Josephson tunnel junctions and two-level systems

Josephson tunnel junctions are formed by two superconducting electrodes separated by a very thin (∼1 nm) insulating barrier. In this configuration, the collective superconducting order of one electrode (parameterized by a phase φ₁) coherently connects with that of the other electrode (φ₂) via the elastic tunneling of Cooper pairs through the barrier (see Figure 2). As Josephson first showed more than 50 years ago, the resulting supercurrent, I, and junction voltage, V, are related to the superconducting phase difference, φ = φ₁–φ₂, across the junction:^{Reference Josephson33}

$$I = {I_C}\sin \varphi$$

$$V = {{{\Phi _0}d\varphi } \over {2\pi \,dt}}$$

Here, I _C is the junction critical current, determined by junction area and barrier transparency, and Φ₀ = h/2e is the superconducting magnetic flux quantum. Combining these using a standard relation for inductance, V = L _J (dI/dt), one can identify a Josephson inductance,

$${L_J} = {{{\Phi _0}} \over {2\pi {I_C}\cos \varphi }}$$

which is nonlinear in φ. The parallel-plate-like structure of the Josephson junction electrodes results in a self-capacitance, C _J, which, when combined with the nonlinear L _J, makes for an anharmonic oscillator (see Figure 1). It is this feature of the Josephson junction that can turn a harmonic superconducting circuit into a uniquely addressable qubit.

The most widespread and robust process for fabricating Josephson junctions involves the diffusion-limited oxidation of an aluminum base layer to form a relatively uniform amorphous oxide tunnel barrier.^{Reference Gurvitch, Washington and Huggins34} The process is widespread, despite the fact that for nearly 30 years this approach has been known to produce barriers with measureable densities of defects.^{Reference Rogers and Buhrman35,Reference Wakai and Van Harlingen36} These defects were thought to be either localized states in the insulator’s bandgap that can temporarily trap electrons, or as tunneling “hot-spots” that incoherently shunt the tunnel junction. Both mechanisms were thought to be responsible for 1/f -type noise (e.g., voltage noise across the junction or flux noise in a superconducting quantum interference device [SQUID]). These noise sources are now considered to be of secondary importance when compared with those defects that form microscopic two-level systems (TLSs).

In the context of superconducting qubits, a TLS is a localized low-energy excitation that is predominantly found in noncrystalline dielectric materials. Microscopically, a TLS is generally visualized as an ion or electron that can tunnel between two spatial quantum states. These systems occur due to either defects in the crystal structure or the presence of polar impurities such as OH^–. Because of the charged nature of the TLS, it has a dipole moment that will interact with electromagnetic fields. Intentionally or otherwise, dielectric materials are found in every type of electronic device, and superconducting qubits are no exception. Because qubits operate at very low temperatures and powers, TLSs can significantly impact their performance.

The impact of TLSs on qubit coherence was not fully appreciated until the observation by Simmonds et al. in 2004 of individual spurious microwave resonators in a phase qubit.^{Reference Simmonds, Lang, Hite, Nam, Pappas and Martinis37} These resonators were observed in qubit spectroscopy measurements as avoided level crossings—the energy-level repulsion that occurs, proportional to the coupling strength, when two coupled quantum systems become degenerate (have the same energy splitting) and hybridize (see Figure 4c). These spurious resonators were observed to limit both the qubit’s coherence and measurement fidelity. Cooper et al. later used a similar device to demonstrate quantum-coherent oscillations between a qubit and TLS, and observed that the coherence times of a TLS could be comparable to (or even longer than) those of the qubits available at that time.^{Reference Cooper, Steffen, McDermott, Simmonds, Oh, Hite, Pappas and Martinis38} Shortly thereafter, Plourde et al. observed TLSs in a flux qubit.^{Reference Plourde, Robertson, Reichardt, Hime, Linzen, Wu and Clarke39}

