24.1.1 History of Single Ion Trapping

The idea for dynamic trapping in alternating fields was conceived by Paul et al. in 1953 (1) and rewarded with the Nobel prize in 1989 (2). As static electric fields do not allow for trapping of charged particles, his invention was to employ an oscillating quadrupole field, which can result in bounded and stable trajectories. This realized a possibility to confine charged particles in deep potential at long trapping times, which has led to experimental progress in many fields of physics. While Paul was probably inspired by the alternating focusing and defocusing elements typically used in storage rings for high‐energy and nuclear‐physics research, his invention finally initiated a rapid development in ultra‐cold atomic and molecular quantum physics. A further driving force of this field was the development of tunable laser sources, which led to the invention of laser cooling for neutral atoms by Hänsch and Schawlow (3), and at the same time for trapped ions by Wineland and Dehmelt (4).

Experiments with single trapped and laser‐cooled ions started as early as 1980, when the fluorescence of one single laser‐cooled Ba ion was observed (5). At that time, research was motivated by two goals: To beat the existing limitations in the spectroscopic precision of atomic clock transitions, and to manipulate and observe a single atomic system in well‐controlled interaction with an optical laser field – Gedankenexperimente of quantum optics became experimental reality. This triggered a stimulating contact between theory and experiments in quantum optics and atomic physics. Highlights of that era include the demonstration of quantum jumps with a single ion (6,7), the investigation of photon antibunching in emitted resonance fluorescence light (8), and the demonstration of the coherent dynamics of a driven two‐level system, realizing the famous Jaynes–Cummings model. In this model of a two‐level atomic system coupled with the equidistant ladder of states of a harmonic oscillator, the interaction of both systems leads to a periodic exchange of excitation, known as Rabi oscillations. The spin‐1/2 system in nuclear magnetic resonance experiments by Felix Bloch, Isidor Rabi, and Edward Hahn is equivalent to an atomic two‐level system. But, only very few systems at that time reached the conditions of “strong coupling,” where the coherent interaction strength exceeds the dissipative rates such as spontaneous atomic decay: Only in 1990, experiments with Rydberg atoms interacting with superconducting cavities that support a set of quantized electromagnetic field modes (9–11) could reach this regime, at the same time, the equivalent coherent Jaynes–Cummings dynamics was demonstrated with a laser‐driven single ion trapped in a harmonic potential resulting in quantized eigenmodes of vibration (12). Laser cooling of a single trapped ion into the vibrational ground state of motion was demonstrated (13), and this degree of control for both the motional degree of freedom and the two‐level system paved the way for seminal quantum experiments: Using a single trapped ion, the NIST, Boulder group led by David Wineland was able to create a Schrödinger cat state of a trapped ion's oscillatory motion (14). Serge Haroche and his team were able to prepare a Schrödinger cat state of the photon field of a cavity and observe its successive decay – decoherence due to coupling to the environment (15). The 2012 Noble prize for physics was awarded to both of them “for ground‐breaking experimental methods that enable measuring and manipulation of individual quantum systems” (16). This groundbreaking success stimulated both theoretical concepts and proposals as well as experimental progress for quantum computing with trapped ions.

24.1.2 History of Quantum Computing

The very idea of quantum computing, when pioneered by Manin (17,18), Feynman (19,20), and later by Deutsch (21) in the 1980s, was known to only a small number of insiders. The interest in quantum computing strongly increased when Peter Shore invented the factorization algorithm (22), as the important implications for modern data encryption systems were obvious. In 1994, at the occasion of the 14th International Conference on Atomic Physics, Ekert (23) brought this factorization algorithm (24) into discussion. A quantum bit, or qubit, allows for encoding superposition of information in an atomic two‐level system. Logic gate operations (quantum gates) control the state of single or multiple qubits. Such operations can generate multiqubit entanglement. The need for a clear concept of an experimental platform based on quantum optics fostered many theoretical and experimental research activities. And precisely, this first “blueprint” of a future quantum computer was given by Ignacio Cirac and Peter Zoller in their seminal proposal for a quantum‐logic two‐qubit gate in 1995: Each ion in a linear crystal of ions stores one bit of quantum information in two long‐lived electronic levels, referred to as and . Quantum gates are implemented by laser‐ion interactions (25). The logic state of such a qubit can thus be expressed as a general superposition with complex amplitudes for which holds. Experimentally, a first quantum logic operation on a single ion was shown (26), and in 2003, the proposed CNOT gate operation was demonstrated with a pair of ions that represent the control‐ and the target‐qubit (27). The Cirac–Zoller proposal stimulated a series of experiments using trapped ions for quantum information processing. We can claim today that the principle of a QC is proven and that trapped ions are indeed a pioneering experimental platform for its development. This experimental development also helped to prevail against an initially overwhelming criticism concerning any experimental realization of a quantum computer (28).

