PDF Quantum communication, computing, and measurement 2

Free download. Book file PDF easily for everyone and every device. You can download and read online Quantum communication, computing, and measurement 2 file PDF Book only if you are registered here. And also you can download or read online all Book PDF file that related with Quantum communication, computing, and measurement 2 book. Happy reading Quantum communication, computing, and measurement 2 Bookeveryone. Download file Free Book PDF Quantum communication, computing, and measurement 2 at Complete PDF Library. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Here is The CompletePDF Book Library. It's free to register here to get Book file PDF Quantum communication, computing, and measurement 2 Pocket Guide.

Entanglement of Formation; W. Giedke, et al. Sohma, et al. Feng, A. Barnett, et al.

Recommended for you

Latest Developments in Quantum Tomography; G. Raymer, et al. Vasilyev, et al. Homodyning Bell's Inequality; G. D'Ariano, et al. Hydrodynamical Quantum State Reconstruction; L. Hradil, et al. Accuracy in Quantum Homodyne Tomography; G. For integer factorization, trial division, the Rho method, the elliptic curve method are common algorithms. Fermat's method, the quadratic- and rational-sieve, leads to the general number field sieve NFS algorithm for integer splitting.

For large integers, the problem becomes intractable for classical computers. The many different guesses of the NFS algorithm are analogous to hitting the log using a dulled axe; after subexponential many tries, the log is cut by half. However, using a sharper axe allows you to split the log faster.

This sharpened axe is the quantum algorithm proposed by Shor in This approach has some issues. A large-enough quantum computer can efficiently break RSA for current instances. This sounds like a catastrophic story where all of our encrypted data and privacy are no longer secure with the advent of a quantum computer, and in some sense this is true. For example, a report shows experiments on the factorization of a bit number using 94 qubits, they also estimate that qubits are needed for factoring a bit number. Hence, there numbers indicates that we are still far from breaking RSA.

What if we increment RSA key sizes to be resistant to quantum algorithms, just like for symmetric algorithms? Bernstein et al. So, for public-key algorithms, increasing the size of keys does not help. In their report, the cost of factoring bit integers is estimated to take a few hours using a quantum machine of 20 million qubits, which is far from any current development. Something worth noting is that the number of qubits needed is two orders of magnitude smaller than the estimated numbers given in previous works developed in this decade.

Under these estimates, current encryption algorithms will remain secure several more years; however, consider the following not-so-unrealistic situation. Information currently encrypted with for example, RSA, can be easily decrypted with a quantum computer in the future. Now, suppose that someone records encrypted information and stores them until a quantum computer is able to decrypt ciphertexts. Although this could be as far as 20 years from now, the forward-secrecy principle is violated. A year gap to the future is sometimes difficult to imagine.

How does this impact the security of your personal information? What if the ciphertexts were company secrets or business deals? Although the current capacity of the physical implementation of quantum computers is far from a real threat to secure communications, a transition to use stronger problems to protect information has already started. This wave emerged as post-quantum cryptography PQC. The core idea of PQC is finding algorithms difficult enough that no quantum and classical algorithm can solve them.

A recurrent question is: How does it look like a problem that even a quantum computer can not solve? These so-called quantum-resistant algorithms rely on different hard mathematical assumptions; some of them as old as RSA, others more recently proposed. For example, McEliece cryptosystem, formulated in the late 70s, relies on the hardness of decoding a linear code in the sense of coding theory. As of , the NIST started an evaluation process that tracks possible alternatives for next-generation secure algorithms.

From a practical perspective, all candidates present different trade-offs in implementation and usage. An initial round collected 70 algorithms for deploying key encapsulation mechanisms and digital signatures. As of early , 28 of these survive and are currently in the analysis, investigation, and experimentation phase.

Cloudflare's mission is to help build a better Internet. As a proactive action, our cryptography team is preparing experiments on the deployment of post-quantum algorithms at Cloudflare scale. Watch our blog post for more details.


  • Quantum Computing in the NISQ era and beyond – Quantum.
  • T. S. USUDA : Publications 2?
  • Passar bra ihop.
  • Almost Like Being in Love: A Novel?

In anticipation of wide-spread quantum computing, the transition from classical public-key cryptography primitives to post-quantum PQ alternatives has started Today, we are excited to announce Cloudflare's Ethereum Gateway, where you can interact with the Ethereum network without installing any software on your computer Product News. Cloudflare Network. Deep Dive. Life Cloudflare. This post introduces quantum computing and describes the main aspects of this new computing model and its devastating impact on security standards; it summarizes some approaches to securing information using quantum-resistant algorithms.

Due to the relevance of this matter, we present our experiments on a large-scale deployment of quantum-resistant algorithms. Our third post introduces CIRCL , open-source Go library featuring optimized implementations of quantum-resistant algorithms and elliptic curve-based primitives. What is Quantum Computing?

Superposition One of the most exciting properties of quantum computing that provides an advantage over the classical computing model is superposition. Quantum computing — A qubit stores a combination of two or more states. Measurement In a classical computer, the values 0 and 1 are implemented as digital signals.

The qubit state is analogous to a point on a unitary circle. Another way to think about superposition and measurement is through the coin tossing experiment. Quantum Gates A logic gate represents a Boolean function operating over a set of inputs on the left and producing an output on the right. The NOT gate is a single-bit operation that flips the value of the input bit. The X quantum gate interchanges the amplitudes of the states of the input qubit. Reversibility An operation is reversible if there exists another operation that rolls back the output state to the initial state.

Composed Systems In quantum mechanics, a single qubit can be described as a single closed system: a system that has no interaction with the environment nor other qubits. Entanglement According to the theory behind quantum mechanics, some composed states can be described through the description of its constituents. Bell states are entangled qubit examples The entanglement phenomenon was pointed out by Einstein, Podolsky, and Rosen in the so-called EPR paradox. All Alice has to do is operate on her qubit according to the value of the message and send the resulting qubit to Bob.

Superdense coding protocol. The quantum teleportation protocol allows Alice to transmit a qubit to Bob without using a quantum communication channel. Alice measures the qubit to send Bob and her entangled qubit resulting in two bits. Quantum teleportation protocol. Quantum Parallelism Composed systems of qubits allow representation of more information per composed state.

Quantum Computers The theory of quantum computing is supported by investigations in the field of quantum mechanics. The DiVincenzo Criteria require that a physical implementation of a quantum computer must: Be scalable and have well-defined qubits. Be able to initialize qubits to a state.

Have long decoherence times to apply quantum error-correcting codes. Decoherence of a qubit happens when the qubit interacts with the environment, for example, when a measurement is performed. Attaching a quantum processor to a classical computer allows it to utilize all of its features without needing to start entirely from scratch. The rest of this chapter reviews the current candidate qubit technology choices upon which to base a quantum computer. For the two furthest developed quantum technologies, superconducting and trapped ion qubits, this discussion includes details of the qubit and control planes in use in prototypical computers at the time of publication of this report , the current challenges that must be overcome for each technology, and an assessment of the prospects for scale-up to very.

The review of other emerging technologies provides a sense of their current status, and potential advantages if they are developed further. The first quantum logic gate was demonstrated in using trapped atomic ions [ 1 ], following a theoretical proposal earlier in the same year [ 2 ]. Since the original demonstration, technical advances in qubit control have enabled experimental demonstration of fully functional processors at small scale and implementation of a wide range of simple quantum algorithms.

Recommended for you

Despite success in small-scale demonstrations, the task of constructing scalable and quantum computers considered viable by current computing industry standards out of trapped ions remains a significant challenge. Unlike the very large scale integration VLSI of transistors enabled by the integrated circuit IC , building a quantum computer based upon trapped ion qubits requires integration of technologies from a wide range of domains, including vacuum, laser, and optical systems, radio frequency RF and microwave technology, and coherent electronic controllers [ 3 - 5 ].

A path to a viable quantum computer must address these integration challenges. A trapped ion quantum data plane comprises the ions that serve as qubits and a trap that holds them in specific locations. Appendix B provides a technical overview of current strategies for constructing a trapped ion quantum data plane and its associated control and measurement plane. Based on the high-fidelity component operations demonstrated to date, small-scale ion trap systems have been assembled where a universal set of quantum logic operations can be implemented on a qubit system in a programmable manner [ 6 - 9 ], forming the basis of a general-purpose quantum computer.

Not surprisingly, at percent for two-qubit gates, the error rates of individual quantum logic operations in these fully functional qubit systems lag behind the 10 —2 to 10 —3 range [ 10 , 11 ] for state-of-the-art demonstrations of two-qubit systems, pointing to the challenge of maintaining the high fidelity across all qubits as the system.

Nonetheless, the versatility of these prototype systems has enabled a variety of quantum algorithms and tasks to be implemented on them. All of the prototype general-purpose trapped-ion quantum computer systems demonstrated to date consist of a chain of 5 to 20 static ions in a single potential well. In these machines, each single qubit gate operation takes 0. Each ion in the chain interacts with every other ion in the chain due to the strong Coulomb interaction in a tight trap through motional degree of freedom that is shared among the ions.

