a Pauli operator on the qubit state and the applied radiation has frequency ω ↓↑ + δ , wavevector ~k , Rabifrequency Ω, and phase φ L at the position of the ion, ~x . For an ion initially prepared in |↓i , the final state Ψ( t )after an interaction time t with δ = 0 is given by Ψ( t ) = cos(Ω t ) |↓i − ie iφ L sin(Ω t ) |↑i . By varying the duration of theapplied radiation, one observes a sinusoidal oscillation of the ion fluorescence with period π/ Ω. A radiation pulsethat induces a total Rabi angle Ω t = π converts |↓i to |↑i and vice versa. Such a pulse is called a π pulse, and onelikewise speaks of π/ We use the conventions of [13] in defining Ω and φ L . order of 100 W cm − to reach Rabi frequencies of hundreds of kHz, so the laser only needs to have a power of a fewmW.As will be seen below, it is convenient to use lasers to manipulate hyperfine as well as optical qubits. Usually onedrives two-photon stimulated Raman transitions between hyperfine levels, with the Raman laser beams far detunedfrom any allowed transition so that no spontaneous photon emission can occur. On two-photon resonance, adiabaticelimination of the excited state gives an effective Hamiltonian that has the same form as the Rabi Hamiltonian,Eq. (1) [13, 29]. To simplify the expression for the Raman Rabi frequency Ω R , we consider the case where onlyone atomic excited state contributes to the two-photon transition amplitude. The frequency difference between theRaman beams is much less than the detuning ∆ of the Raman laser beams from the excited state. Writing theresonant Rabi frequencies of the two Raman laser beams as Ω and Ω , we find Ω R ≡ Ω / ∆. By conservation ofmomentum, we find that ⇀ k in Eq. (1) is replaced by the difference ⇀ k − ⇀ k between the Raman beam wavevectors.Likewise φ L is replaced by the difference ϕ − ϕ between the phases ϕ , ϕ of the Raman beam electric fields,evaluated at the position of the ion. The ideal Rabi flopping behavior of optical and hyperfine qubits is identical,but in practice laser frequency noise affects the single-photon coupling of optical qubits much more than it affectsstimulated Raman coupling of hyperfine qubits, because any common-mode frequency noise of the Raman laserbeams cancels out. At a typical excited state detuning of ≈
100 GHz, a laser intensity of about 100 W cm − is againsufficient for Rabi frequencies of hundreds of kHz. C. Coherent control of ion motion
So far we have described initialization of our quantum register, operations on single qubits, and detection of theregister state. However, one more ingredient is needed to perform universal quantum logic: a gate that entangles twoparticles [51, 52, 53]. In all deterministic two-ion gates that have been experimentally demonstrated, one indirectlycreates a qubit-qubit interaction by transferring quantum information through a motional mode of the ion crystal.After Doppler cooling, the residual thermal excitation of the ions’ quantized motion still has a mean occupationnumber n ≈ −
100 for typical axial trap frequencies of 0.1 - 1 MHz. High-fidelity quantum logic operations requirefurther cooling of the ion motion to nearly n = 0, which is generally accomplished by resolved-sideband cooling[54, 55, 56], although other methods have been demonstrated [57].Resolved-sideband cooling operates on the qubit transition of Fig. 3, rather than the detection transition. Becausethe qubit transition is so long-lived, all the motional modes of the ion crystal have frequencies much larger than thetransition linewidth. In this case, the ion motion along the laser beam gives rise to sideband transitions offset fromthe |↓i - |↑i transition frequency ω ↓↑ by multiples of the trap frequencies, as can be seen from Eq. (1). The numberof observed sidebands increases with ion temperature, as expected classically. In the following, we need only considerthe first-order “red” and “blue” sidebands of each motional mode, which occur at frequencies ω ↓↑ − ν and ω ↓↑ + ν for a motional mode of frequency ν .Red sideband excitation, combined with the initialisation process (see Fig. 3), provides a convenient mechanismfor cooling to the ground state. When the ion crystal makes a red sideband transition, it loses one phonon in thecorresponding motional mode. Initialisation heats the crystal by an amount corresponding to the recoil energy ofa single photon, but the recoil energy is much smaller than the energy splitting of motional levels, so the motionalmode readily reaches the quantum ground state. Single ions have been cooled to the ground state of motion alongthe trap axis with residual thermal excitation h n i of 0.001 quanta [49]; in other words, the probability of finding theion outside the motional ground state was 0.1%. Three-ion crystals have been cooled to the axial ground state with99% efficiency [10] by successive cooling on the three motional modes.To describe coherent coupling between an ion spin and a motional mode cooled to the ground state, we return tothe Rabi Hamiltonian Eq.( 1), now considering ⇀ x as the quantum position operator of the ion. If the laser wavevectoris parallel to the trap axis, only the sidebands for motion along ˆ z are observed; many experiments use this geometryand we assume it for simplicity. It is easy to quantize the normal modes of the ion motion, since each mode is just FIG. 6: Experimentally measured Rabi oscillations on the blue sideband as a function of laser pulse duration, for the initialstate |↓i| n = 0 i . After [58]. a simple harmonic oscillator. Writing the operator for small displacements of the i th ion as x i and the conjugatemomentum as p i , we define the annihilation operator a k for the k th mode of N ions in the usual way: a k = r πN mν k N X i =1 v ( i ) k (cid:18) x i + i πN mν k p i (cid:19) (2)where ν k is the mode frequency, ⇀ v k is the normalized eigenvector of ion motional amplitudes for the mode, and m is the ion mass. The dynamics is readily expressed in terms of the Lamb-Dicke parameter η ≡ k z z , where z = (2 m ion ω z ) − / is the zero-point wavepacket spread of a single ion along the trap axis and ⇀ k is the laserwavevector. In the multi-ion case, generalised Lamb-Dicke parameters apply to the various normal modes of the ioncrystal; in particular, the center-of-mass mode for N ions always has η COM = η/ √ N . For low-error quantum logic,one must arrange that η k ≪ h n k i η k ≪ k and n k is the number operator of the i th mode.The logic lasers in ion-trap QIP experiments are either focused onto one ion at a time, or they illuminate all ionsequally. In the former case, the Rabi Hamiltonian Eq. (1) near a spin-motion resonance can be approximated as [29] H = e iφ L Ω σ j, + + h.c. δ = ω ↓↑ carrier (3)= η k e iφ L Ω σ j, + a k + h.c. δ = ω ↓↑ − ν k red sideband (4)= η k e iφ L Ω σ j, + a † k + h.c. δ = ω ↓↑ + ν k blue sideband (5)where a k is the annihilation operator of the k th normal mode and ⇀ σ j is the Pauli spin operator of the j th ion. For thecase of equal illumination, the individual spin operator σ j is replaced by the collective spin operator ⇀ J ≡ P j e iφ L,j ⇀ σ j .The collective spin is just the coherent sum of the individual ion spin operators ⇀ σ j .From Eq. (5), we see that coherent driving on a motional sideband induces Rabi flopping dynamics between thecollective spin state and a particular motional mode. The sideband Hamiltonian is that of the Jaynes-Cummingsmodel, familiar from quantum optics. By driving on the blue sideband of a single ion, one coherently couplesthe states |↓i| n i and |↑i| n + 1 i , as demonstrated experimentally in [58] (see Fig. 6). A π/ |↓i| i to the superposition state |↓i| i + |↑i| i . (Hereafter we omit normalisation factors forwavefunctions.) This state exhibits entanglement between spin and motion and the π/ {| n = 0 i , | i} [59]. Many interesting QIP tasks can be performed using only the spin/motion quantum states of a single ion, butthese have been thoroughly described elsewhere [60]. In this review, we concentrate on the use of the spin/motioncoupling as a means to create spin/spin logic gates.In dealing with hyperfine qubits, one must remember that a Raman transition has an effective wavevector ⇀ ∆ k = ⇀ k − ⇀ k that is the difference of two optical wavevectors. The effective wavelength for a Raman transition isusually in the optical range, although the transition frequency is in the microwave range. The effective wavevector ⇀ ∆ k only becomes small if the Raman laser beams are nearly copropagating. A common geometry for operations onhyperfine qubits uses Raman beams with an angular separation of 90 ◦ and with ⇀ ∆ k parallel to the trap axis ˆ z . Then0the Raman process only couples to motion along the ˆ z axis, leading to the simplification of the sideband spectrumnoted above. D. Universal quantum logic and QIP protocols
A wide variety of two-ion gates can be implemented using the collective motion, from a controlled-NOT [61, 62] toa controlled phase gate [63, 64], or any of a class of other two-qubit gates [65, 66, 67]. These gates are deterministic,i.e., they execute a desired unitary operation at a desired moment (apart from technical limitations). Althoughdeterministic gates are not absolutely necessary for large-scale QIP [68], access to deterministic gates is a keyadvantage of ion-trap quantum computing in the near term and drastically reduces the physical resources requiredto construct a large ion-trap QIP device.Here is a simple example of a quantum gate that uses the shared motion to entangle two ions, along the lines of Ciracand Zoller’s original proposal [61]. This example requires the presence of an additional qubit transition from |↓i to anauxiliary state | e i . Suppose we use a π/ |↓ i + |↑ i )( |↓ i + |↑ i ) | n k = 0 i ,where n k is the number operator on the k th motional mode. The gate then proceeds as follows:1. One performs a π pulse on the red sideband of ion 1. The red sideband has no effect on |↓ i| i , so one obtainsthe state ( |↓ i| i + |↓ i| i )( |↓ i + |↑ i ).2. One performs a 2 π pulse on the red sideband of ion 2, but on the auxiliary |↓i - | e i transition. Only the |↓ i| i state is affected by this pulse, and it is transformed to −|↓ i| i by the action of H Rabi (Eq. 1). The resultingstate is |↓ ↓ i| i + |↓ ↑ i| i − |↓ ↓ i| i + |↓ ↑ i| i .3. One performs another π pulse on the red sideband of ion 1. Again |↓ i| i is unaffected, so the state becomes |↓ i ( |↓ i + |↑ i ) + |↑ i ( |↓ i − |↑ i ). We can see that this state is entangled by reversing the π/ |↓ ↑ i − |↑ ↓ i , the Bohm-EPR state [69, 70].A more widely used class of two-qubit gates is based on the geometric phase (Berry’s phase) associated withtransporting the two-ion state around a closed path in the phase space of the motional mode used for logic [63, 71].For instance, simultaneous application of the sideband Hamiltonians of form J x a k , J x a † k produces a unitary evolutiondriven by the commutator J x . Such gates have been used in experiments to realize the entangling operators σ x σ x [66] and σ z σ z [64]. They do not require individual laser addressing of the ions and so they are widely used in QIPexperiments. They are also relatively insensitive to the motional state, an important advantage for robust quantumcomputing experiments because of both imperfect initialisation and ion heating (see Sec. ?? ). In some cases onecan generalise the two-qubit gate to provide a direct multiqubit interaction [71, 72], and up to six ions have beenentangled in this way [48]. In a more algorithmic approach, entangled states of up to eight ions have been createdusing a sequence of controlled-NOT operations [73].By combining vibrational multiqubit gates with the detection and single-qubit logic techniques outlined above, itis currently possible to construct an ion-trap quantum computer with up to four qubits and capable of implementingtens of logic gates in a computation. Many quantum computing algorithms have been demonstrated using trappedions, including the Deutsch-Josza algorithm [15], the Grover search algorithm [18], and the semiclassical Fouriertransform [7], which is the engine of Shor’s efficient factoring algorithm [1]. Likewise, several quantum communicationprotocols have been implemented, among them teleportation [10, 16], dense coding [17], and entanglement purification[19].These experiments meet the most stringent requirements for QIP with small numbers of qubits. Conceptually,the system is nearly ideal. Every qubit and all relevant motional modes are initialised with low error. Quantumgates with small errors can be performed on demand, and the long qubit coherence time allows the application of acomplex sequence of gates. The final state is easy to read out with low error. The quantum states generated in theseexperiments clearly show entanglement, ruling out classical explanations for the behavior of the quantum register.1The QIP protocols that have been demonstrated with trapped ions by no means exhaust the theoretical proposalsthat can be realised in this system. Many fascinating phenomena of quantum mechanics remain to be elucidatedwith small ion-trap quantum registers, and alternate approaches to QIP, such as one-way quantum computing [74],can also be explored with trapped ions. Quantum simulations of physical systems, especially in quantum field theory,are an attractive near-term goal for ion-trap QIP. At the same time, it is generally admitted that a new technicalapproach is needed to scale up ion-trap quantum registers to hundreds of ions and thousands of logic gates. Largeion crystals exhibit a dense sideband spectrum, and unwanted sideband excitations degrade logic gate performanceunacceptably for more than ∼
10 ions. Most experimental efforts toward large-scale ion QIP now use an array ofinterconnected ion traps, in a scheme discussed below (Sec. V A).
