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D Other Approaches to Building Qubits 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. These technologies are much less developed and are still focused on creating single-qubit and two-qubit gates. Scale-up issues for these technologies have many similarities to those faced by ion traps and by superconductors. The rest of this appendix will briefly discuss these methods. D.1 PHOTONIC QUANTUM COMPUTATION Photons have some properties that make them extremely attractive for use in quantum computers: photons interact relatively weakly with their environment and with each other. This is the reason that photons can travel quite far in many materials without being scattered or absorbed, giving photonic qubits good coherence properties and making them useful for transmitting quantum information over long distances [1]. Thus, research and development in this area is important for enabling long-distance quantum communication channels even if other technologies turn out to be preferable for large-scale computing applications. Development of photonic quantum manipulation capabilities has potentially transformative applications for quantum sensing and quantum communication. Experiments probing quantum entanglement of photons have a long history, dating back to the earliest experiments looking for violations of Bellâs theorem1 in the 1970s [2]. Over the past several decades, methods have been developed that overcome many of the impediments to creating, manipulating, and measuring many-photon entangled states. This section describes briefly these advances, the remaining challenges that must be overcome to develop error-corrected photonic processors, and the ultimate limits to scale-up. In many ways, photons are excellent qubits; single-qubit gates can be performed using standard optical devices such as phase shifters and beamsplitters, and as mentioned earlier they interact weakly with matter and with each other, giving them good coherence. But their strengthâweak interactionsâ also causes a major hurdle to the development of photonic quantum computers, since two-qubit gates become difficult to create. Two strategies for overcoming this issue are described in this section. 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 two-qubit gate. A second approach, which uses optically active defects and quantum dots2 that interact strongly with photons to induce strong 1 Bellâs theorem says that âIf [a hidden variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local.â In essence, it suggests that a nonquantum physical theory that explains quantum mechanical phenomena such as entanglement would refute the current understanding of quantum physics. John Bell, 1987, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, p. 65. 2 Also referred to as ânanoparticles,â quantum dots are small clusters of atoms with a crystalline structure whose physical properties are quite different from the properties of the elements involved in either atomic or bulk form. Quantum dots exhibit unusual propertiesâfor example, the wavelength of light they absorb or emit may be tuned through engineering of their size. PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-1

effective interactions between photons, is discussed in Section 5.4.3.1 on optically gated semiconducting qubits. In photonic quantum computing, typically the qubits are individual photons, with the two different photon polarizations (up-down and left-right) serving as the two qubit states. Single qubit gates can be implemented with standard passive optical components used to rotate the polarization, but two- qubit gates require a low-loss nonlinearity, which is difficult to achieve [3]. As described in the trapped ion section of Chapter 5, coincident measurements on two output ports of a beamsplitter create a strong effective nonlinearity and implement a two-qubit gate [4], but the gate is probabilistic. Fortunately, the gate signals when it was successful (photons are detected on both detectors), which means that algorithms can be implemented, but the timing requirements are complex, and a steady source of suitably initialized photons is needed. More recently, measurement-based quantum computing schemes, in which a highly entangled âcluster stateâ is constructed before the start of the computation and the computation itself is implemented by performing measurements, have attracted substantial interest [5]. Many of the technical developments needed to implement photonic quantum computing have been achieved over the past several years. Photonic chips continue to improve, and photon loss rates both within photonic elements and at interfaces are approaching the values needed to be able to implement quantum error correction. Very high efficiency photon detectors have been developed [6], which are key to the implementation of error correction. These nanowire-based detectors operate at helium temperatures (about 4 K), so cooling to this temperature will be required, but as described earlier, such cooling is expected to be entirely feasible. Assuming continued progress in reducing photon loss rates, the main hurdle toward the fabrication of devices of moderate size is to develop a source that generates at a high rate triplets of entangled photons [7]. Sources of triplets of entangled photons exist [8], but the rate at which entangled photon triplets are generated would need to be increased substantially for this strategy to enable large scale computations. As of 2018, the largest entangled and fully connected system of qubits was demonstrated using three degrees of freedom on each of six photons [9], although this method faces its own challenges and is unlikely to scale. Ultimate scalability: 