24.1.3 Recent Milestones in Ion Trap Quantum Computing

Since the aforementioned pioneering experiment, the field has seen rapid progress, with many new proposals and demonstrations for quantum gate operations. Highlights by the Ion storage group at NIST Boulder and the team at Innsbruck University led by Rainer Blatt have been the unconditional teleportation with massive and long‐lived carriers of qubits (29–31), Bell tests to support quantum theory against local‐realistic theories (32) and multiparticle entanglement with up to 14 qubits (33). Furthermore, novel types of quantum gate operations, either with microwaves (34–37) or with short laser pulses (38) have been shown. Recently, the Shor factoring algorithm was realized (39), and a topologically protected qubit (40) based on seven physical qubits (ions) has been demonstrated. A set of methods for scalable ion trap quantum information processing was demonstrated (41). Today, two‐qubit entangling gate fidelities of 0.999 ± 0.001 (42,43) are realized, such that quantum error correction beyond proof‐of‐principle demonstrations seems to be within reach.

Trapped ions feature several advantages which make them one of the leading approaches for a realization of a future quantum computer (QC):

• In an ion trap, before a quantum algorithm starts, the quantum state of each ion in the crystal can be prepared such that the register of qubits, a quantum register, is initialized. The necessary techniques for this are laser cooling and optical pumping. Both techniques have been established already for the purpose of high‐resolution laser spectroscopy. More recently, sympathetic cooling methods have been developed, such that the motional state of a quantum register may be cooled without affecting its coherence.
• Ions can be stored in a linear Paul trap, or a segmented multizone trapping device, which use the charge of the ions as a “handle” via interaction with electric fields. Therefore, the trapping potential is very deep and tight, without cross‐talk to the internal electronic states – which are employed to store qubit information. Cirac and Zoller deduced that in an experiment with a trapped ion crystal serving as a quantum register, with the ions being held in a Paul trap under ultra‐high vacuum conditions, one could avoid undesired coupling of this quantum register to the environment. Indeed, measured qubit coherence times exceed the time for single qubit operations by more than five orders of magnitude.
• For the necessary quantum gate operations, ions are coherently manipulated by laser radiation. For two‐qubit gates, the coupling to a common vibrational mode – termed quantum bus – is employed. Various two‐qubit gate operations are proposed and a number of them have been realized experimentally (see Section 24.4.5).
• Fairly unique for trapped ions is the asset of a very high readout efficiency and fidelity. The technique of “electron shelving” dates back to Dehmelt (44). It allows to scatter a sufficiently large number of photons off an ion and to observe at least some fraction of them as clicks on a detector, if the ion is in qubit state , while no photons are scattered if the ion is in state . A single‐shot readout fidelity of better than 99.93 has been reached (45).

Today, the central challenges in trapped ion quantum computing are (i) developing architectures for scalable quantum computing with trapped ions and (ii) optimizing the gate fidelity and speed, and (iii) implementing quantum error correction schemes such that a large number of operations are feasible.

This article is organized as follows: After a discussion of linear Paul traps and quantized eigenmodes of ion crystals in the harmonic trap potential, we explain the operations on a static linear qubit registers. Here, tightly focused laser beams allow for single qubit addressing. Several highlights of ion trap quantum computing have been realized with this architecture. Modern segmented trap devices are aiming for scalable quantum computing toward a much larger number of trapped ions in reconfigurable quantum registers. This architecture has been coined quantum charged coupled device ( CCD ) (46). In the following, we discuss ion–laser and ion–microwave interactions and give examples for ion species used for qubits. We outline single‐qubit gate operations and a number of different two‐qubit gate schemes. The experimental realization of the Cirac–Zoller gate is described. Quantum logic operations have been combined in different ways to establish quantum algorithms. In order to exemplify the realization of elementary quantum algorithms, we focus on quantum teleportation. The article will sketch the most recent highlights and discuss the future challenges.