IV. ERRORS AND ERROR HANDLING IN ION-TRAP QIP
Large-scale QIP requires long-term storage of quantum information and low-error quantum gates. Errors inclassical digital computing are relatively easy to detect and correct by simultaneously processing many copies of thesame information. In principle, each logical bit in a classical CPU can be stored in the position state of a singleelectron, but the CPU actually uses a macroscopic number of electrons, all of which occupy the same state, to storeeach bit. Implicitly, classical computing relies on the availability of many copies of the same state. However, thedestructive nature of quantum measurement makes it impossible to reliably copy a single instance of a quantum state[75], requiring a radically different approach to error handling. The sensitivity of QIP to physical errors is currentlyaddressed by 1) purely QIP-based error-handling protocols that do not depend on the qubit implementation, 2)careful engineering of qubit encoding and gate operations to take advantage of the qubit physics, and 3) brute-forcereduction of sources of technical noise.In order to describe the effect of errors mathematically, the density matrix ρ supplants the wavefunction as thedescription of the quantum state of the system, so as to include both quantum coherence and classical (incoherent)randomness on an equal footing [76]. In a rigorous formalism, the Hamiltonians and unitary operators of fullycoherent quantum dynamics are replaced by superoperators [77]. However, one can often model the effect of a noisesource on QIP more simply by assuming that a classical control parameter, like laser power or trap frequency, isvarying in a random fashion, and taking an appropriate ensemble average over the realisations of the random cases[13]. A. QIP error handling
QIP-based error handling, and especially quantum error correction (QEC) [78, 79], is widely believed to becrucial for the operation of large-scale quantum computers. Without error handling, QIP would consist simply ofcontrolling the quantum evolution of a system so that the initial state encoded a problem, and the final state, asolution. However, complex physical systems usually display chaotic dynamics, i.e., an exponential sensitivity toerrors in the initial conditions, so that the maximum length of an accurate computation scales as the log of the errorrate. In both classical computing and QIP, frequent and repeated error correction allows accurate computation toproceed indefinitely. This property, called fault tolerance, requires gate and memory errors to remain below certainthreshold values; otherwise the process of repeated error correction induces more errors than it repairs. For particularerror-handling architectures with a high computational overhead, theoretical estimates of the threshold error canrange as high as a few percent [80], but gate errors below 10 − appear computationally desirable, though technicallychallenging. A basic QEC protocol has been demonstrated with trapped ions, the detection and correction of anion spin-flip [9], but the gate operations involved in QEC did not reach fault-tolerance, so repeated QEC was notpossible. Fault-tolerance necessarily requires an attack on the physical sources of decoherence (Sec. IV C).2Decoherence-free subspace (DFS) encoding, a useful complement to QEC, arises in the context of collectivedecoherence processes, i.e., those processes which have the same effect on each qubit. One of the most prominentdecoherence mechanisms in ion-trap QIP is the collective dephasing caused by fluctuating magnetic fields. Since theion string is very small compared to the spatial wavelength of these magnetic fields, the interaction Hamiltonian isjust proportional to J z . All states of the quantum register with the same number of spins in |↑i are degenerate witheach other. Such a degenerate subspace is called a decoherence-free subspace (DFS) [81, 82, 83]. Any superposition ofstates in this DFS is protected from the collective dephasing. The method has been demonstrated to improve memorylifetime in ion-trap QIP under ambient conditions [8]. The J x multiqubit gate, available in ion-trap QIP, enablesuniversal quantum computation on qubits encoded in the dephasing DFS [20]. The DFS concept can be generalisedto protection against collective amplitude noise [81, 83, 84], but this kind of noise does not figure prominently incurrent ion-trap QIP experiments. B. Diagnostics
To apply any error-reduction method effectively, we must clearly understand the methods of quantum statecharacterisation and their application in ion-trap QIP. A simple diagnostic for single-qubit logic gates is simply toobserve the Rabi oscillations of the initial state |↓i under increasing laser pulse duration (see Fig. 6). The slow decayof the oscillation amplitude permits an estimate of the maximum feasible number of single-qubit logic gates, whilethe form of the decay envelope can help to discriminate between various sources of technical noise [13, 29]. To samplememory errors independently from gate errors, one implements Ramsey interferometry [85] by applying a carrier π/ T , and applying another carrier π/ |↓i oscillates sinusoidally with the phase, and the contrast of the oscillations as a function of T measures the overallerror of the Ramsey sequence. Generally T is much longer than the π/ T can be attributed purely to memory error.Quantum state tomography is the general QIP method for characterisation of quantum states [3]. In this method,one measures the quantum state in several noncommuting measurement bases in order to reconstruct the densitymatrix. For a single qubit, one can obtain a complete reconstruction by measuring the spin state along σ x , σ y ,and σ z . In ion-trap experiments, one implements the measurement along σ x by performing a π/ φ = 0and subsequently measuring in the usual σ z basis; a similar procedure gives the measurement along σ y . Densitymatrices for experimentally produced two-qubit states can be readily obtained in this manner [86]. Quantum processtomography extends the tomographic idea to reconstruction of quantum gates. The superoperator correspondingto an experimental application of a gate is fully described by the action of the gate on the qubit basis states.For a two-ion gate, one performs quantum state tomography of the final state produced from |↓↓i , then for thatproduced from |↓↑i , and so on. A second tomographic reconstruction from the final-state density matrices gives thegate superoperator. The diagnostic use of process tomography has yielded experimental improvements for two-ionentangling gate implementations and also enables comparisons between various gate schemes [87].Unfortunately, quantum state tomography is resource-intensive. Measuring the state populations along σ z to 1%accuracy requires on the order of 10 experimental repetitions, even for quantum-limited detection. Tomographyof a general N -particle state requires measurement along O ( n ) noncommuting measurement bases. For eight ions,this poses a significant challenge [73], as illustrated in Fig. 7 . Quantum process tomography requires even moreresources, scaling as O ( n ) for an n -qubit gate. Although theoretical efforts are underway to simplify tomography oflarge quantum systems, many other diagnostic methods currently supplement tomographic analysis in ion-trap QIP.Multi-qubit states often lend themselves to a partial, but highly informative, characterisation by specially designedmeasurements. A classic benchmark for verifying “quantumness” of a two-qubit state is the violation of a Bellinequality [88]. Bell inequalities give upper bounds on classical two-particle correlation functions, but quantummechanics predicts that higher correlation values should be possible, and indeed they are observed in experimentson two ions [43]. Because the assumptions required to derive a Bell inequality are very general, experimental testsof Bell inequalities show that the counterintuitive features of quantum mechanics are essential for a correct physicaltheory.3 FIG. 7: Partial data from a tomographic analysis of eight ion qubits prepared in the W-state |↑↓ . . . ↓i + |↓↑ . . . ↓i + . . . + |↓↓ . . . ↑i [73]. Only the absolute value of the density matrix elements is shown. Corresponding data on the phase of the density matrixelements (not shown) completes the tomography dataset. Figure courtesy of R. Blatt and H. H¨affner, U. Innsbruck. In some cases, a special measurement is scalable in that it remains useful for entangled states involving more andmore qubits. For instance, the state |↓↓ . . . ↓i + |↑↑ . . . ↑i has been recently produced for up to six ions [48]. Ideally, thedensity matrix has only four nonzero elements: ρ ↓ N , ↓ N , ρ ↑ N , ↑ N , ρ ↓ N , ↑ N , and ρ ↑ N , ↓ N = ρ ∗↓ N , ↑ N . The diagonal populationelements ρ ↓ N , ↓ N , ρ ↑ N , ↑ N are readily measured by the usual detection methods. To measure the experimental value ofthe coherence ρ ↓ N , ↑ N , one applies a carrier π/ φ . One thendetects the number N ↓ of ions in |↓i and computes the parity Π( φ ) ≡ ( − N ↓ . Assuming an ideal carrier pulse, onefinds Π( φ ) = 2 | ρ ↓ N , ↑ N | cos N φ (6)so one extracts the desired coherence element of the density matrix by measuring the parity as a function of phase.The deviation of | ρ ↓ N , ↑ N | from its ideal value of 0.5 provides useful bounds on a wide variety of error processes. Onecan also place limits on other sources of gate error by Fourier analysis of Π( φ ). C. Physical errors
The physical error sources during quantum logic operations are quite different from the sources of “memory”errors incurred during the times when an ion is sitting idle. In general, logic errors dominate, and their sources arequite different for optical than for hyperfine qubits, while the sources of memory errors are common to both kindsof qubit. The minimum gate error reported in the refereed literature is 3%, obtained with a geometric phase gateoperating on a Raman transition between hyperfine states [64]. For a hyperfine qubit, spontaneous emission fromthe excited state in the Raman process dominates the gate error [64]. Spontaneous emission error can be reduced toan arbitrarily low value by increasing the excited-state detuning, at the cost of increased Raman laser power [89].Theoretical calculations for a number of ion species show that laser intensities of perhaps 10 W cm − and detuningsof 10 THz are required for error rates on the order of 10 − [90]. For an optical qubit, the dominant source of gateerror is usually laser frequency noise [62], which is purely technical in origin. Even so, an ultrastable laser linewidthon the order of 100 Hz can contribute as much as 10% error [86]. A recent preprint from the Innsbruck group givesan error of 7 × − , obtained by a geometric phase gate operating on an optical qubit [91], not impossibly far fromthe target error rate of 10 − commonly assumed for large-scale ion-trap QIP architectures [34, 92].Although heating of the ion motion is not a major error source during a single gate operation, recooling the ionswould require spontaneous scattering of laser light and might easily destroy the quantum coherence. In current QIP4 Beryllium (NIST)Cadmium (Michigan)Barium (IBM)Mercury (NIST)Ytterbium (PTB)Calcium (Innsbruck,Oxford) H ea t i ng r a t e [ quan t a pe r s e c ond ] Oxford) H ea t i ng r a t e [ quan t a pe r s e c ond ]
20 50 200 500100 100011010 H ea t i ng r a t e [ quan t a pe r s e c ond ] Distance to nearest trap electrode [ m m]
20 50 200 500100 1000
FIG. 8: Heating rate measurements for ion traps of different sizes d , expressed in units of motional quanta per second.The measurements were taken over the last ten years by researchers around the world, using a variety of macroscopic andmicrofabricated traps, and follow a general d − power law. The large blue dots represent data taken with a single trap withmovable electrodes, and clearly rule out the d − scaling expected for Johnson (thermal) noise [95]. The large scatter of thedata is only partly understood. Figure courtesy of C. Monroe, U. of Maryland. experiments, the thermal excitation continues to accumulate and can degrade the two-qubit logic gates toward theend of a long gate sequence, so significant efforts have been expended to understand and eliminate ion heating.Unfortunately, the exact mechanism of ion heating remains elusive, although some features of the experimental datacan be understood phenomenologically. The heating almost certainly arises from fluctuating electric fields that arenearly spatially uniform over the extent of a few-ion crystal [93, 94]. However, the observed heating rates indicatefluctuating fields that are much stronger than those expected from thermal noise [14], and the inferred spectraldensity of electric field noise scales approximately as 1 /f , rather than the constant spectral density of thermal noise.As shown in Fig. 8, the heating rate scales with trap size d as approximately d − ; this result has been rigorouslyconfirmed using a trap with movable electrodes [95]. These observations are consistent with the idea that the heatingis driven by the movement of small patches of charge on the trap electrodes, perhaps caused by adsorbed gas [13, 14].It also appears that contamination of the electrodes by the atomic beam used for loading can substantially increasethe heating rate [14, 96, 97], perhaps accounting for some of the scatter of the data in Fig. 8. Recent results showthat the ion heating is suppressed by several orders of magnitude at cryogenic temperatures [98, 99], though thethermal noise limit has not yet been observed.Memory errors are well understood in trapped-ion QIP. The dephasing of a two-level atomic system is thefundamental limitation to the precision of atomic clocks. The efforts of the metrology community over the last fewdecades have already uncovered, measured, and found ways to minimise the sources of decoherence that apply toan ion-trap quantum memory, and these methods are rapidly being adapted for trapped-ion QIP. The error ratesfor memory implementations depend on the precise choice of ion energy levels for representing the qubit states. Afundamental limit to the memory lifetime of an optical qubit is given by the radiative decay rate, of order 1 secondin current experiments, while