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, the number of photons, and hence the number of qubits, in a photonic device cannot be made as large as in systems with qubits that can be localized in space. However, arrays with many thousands of qubits are expected to be possible [10]. In addition, the technology will be crucial for developing switching networks that will enable quantum communication on large scales. D.2 NEUTRAL ATOM QUANTUM COMPUTATION Rather than creating an array of ions and using the charges on the ions to hold them in place, one can use lasers to create an array of optical traps that confine neutral atoms [11,12]. This approach has technological similarities to ion trap quantum computation, and uses optical and microwave pulses for qubit manipulation, with the potential for making individual arrays with up to a million qubits. Neutral atom technology may be extremely useful for providing an interface between photons and other types of qubits, including superconducting qubits [13]. To date, arrays of about 50 atoms have been made, and a 51-atom quantum simulator has been demonstrated [14]. Assuming a typical 5-micron spacing, 104 atoms can be trapped in a 0.5 mm two-dimensional (2D) array, and a million atoms can be trapped in a 0.5 mm three-dimensional (3D) array. The qubit states are the energy levels of an alkali atom (often rubidium or cesium), there is one atom per trap, and the qubit manipulation and readout are performed optically. Like trapped ion systems, lasers are used to cool the atoms to micro-Kelvin temperatures, and then these very cold atoms are loaded into optical traps in a vacuum system. Another laser is used to initialize the state of the qubit, logic gates are carried out via a combination of optical and microwave PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-2

fields, and the output is detected via resonance fluorescence [15]. In this system there are a number of challenges just to create the starting state of the system: ï· Light-assisted collisions during laser cooling tend to cause atoms to pair and leave the trap. Vacancies complicate the arrayâs use as a quantum computer. However, recently methods have been developed that take traps with vacancies and reconfigure them to create traps with full occupancy, so this difficulty is not insurmountable. ï· Neutral atoms are vulnerable to being knocked out of their traps by collisions with residual background gas atoms. In standard systems, these collisions occur about once every 100 seconds per atom. Lifetimes exceeding tens of minutes are possible in cryogenic vacuum systems. Eventually, error correction schemes will need to be employed to deal with this infrequent loss. Atom reloading from an auxiliary reservoir of precooled atoms, which provides a path toward continuous operation, has been demonstrated on a small scale [16]. ï· Currently, sideband laser cooling has been used to get about 90 percent of the atoms in a trap in their absolute 3D vibrational ground state. This is cold enough for most quantum computing schemes, but it is believed that the cooling can be improved significantly; theoretical cooling limits approach 100 percent ground state occupation. Because single-qubit gate times range from a few to a few hundred microseconds, in principle, on the order of 105 operations can be performed within the longest demonstrated decoherence times (these are best-case numbers). Single-qubit gates of low error rates (down to 0.004) have been demonstrated;[17] in experiments the fidelity is limited by inhomogeneities in the microwave field, variability in the trap-induced shifts in qubit transition frequencies, and errors arising from imprecision or imperfections in the laser beam that affect nontargeted sites [18]. The strategies for two-qubit gates are again similar to those for trapped ions. One method requires moving the desired atoms close together; since the atoms are neutral, the spacing must be small, so accurate enough control of moving traps and the motional states of the atoms is challenging. The other method is to temporarily excite the atoms to highly excited Rydberg states (where an electron is very weakly bound to the atom), in which they have strong mutual dipolar interactions. This second approach has been pursued by several groups. Theoretical calculations predict that an entanglement error rate of 0.01 percent should be achievable; as of mid-2018, entanglement error rates of 3 percent have been achieved [19]. Known sources of infidelity such as heating of the atoms and the finite radiative lifetimes of the Rydberg states in current experiments are not sufficient to explain this large value, but it is known that fluctuating background electric fields due to atoms and molecules adsorbed on the container surfaces could yield greater infidelity during two-qubit gates because of the large susceptibility of the Rydberg atoms. This problem could be addressed by the development of appropriate surface coatings. Experimental improvement of the two-qubit gates is critical for this technology to be competitive with superconducting and ion trap qubits. Ultimate scalability: The trapping mechanism for neutral atoms is different than for trapped ions, but this platform will use similar control and measurement planes. The vision for scaling beyond the number of qubits that can be controlled in a single array is to connect multiple arrays using photonic entanglement, again following the architecture that is being developed for trapped ion systems. D.3 SEMICONDUCTOR QUBITS Semiconductor qubits can be divided into two types, depending on whether they are manipulated optically or electrically. 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, a technology that is very similar to that used for current PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-3