24.2.1 Stability Diagram of Dynamic Trapping

We focus first on the case of a two‐dimensional trap and discuss the parameter range for dynamic trapping. To confine the ions in 2D, we apply an rf‐voltage and an (optional) dc‐voltage to the trap electrodes. Near the trap axis for , this gives rise to an alternating electrostatic potential of the form

24.1

where denotes the distance between the trap axis and the surface of one of the electrodes. The equations of motion in dimensionless form resulting from 24.1 are the Mathieu equations,

24.2

24.3

where and with . The general solution of Eqs. 24.2 and 24.3 can be given as an infinite series of harmonics of the trap frequency (47). For the appropriate choice of parameters and , the ion trajectory is bounded in space and momentum: dynamical trapping is achieved, see Figure 24.1. If the conditions hold, an analytical approximate solution to the equations of motion can be given. It consists of a harmonic secular motion (macromotion) at frequencies with a superimposed micromotion at the trap drive frequency ,

24.4

The amplitude and the phases depend on the initial conditions, and the secular frequencies are given by

24.5

24.2.2 3D Confinement in a Linear Paul Trap

Axial confinement is provided by an additional static potential applied along the ‐axis using additional axial electrodes. This gives rise to an electrostatic harmonic potential well along the ‐direction, which is characterized by the longitudinal trap frequency

24.6

Here, z₀ is half the length between the axial confinement electrodes, and is a factor of order unity which accounts for the specific electrode geometry. Values of can be obtained either numerically or, in some cases, analytically. For the macroscopic ion trap in Figure 24.2a), a voltage of  = 2000 V applied to the tips gives rise to an axial trap frequency of 1.4 MHz for . Under typical operating conditions, radial trap frequencies of about 5 MHz are achieved. Until today, groundbreaking QC demonstration experiments with linear static qubit registers have been carried out by the Innsbruck group in this device.

The resulting (pseudo)potentials in all spatial directions are harmonic, and the motion of a trapped ion is accurately described by a quantum harmonic oscillator with frequencies . It is a major advantage of linear traps that the radial and axial trap frequencies can be adjusted freely and independently by tuning the applied voltages. Further details of the calculation of the stability diagram for 3D linear traps are given in Ref. (49).

24.4.1 Trap Architectures

The scalability perspective for trapped ions was opened up with the seminal proposal for a quantum CCD (46), where – similar to a CCD in modern cameras – ions are moved within a distributed architecture by changing control voltages on electrodes. The underlying devices are segmented ion traps, that is, devices which consist of a multitude of trapping electrodes, arranged in a geometry which allows for trapping ions at different locations and for shuttling the ions between these zones. In the last decade, such traps were developed and successfully demonstrated by several research groups, and the technology and required methods have reached a maturity which already allows for conducting state‐of‐the‐art experiments and quantum algorithms with few‐qubit systems. As segmented ion traps are produced using micro‐fabrication techniques, this also offers the possibility of miniaturization, which is a natural requirement for scalability. A common challenge for miniaturized trap lies in anomalous heating: Due to the increased proximity of the ion to surfaces, electric field noise generated by these leads to undesired ion motion leading to thermalization with the environment. The experimentally heating rates lie orders above the limit given by Josephson noise (59), which leads to the conclusion that the noise is generated by surface contaminants or structural defects within the metal layers. As pointed out in Section 24.4.5 , all entangling gate schemes known so far rely on control over the ion motion in the quantum regime. Therefore, significant research efforts are devoted to understanding and mitigating these effects, for example, by using ion traps in cryogenic environments (60–62) or in‐situ surface cleaning via ion bombardment (63,64).

Currently, segmented microstructured ion trap fall into two different categories.

24.4.2 Ion Shuttling

Ion shuttling operations in segmented trap are performed in essentially the same way as photo‐induced charges are transferred for readout on a CCD chip: By sweeping the control voltages applied to neighboring trap electrodes such that the resulting confining electrostatic potential well is moved, the confined ions are moved as well. In the frictionless case of trapped ions, this is physically equivalent to moving the support of a pendulum. The challenge behind these operations lies in the requirement to perform these operations fast – on the timescale set by the trap frequencies – in order to not compromise the quantum computer operation by excessive overhead. However, the excitation of ion motion which persists after the shuttling is to be avoided, as it would deteriorate the fidelity of subsequent gates. Thus, fast shuttling operations pose stringent requirements on control, especially in terms of the signal integrity of the utilized voltage ramps. To that end, signal generators have been developed which provide arbitrary waveform generation for a large number of up to 60 independently controllable output channels, which are updated in real time at rates which exceed typical trap frequencies, that is, more than 10 samples per second. At the same time, these signal generators feature excellent noise characteristics permitting the desired degree of control (70,71).

Following initial demonstrations of shuttling operations (72), diabatic shuttling at duration of a few trap periods has been accomplished at residual motional excitations below the single‐quantum level (73,74). Furthermore, more complex movement operations such as separation and merging of ion crystals have been shown ( 73,75). These operations are significantly more challenging, as they involve the transition between a common single‐well potential and a double‐well potential. This leads through a situation with low‐confinement along the movement axis. Precise calibration and optimized voltage waveforms are required (76) to avoid strong excitation either from increased thermalization rates at low trap frequencies or from oscillatory excitation due to insufficient control over the process.