This interaction can be leveraged to realize quantum logic gates between nonadjacent ions, leading to dense connectivity among the qubits in a single ion chain. An alternative approach is to induce a two-qubit gate between an arbitrary pair of ions in the chain by illuminating specific ions with tightly focused and carefully tailored control signals, such that only the desired ions move—many control signals are used to make the force on all the other ions cancel out [ 19 ].

Using either approach, one can realize a general-purpose quantum processor with fully connected qubits [ 20 ], meaning that two-qubit gates may be implemented between arbitrary pairs of qubits in the system [ 21 ]; these capabilities are expected to scale to over 50 qubits in a relatively straightforward way [ 22 ]. It is likely that some early, small-scale quantum computers qubits based on ion traps will become available by the early s.

Like current machines, these early demonstration systems are likely to consist of a single chain of ions and feature unique all-to-all connectivity among the qubits in the chain, efficiently implementing any quantum circuit with arbitrary circuit structures. However, many conceptual and technical challenges remain toward a creating a truly scalable, fault-tolerant ion trap quantum computer.

Examples of such challenges include the difficulty of isolating individual ion motions as chain length increases, the number of ions one can individually address with gate laser beams, and measuring individual qubits. Further scaling of trapped ion quantum computers to well beyond the sizes necessary for demonstrating quantum supremacy.

Such shuttling requires a complex trap with multiple controllable electrodes.

Because the quantum information is stored in the internal states of the ion, which have been shown to be unaffected by shuttling between chains in small experiments, this approach does not contribute to any detectable decoherence [ 24 ]. Recent adoption of semiconductor microfabrication techniques has enabled the design and construction of highly complex ion traps, which are now routinely used for sophisticated shuttling procedures.

This technology could potentially be used to connect multiple ion chains on a single chip, enabling for an increase in scale—provided that the controllers necessary to manipulate these qubits can be integrated accordingly. Even if this ion shuttling is successful on a single chip, eventually the system will need to be scaled up further.

Two approaches are currently being explored: photonic interconnections, and tiling chips. A strategy for connecting multiple qubit subsystems into a much larger system is to use quantum communication channels. One viable approach involves preparing one of the ions in a subsystem in a particular excited state and inducing it to emit a photon in such a way that the quantum state of the photon for example, its polarization or frequency is entangled with the ion qubit [ 25 , 26 ]. When both output ports simultaneously record detection of a photon [ 27 ], it signals that the two ions that generated the photons have been prepared in a maximally entangled state [ 28 , 29 ].

This protocol entangles a pair of ion qubits across two chips, without the ion qubits ever directly interacting with each other. Although the protocol must be attempted many times until it succeeds, its successful execution is heralded by an unmistakable signature both detectors registering photons , and can be used deterministically in ensuing computational tasks—for example, to execute a two-qubit gate acting across chips [ 30 ]. This protocol was indeed demonstrated first in trapped ions [ 31 ] followed by other physical platforms [ 32 - 34 ]. Given the continued improvement.

This approach opens up the possibility of using existing photonic networking technology, such as large optical cross-connect switches [ 37 ], to connect hundreds of ion trap subsystems into a network of modular, parallel quantum computers [ 38 - 40 ]. An alternative approach to the scaling beyond a single-ion trap chip is to tile all-electrical trap subsystems to create a system where ions from one ion trap chip can be transferred to another chip [ 41 ].

This shuttling across different integrated circuits requires careful alignment of shuttling channels and special preparation of the boundaries of these integrated circuits, which has not yet been demonstrated. In this proposal, all qubit gates are carried out by microwave fields and magnetic field gradients, free from the off-resonant spontaneous scattering and stability challenges associated with the use of laser beams [ 42 ]. While this integration approach remains entirely speculative at this point, this approach has the potential benefit of relying only on mature microwave technology and electrical control for the critical quantum logic gates, rather than using lasers and optics, which require much higher precision components.

For trapped ions, necessary technology developments toward scalable quantum computer systems include the ability to fabricate ion traps with higher levels of functionality, assemble stabilized laser systems with adequate control, deliver electromagnetic EM fields that drive the quantum gates either microwave or optical to the ions with sufficient levels of precision to affect only the qubit being targeted preferably allowing multiple operations at a time , detect the qubit states in parallel without disturbing the data qubits, and program the control EM fields that manipulate the ion qubits so that the overall system achieves sufficient fidelity for the practical application needs.