classical computing electronics. Optically gated semiconducting qubits can be used to implement strong effective interactions with photons, which greatly enhances the capabilities of photonic qubitsâfor example, by being a mechanism for implementing a quantum memory for optical photons. Electrically gated semiconducting qubits are attractive because the methods used to fabricate and control them are quite similar to those used in classical computing electronics, potentially enabling the large investments that have enabled the tremendous scalability of classical electronics to facilitate the scaling of quantum information processors. D.3.1 Optically Gated Qubits in Crystals An optically gated semiconductor qubit is a system in a semiconductor (typically either a defect in a crystal or a quantum dot in a host material) whose optical response depends on the quantum state of that defect/dot. Defect and quantum dot systems have somewhat complementary strengths and weaknesses, but also have many commonalities. Qubits constructed from optically active impurities or quantum dots in semiconductors provide a means of introducing strong nonlinearities into photonic approaches and also have the potential to be transformative for communication and sensing applications. A defect system that has been the focus of intense interest is the nitrogen-vacancy (NV) center in diamond [20,21]. This defect, which consists of a nitrogen atom substituting for a carbon together with a vacancy, is a paramagnetic center that can be manipulated and measured optically. Initialization, manipulation, and measurements of individual NV centers have been demonstrated [22]. Quantum manipulation has been demonstrated of defect centers in other materials, including vacancies in silicon carbide [23]. Remarkably, quantum coherence in these systems can persist at temperatures as high as room temperature [24]. Because of their quantum coherence at high temperatures and their good biocompatibility, optically active defect centers in semiconductors are expected to have important applications as quantum sensors [25], including for biological applications [26]. Two-qubit gates between these qubits either requires them to be extremely close together [27] (tens of nanometers), which makes optical addressing of the detects extremely hard, or requires them to be coupled using photons [28]. Using photons allows the qubits to be spaced meters apart, but because the interaction between the defects and photons tends to be weak, entanglement-generating gates tend to be slow (typically, many attempts at the entangling operation must be made before one succeeds). While successful gate operation is heralded, the slow entanglement rates complicate attempts to create entanglement between large numbers of qubits. Optically active quantum dots also have been demonstrated to have promise for applications requiring quantum coherence. Two-qubit gates have been implemented using tunnel couplings between quantum dots [29], and strong coupling between photons and quantum dots has been achieved [30], which is promising for the development of high-fidelity photon-mediated two-qubit gates. Qubit speeds in these systems tend to be very fast, but decoherence rates are also fast. The strong coupling between quantum dots and photons makes them attractive as a mechanism for integration with photonic quantum computing, enabling creation of entangled states of three photons [31] and enabling the implementations of quantum memories for photonic circuits [32]. Materials development will be key to improving optically gated semiconducting qubits. For defect centers in semiconductors, it would be extremely useful to find a defect-material combination in which the couplings between the defect and crystal lattice excitations are very weak, so that essentially all the optical decays do not transfer energy to the crystal lattice. While there has been some important work showing the importance and demonstrating the promise of theoretical techniques for predicting robust qubits in new materials [33], more needs to be done in this area. It is also important to increase the relatively weak coupling between photons and the defects; much recent progress has been enabled by improving control of the optical fields to increase coupling, and further improvements should be possible. This also relates to exploring mechanisms for spin decoherence and strategies to increase quantum coherence times [34]. For quantum dots, a major limitation currently arises from the difficulties in PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-4

developing well-controlled and reproducible fabrication methods: because the optical properties of a quantum dot depend on its size and shape, uniform and predictable quantum dot sizes are critical. Ultimate scalability: Because the requirements of ensuring optical access to be able to address each qubit individually places significant constraints on qubit densities, scale-up to very large numbers of qubits in these systems will be challenging. However, they are likely to be very important as interconnects, providing a method to interface material-based qubits with optical photons that can maintain coherence over extremely long distances [35]. In addition, because of sensing applications, systems with moderate numbers of qubits are likely to be important commercially, providing a means of establishing commercial viability of quantum systems as they are scaled up to sizes relevant for information processing applications. D.3.2 Electrically Gated Semiconductor Qubits Electrically gated semiconducting quantum computing technologies have the potential to scale up to extremely large number