hyperfine qubits have radiative lifetimes comparable to the age of the universe.The dominant source of memory error in current experiments is generally dephasing by ambient magnetic fieldnoise, which shifts the qubit energy levels through the Zeeman effect. Even when the quantum information is encodedin a DFS, the residual error caused by fluctuating magnetic field gradients limits memory time [11]. Magnetic fielddephasing can be suppressed by several orders of magnitude by choosing a qubit transition with vanishing first-orderZeeman shift, a fact well known by the metrology community and explicitly demonstrated in the QIP context [100].A two-ion gate compatible with such a “clock” transition has been demonstrated [101] and has been found to beresistant to laser phase fluctuations [67]. A DFS made with clock-state qubits might offer entangled states withlifetimes far in excess of the current record of 20 seconds [11] and on to the 10 minute coherence times demonstratedfor ion-trap atomic clocks [12].5 V. TOWARD LARGE-SCALE ION TRAP QUANTUM COMPUTINGA. The QCCD architecture
As we have seen, one can use a small number of trapped ions to construct a quantum register. However,manipulating a large number of ions in a single trap presents immense technical difficulties, and scaling argumentssuggest that a single trap can only support computations on tens of ions [13, 102, 103]. To build up a large-scalequantum computer, Wineland and co-workers proposed a “quantum charge-coupled device” (QCCD) architectureconsisting of a large number of interconnected ion traps [13, 20]. By adjusting the operating voltages of these traps,one can confine a few ions in each trap or shuttle ions from trap to trap. In any particular trap, one can manipulatea few ions using the methods already demonstrated, while the connections between traps allow communicationbetween sets of ions. Since both the speed of quantum logic gates [104] and the shuttling speed are limited by thetrap strength, shuttling ions between memory and interaction regions can consume an acceptably small fraction of aclock cycle. The QCCD architecture is not only a dream: the NIST group used small QCCD devices to demonstratemany of the protocols discussed in Sec. III D [7, 10, 17, 19].Figure 9 shows a schematic of an ion-trap array used in the QCCD architecture. Trapped ions storing quantuminformation are held in the memory regions. To perform a logic gate, one moves the relevant ions into an interactionregion by applying appropriate voltages to the electrode segments. In the interaction region, the ions are held closetogether, enabling the Coulomb coupling necessary for entangling gates. Lasers are focused through the interactionregion to drive the gates. Detection can occur in the interaction regions or in separate parts of the trap array.The QCCD array can be considered as a network of single-trap QIP devices, so that the architecture is essentiallymodular. The techniques used for quantum logic are those already demonstrated for single-trap quantum registers,so many of the problems of a large-scale QCCD device, for instance the problems of fabricating large trap arrays andof addressing the many interaction regions with laser beams, are easily understood in terms of classical physics andengineering [34].One can create the trapping and transport potentials needed for the QCCD using a combination of radiofrequency(RF) and quasistatic electric fields. Figure 9 shows a conceptual picture, including only the electrodes that supportthe quasistatic fields. By varying the voltages on these electrodes, we can perform ion transport and confinementalong the local trap axis, which lies along the arrows in Figure 9. Two more layers of electrodes lie above and belowthe static electrodes. Applying RF voltage to the outer layers confines the ions transverse to the local trap axis, justas for the standard linear trap. This geometry allows stable transport of the ions around T - and X -junctions, so onecan build complex, multiply connected trap structures.The first experiments toward the QCCD scheme used a trap with six axial electrode segments, giving two trappingregions separated by 1.2 mm. Single ions were reliably transported between the trapping regions in ∼ µ s, withlittle loss of qubit coherence and motional heating much less than 0.1 quantum per round trip [97]. Mastering themore complex task of splitting an ion crystal with low heating [10] enabled the NIST group to demonstrate someof the three- and four-qubit QIP protocols discussed in Sec. V A. The extension of these linear trap arrays to T -and X -junctions is challenging because residual RF fields cause unwanted trapping potentials along the ion transportdirection. Transport around a T -junction has been achieved with less than 2% error, at the cost of a massive heatingof the ion motion to a few thousand kelvin [105]. In principle, these residual RF fields can be eliminated by carefultrap design. Recent work indicates that transport speed can be increased by an order of magnitude using carefulshaping of the voltage waveforms that drive the trap electrodes [106], but it remains to be seen whether this techniqueleads to substantial heating from the quantum ground state. The QCCD architecture is named in analogy to the ubiquitous charge-coupled device (CCD) camera, which uses shuttling of electronsfor image readout. interaction regionmemoryregion electrodesegments FIG. 9: Schematic of the QCCD architecture for large-scale ion-trap QIP, after [20]. Ions (dots) are confined to the local trapaxis by RF potentials and are shuttled between memory, interaction, and detection regions by quasistatic potentials applied tothe DC electrodes (shaded boxes). For simplicity, only the DC electrodes are shown; the RF electrodes can be implemented ina variety of geometries (see text).
Even with the best trap design, it is hard to imagine that the ions will remain in the motional ground stateindefinitely, especially under transport. Although laser cooling of the qubits cannot occur during QIP, sympatheticcooling of the qubit ions by another ion species has been experimentally demonstrated [107, 108]. Confining bothspecies in the interaction region allows use of the cooling species as a heat sink, with the Coulomb interactionproviding energy transfer from the qubit ions. No decoherence need occur if the laser wavelengths relevant to thecooling species are sufficiently far detuned from the transitions of the qubit species.Additional decoherence mechanisms for the qubit states also arise from ion transport. For instance, the spatialvariations of the magnetic field strength along the transport path shift the qubit energy levels through the Zeemaneffect, so that, e.g., |↓i + |↑i → |↓i + e iα |↑i with a phase α depending on the transport path. Current experimentscompensate for these phases using spin-echo refocusing, i.e., by swapping the states |↓i and |↑i in an appropriate wayduring ion transport.7 FIG. 10: Integration of a trap array with a microfabricated array of phase Fresnel lenses. The lenses collimate ion fluorescencefrom the individual detection zones for reimaging onto detectors or optical fibers.