If these challenges are met, one will be able to take advantage of the strengths in trapped ions: some of the best performances of all physical systems in representing a single qubit, thanks to the fact that these qubits are fundamentally identical as opposed to those which are manufactured , and the high fidelity of qubit operations at small experimental scales.

Like current silicon integrated circuits, superconducting qubits are lithographically defined electronic circuits. Their compatibility with. Appendix C provides a technical overview of current strategies for constructing a superconductor quantum data plane and its associated control and measurement plane.

In the context of digital quantum computation and quantum simulations, the present state-of-art for operational gate error rate is better than below 0. Based on these developments, superconducting qubit circuits with around 10 qubits have been engineered to demonstrate prototype quantum algorithms [ 48 , 49 ] and quantum simulations [ 50 , 51 ], prototype quantum error detection [ 52 - 55 ], and quantum memories [ 56 ], and, as of , cloud-based 5-, , and qubit circuits are available to users worldwide. However, the error rates are higher in these larger machines—for example, the 5-qubit machines available on the Web in have gate error rates of around 5 percent [ 57 , 58 ].

In the context of quantum annealing, commercial systems exist with over 2, qubits and integrated cryogenic control based on classical superconducting circuitry [ 59 , 60 ]. These are the largest qubit-based systems currently available, with two orders of magnitude times more qubits than current gate-based QCs. To achieve this scale machine required careful design trade-offs and significant engineering effort. The decision to integrate the control electronics with the qubits enabled D-Wave to rapidly scale the number of qubits in their system, but also results in the qubits being built in a more lossy material.

They purposely traded off qubit fidelity for an easier scaling path. Thus, the coherence times of the qubits in these machines are over 3 orders of magnitude worse than those in current gate-based machines, although this is expected to be less of a limitation for quantum annealers than for gate-based machines. Progress in gate-based machines has emphasized the optimization of qubit and gate fidelities, at sizes limited to on the order of tens of qubits. Since the first demonstration of a superconducting qubit in , the qubit coherence time T 2 in gate-level machines has improved more than five orders-of-magnitude, standing at around microseconds today.

This remarkable improvement in coherence arose from reducing energy losses in the qubit through advances in materials science, fabrication engineering, and qubit design by groups worldwide.

The Quantum Menace

The current approach, using room temperature control and measurement planes, with multiple wires per qubit, should scale to around 1, physical qubits [ 61 ]. This section reviews the factors that cause this limit, and then discusses what is currently known about the path to even larger machines. Many factors will limit the size of machine that can be achieved by simply scaling up the number of qubits placed on a single integrated circuit.

These include the following:. This areal connection will need three-dimensional 3D integration schemes using flip-chip bump-bonding and superconducting through-silicon vias, technologies that are being developed to connect high-coherence qubit chips with multilayer interconnect routing wafers [ 63 , 64 ]. First, qubit fidelities need to be improved to provide the lower error rates needed to support practical quantum error correction.

In addition, as the size of the computer increases to millions of qubits and beyond, advanced process monitoring, statistical process control, and new methods for reducing defects relevant to high-coherence devices will be required to assess and improve qubit yield. Just as fabrication tools have been specialized to target specific, advanced complementary metal-oxide semiconductor CMOS processes, it is likely that specialized tools that target specific qubit-fabrication processes will need to be developed to enhance yield and minimize fabrication-induced defects that cause decoherence.

Wafer real estate is another consideration for larger machines. Assuming qubit unit cells with repeat distance critical dimensions of 50 microns state-of-the-art today [ 65 ], a large integrated circuit of 20 mm by 20 mm. If one used an entire mm wafer for one processor, the wafer could hold around , qubits.

Blind topological measurement-based quantum computation

While that is sufficient for the near future, reducing the qubit unit cell critical dimension while retaining coherence and controllability will increase qubit density and enable larger numbers of qubits on a single mm wafer. Moving to wafer-size integrated circuits requires creating a new package.

The qubits are generally around 5 GHz, which corresponds to a free-space wavelength of around 60 mm. The wavelength is further reduced in the presence of dielectrics like the silicon wafer. Using the rule of thumb that a clean microwave environment requires dimensions less than one-quarter of a wavelength, it is clear that further research is needed before large high-quality packages can be built. Controlling more than a thousand qubits will require a new strategy for the control and measurement plane.

