Abstract
The first generation of quantum computers are on the horizon, fabricated from quantum hardware platforms that may soon be able to tackle certain tasks that cannot be performed or modelled with conventional computers. These quantum devices will not likely be universal or fully programmable, but specialpurpose processors whose hardware will be tightly codesigned with particular target applications. Trapped atomic ions are a leading platform for firstgeneration quantum computers, but they are also fundamentally scalable to more powerful general purpose devices in future generations. This is because trapped ion qubits are atomic clock standards that can be made identical to a part in 10^{15}, and their quantum circuit connectivity can be reconfigured through the use of external fields, without modifying the arrangement or architecture of the qubits themselves. In this forwardlooking overview, we show how a modular quantum computer with thousands or more qubits can be engineered from ion crystals, and how the linkage between ion trap qubits might be tailored to a variety of applications and quantumcomputing protocols.
Quantum information processors have the potential to perform computational tasks that are difficult or impossible using conventional modes of computing.^{1,2,3} In a radical departure from classical information, the qubits of a quantum computer can simultaneously store bit values 0 and 1, and when measured they probabilistically assume definite states. Many interacting qubits, isolated from their environment, can represent huge amounts of information: there are exponentially many binary numbers that can coexist, with entangled qubit correlations that can behave as invisible wires between the qubits. Even in the face of Moore’s Law, or the doubling in conventional computer power every year or two, the complexity of massively entangled quantum states of just a few hundred qubits can easily eclipse the capacity of classical information processing.^{4} There are but a few known applications that exploit this quantum advantage, such as Shor’s factoring algorithm,^{5} and future quantum information processors will likely be applied to specialpurpose applications. On the other hand, a quantum computer has not yet been built; therefore, new quantum applications and algorithms will likely follow from the evolution and capability of quantum hardware.
In the 20 years since the advent of Shor’s algorithm^{5} and the discovery of quantum error correction,^{6,7,8} there has been remarkable progress in demonstrations of entangling quantum gates on <10 qubits in certain physical systems. Current efforts aim to scale to hundreds, thousands or even millions of interacting qubits. Unlike the classical scaling of bits and logic gates, however, large quantum systems are not comparable to the behaviour of just a few qubits. Just because 2 or 4 qubits can be completely controlled with negligible errors does not mean that this system can readily scale to >100 qubits.
In the last few years, two particular quantum hardware platforms have emerged as the leading candidates for scaling to interesting numbers of qubits: trapped atomic ions^{9,10} and superconducting circuits.^{11,12,13} These technologies will likely both be built out in coming years, and may find complementary uses. Superconducting circuitry exploits the significant advantages of modern lithography and fabrication technologies: it can be integrated on a solidstate platform, and many qubits can simply be printed on a chip. However, they suffer from inhomogeneities and decoherence, as no two superconducting qubits are the same, and their connectivity cannot be easily reconfigured without replacing the chip or modifying the wires connecting them within a very lowtemperature environment. Trapped atomic ions, on the other hand, feature virtually identical qubits, and their linkage can be reconfigured by modifying externally applied electromagnetic fields. However, atomic qubit switching speeds (1–100 μs) are generally much slower than that of solidstate devices (1–100 ns), and the development of engineering infrastructure for trapped ion quantum computers and the mitigation of noise and decoherence from the applied control fields is just beginning.
In this paper, we anticipate the upcoming engineering efforts on trapped atomic ions for quantum computing, and highlight their reconfigurable quantum circuit connectivity as a flexible platform to be applied to a wide range of potential quantum applications. This path to scaling to thousands or more qubits will almost certainly involve the concept of architectural codesign,^{14} where algorithms and applications are invented alongside the development of trapped ion hardware, and the laboratory engineers fabricate an ion trap system that is welladapted to certain types of quantum circuit applications.