24.4.3 Ion–Light Interaction

As the relevant theory is outlined in Section 23 and in review papers (77,78), we concentrate on examples of carrier and sideband Rabi oscillations to demonstrate the Jaynes–Cummings dynamics. The coupling between two electronic states and is mediated by a light field, characterized by the Rabi frequency and resulting in the Hamiltonian . If the laser frequency fulfills the resonance condition for the bare electronic transition , the interaction is not affecting the vibrational modes (carrier transition). Vibrational modes can also be coherently excited by the laser–ion interaction. The exchange of momentum between the ion crystal and the optical field is governed by the Lamb‐Dicke factor. For a single ion and coupling to a single vibrational mode, this is given by ratio of the ion's position‐space ground‐state wavefunction and the wavelength of the driving radiation:

24.14

where is the ion mass, is the eigenfrequency of the respective normal mode, and is the effective wavenumber of the optical field, where an angle between between the direction of vibration of the normal mode and the propagation direction of the light is taken into account.

Including coherent coupling to the motion, the carrier Rabi frequency remains almost unchanged for small values of . However, if the laser excitation frequency is tuned to , that is, it is red detuned with respect to the carrier frequency by trap frequency, , we realize the Hamiltonian of the Jaynes–Cummings type. For the ion initially prepared in the Fock state of the harmonic motion, the red sideband Rabi frequency is given by to . Blue laser detuning by the trap frequency, with , realizes the anti‐Jaynes–Cummings Hamiltonian with . For resolving the carrier and sideband transitions spectroscopically, we require an optical transition between long lived electronic states and with a linewidth much smaller than the trap frequency.

In experiments, one particular vibrational mode is selected as the quantum bus mode. Criteria for this selection are a low motional heating rate, that all ions to be addressed participate in the collective vibration, and sufficient spectral separation from other modes. For experiments that demonstrate the interactions for quantum gates, a single ion is kept in a Paul trap and the center‐of‐mass mode at is used. An experimental four‐step‐sequence is applied:

1. The ion is laser cooled into, or near to the ground state of vibration .
2. The electronic state is initialized to the ground state by optical pumping.
3. Laser light is applied with a fixed laser frequency, phase and intensity and interaction time . Thus, a superposition of electronic qubit basis states is prepared.
4. The ion is exposed to light resonant with a dipole transition. State‐dependent resonance fluorescence is observed only if the ion is projected into . If light is detected, the ion is measured to have been in state , while the detection of no fluorescence means the ion is measured to have been in state .

Finally, the probability for upper state population is revealed as the average of all measurement outcomes in the above sequence (a)–(d) for a large number of repetitions.

Figure 24.6a exemplifies single qubit (carrier) rotations denoted by , see Section 23, where the pulse area is the product of pulse duration and Rabi frequency, and is controlled by the phase of the laser field. Rotations on the blue sideband Rabi are denoted by , and this operation is used to coherently transfer quantum information from the electronic degree of freedom onto the quantum bus vibrational mode. In the following section, we discuss possible qubit candidates and the corresponding experimental realizations.

24.4.4 Levels and Transitions for Typical Qubit Candidates

Although an ion trap is deep and capable of holding every kind of atomic ion, only a few atomic species are actually suitable for QC experiments. These ions should exhibit energy levels appropriate for the implementation of a stable two‐level system with long‐lived qubit levels and , and the ion has also need to have a closed transition to a short‐lived excited state to allow for laser cooling and efficient fluorescence detection. The “ideal ion” typically has one electron in the outermost shell (hydrogen‐like electronic structure) and a correspondingly simple electronic level structure. The two‐level qubit system can either be provided by two hyperfine ground states, by Zeeman sublevels or by a long‐lived metastable electronic state. Prominent examples are the hyperfine‐qubit in Be (pioneered by the NIST, Boulder group), the so‐called optical qubit in Ca , where the ground state S and the optically excited metastable level D are used (pioneered by the Innsbruck group) or the spin‐qubit, where information is stored in the two Zeeman sublevels of the Ca S ground state (pioneered by the Oxford group and currently used by the Mainz group). The level schemes of Ca and Be are shown in Figure 24.7. Widely employed qubit implementations use the hyperfine ground states of Yb or Mg , and the isotope Ca (36,79). Qubit manipulations via stimulated Raman transitions feature the advantage that both required light fields can be derived from one single laser source, such that the differential phase fluctuations of both beams can be kept very small at moderate experimental effort. If the detuning of these beams from the resonance is chosen large enough, spontaneous emission from the off‐resonantly excited P and P states is suppressed and the coherence of the qubits is hardly affected (80). Additionally, the hyperfine and the spin‐qubits work with electronic ground states that do not show spontaneous decay. For this case, the decoherence is dominated by magnetic field fluctuations, but using magnetic shielding (81) or magnetic‐field insensitive clock states, the coherence times can exceed a few seconds. Microwave qubit operations on hyperfine qubits have been demonstrated as an alternative to optical qubit manipulations in Yb , Ca , and Be ( 34– 37).