This control logic will need to be introduced using either 3D integration to connect the qubit plane with this local control plane or fabricated monolithically but must be done so without compromising qubit coherence and gate fidelity. Of course, this means that this logic will operate at very cold temperatures, either at tens of milli-Kelvins, or at 4 K. Operating at 4 K is much easier, since the capacity for heat dissipation is larger, and it saves on the wire count from room temperature to 4 K, but it still requires extensive control wiring to continue down to the base-temperature stage in the cryostat.

While there are technologies that could operate at these temperatures, including cryogenic CMOS, single-flux quantum SFQ , reciprocal quantum logic RQL , and adiabatic quantum flux parametrons, significant research will be needed to be create these designs at scale, and then determine which approaches are able to create a local control and measurement layer that supports the needed high-fidelity qubit operations. Even if one is able to scale to mm wafers, a large quantum computer will need to use a number of these subsystems, and with high probability, the optimal size of the subsystem will be modules smaller than that.

Thus, there will be a need to connect these subsystems to each other with some kind of quantum interconnect. There are two general approaches that are currently being pursued. One assumes that the interconnection between the modules is at milli-Kelvin temperatures, so one can use microwave photons to communicate. This involves creating guided channels for these photons, interconverting quantum information between a qubit and a microwave photon, and then converting the quantum information back from that photon to a second, distant qubit.

The other option is to couple the qubit state to a higher energy optical photon, which requires a high-fidelity microwave-to-optical conversion technique. This is an area of active research today. Since many technical challenges remain in scaling either trapped ion or superconducting quantum computers, a number of research groups are continuing to explore other approaches for creating qubits and quantum computers.

These technologies are much less developed, and are still focused on creating single qubit and two qubit gates.

The Imminent Threat of Quantum Algorithms

Appendix D provides an introduction to these approaches, which is summarized in this section. Photons have a number of properties that make them an attractive technology for quantum computers: they are quantum particles that interact weakly with their environment and with each other. This natural isolation from the environment makes them an obvious approach to quantum communication. This base communication utility, combined with excellent single-qubit gates with high fidelity means that many early quantum experiments were done using photons.

One key challenge with photonic quantum computers is how to create robust two-qubit gates. Researchers are currently working on two approaches for this issue. In linear optics quantum computing, an effective strong interaction is created by a combination of single-photon operations and measurements, which can be used to implement a probabilistic two-qubit gate, which heralds when it was successful. A second approach uses small structures in semiconductor crystals for photon interaction, and can also be considered a type of semiconductor quantum computer.

Work on building small-scale linear photon computers has been successful, and there are a number of groups trying to scale up the size of these machines. Because the photons used in photonic quantum computing typically have wavelengths that are around a micron, and because the photons move at the speed of light and are typically routed along one dimension of the optical chip, increasing the number of photons, and hence the number of qubits, to extremely large numbers in a photonic device is even more challenging than it is in systems with qubits that can be localized in space.

However, arrays with many thousands of qubits are expected to be possible [ 66 ]. Neutral atoms are another approach for qubits that is very similar to trapped ions, but instead of using ionized atoms and exploiting their charge to hold the qubits in place, neutral atoms and laser tweezers are.

Quantum Information and Measurement

Like trapped ion qubits, optical and microwave pulses are used for qubit manipulation, with lasers also being used to cool the atoms before computation. In , systems with 50 atoms have been demonstrated with relatively compact spacing between the atoms [ 67 ]. These systems have been used as analog quantum computers, where the interactions between qubits can be controlled by adjusting the spacing between the atoms. Building gate-based quantum computers using this technology requires creating high-quality two-qubit operations and isolating these operations from other neighboring qubits.

As of mid, entanglement error rates of 3 percent have been achieved in isolated two-qubit systems [ 68 ]. Scaling up a gate-based neutral atom system requires addressing many of the same issues that arise when scaling a trapped ion computer, since the control and measurement layers are the same. Its unique feature compared to trapped ions is its potential for building multidimensional arrays. Semiconductor qubits can be divided into two types depending on whether they use photons or electrical signals to control qubits and their interactions. Optically gated semiconductor qubits typically use optically active defects or quantum dots that induce strong effective couplings between photons, while electrically gated semiconductor qubits use voltages applied to lithographically defined metal gates to confine and manipulate the electrons that form the qubits.

While less developed than other quantum technologies, this approach is more similar to that used for current classical electronics, potentially enabling the large investments that have enabled the tremendous scalability of classical electronics to facilitate the scaling of quantum information processors. Scaling optically gated qubits requires improved uniformity and requires accommodation of the need to individually address optically each qubit. Electrically gated qubits are potentially very dense, but material issues have limited the quality of even single-qubit gates until recently [ 69 ].