Ion trap qubits and wires
Atomic ions can be confined in free space with electromagnetic fields supplied by nearby electrodes. The linear radio frequency (r.f.) trap is the typical choice for quantum information applications.^{9,10} When the ions are lasercooled to bottom of the trapping potential, they form a linear crystal of qubits, with the Coulomb repulsion balancing the external confinement force, as shown in Figure 1a. Ions are typically loaded into traps by generating neutral atoms of the desired element and ionising the atoms once in the trapping volume. Ion trap depths are usually much larger than room temperature, so rare collisions with background gas do not necessarily eject the ion from the trap, but they can temporarily break up the crystal and scramble the qubits. Under typical ultrahigh vacuum conditions, these qubit interruptions occur roughly once per hour per ion, but cryogenic vacuum chambers can reduce the collision rate by orders of magnitude, where qubits may last months or longer. An alternative ion trap geometry is the Penning trap, which can confine large twodimensional (2D) crystals for quantum information applications.^{15,16} However, the addressing and control of individual qubits and the binding strength of internal modes of motion may pose difficulties, and this platform is thus not considered here.
Qubits stored in trapped atomic ions are represented by two stable electronic levels within each ion, often represented as an effective spin with the two states $\downarrow \u3009$ and $\uparrow \u3009$, corresponding to bit values 0 and 1, respectively. The qubits can be initialised and detected with nearly perfect accuracy using conventional optical pumping and statedependent fluorescence techniques.^{17} This restricts the atomic species of trapped ion qubits to those with simple electronic structure (e.g., those with a single valence electron: Be^{+}, Mg^{+}, Ca^{+}, Sr^{+}, Ba^{+}, Zn^{+}, Hg^{+}, Cd^{+} and Yb^{+}).
Figure 1b,c show the reduced energy level diagram of ^{171}Yb^{+}, where the qubit levels $\downarrow \u3009$ and $\uparrow \u3009$ are represented by the stable hyperfine levels of electron/nuclear spin in the ground electronic state, separated by frequency v_{HF}=12.642812 GHz. Such states form an excellent frequency standard, and coherence times >1,000 s have been observed.^{18,19} The optically excited electronic states $e\u3009$ and $e\prime \u3009$ are themselves split by a hyperfine coupling and separated from the ground states by an optical interval. Laser radiation tuned just below resonance in these optical transitions allows Doppler laser cooling to confine ions near the bottom of the trap. Other more sophisticated forms of laser cooling can bring the ions to nearly at rest in the trap.^{17} When laser beams resonant with both $\uparrow \u3009\leftrightarrow e\u3009$ and $\uparrow \u3009\leftrightarrow e\prime \u3009$ transitions are applied, the ion rapidly falls into the state $\downarrow \u3009$ and no longer interacts with the light field (Figure 1b). This optical pumping technique allows the initialisation of a qubit with essentially $100\%$ fidelity. When a single laser resonant with the transition $\uparrow \u3009\leftrightarrow e\u3009$ is applied, the closed cycling optical transition causes an ion in the $\uparrow \u3009$ state to fluoresce strongly at a rate scaled by the excited state radiative linewidth γ~2π×10 MHz, whereas an ion in the $\downarrow \u3009$ state stays dark, because the laser is far from its resonance (Figure 1c). The collection of even a small fraction of this fluorescence thus allows for the detection of the atomic qubit state with nearperfect efficiency, with integration times as low as ~20 μs.^{20} Other atomic species have similar initialisation/detection schemes.