24.4.5 Multiqubit Entangling Gates

In this section, we explain entangling gate schemes which have been implemented at high fidelity. We first describe the Cirac–Zoller gate scheme because it illustrates how laser‐ion interaction can be employed to generate entanglement.

24.4.5.1 The Cirac–Zoller Scheme

An entangling quantum gate between the internal states of any pair of ions and in a linear string can be achieved by three successive laser‐driven operations, addressing the th, then the th, and finally again the th ion (82). The gate operation relies on the initialization of the ion crystal in the vibrational ground state of the quantum bus mode ( 26 83–85) of the quantum bus mode and individual optical addressing of ions ( 27,86), which are technically demanding requirements. However, the gate operation can be easily understood by looking at the stepwise flow of quantum information:

1. First, the quantum state of the control ion is mapped onto the bus mode by a pulse on the blue sideband of the qubit transition,
2. A controlled‐NOT gate is carried out between the bus mode the target ion, driven by a rotation on an auxiliary transition, which is conditional on the vibrational quantum state (see Figure 24.8a))
3. The state of the bus mode is mapped back onto the control ion by a second pulse on the blue sideband, such that the final motional state is disentangled with the state of the qubits and the state of the control qubit remains unaffected.

24.4.5.2 Experimental Realization of the Cirac–Zoller Gate

The operation of mapping quantum information between the control qubit and the bus mode are simple pulses. We therefore focus on the central controlled NOT operation between the bus mode and the target ion. This operation is further decomposed into two Ramsey rotations on the carrier transition of the target qubit, which map a phase accumulated between the two pulse onto resulting population. Between the Ramsey pulses, a conditional phase is accumulated in the course of a controlled‐phase gate. Defining the computational subspace by , this controlled‐phase gate can be described by a diagonal unitary evolution matrix . Unlike the original proposal (82), which requires a ‐rotation on an auxiliary transition, the experiment ( 27,88) uses a blue sideband excitation leading to pairwise coupling between the states except for the state , see Figure 24.8b). In this case, the evolution of the controlled phase gate in the relevant subspace reads . For every basis state for which the controlled‐phase gates yields a resulting phase factor of , the state of the target qubit is not flipped with respect to its original state after the second Ramsey pulse.

This means that for the controlled phase gate, we perform an effective ‐pulse (48) on the two two‐level systems ( and , which flips the sign of the state for all computational basis states except for ). Since the Rabi frequency depends on , we need to compensate for this by utilizing a composite‐pulse sequence (87) instead of a single blue sideband pulse. Up to an overall phase factor, this transformation yields the desired controlled phase gate. The sequence is composed of four blue sideband pulses and can be described by

24.15

For an intuitive picture of we plot the evolution of the Bloch vector in Figure 24.9 ⁰¹. This phase gate is transformed into a controlled‐NOT operation if sandwiched between two ‐carrier pulses on the target ion, .

We realize this gate operation ( 27, 88) with a sequence of laser pulses. A blue sideband ‐pulse, , on the control ion transfers its quantum state to the bus mode. Then we apply the controlled‐NOT operation to the target ion. Finally, the bus mode and the control ion are reset to their initial states by another ‐pulse on the blue sideband. The gate reaches a fidelity⁰² of 0.71 ± 0.03 ( 27, 88). If the control qubit is initialized in a superposition state and the target qubit in , the controlled‐NOT operation generates an entangled state . Later, the gate operation was improved and a full process tomography (89) was carried out, yielding a process fidelity of 0.926 ± 0.006.

24.4.5.3 The Sørensen–Mølmer Gate Scheme

Mølmer and Sørensen (90–92), and in a different formulation Milburn (93), proposed a gate scheme which does not require perfect ground state cooling. Instead, only the cooling of the ion crystal into the Lamb‐Dicke regime is necessary, such that 1, where is the mean thermal phonon number and the Lamb‐Dicke factor. The authors assume an even number of ions, which are homogeneously illuminated by a bichromatic laser field with laser frequencies of opposite detunings with respect to the red and blue sideband frequencies, that is, (see Figure 24.10). The initially prepared state undergoes a sinusoidal Rabi oscillation to at an effective Rabi frequency . In the weak excitation regime , intermediate levels with vibrational numbers other than , that is, and , are not populated. The effective Rabi frequency reads . It appears that the ions only absorb photons simultaneously from the bichromatic laser field as the ions share the same vibrational mode. While the absorption of a single photon is suppressed due to the frequency mismatch , the coupling of the ions to the common vibration mode allows a mutual compensation of this frequency mismatch. As a consequence, the Sørensen–Mølmer scheme works with any even number of ions in a string. For two ions, an effective spin‐spin interaction is realized, which is an entangling interaction. If the gate evolution is stopped at , one has generated an entangled state of the electronic components of the ions only, while the state of the vibrational mode is disentangled from the qubit state. This results in a unitary transformation