B. Implementing the QCCD
The QCCD architecture requires large-scale system integration of widely disparate electronic and optical techniquesin an ultra-high vacuum environment. Microfabrication of trap arrays has been discussed above in Sec. II. Low-noise,rapid ion transport demands parallel delivery of precisely shaped arbitrary voltage waveforms to all trap electrodes,a need currently met by computer-controlled digital-to-analog converters [105, 106]. However, this technology isdifficult to scale past a few tens of electrodes and integration of classical control circuitry with the trap array appearsessential in the long run [34]. Such integration would ideally enable programming of ion transport sequences bysimple digital commands from an external computer. A recently demonstrated silicon-based surface trap [109],fabricated using a CMOS-compatible process, represents a first step toward integration of classical circuitry with theQCCD.Highly parallel quantum logic and initialisation operations require that an array of laser beams be delivered tothe trap sites through switching or repositioning. Currently, laser addressing is achieved by changing the deflectionangle of an acousto-optic modulator [37]. In recent work, a micromirror array fabricated by a microelectromechanicalsystems (MEMS) process was used to steer a near-infrared laser beam, allowing laser addressing on a 5 × µ m with a switching time of ∼ µ s [110]. Similar MEMS micromirrors have been fabricated with highreflectivity at the UV wavelengths relevant to ion QIP [111]. These devices operate outside the UHV chamber,allowing wide latitude in the choice of fabrication technology.Methods for highly parallel detection of ion fluorescence have not been explored much. The objective lenses usedin current experiments have apertures ∼
50 mm and a useful field of view of < >
10 mm for lithographic exposures, but these lenses are the result of a massive engineering effort, concentratedat a few specific wavelengths, that would be difficult to replicate in the QIP community. Alternatively, one can realiseparallel detection using an array of light collection regions, one for each detection site. The author and co-workershave proposed using a microfabricated array of Fresnel lenses for this purpose [112], as shown in Fig. 10. In [112]a single phase Fresnel lens with f = 3 mm was designed for the 369.5 nm Yb + transition and was fabricated bylithography and etching of fused silica, in a scalable and UHV-compatible process. From optical tests, we predicta fluorescence collection efficiency of 4% for this lens, several times higher than current multi-element lens systems[113] and nearly sufficient for fault-tolerant QIP [92]. C. Quantum interfacing of ions and photons
Since trapped ions are essentially stationary, quantum communication between ion-trap QIP devices will requirethe coherent transfer of quantum information between ion qubits and an altogether different physical qubit imple-mentation that is better suited to long-distance travel. Single photons are a natural candidate for this quantuminterface, in view of the well-developed QIP techniques in linear optics, and since ion QIP is already heavily reliant8on optical technology. Optical quantum communication among ion-trap QIP devices may prove a useful route tolarge-scale quantum computing [114]. A near-term application for ion-trap QIP is the construction of a quantumrepeater for optical quantum cryptography over long distances [115].So far, the most complex ion-photon networking tasks have been performed by probabilistic detection of ionfluorescence. This method relies on the entanglement between the final state of a spontaneously emitting ion and thepolarisation of the emitted photon, first demonstrated in [116]. If the single photons emitted in fluorescence fromtwo widely separated ions are made to interfere at a beamsplitter, a subsequent coincident detection of the photonsprojects the ions into an entangled state. Remote entanglement of two ions in independent traps separated by ameter has been achieved [113] and shown to violate a Bell inequality [117].Figure 11 illustrates the protocol for ion-photon entanglement used in [116]. A single ion with S / groundstate and nuclear spin 1/2 is initially prepared in the | F, m f i = | , i hyperfine level. Laser excitation to the P / | F ′ = 2 , m F ′ = 1 i state is followed by spontaneous emission of a single photon. If the photon has π polarisation,the final atomic state must be | F, m F i = | , i , while for σ + polarisation the final state must be | , i . In the planeperpendicular to the quantisation axis, an emitted π ( σ + photon is vertically (horizontally) polarised. The statisticsof ion and photon measurement results under single-qubit rotations reveal the nonclassical correlations. This protocolis readily extended to remote ion-ion entanglement through the nonclassical interference of emitted photons at abeamsplitter [113].Several experiments have taken steps toward deterministic coupling between ion and photon quantum states byusing high-finesse resonators to collect a large fraction of the photons spontaneously emitted by a single ion. Thesubwavelength confinement of the ion allows fine control over the ion-photon coupling [118] and selective excitationof the vibrational sidebands that the ion imposes on the cavity mode [119]. Ion-cavity systems have been used todemonstrate stable probabilistic single-photon generation over tens of minutes [120], illustrating their suitability asrobust building blocks for QIP. However, in current experiments the single-ion cooperativity parameter C = g / (2 κγ )is on the order of one, so the coherence of the ion-photon coupling is low. Here g is the Rabi frequency of the ionwhen a single photon is present in the cavity, κ is the cavity decay rate, and γ is the decay rate of the ion transition.Proximity effects pose a major obstacle to increasing C . The short ( . µ m) resonators used for neutral-atomcavity experiments achieve high C largely because of their small mode volume. Since the single-photon energyremains constant, the electric field of a single photon grows, and so does g . This path is not available in ion-trapexperiments, making high C solely reliant on high mirror reflectivity. VI. CONCLUSION
All physical implementations of QIP are currently in their infancy. None has yet clearly demonstrated the capacityfor fault-tolerant quantum computing or the possibility of outcompeting classical computers in any way. Nevertheless,ion-trap experiments fulfill the requirements for effective small-scale QIP. Ion qubits can be initialised and read outon demand, and the deterministic quantum logic gates afforded by the motional modes enable universal quantumlogic. The low error in these basic QIP operations has allowed successful tests of many simple QIP protocols.Trapped ions appear to be an attractive system for large-scale quantum computing under the QCCD architecture.Experimental progress toward the QCCD has already advanced the state of the art in small-scale QIP. The ion-trapcommunity is successfully addressing the technical problems of scaling, namely trap array fabrication, fast laserbeam steering, and parallel detection. Demonstrations of quantum interfacing of ions with photons promise opticalquantum communication between QCCD devices. With these advantages, trapped ions will likely occupy a prominentrole in QIP for many years to come.9 |m F = 1> P |F = 1> s + (H) p (V) S |F = 1> |m F = 1>|m F = 0> FIG. 11: Protocol for generating probabilistic ion-photon entanglement used in [116]. Laser excitation is followed by spontaneousemission of a single photon whose polarisation is correlated with the final atomic state.