Linking atomic qubits with the Coulomb interaction
The motion of many trapped ions is coupled through their mutual Coulomb repulsion, so the qubits can therefore be linked by relating the internal qubit states to the external motion of the ions, as depicted in Figure 2a. This can be accomplished by applying qubit statedependent optical or microwave dipole forces to the ion(s).^{9,17,21,22,23} To see how this type of motional data bus works, we assume that a given ion responds to an external field $\mathcal{E}$ by experiencing an equal and opposite energy shift $\mathrm{\Delta}E=\pm \mu \mathcal{E}$ with effective dipole moment μ that depends upon the qubit state through selection rules. When the field $\mathcal{E}\left(x\right)$ varies with position x, there is a qubitstatedependent force of the form ${F}_{x}=\mu \mathcal{E}\prime \left(x\right){\stackrel{\u02c6}{\sigma}}_{z}$, where $\mathcal{E}\prime \left(x\right)$ is the field gradient and ${\stackrel{\u02c6}{\sigma}}_{z}$ is the Pauli matrix corresponding to the angular momentum of the qubit effective spin. Here we neglect higher order field gradients, which is justified when the ion is lasercooled to much less than the characteristic length scale (or wavelength) of the applied field. For planewave radiation coupled with wavevector k and amplitude ${\mathcal{E}}_{0}$, ${F}_{x}=\hslash k\mathrm{\Omega}{\stackrel{\u02c6}{\sigma}}_{z}$, where ħ is Planck’s constant, and the Rabi frequency $\mathrm{\Omega}=\mu {\mathcal{E}}_{0}/\hslash $ parametrises the fieldqubit coupling. (For twophoton optical Raman couplings, the effective wavevector k is given by the wavevector difference between the two beams.^{17} The use of such Raman couplings between ground states also gives the practical advantage of being insensitive to the absolute optical phase at the position of the ion, instead depending upon the microwave or r.f. phase difference between the beams.^{24})
As this force acts differently on the two qubit states, it couples the qubit state to the collective motion of N ions, with characteristic speed ${R}_{\mathrm{gate}}=\mathrm{\Omega}\sqrt{{\omega}_{R}/\omega}$, where ${\omega}_{R}=\hslash {k}^{2}/\left(2Nm\right)$ is the recoil frequency of the ion crystal associated with field momentum ħk, Nm is the total mass of the ions and ω the frequency of harmonic oscillation of collective motion along the x direction. When this mapping affects multiple ions, entangling gates can be operated between separated ions, mediated through the motion. There are many protocols for the creation of controlledNOT and other gates using this coupling to the collective motion of the ions.^{9} Current experiments with a few ions have realised entangled state fidelities of greater than 99.9%^{25,26} and operate in the range R_{gate}/2π~10–100 kHz, although with ultrafast highintensity optical fields it may be possible to operate gates in the GHz range.^{27}
As the number of ions N in the crystal grows, the gate speed slows down as ${R}_{\mathrm{gate}}\sim 1/\sqrt{N}$. For large crystals, there will also be crosstalk between the many modes of collective motion. Background errors such as the decoherence (heating) of the motional modes^{28} or fluctuating fields that add random phases to the qubits will become important at longer times, thus there will be practical limits on the size of a single crystal for the performance of faithful quantum gates. Through the use of individual optical addressing of ions^{29,30} and pulseshaping techniques,^{31} these errors should not be debilitating for the full control of single crystals ranging from N=10–100 qubits.
In order to scale beyond ~50 trapped ion qubits, we can shuttle trapped ions through space in order to couple spatially separated chains of ions, in a multiplexed architecture called the quantum chargecoupled device (QCCD)^{23,32,33} and depicted in Figure 2b. The QCCD architecture requires exquisite control of the atomic ion positions during shuttling and may require additional atomic ion species to act as ‘refrigerator’ ions to quench the excess motion from shuttling operations.^{34} Rudimentary versions of the QCCD idea have been employed in many quantum information applications such as teleportation and small quantum algorithms,^{9} and recent experiments have shown the reliable, repeatable and coherent shuttling of ion qubits over millimetre distances in microsecond timescales^{35,36} and through complex 2D junctions.^{37,38,39,40} The QCCD approach will help usher the development of trapped ion quantum computers with perhaps 50–1,000 qubits. However, scaling to many thousands or more qubits in the QCCD may be challenging because of the complexity of interconnects, diffraction of optical beams and the extensive hardware required for qubit control. In addition, as the number of qubits grows sufficiently large, the QCCD architecture will suffer from the complexity of moving quantum information arbitrarily through the entire system. Regardless of the speed of nearestneighbour elementary shuttling operations, the QCCD shuttling complexity over large distances may eventually limit the quantum processing speed. More generally, it is likely that a largescale quantum computer will require connections between distant qubits and a method for producing entanglement that is independent of distance.^{41}
Linking atomic qubits with photons
To scale beyond the QCCD in a modular architecture, it will likely become necessary to link separate registers of trapped ion chains with photonic interfaces.^{42,43} This allows quantum gates to be performed between any qubits in the processor, regardless of their relative location,^{41,44,45} while supporting faulttolerant error correction even in the face of photonic interconnects that succeed with small probability per attempt.^{44,45,46}