24.16

The two ions are in an entangled state after the laser pulse. However, the weak excitation regime implies a low Rabi frequency, and the evolution at becomes correspondingly slow. This leads impractically long gate durations of typically a few milliseconds. To illustrate this, a calculation of the dynamics of the evolution of the populations and according to (92) is displayed in Figure 24.11.

The scheme was significantly improved shortly after the initial proposal, when Sørensen and Mølmer realized that the gate operation could also be driven much faster. However, with faster evolution of and and a larger , intermediate levels with vibrational numbers are populated, and in general the vibrational quantum number no longer remains unaffected by the evolution. The internal electronic states are therefore entangled with the ion motion during the course of the gate operation. For a successful gate operation, we have to make sure that the vibrational mode returns back to its initial state at the end of the gate operation. This corresponds to a closed circle in the phase space of the gate mode. As shown in (91, 92), the interaction time has to be adjusted to fulfill with . Figure 24.12a) shows the measured population evolution of such a fast bichromatic two‐ion entangling gate. Compared to the simulation in Figure 24.11, the Rabi frequency is increased, which allows the generation of an entangled state after 0.27 ms. Effects that limit the gate fidelity are discussed in (92). The fast Sørensen–Mølmer entanglement operation was realized for Ca ions with a fidelity of 0.993 ± 0.003 (94) on the optical qubit, see Figure 24.12. Further refinement of the method, for example, by shaping the bichromatic laser field, is outlined in (95), and the operation for thermally excited ions even at Doppler cooling temperatures was demonstrated (96). Recently, bichromatic gates have been demonstrated on Be ions at a gate error as low as by the NIST ion trapping group (43).

24.4.5.4 The Geometric Phase Gate

Holonomic quantum computing has been discussed in Section V D, and a proposal for single and two‐qubit phase gates exists for the case of trapped ions (97). Here, depending on the global state configuration of a set of ions, ion motion is transiently excited such that the corresponding trajectories in phase space are closed. This yields a state‐dependent Berry phase given by area enclosed by the trajectories. Thus, these gate operations are robust, since small variations of the actual path from the desired one do affect the accumulated Berry phase to first order.

Experimentally, such a geometric two‐ion geometric phase gate has been realized first by the NIST group (98). Two Be ions are held in a linear trap and are exposed to off‐resonant laser beams (see Figure 24.7) each at an angle of 45 with respect to the trap axis, such that the resulting difference vector points along the axial direction (98). Up to a small detuning , the frequency difference of both Raman beams is set close to the breathing mode frequency. For appropriate choice of the laser polarizations, this gives rise to alternating optical polarization along the trap axis. This in turn causes an ac Stark shift which oscillates in space along the trap axis and in time, near the breathing mode frequency. This leads to a near‐resonant optical dipole force, which can coherently excite breathing mode oscillations. The sign and magnitude of the force depends on the global spin configuration, as expressed by the parity operator (see Figure 24.13). If the interion distance is chosen to fulfill with integer , the optical field at the positions of both ions is identical, and the optical force acts on both ions in opposite directions if the spin configuration is odd, that is, for the states and . For even configurations, and , the forces on both ions act in the same direction and the breathing mode is not excited. Hence, we obtain excitation of breathing mode oscillation only for odd state configurations. As the difference frequency of both driving beams is deviates from the mode frequency by a detuning , the force becomes out‐of‐phase with the oscillation and counteracts it after some time. Therefore, the breathing mode oscillation is of transient nature and vanishes after the gate time . This is analogous to a classical pendulum which is excited slightly off resonance. Its oscillation amplitude initially grows, and the phase difference between external drive and pendulum eigenfrequency is accumulated, such that it is de‐excited again. As a consequence the phase space trajectory is a closed circle for odd state configurations. This leads to accumulation of a state‐dependent geometric phase, which is given by the phase‐space area enclosed by the respective trajectories. If, for a given detuning , the force magnitude is adjusted such that the enclosed area corresponds to a differential phase of , this realizes a unitary transform corresponding to a controlled‐phase gate:

24.17

The gate operation does not require perfect ground state cooling, as the trajectories in phase space close regardless of their initial vibrational quantum states, which leads to the required final disentanglement between qubit state and motional state. The gate speed – in contrast to the Cirac–Zoller gate scheme – is not limited by off‐resonant carrier excitations (99). Today, measured gate fidelities reach 0.999(1) (42) and are limited mainly by the spontaneous photon scattering. Increasing the detuning of the gate drive fields from the corresponding atomic transition leads to loss of drive strength scaling with , but suppression of photon scattering with . Therefore, increasing both the drive laser power and the detuning can yields increased gate fidelity.