While high density may enable a very large number of qubits to be integrated on the chip, it exacerbates the problem of building a control and measurement plane for these types of qubits: providing the needed wiring while avoiding interference and crosstalk between control signals will be extremely challenging. The final approach to quantum computing discussed here uses topological qubits. In this system, operations on the physical qubits have extremely high fidelities because the qubit operations are protected by topological symmetry implemented at the microscopic level: error correction is done by the qubit itself.

This will reduce and possibly eliminate the overhead of performing explicit quantum error correction. While this would be an amazing advance, topological qubits are the least developed technology platform. In mid, there are many nontrivial steps. Once these structures are built and controlled in the lab, the error resilience properties of this approach might enable it to scale faster than the other approaches. Many qubit technologies have significantly improved over the past decade, leading to the small gate-based quantum computers available today.

For all qubit technologies, the first major challenge is to lower qubit error rates in large systems while enabling measurements to be interspersed with qubit operations. As mentioned in Chapter 3 , the surface code is currently the primary approach to error correction for systems with high error rates. Current systems are limited by two-qubit gate error rates, which is still above the surface code threshold for the larger systems available today; error rates of at least an order of magnitude better than threshold are required if quantum error correction is to be practical.

Clearly, improving physical qubit fidelity—through improvements to fabrication and control—is paramount to demonstrating logical qubits or even a machine with physical qubits that can cascade an interesting number of qubit operations before losing coherence. The next challenge is to increase the number of qubits in the quantum computer. In fact, by mid a number of companies have announced ICs that contained order of 50 qubits, but as of this writing there are no published results benchmarking the functionality or error rates of these systems.

Unlike conventional silicon scaling, where creating the manufacturing process for the more complex integrated circuit set the pace of scaling, for quantum computing, scaling will be dictated by the degree of difficulty in obtaining low error rates with these larger qubit systems, a task that requires joint optimization. The next generation are likely to use linear ion traps, which will scale to the order of qubits. Further scaling will require another change to the trap design to enable shuttling of ions between different groups, which should also allow more flexible qubit measurements.

At some point in increasing the number of qubits in a quantum processor or chip, the scaling will become easier using a modular approach, where a number of chips are linked together to create a larger machine rather than creating a larger chip. A modular design will require the development of a fast, low error rate quantum interconnection between the modules; with photonic connections the most promising due to their speed and fidelity. While the component technologies and baseline protocols for realizing some of these integration strategies have already been demonstrated, system-scale demonstration with practical levels of performance remains a major challenge.

As a result of the challenges facing superconducting and trapped ion quantum data planes, it is not yet clear if or when either of these technologies can scale to the level needed for a large error corrected quantum computer. Thus, at this time, the viability of other, currently less-developed quantum data plane technologies cannot be ruled out, nor can the possibility that hybrid systems making use of multiple technologies might prevail. Monroe, D. Meekhof, B.

King, W.

Measuring Quantum Physics in a Quantum Annealer

Itano, and D. Wineland, , Demonstration of a fundamental quantum logic gate, Physical Review Letters Cirac and P. Zoller, , Quantum computations with cold trapped ions, Physical Review Letters Monroe and J. Kim, , Scaling the ion trap quantum processor, Science Brown, J. Kim and C. Monroe, , Co-designing a scalable quantum computer with trapped atomic ions, npj Quantum Information Kim, S.

Crain, C. Fang, J. Joseph, and P. Hanneke, J. Home, J. Jost, J. Amini, D. Leibfried and D. Wineland, , Realization of a programmable two-qubit quantum processor, Nature Physics Schindler, D. Nigg, T. Monz, J. Barreiro, E. Martinez, S. Wang, S. Quint, M. Brandl, V. Nebendahl, C. Roos, M. Chwalla, M. Hennrich, and R. Blatt, , A quantum information processor with trapped ions, New Journal of Physics Debnath, N. Linke, C. Figgatt, K. Landsman, K. Wright, and C. Monroe, , Demonstration of a small programmable quantum computer with atomic qubits, Nature Friis, O.

Marty, C. Maier, C. Hempel, M. Holzapfel, P. Jurcevic, M. Plenio, M. Huber, C. Roos, R. Blatt, and B. Gaebler, T. Tan, Y. Lin, Y. Wan, R. Bowler, A. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, and D. Ballance, T. Harty, N.