A pair of trapped ion qubit modules (elementary logic units or ELUs) can be entangled with each other using propagating photons emitted by a subset of ions from each register, designated to be ‘communication qubits’. As shown in Figure 2c, the communication qubit is driven to an excited state with fast laser pulses (typically ~ps) so that at most one photon emerges from each qubit. Following appropriate radiative selection rules, there are multiple decay paths, resulting in entanglement between a photonic qubit (e.g., polarization, frequency) and the atomic qubit. When photons collected from two separate communication qubits are modematched and interfered on a 50/50 beamsplitter (Figure 2d), coincident detection of single photons on the output modes of the beamsplitter herald a Bellstate of the photons and thus the creation of entanglement between the memory qubits through entanglement swapping.^{47,48} For hyperfine atomic qubits, the optical path length of this interferometer need only be stabilised to well within the wavelength corresponding to the qubit frequency difference (~mm). The mean connection rate of this photonic interface is $R{\left(F{\eta}_{D}\right)}^{2}/2$, where F is the fraction of light collection from each ion emitter, η_{D} is the singlephoton detector efficiency and R is the repetition rate of the initialisation/excitation process limited by the emission rate γ (an alternative singlephoton approach involving the weak excitation of the ions^{49,50} suffers from optical path length instabilities, and, as the light collection improves, the performance of this alternative protocol is inferior to the twophoton scheme discussed in the text.). For typical atomic transitions into free space with γ/2π~10 MHz, light collection fraction F~1–10% and detector efficiency η_{D}~20%, we find typical connection rates of ~100 Hz,^{51} but this could be markedly improved with integrated photonics, as discussed below. Although this ion–ion entanglement source utilising photonic channels is probabilistic, the detected photons announce when it does succeed, and thus the heralded entanglement of the trapped ions can be subsequently used for deterministic quantum information applications.^{52} Moreover, by performing such operations in parallel on many pairs, a FIFO (first in, first out) buffer can provide a synchronous stream of entangled ion pairs between the trapped ion modules that can be used as needed, thus eliminating the probabilistic nature of the connection.
In practice, the communication qubit must be wellisolated from the memory qubits so that scattered light from the excitation laser as well as the emitted photons themselves do not disturb the spectator qubit memories. It may be necessary to physically separate (shuttle) the communication qubit away from the others, invoking techniques from the QCCD approach, but ultimately using two different atomic species can eliminate this crosstalk,^{53,54,55} such as ^{171}Yb^{+} for memory qubits and ^{138}Ba^{+} for communication qubits. Here the communication qubits are connected through the photonic channel, and then mapped to neighbouring memory qubits through Coulomb gates as described above.
In Figure 2e, we show a concept of a largescale modular trapped ion quantumcomputing architecture, involving individual ELU modules that host Coulombbased quantum links within the module and can be wired to other ELU modules through photonic connections as described above. By using a nonblocking optical crossconnect switch,^{56,57} the connectivity between the entire sample of qubits can be extended in order to scale up to very large numbers of ELU modules, potentially to thousands or millions of qubits.
Integration technologies for trapped ion quantum computers
Unlike classical solidstate circuits where largescale integration of complex information processors is readily available, practical implementation of a trappedion quantum processor will require development of new integration technologies and system engineering approaches. In this section, we will describe the current efforts towards such technology development.
Chip traps and optical control of qubits
Reliable and reproducible fabrication of many identical ELU modules starts with the ion trap itself. Ion trap electrode structures can be fabricated by lithographically etching semiconductor platforms such as Si/SiO_{2} wafers and metallising the electrodes, with positions defined to submicrometer precision. The electrodes must hold high static and r.f. electrical potentials, with excellent insulating barriers between the electrodes, all in an ultrahigh vacuum (UHV) environment. The trap structure must also be optically open, and allow highpower laser beams to cross near the electrode surfaces to affect Coulomb gates or photonic couplings, without causing excessive light scattering. Finally, the ions must be spatially separated from other qubit memories during initialisation and measurement processes, requiring precise control over the electrical potentials over space and time in order to shuttle ions throughout the trapping regions.
An approach to create the necessary electromagnetic potential to trap an ion above the surface of a chip was first suggested in 2004^{58,59} and demonstrated in 2006,^{60} along with the first monolithic semiconductor ion trap.^{61} The design and fabrication of complex surface traps using silicon microfabrication processes has now matured, with examples of the Sandia highoptical access (HOA) trap^{62} and the GTRI/Honeywell ballgrid array (BGA) trap^{63} shown in Figure 3. Recent experiments have demonstrated highperformance qubit measurement^{20} and singlequbit quantum gates^{64,65,66} in such microfabricated surface traps that outperform conventional manually assembled macroscopic traps. The ability to design and simulate the electromagnetic trapping parameters prior to fabrication provides an attractive path to developing complex trap structures that are both repeatable and produced with high yield.