of qubits, because of the qubitsâ small size and because of the use of fabrication methods very similar to those used in classical electronics. Electrically gated semiconducting qubits are defined and manipulated by applying voltages to lithographically defined metal gates on semiconductor surfaces [36]. The fabrication and lithographic methods are very similar to those used in classical electronics, and the similarity of methods makes it plausible that the large investments that have been made to enable scale-up of classical electronics can be leveraged to facilitate scale-up to very large numbers of qubits. However, in this platform a significant amount of materials and technique development was required to be able to construct even single qubits, and high-fidelity single-qubit gates have been achieved only relatively recently [37]. Over the past few years, high-fidelity single-qubit gates have been implemented by several groups, and there has been substantial recent progress toward the implementation of high-fidelity two-qubit gates [38], and very recently quantum algorithms have been implemented on a programmable two-qubit quantum processor [39]. A key enabler of these recent advances was the development of new materials systems and lithographic methods that have enabled experimenters to overcome limitations of previous materials platforms and lithography strategies. The first electrically gated semiconducting qubits were fabricated in heterostructures of gallium arsenide and aluminum gallium arsenide [40], but in this materials system the decohering effects of the nuclear spins in the host material greatly complicated the implementation of high-fidelity gate operations. The development of qubits in silicon-based structures [41-43] has greatly reduced decoherence from nuclear spins, because natural silicon has an abundant zero-spin nuclear isotope, and isotopically enriched silicon in which more than 99 percent of the nuclei have spin zero has recently become available, which has led to further substantial increases in the coherence times [44]. Another important development was the development of new device designs that enabled more compact gate patterns and also enabled a transition from doped to accumulation-mode devices. These changes enabled the fabrication of devices with small (~25 nm) dots with reasonable device yields. The current challenge for the field is the development of reliable and high-fidelity two-qubit gates. Current two-qubit gate error rates [45-48] are about 10 percent, and further improvements are needed to achieve fault-tolerant operation. Currently, charge noise in these devices limits gate coherence, but recent work points to strategies that are expected to enable high-fidelity gating in the near term [49- 52]. Recent progress has been rapid, but it is constrained by the mediocre fabrication yields in current university-based fabrication facilities of the complex multilayer gate patterns separated by very thin oxide layers. Fabrication yields are expected to improve rapidly with the recent entry into this area by industry, including HRL Laboratories and Intel, and by participation by Department of Energy (DOE) labs such as Sandia National Laboratories. In principle, electrically gated semiconducting qubits have the potential for scalability to billions of qubits, because the methods used for fabrication are so similar to those used for classical electronics PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-5

and the qubit footprints are substantially less than a square micron. In practice, in addition to developing two-qubit gates with the requisite fidelities, measurement fidelities need to be improved and the measurement methods need to be made compatible with large-scale qubit arrays. Also, because of similarities in the cooling requirements, the control strategies, and the frequency range of the qubit control voltages with those of superconducting qubits, it will be necessary to overcome crosstalk and fanout issues similar to those faced by the superconducting qubit community. These issues will be especially challenging in this system, since the small spacing between the qubits will exacerbate the coupling between wires, and it will be harder to create scalable control/measurement layers that can interface with the qubits. D.4 TOPOLOGICAL QUBITS Development of topological quantum computing architectures is an approach for constructing qubits that could plausibly achieve extremely low intrinsic error rates so that implementation of error correction using logical qubits would not be necessary or would at least enable error correction with substantially less overhead. If successful, this approach would greatly reduce the number physical qubits needed to achieve the computational power to solve problems that are not tractable on classical computers compared to other approaches. Thus, it could be a promising path to scaling for a quantum computer. Topological quantum computation enables operations on the physical qubits to have extremely high fidelities because the qubit operations are protected by topological symmetry implemented at the microscopic level. Topological protection of quantum information is also the basis underlying the surface code, so one can view topological quantum computation as the implementation of the error-correction mechanism into the microscopic physics instead of by application of an error-correction algorithm on nontopological qubits. The potential to achieve the extremely high fidelities required to solve commercially interesting problems that are intractable on classical computers without the need to incur the large overheads involved in error correction is a strong motivation for the significant investments in this strategy