Qubit operations as this geometric or the Mölmer & Sörensen gate, see Section 24.4.5.6, are ideal in combination with the quantum CCD architecture which is used in segmented micro traps, and which requires a high overhead in ion shuttling operations, see Section 24.4.2. In order to mitigate any effect of residual motional excitation of the phonon number in direction of the trap axis, thus in ‐direction, we have implemented the geometric gate on the radial rocking mode such that a high fidelity is maintained even for extended algorithms with a few hundred ion reconfiguration operations (100).

24.4.5.7 Quantum Computing Architectures

A future large‐scale universal quantum computer will necessarily rely on quantum error correction, which increases the number of required ions for redundant qubit storage and operations. Additionally, ancilla ions for readout of error syndromes are needed, such that in total on the order of – ions may be necessary to demonstrate an actual quantum supremacy. This leads to the question of how an architecture hosting a sufficient number of ions can be technologically realized, while retaining excellent control.

The quantum CCD approach (46) is the first attempt to address this question. While in this proposal, gate operations are driven by laser pulses, an interesting alternative would be using microwave pulses (108). However, in both cases we face a large overhead of ion shuttling operations, such that it may be of interest to investigate different means to couple qubits which are stored at different sites. Here, ion–photon interfaces which have the potential to bridge large distances within one quantum processing unit or even between spatially separated processors (109,110). This approach was stimulated by cavity‐QED experiments (111–114). Even today, realizing strong coupling of a single ion to an optical cavity mode remains technically challenging. Recently, however, such interfaces have been largely improved and lead to controlled photon–ion entanglement and ion–ion entanglement (115–117). Another option for realizing large scale QC is the electric coupling either via antenna structures (118) or directly. Such coupling has been demonstrated for small ion crystals (119,120) and stimulated some further experimental investigations for two‐dimensional arrays of ions (121,122). Obviously, and for all cases, the challenge lies in the manufacturing and control of such complex architectures.

24.4.6 Quantum Teleportation

In 2004, more than a decade after the proposal (123), deterministic quantum teleportation with matter has been demonstrated in two different laboratories ( 29,30). For the teleportation with light, we refer the reader to Section IV. The quantum teleportation algorithm for a qubit from Alice to Bob is based on five consecutive steps:

1. An entangled pair of qubits is created and distributed, the particle with index 2 to Alice and the other one to Bob.
2. Alice holds a single qubit in an unknown quantum state , which is to be teleported to Bob. The overall quantum state of the three qubit system reads .

   The two particles with index 1 and 2 belong to Alice, and she performs the ‐gate and the operation (see Figure 24.13(b)) to rotate the basis for her qubits from the computational basis states into the Bell basis states denoted by and .

3. The overall quantum state can be written as a sum of four terms
   24.18

   Alice performs a measurement in the Bell basis on both her qubits, particles 1 and 2. Thus, she projects her particles of into one of the possible Bell states, each with 1/4 probability outcome. If Alice's measurement result is , Bob's qubit is already in the desired quantum state .

   If she finds a different outcome, for example, , Bob has recovered the quantum state which can be transformed into by a single qubit rotation in his particle with index 3. Using the Pauli operators which generate rotations about the different axes of the Bloch sphere we obtain,

   24.19

   with and . Note, that only and rotations are necessary.

4. Therefore, Alice sends the outcome of her measurement, two bits of classical information to Bob, such that
5. Bob is able to perform the correct single qubit rotation on his particle to obtain back the original state . Note, that no quantum information is duplicated but that the entire information about the state has fully disappeared at Alice's side.

The protocol, see Figure 24.14 (b), has been realized (30) with three ions in the linear trap shown in Figure 24.2a). The quantum algorithms consist of more than 30 laser pulses. The outcome of the deterministic teleportation is revealed by inverse reconstruction and shows a fidelity of 75 (30). Subsequently, this result has been improved to 83%, and the analysis of the algorithm has been completed by applying process tomography on the teleported output state ( 25 124–126). Teleportation with a fidelity of 78 was demonstrated with three ions in a linear segmented ion trap by the NIST group (29). For quantum gate operations among two ions, for single qubit rotations, and for the readout of single qubits without affecting the other qubits, the required ions are singled out from the linear crystal via ion shuttling, and transported into a processor section of the ion trap where the laser‐ion interactions are driven. Thus, quantum teleportation has become the first algorithm to demonstrate the benefits of segmented and miniaturized Paul traps.