Once atomic ion qubits are produced and confined to standard semiconductor trap structures, interactions between arbitrary subsets of the qubits in a single ELU or reduced sets of ions between ELUs must be gated and controlled in order to perform the desired calculations, simulations or quantum circuit. For local Coulomb gates as described above, we require a fixedfrequency offresonant laser to provide dipole forces and a laser beam distribution technology that drives the desired interactions to operate a programmable and reconfigurable quantum computer. Moreover, in order to accommodate shuttling of ions, the laser sources may require synchronisation with the shuttling protocol and in some cases the trap r.f. fields. In the ^{171}Yb^{+} system, lasers at 355 nm are ideal for gate operations, with reliability owing to their widespread use in conventional UV lithography. For optical beam delivery, recent developments in micromirror technology^{67} and multichannel acoustooptic modulator (AOM) technology developed for the optical communication and semiconductor fabrication industries^{68} are attractive solutions, and in the coming years these devices will be tightly integrated with trapped ion systems.
Compact lasers and vacuum system technology
A largescale ion trap quantum computer requires several tunable laser systems to match electronic resonances in the atomic ion, with optical frequencies stable and accurate to better than 10 kHz, a fractional precision of ~10^{−10}. Traditionally, a significant effort is dedicated to laser stabilisation, with individual optical components on an optical table utilising long optical path lengths. Such a large footprint invariably drifts because of environmental changes (temperature, humidity, air pressure and so on), and requires constant adjustments to keep the system operational. Compact and stable tunable semiconductor lasers have recently been developed that provide the narrow linewidths necessary for initialising and reading out trapped ion qubits.^{69,70,71} The complete optical system including the frequency stabilisation can be assembled on a compact optical breadboard or a microfabricated optical bench.^{72} Following the footsteps of laser integration in modern dense wavelengthdivision multiplexed optical communication systems,^{73} it is feasible to design and assemble a stabilised laser system where all lasers necessary for running an iontrapping experiment are packaged in a compact box that fits on an instrument rack, with fibre optic delivery to the ion trap chip.
Trapped atomic ions are suspended in a UHV environment, where collisions with background molecules should be minimised for sustained operation. Although careful assembly of clean UHV chamber can help create such an environment, ultimate vacuum environment might require lowtemperature operation (<10 K).^{74} Closedcycle cryogenic technology will markedly reduce the volume and operational burden of a UHV environment, while improving the vacuum conditions for operating the trapped ion quantum computer. Figure 4 shows an example of a compact vacuum environment created on a ceramic package that holds the surface trap. After the trap is dieattached and wirebonded to the ceramic package, a sealed cover is assembled in a UHV environment (Figure 4a). The cover provides all optical access necessary to operate an ion trap, an ion source that utilises laser ablation technique to load the trap, and getter material that will efficiently pump any residual gas molecules at low temperatures. This compact ceramic package can be installed in a closedcycle cryostat (Figure 4b) and cooled down to cryogenic temperatures (~5 K) to provide the operating environment for the ion trap quantum processor. Figure 4c shows an example of a compact optical layout sufficient to trap a chain of ions in the cryogenic environment, onto which further qubit control optics could be assembled. This approach is compatible with recent development in surface treatment techniques for ion traps that are shown to substantially reduce anomalous heating, which may be necessary for highfidelity operation of multiqubit gates.^{75,76}
Photonic technology
Integrated optical technology will be critical for a largescale trapped ion quantum computer. Although efficient light collection and detector arrays will be necessary for the measurement of many trapped ion qubits through statedependent fluorescence, it will be crucial for the singlephoton linking of ELU modules as discussed above. With high numericalaperture collection optics, ~10% of the emitted photons can be collected,^{51} and more could be extracted through an optical cavity^{77,78} integrated with the ion trap.^{79} Highly efficient photonic Bellstate detectors with nearideal modematching can be realised in fibre or waveguide beamsplitters^{80} and nearunit efficiency photon detectors.^{81,82} Taken together, these advances may allow the linking of two ELUs to approach the speed of local Coulombbased gates (~10 kHz).