for quantum computation by companies like Microsoft. However, the committee notes that the technology is significantly less developed than the others described in this report: there are nontrivial steps to demonstrating even the capability of single-qubit operations experimentally at the time of the writing of this report (2018) [53]. To implement topological quantum computation, one must construct a system in which there is a large number of degenerate ground states that cannot be obtained from each other from local changes. A simple example relevant to current efforts to implement topological quantum computation experimentally is shown in Figure D.1. This figure illustrates that there are systems of spinless fermions whose ground states can be viewed as collections of Majorana fermions paired on neighboring sites, with two âleftoverâ sites on the ends. The unpaired Majorana fermions can be arbitrarily far apart, and recombining them requires modifying the quantum state of the entire length of the system, which makes the excitations extremely resistant to local perturbations. The interest in developing materials systems that can support Majorana zero modes was sparked by Kitaevâs work (2003) showing that a quantum computer can be constructed if these topological excitations could be constructed and manipulated appropriately [54]. Much work has been done to make the construction of an appropriate system more feasible experimentally, with recent work demonstrating that quantum computation can be implemented if arrays of nanowires of a material with strong spin-orbit coupling that are strongly coupled to superconducting films where the single-particle excitations are highly suppressed can be constructed and measured [55]. While experimental demonstration of nontrivial manipulation of Majorana zero modes has not yet been achieved, the evidence that such nanowires have excitations at the nanowire ends that exhibit interactions that decay exponentially with the wire length is very strong [56]. Given a well-controlled materials system that supports Majorana zero modes, there is expected to be a reasonably straightforward experimental path toward the demonstration of the performance of a nontrivial qubit operation [57]. However, substantial materials and fabrication PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-6

challenges remain in order to do so. Some of the complexities that must be dealt with are that the excitations on the superconducting nanowires are measured via coupling to nonsuperconducting quantum dots, and the couplings between these dissimilar systems must be well controlled and tunable to implement the necessary operations. FIGURE D.1 Schematic of a one-dimensional (1D) system supporting Majorana zero modes. Each spinless fermion decomposes into two Majorana fermions, one on each site (denoted by Î³âs and shown in red). The Majoranas pair in the bulk (denoted by the thick lines connecting them), leaving two zero- energy Majorana modes at the ends of the chain. The large spatial separation between the two ends underlies the resistance to decoherence of quantum computing implemented in this architecture. SOURCE: J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M.P.A. Fisher, 2011, Non-Abelian statistics and topological quantum information processing in 1D wire networks, Nature Physics 7(5):412-417. Once successful nontrivial manipulation and measurement of Majorana zero modes has been demonstrated experimentally, it will be possible to determine whether the excellent fidelities that have been predicted theoretically are indeed achieved in experiment. If experimentally measured fidelities do indeed improve exponentially with the length of the nanowires on the expected length scale of microns, then nanowires with modest lengths could yield gates with extremely high fidelities. It should be noted that, similar to proposed implementations of the surface code, implementation of Clifford gates is expected to be significantly more straightforward than the realization of an additional gate (often called the âT gateâ) necessary to implement universal quantum computation. Recent theoretical work predicts that high-fidelity T gates are achievable using the same hardware architecture as that used for the Clifford gates [58], but the implementation of these gates is an additional step necessary for the implementation of a universal quantum computer using this technology. As discussed above, significant materials, fabrication, and measurement challenges must be overcome to demonstrate even single-qubit gates of a topological quantum computer. However, the possibility of being able to implement extremely high fidelity gates that do not require error correction, or require very little error correction, is strong motivation to pursue this approach, partly because of the challenges that arise in the implementation of quantum error correction and partly because the necessary processor sizes would be much smaller than those needed for error-corrected architectures. NOTES [1] J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, J.-W. Pan, 2017, âSatellite-based entanglement distribution over 1200 kilometers,â Science, 356:1140-1144. [2] S.J. Freedman and J.F. Clauser, 1972, âExperimental test of local hidden-variable theories,â Physical Review Letters, 28:938-941. [3] J.W. Silverstone, D. Bonneau, J.L. OâBrien, and M.G. Thompson, 2016, âSilicon quantum photonics,â IEEE Journal of Selected Topics in Quantum Electronics, 22: 390-402. [4] E. Knill, R. Laflamme, and G. J. Milburn, 2001, âA scheme for efficient quantum computation with linear optics,â Nature, 409: 46-52. PREPUBLICATION COPY â SUBJECT TO FURTHER EDITORIAL CORRECTION D-7

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