24.4.7 Selected Recent Highlights

As the invention of Shor's factoring algorithm fostered the development of quantum computing, it is intriguing to the progress on its actual realization, recently performed for the case and based on scalable algorithmic building blocks (39). The problem of finding the prime factors of an integer can be mapped to the problem of finding the period for integers , which is the smallest integer for which modular exponentiation yields zero, that is, From the resulting period , one determines prime factors of as the greatest common divisor of and . In the experiment (39), has been factorized by using a linear ion crystal with five ions in the trap Figure 24.2. It is crucial that the algorithm has been demonstrated without using precompiled gate sequences for the modular exponentiation operation. One ancilla and four register qubits are required for the modular exponentiation for a suitable base , and , to find its period. The measurement results for base clearly show a period with a squared statistical overlap larger than 0.90 for all cases. This is the outcome of a large number of experimental repetitions. However, Shor's algorithm should allow for deducing the period with high probability already from a single‐shot measurement. In the experiment this is achieved for all with a probability greater than 0.5, therefore, after only eight single‐shot repetitions, one would reveal the correct periodicity at a probability exceeding 0.99 (39).

Remarkable progress toward fault‐tolerant quantum computing has been realized by the demonstration of a topologically encoded qubit (40). One logical qubit is encoded in seven physical trapped‐ion qubits. The corresponding stabilizer operators and are defined on three different subsets (plaquettes), each consisting of four ions. State preparation via plaquette‐wise entangling gates and stabilizer readout operations are realized by a global laser beam, such that it is required to hide the other ions from the laser interaction. A tightly focused laser beam is used to transfer these into the metastable D state, such that out of the static crystal of seven ions, only the required sub‐set of qubits is affected by the gate operation.

The technique of individual single ion addressing with a tightly focused laser beam acting on a static linear ion crystal is also used to implement a universal programmable quantum processor (127). All trapped ion qubits are fully connected by the common mode of vibration such that pairwise entangling gates between all possible pairs are realized, which yields a crucial advantage over QC architectures with nearest‐neighbor or star‐shaped coupling topology. Gate operations reach a fidelity of 0.98. A quantum Fourier transformation performed with 15 gate operations on the five qubits and achieves an average fidelity of 0.619 ± 0.005s. Note, that such static ion crystal approach has been used for quantum simulation of interacting spin systems (128)

Such highlights obtained with static linear ion qubit registers are complemented by experiments toward a scalable re‐configurable quantum CCD. The first demonstration of a complete methods set comprises individual addressed single qubit preparation and state readout, single and two‐qubit laser‐driven gate operations, in combination with a spatial separation of two‐ion crystals, and their recombination as well as transport operations for single ions and small crystals (41), 129. Process tomography reveals a fidelity of 0.987 ± 0.003. The set of operations was completed by the ion‐SWAP operation for two‐ and three‐ion crystals, where a mean process fidelity of 0.995 ± 0.005 is obtained. Only in this way, any of the ions in the linear arrangement can be coupled with any other ion, regardless of the initial positions (130), without using complicated ‐ or ‐junctions in segmented traps, see Figure 24.2. Even sequences with about 250 shuttling on a four ion qubit register, and several laser‐driven geometric phase gate operations have been realized (100) and multipartite entanglement with fidelity of about is achieved.

For algorithms with even higher complexity and thus even more overhead in terms of register reconfigurations operations, the technique of sympathetic cooling is required: An auxiliary ion of a different species is used for ground state cooling of the relevant collective vibrational modes of oscillation of the mixed crystal, thus cooling also the qubit ion(s) without affecting stored quantum states. This way, the quantum gate operations on the reconfigured qubit at persistent high fidelity are enabled. Mixed ion crystals can serve for another crucial purpose: Quantum error correction relies on the readout of error syndromes, but obviously this has to be performed without affecting the stored quantum information. As any readout scheme relies on state‐dependent resonance fluorescence, it is hardly avoidable in a realistic setting that resonant scattered light affects memory qubits in an undesired way – unless a different species with different resonance wavelengths is employed for readout. As an important step toward quantum non‐demolition stabilizer readout, multi‐element logic gates for trapped ion qubits have been realized for mixed crystals of Be and Mg (131), of Ca and Ca (132) or of Ca and Sr . Furthermore, quantum logic based optical frequency standards, see Section 36, rely on similar interspecies entanglement gate operations.

Observing the rapid and substantial progress of ion trap quantum computing, overcoming technological barriers, and controlling the required complex experimental environments increasingly well, we are confident that the concepts for scalability to a large number of qubits are sustainable. Trapped ions stay among the most promising platforms for experimental quantum computing.