For large numbers of optically networked trapped ions with many optical communication qubits, multiplexed photonic circuit elements will be necessary. Nonblocking and transparent optical crossconnect switches with many input/output ports, developed for conventional optical communication networks and data centres,^{56,57} are wellsuited for this task. Transparent optical switches establish an optical path between select input and output ports by using passive optical elements such as tilting mirrors^{83} and can guide single photons that are entangled with the trapped ion qubits to form quantum links. These devices can also be reconfigured in real time to make parallel connections between multiple ELUs.
The extension of the above integrated photonics technology, including detectors and waveguides to optical crossconnect switches, to the visible and blue portions of the spectrum where atomic ions respond will be highly valuable to the trapped ion quantum optical network. Alternatively, noiseless photonic conversion technology from visible/blue to infrared and telecom bands will have an important role, especially for longdistance quantum communication network applications.
Hardware and software for scalable controllers
Although these integration technologies have a crucial role in developing compact, standalone ion trap quantum hardware, a scalable controller system is needed to run such a system. The controller system consists of hardware needed to (1) maintain the operation of frequencystabilised laser systems, (2) manage the ion position by control of r.f. and static voltages, (3) measure and processemitted photons with photon detectors and associated readout circuits, and (4) apply laser pulses that are generated by a digital system to prepare, measure and manipulate qubits. These controller systems must be designed to precisely track the amplitude and phase of the qubits used in the informationprocessing task. It must be accompanied by control software that the user can program to instruct the quantum hardware to carry out the desired task. Both the hardware and the software for the controller should be designed for modularity and expandability, consistent with a fully coherent control of all qubits in the system. The practical scalability of the ion trap processor may ultimately be limited by the scalability of the interface between such a classical controller and the ion qubits, or how many ion qubits the controller can manage. A careful design of such controller system amounts to the ‘operating system’ for the ion trap quantum processor.
Applications and opportunities for the trapped ion quantum processor
Quantum algorithms, applications and errorcorrecting codes are usually designed independently of the underlying hardware, and therefore do not respect the underlying geometry of the physical system. The modular ion trap architecture has a flexible and reconfigurable connectivity that allows for the realisation of arbitrary geometries with a minimal number of swaps and/or teleportation steps. In this section, we examine the opportunities that such a hardware affords, beyond conventional gatemodel quantum applications such as Shor’s factoring algorithm.
Topology of interactions
Although the spatial geometry of a crystal of lasercooled trapped ions is typically onedimensional, the interaction graph between qubits within a single ELU module can be fully connected and have high or undefined dimensionality. Owing to the strong longrange Coulomb interaction between the ions, quantum gates can be realised directly between distant pairs of qubits in the chain.^{84} Furthermore, multiple twoqubit gates can be performed in parallel on the chain, and multiqubit or even global entangling gates can be performed by carefully controlling the intensity and spectrum of the lasers on all ions.^{85,86} Such multiqubit operations are useful for the generation of certain entangled states^{9,22}, and for the simulation of longrange global Ising interactions between the spins.^{87,88} In the context of such multiqubit operations, additional singlequbit gates can be used to remove links in the graph^{89} and generate arbitrary circuits.
In the larger ion trap quantum computer, such highly connected ELU modules are linked to other modules through photonic channels, as discussed above.^{44} This higher level graph is determined by the density of photoncoupled ions (Figure 5) and is itself dynamic and reconfigurable, leading to great advantages in the use of this type of hierarchical architecture for a host of quantum applications, and may even suggest algorithm structures that have not yet been discovered.
Quantum simulation
Quantum simulators exploit a standard wellcontrolled quantum system to emulate model Hamiltonians that cannot easily be understood or solved numerically.^{1} The global entangling operations between trapped atomic ions are naturally suited to simulate hard quantum problems, some that may not necessarily correspond to a physical system (such as graphs that have nonlocal structures), and others with local interactions such as quantum magnetic Ising models and many body spin physics. For instance, when executing statedependent forces as discussed above, the applied field can be adjusted to simulate variablerange Ising models with interaction strength falling off with a power law 1/r^{α} as the physical distance r increases, where the exponent can be tuned between α=0 (infiniterange) and α=3 (dipoledipole).^{15,87,90} In addition, digital quantum simulation techniques, which apply a series of distinct control Hamiltonians in discrete time steps, can be applied to generate arbitrary spin models^{91} and to control the underlying graph structure.^{89} Iontrap quantum simulations could assist our understanding of models of exotic materials (such as hightemperature superconductors), or even stimulate the search for new material properties that have not yet been observed.