Acknowledgments
I thank Rainer Blatt, Wolfgang Lange, Chris Monroe, and David Wineland for helpful discussions. This work wassupported under Australian Research Council grants DP0773354 (Kielpinski) and FF0458313 (Wiseman) and by theUS Air Force under grant FA4869-08-1-4005. [1] P. W. Shor,
Proc. 35th Ann. Symp. Found. Comp. Sci. (IEEE Computer Society, Los Alamitos, CA, 1994), p. 116.[2] L. K. Grover, Phys. Rev. Lett. , 325 (1997).[3] M. A. Nielsen and I. A. Chuang, Quantum Information and Quantum Computation (Cambridge Univ. Press, Cambridge,UK, 2000).[4] R. Clark, ed.,
Experimental Implementation of Quantum Computation ’01 (Rinton, Princeton, NJ, 2001).[5] T. P. Spiller, W. J. Munro, S. D. Barrett, and P. Kok, Contemp. Phys. , 407 (2005).[6] D. Kielpinski, J. Opt. B , R121 (2003).[7] J. Chiaverini, J. Britton, D. Leibfried, E. Knill, M. D. Barrett, R. B. Blakestad, W. M. Itano, J. D. Jost, C. Langer,R. Ozeri, et al., Science , 997 (2005).[8] D. Kielpinski, V. Meyer, M. A. Rowe, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Science , 1013(2001).[9] J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett, R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost, E. Knill,C. Langer, et al., Nature , 602 (2004).[10] M. D. Barrett, J. Chiaverini, T. Schaetz, J. Britton, W. M. Itano, J. D. Jost, E. Knill, C. Langer, D. Leibfried, R. Ozeri, et al., Nature , 737 (2004).[11] H. H¨affner, F. Schmidt-Kaler, W. H¨ansel, C. F. Roos, T. K¨orber, M. Chwalla, M. Riebe, J. Benhelm, U. D. Rapol,C. Becher, et al., Appl. Phys. B , 151 (2005).[12] J. J. Bollinger, D. J. Heinzen, W. M. Itano, S. L. Gilbert, and D. J. Wineland, IEEE Trans. Instr. Meast. , 126 (1991).[13] D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. NIST , 259 (1998).[14] Q. A. Turchette et al., Phys. Rev. A , 063418 (2000).[15] S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher, J. Eschner, H. H¨affner, F. Schmidt-Kaler, I. L. Chuang, and R. Blatt,Nature , 48 (2003).[16] M. Riebe, H. H¨affner, C. F. Roos, W. H¨ansel, J. Benhelm, G. P. T. Lancaster, T. W. K¨orber, C. Becher, F. Schmidt-Kaler,D. F. V. James, et al., Nature , 734 (2004).[17] T. Schaetz, M. D. Barrett, D. Leibfried, J. Chiaverini, J. Britton, W. Itano, J. D. Jost, C. Langer, and D. J. Wineland,Phys. Rev. Lett. , 040505 (2004).[18] K.-A. Brickman, P. C. Haljan, P. J. Lee, M. Acton, L. Deslauriers, and C. Monroe, Phys. Rev. A , 050306 (2005).[19] R. Reichle, D. Leibfried, E. Knill, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J.Wineland, Nature , 838 (2006).[20] D. Kielpinski, C. Monroe, and D. J. Wineland, Nature , 709 (2002).[21] P. K. Ghosh, Ion Traps (Clarendon, Oxford, 1995).[22] R. J. Hendricks, D. M. Grant, P. F. Herskind, A. Dantan, and M. Drewsen, Appl. Phys. B , 507 (2007).[23] D. R. Leibrandt, R. J. Clark, J. Labaziewicz, P. Antohi, W. Bakr, K. R. Brown, and I. L. Chuang, Phys. Rev. A ,055403 (2007).[24] N. Kjærgaard, L. Hornekær, A. Thommesen, Z. Videsen, and M. Drewsen, Appl. Phys. B , 207 (2000).[25] C. Balzer, A. Braun, T. Hannemann, C. Paape, M. Ettler, W. Neuhauser, and C. Wunderlich, Phys. Rev. A , 041407(2006).[26] J. D. Prestage, G. J. Dick, and L. Maleki, J. Appl. Phys. , 1013 (1989).[27] M. G. Raizen, J. M. Gilligan, J. C. Bergquist, W. M. Itano, and D. J. Wineland, J. Mod. Opt. , 233 (1992).[28] S. R. Jefferts, C. Monroe, E. W. Bell, and D. J. Wineland, Phys. Rev. A , 36 (2006).[31] D. J. Wineland et al., Proc. 17th Intl. Conf. on Laser Spectroscopy (Academic Press, 2006).[32] J. Chiaverini, R. B. Blakestad, J. Britton, J. D. Jost, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, Quant. Info.Comp. , 419 (2005).[33] K. R. Brown, R. J. Clark, J. Labaziewicz, P. Richerme, D. R. Leibrandt, and I. L. Chuang, Phys. Rev. A , 015401(2007).[34] J. Kim et al., Quant. Info. Comp. , 515 (2005).[35] A. Steane, Appl. Phys. B , 623 (1997).[36] D. F. V. James, Appl. Phys. B , 191 (1998).[37] H. C. N¨agerl, D. Leibfried, H. Rohde, G. Thalhammer, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. A ,145 (1999).[38] D. Kielpinski, M. Cetina, J. A. Cox, and F. X. K¨artner, Opt. Lett , 757 (2006).[39] H. G. Dehmelt, Bull. Am. Phys. Soc , 60 (1975).[40] D. J. Wineland, J. C. Bergquist, W. M. Itano, and R. E. Drullinger, Opt. Lett , 245 (1980).[41] T. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek, Phys. Rev. Lett. , 1696 (1986).[42] M. J. McDonnell, J.-P. Stacey, S. C. Webster, J. P. Home, A. Ramos, D. Lucas, D. N. Stacey, and A. Steane, Phys. Rev.Lett. , 153601 (2004).[43] M. A. Rowe, D. Kielpinski, V. Meyer, C. A. Sackett, W. M. Itano, C. Monroe, and D. J. Wineland, Nature , 791(2001).[44] M. Acton, K.-A. Brickman, P. C. Haljan, P. J. Lee, L. Deslauriers, and C. Monroe, Quant. Info. Comp. , 120502 (2007).[48] D. Leibfried, E. Knill, S. Seidelin, J. Britton, R. B. Blakestad, J. Chiaverini, D. B. Hume, W. M. Itano, J. D. Jost,C. Langer, et al., Nature , 639 (2005).[49] C. Roos, T. Zeiger, H. Rohde, H. C. N¨agerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. , 4713 (1999).[50] L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1975).[51] D. Deutsch, A. Barenco, and A. Ekert, Proc. Roy. Soc. London A , 669 (1995).[52] S. Lloyd, Phys. Rev. Lett. , 346 (1995).[53] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter,Phys. Rev. A , 3457 (1995).[54] D. J. Wineland and W. M. Itano, Phys. Rev. A , 1521 (1979).[55] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. , 403 (1989). [56] C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, Phys. Rev. Lett. ,4011 (1995).[57] C. F. Roos, D. Leibfried, A. Mundt, F. Schmidt-Kaler, J. Eschner, and R. Blatt, Phys. Rev. Lett. , 5547 (2000).