Determining the equilibrium behaviour (ground state) of spins subject to an arbitrary Ising coupling graph and local field terms is an NPhard problem.^{92} The quantum adiabatic algorithm attempts to find the ground state by starting in a strong transverse field and then adiabatically switching it off.^{93,94} As the minimum energy gap to the excited states is not known for the most difficult problems, ‘adiabatic’ in this case means slow enough to avoid excitations. The quantumannealing algorithm is similar in style, but interaction with a thermal bath complicates the dynamics.^{95,96} Although a sufficiently cold bath could improve the possibility of finding the ground state, this is not always the case. Largescale superconducting systems fabricated by DWave have generated significant controversy,^{97} and the results to date are consistent with simulations of an opensystem quantum annealer having limited use.^{95} The main limitations are insufficient precision and stability in the coupling parameters, twoqubit couplings of fixed type, a simple 2D static network structure and a temperature that is much larger than the characteristic energy scales in the system. As the gaps are expected to shrink polynomially with the system size for good problems, the DWave system thus relies on thermal effects. Although trapped ion qubit couplings are roughly 1,000 times smaller than superconducting systems in absolute terms, ion trap architectures promise significant advantages: higher coupling precision, twoqubit couplings of different forms, arbitrary and reconfigurable network structure and an effective zero temperature environment.^{9} Although key challenges remain in such an ion trap quantum adiabatic processor, such as stable laser delivery and the engineering challenges of fabricating largescale ion trap chips as discussed above, trapped atomic ions are wellsuited to the generic problem of quantum adiabatic algorithms.
Machine learning and the Boltzmann machine
Many models of artificial intelligence or machine learning are inspired by natural neural networks. In a model where the communication between neurons is bidirectional, the problem of determining the state of output neurons can be mapped to calculate the thermal distribution of an Ising model.^{98} The process of learning is the strengthening and weakening of connections between neurons, so that the output neurons optimally classify the signal received by the input neurons. In the model, learning is achieved by tuning the Ising model parameters to optimise the classification. This model of learning, known as a ‘Boltzmann machine’,^{99} can be simulated using an adiabatic/annealing quantum protocol wellsuited to the ion trap architecture.
The Boltzmann machine is borrowed from the field of machine learning and uses the thermal distribution of an Ising model to make classifications.^{99} These methods have had a recent renaissance because of increased computational power and large data sets making deep learning both practical and useful. The optimisation procedure involves inputting training data and observing the classifier labels. The Ising couplings between the spins are adjusted until the machine generates an optimal classification. Quantum annealing or the quantum adiabatic method can be used to determine the classification of the data. This may result in a different set of optimal coupling values that can also be tested on classical machines. Classical algorithms tend to use layers of spins where each layer can have arbitrary connections with the next layer: two layers is a reduced Boltzmann machine and many layers is a deep reduced Boltzmann machine. A full Boltzmann machine allows for connections between any spin. Quantum algorithms are predicted to speed up the tuning of the machine and have already provided insight into new classical algorithms.^{100,101} Both full and reduced Boltzmann machines can be implemented with our modular system. The natural connectivity suggests a different class of Boltzmann machines where the outputs of one full Boltzmann machine can be teleported to the inputs of a second full Boltzmann machine.
Quantum error correction
A truly scalable quantum computer requires that the computational failure rate does not depend on the system size. Faulttolerant quantum error correction provides a method for generating arbitrarily small logical errors when the physical gate errors are below an accuracy threshold.^{102} The accuracy threshold is defined over a family of codes and a pseudothreshold for a specific code is the error rate per physical gate below which the logical error rate is smaller than the physical error rate. A full discussion of quantum error correction is outside the scope of this article, and we refer the reader to reviews by Gottesman^{103} and Terhal^{104} for more information. Given the importance of quantum error correction as the system size increases, we would like to make a few remarks on performing quantum error correction methods in iontrap hardware.