[58] D. M. Meekhof, C. Monroe, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. , 1796 (1996).[59] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. , 4714 (1995).[60] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. , 281 (2003).[61] J. I. Cirac and P. Zoller, Phys. Rev. Lett. , 4091 (1995).[62] F. Schmidt-Kaler, H. H¨affner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner,and R. Blatt, Nature , 408 (2003).[63] G. J. Milburn, S. Schneider, and D. F. V. James, Fortschr. Phys. , 801 (2000).[64] D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovi´c, C. Langer, T. Rosen-band, et al., Nature , 412 (2003).[65] A. Sørensen and K. Mølmer, Phys. Rev. Lett. , 1971 (1999).[66] C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, M. Rowe, Q. A. Turchette, W. M. Itano,D. J. Wineland, et al., Nature , 256 (2000).[67] P. C. Haljan, K.-A. Brickman, L. Deslauriers, P. J. Lee, and C. Monroe, Phys. Rev. Lett. , 153602 (2005).[68] E. Knill, R. Laflamme, and G. J. Milburn, Nature , 46 (2001).[69] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. , 777 (1935).[70] D. Bohm, Quantum Theory (Dover, 1989).[71] X. Wang, A. Sørensen, and K. Mølmer, Phys. Rev. Lett. , 3907 (2001).[72] K. Mølmer and A. Sørensen, Phys. Rev. Lett. , 1835 (1999).[73] H. H¨affner, W. H¨ansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. K¨orber, U. D. Rapol, M. Riebe, P. O.Schmidt, et al., Nature , 643 (2005).[74] R. Raussendorf and H.-J. Briegel, Phys. Rev. Lett. , 5188 (2001).[75] W. K. Wootters and W. H. Zurek, Nature , 802 (1982).[76] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994), revised ed.[77] D. F. Walls and G. J. Milburn,
Quantum Optics (Springer, New York, 2008), 3rd ed.[78] P. W. Shor, Phys. Rev. A , R2493 (1995).[79] A. M. Steane, Phys. Rev. Lett. , 793 (1996).[80] E. Knill, Nature , 39 (2005).[81] P. Zanardi and M. Rasetti, Phys. Rev. Lett. , 3306 (1997).[82] L.-M. Duan and G.-C. Guo, Phys. Rev. Lett. , 1953 (1997).[83] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. , 2594 (1998).[84] L. M. Duan and G. C. Guo, Phys. Rev. A , 737 (1998).[85] N. F. Ramsey, Molecular Beams (Oxford Univ. Press, London, 1963).[86] C. F. Roos, G. P. T. Lancaster, M. Riebe, H. H¨affner, W. H¨ansel, S. Gulde, C. Becher, J. Eschner, F. Schmidt-Kaler,and R. Blatt, Phys. Rev. Lett. , 220402 (2004).[87] M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K. K¨orber, W. H¨ansel, H. H¨affner, C. F. Roos, and R. Blatt,Phys. Rev. Lett. , 220407 (2006).[88] J. S. Bell, Physics (N.Y.) , 195 (1965).[89] R. Ozeri et al., Phys. Rev. Lett. , 030403 (2005).[90] R. Ozeri et al., Phys. Rev. A , 042329 (2007).[91] J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt (2008), arXiv:0803.2798.[92] A. M. Steane, Quant. Info. Comp. , 171 (2007).[93] D. F. V. James, Phys. Rev. Lett. , 317 (1998).[94] B. E. King et al., Phys. Rev. Lett. , 1525 (1998).[95] L. Deslauriers, P. C. Haljan, P. J. Lee, K.-A. Brickman, B. B. Blinov, M. J. Madsen, and C. Monroe, Phys. Rev. A ,043408 (2004).[96] R. G. DeVoe and C. Kurtsiefer, Phys. Rev. A , 063407 (2002).[97] M. A. Rowe, A. Ben-Kish, B. DeMarco, D. Leibfried, V. Meyer, J. Beall, J. Britton, J. Hughes, W. M. Itano, B. Jelenkovi´c,et al., Quant. Info. Comp. , 257 (2002).[98] L. Deslauriers, S. Olmschenk, D. Stick, W. K. Hensinger, J. Sterk, and C. Monroe, Phys. Rev. Lett. , 103007 (2006).[99] J. Labaziewicz, Y. Ge, P. Antohi, D. Leibrandt, K. R. Brown, and I. L. Chuang, Phys. Rev. Lett. , 013001 (2008).[100] C. Langer et al., Phys. Rev. Lett. , 060502 (2005).[101] P. C. Haljan, P. J. Lee, K.-A. Brickmann, M. Acton, L. Deslauriers, and C. Monroe, Phys. Rev. A , 062316 (2005).[102] R. J. Hughes et al., Phys. Rev. Lett. , 3240 (1996).[103] D. G. Enzer, Experimental Implementation of Quantum Computation ’01 (Rinton, Princeton, NJ, 2001), p. 99.[104] A. Steane, C. F. Roos, D. Stevens, A. Mundt, D. Leibfried, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. A , 042305(2000).[105] W. K. Hensinger, S. Olmschenk, D. Stick, D. Hucul, M. Yeo, M. Acton, L. Deslauriers, C. Monroe, and J. Rabchuk,Appl. Phys. Lett. , 034101 (2006).[106] G. Huber, T. Deuschle, W. Schnitzler, R. Reichle, K. Singer, and F. Schmidt-Kaler, New J. Phys. , 013004 (2008).[107] B. B. Blinov, L. Deslauriers, P. Lee, M. J. Madsen, R. Miller, and C. Monroe, Phys. Rev. A , 040304 (2002).[108] M. D. Barrett, B. DeMarco, T. Schaetz, V. Meyer, D. Leibfried, J. Britton, J. Chiaverini, W. M. Itano, B. Jelenkovi´c, J. D. Jost, et al., Phys. Rev. A , 042302 (2003).[109] J. Britton et al. (2006), quant-ph/0605170.[110] C. Knoernschild, C. Kim, B. Liu, F. P. Lu, and J. Kim, Opt. Lett. , 273 (2008).[111] C. Kim, C. Knoernschild, B. Liu, and J. Kim, IEEE J. Sel. Top. Quant. Elec. , 322 (2007).[112] E. W. Streed, B. G. Norton, J. J. Chapman, and D. Kielpinski, arXiv:0805.2437.[113] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe, Nature ,68 (2007).[114] L.-M. Duan, B. B. Blinov, D. L. Moehring, and C. Monroe, Quant. Info. Comp. , 165 (2004).[115] D. L. Moehring, M. J. Madsen, K. C. Younge, J. R. N. Kohn, P. Maunz, L.-M. Duan, C. Monroe, and B. B. Blinov, J.Opt. Soc. Am. B , 300 (2007).[116] B. B. Blinov, D. L. Moehring, L.-M. Duan, and C. Monroe, Nature , 153 (2004).[117] D. N. Matsukevich, P. Maunz, D. L. Moehring, S. Olmschenk, and C. Monroe, Phys. Rev. Lett. , 150404 (2008).[118] G. R. Guth¨ohrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature , 49 (2001).[119] A. B. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. ,103001 (2002).[120] M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, Nature431