The basic elements of quantum error correction have already been demonstrated in ion trap systems. Redundancy codes have been shown to reduce the damage caused by bitflip^{105} or phaseflip errors.^{106} The seven qubit Steane code^{8} corrects both bitflip and phaseflip errors and is the smallest example of a colour code.^{107} Ions have been encoded into the sevenqubit code^{108}, but errror correction has not yet been performed. The current challenge in the field is to implement a faulttolerant logical qubit memory.
In the near term, two promising implementations for quantum memories in single ELUs are small surface codes^{109} and the Steane logical qubit with Steane ancilla.^{8,110} These small errorcorrecting procedures only require 17–25 total ions per logical qubit to generate faulttolerant circuits that can have error thresholds near 10^{−3}. This error rate is compatible with the best current ion trap gates and measurements^{20,25,26,64,65,66}, and the whole procedure can easily fit within a single ELU. These small codes are only guaranteed to correct single errors, so the total number of reliable, but nonuniversal, operations on many logical qubits will scale as the error threshold divided by the square of the physical error per operation. Fault tolerance requires repeated measurement of error syndromes and ion shuttling or a secondion species will need to be used to reduce unwanted excitation of the data. This constraint is similar but less challenging than protecting the quantum data during generation of entanglement between communication qubits.
In the future, the architecture could support faulttolerant quantum computation. Proposals already exist for implementing the surface code^{46} and the concatenated Steane code^{44} in a modular architecture. There is currently great interest in the surface code due to its high computational threshold around 1%^{111,112} in 2D architectures. Early papers on the surface code emphasised computation near the threshold resulting in large overheads. For a target logical error rate of 10^{−10}, the surface code requires millions of physical qubits per logical qubit when the physical error is 95% of the threshold and a few hundred physical qubits per logical qubit when the error rate is 1% of threshold.^{112} The key feature of our architecture is that there are two length scales where nonlocal interactions are possible: within the ELU and between ELUs. This enables the testing of codes that cannot be easily implemented in a 2D geometry. The increased connectivity allows for a possible reduction in qubit resources by allowing universal gates through gauge fixing in a 3D colour code,^{113} which reduces the overhead due to magic state distillation^{114,115} and allows for the implementation of finite rate codes,^{116,117} which yield more logical qubits per physical qubit as the logical error decreases. We expect that future developments in quantum error correction will also benefit from a nonplanar geometry.
Outlook
Quantum computers will look very different than the semiconductorbased computers of today, just as current solidstate semiconductor devices look nothing like the vacuum tubes, relays and mechanical gears of an earlier era. Athough trapped atomic ion qubits may be seen as exotic today, their exquisite quantum coherence, highperformance quantum logic gates and unmatched connectivity and reconfigurability makes the trapped ion platform a leading candidate for largescale quantum computing. The continued progress in ion trap integration strategies and supporting technologies have the potential to enable practical quantumcomputing machines in a matter of years. We expect this device development to be driven by applications that harness the connectivity and reconfigurability of trapped ion qubits, where quantum computer scientists work closely with physicists and engineers in the codesign of tomorrow’s quantum computer.
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Acknowledgements
This work is supported by the US Army Research Office (ARO) with funds from the IARPA MQCO and LogiQ Programs and the ARO Atomic and Molecular Physics Program, the AFOSR MURI on Quantum Measurement and Verification, the DARPA Quiness Program, the Army Research Laboratory Center for Distributed Quantum Information, the NSF Physics Frontier Center at JQI and the NSF Physics at the Information Frontier program.
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Affiliations
Schools of Chemistry and Biochemistry, Computational Science and Engineering and Physics, Georgia Institute of Technology, Atlanta, GA, USA
 Kenneth R Brown
Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA
 Jungsang Kim
IonQ, Inc., College Park, MD, USA
 Jungsang Kim
 & Christopher Monroe
Joint Quantum Institute, Joint Center for Quantum Information and Computer Science and Department of Physics, University of Maryland, College Park, MD, USA
 Christopher Monroe
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Competing interests
JK and CM are founding scientists of ionQ. The remaining author declares no conflict of interest.
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Correspondence to Christopher Monroe.
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