There are two primary ways by which a TLS in the tunnel barrier can interact with the qubit.^{Reference Ku and Yu40} In the first scenario, the dipole moment of the TLS couples directly to the voltage across the junction.^{Reference Martin, Bulaevskii and Shnirman41} When the TLS and qubit are in resonance, charge fluctuations in the former drive phase oscillations in the latter, and the two quantum systems become coupled. The qubit exchanges energy coherently with the TLS, which subsequently loses that energy via phonon emission, resulting in decoherence of the coupled qubit-TLS system. In the second scenario, low-frequency charge fluctuations in the TLS causes fluctuations in the height of the junction’s tunneling barrier.^{Reference Yu42} This modulation of the barrier height leads to critical current noise, which in turn produces fluctuations in the energy level splitting of the qubit and leads to dephasing. Based on the distribution of TLS-qubit avoided level crossings observed in phase qubits, Martinis et al. concluded that it was the direct interaction of the junction’s electric field with charge fluctuations that caused the spectral splittings.^{Reference Martinis, Cooper, McDermott, Steffen, Ansmann, Osborn, Cicak, Oh, Pappas, Simmonds and Yu43} That is not to say that critical current fluctuations will not matter, but experiments performed to date suggest they are not yet a limiting factor in measured qubit coherence times.^{Reference Van Harlingen, Robertson, Plourde, Reichardt, Crane and Clarke44}

If TLSs exist in the thin insulating oxide that serves as the Josephson tunnel barrier, can they be eliminated, or at least drastically reduced, by improving the oxide quality? Oh et al. showed that, yes, their number can be reduced by growing a crystalline barrier layer.^{Reference Oh, Cicak, Kline, Sillanpaa, Osborn, Whittaker, Simmonds and Pappas45} In general, the conventional approach has been to oxidize the surface of a thin Al layer to form an amorphous AlO_x barrier.^{Reference Gurvitch, Washington and Huggins34} In this process, the thickness of the oxide, and the critical current density of the resulting JJ, is determined by a combination of oxygen gas exposure and diffusion kinetics. In contrast, Oh et al. grew ultra-thin epitaxial Al₂O₃ layers on the (0001) surface of rhenium (chosen for its low free energy of oxidation and near-perfect lattice match with sapphire) and used these crystalline multilayers for qubit fabrication (see Figure 4). When compared with a traditional diffused AlO_x barrier, the epitaxial barrier showed five times fewer TLSs for the same tunnel junction size.

Figure 4. Reduction in two-level system (TLS) density resulting from the use of epitaxial trilayers. (a) Process for the growth of Re/Al₂O₃/Al trilayers with the corresponding electron diffraction images. Epitaxial Re is first deposited at high temperature (850°C) on C-plane sapphire, followed by the reactive deposition of Al₂O₃ at room temperature. The oxide layer is crystallized by annealing the film to 800°C in an O₂ background. Finally, a polycrystalline Al layer is deposited for the top electrode. (b) A cross-sectional schematic showing the various device layers. For phase qubit fabrication, the trilayer is patterned using standard photolithography and selectively etched. Amorphous SiO₂ is used as an insulator, with vias etched to both the top Al electrode and the bottom Re electrode (red oval). A second Al layer is used to complete device wiring. (c) Spectroscopy scans from phase qubits with amorphous (top) and crystalline (bottom) tunnel barriers. By adjusting the bias current, one can change the operating frequency f _μw of the qubit. The spectral splittings occur when the energy-level difference of the qubit matches that of a TLS in the junction barrier. The single-crystal barrier exhibits five times fewer such defects, with a resulting Rabi decay time of 90 ns. Reprinted with permission from Reference 45. © 2006 The American Physical Society.

Unfortunately, little progress has been made beyond this initial reduction in defect density. Instead of pushing for new materials that reduce TLS densities another order of magnitude or more, the superconducting qubit community has largely avoided the TLSs observed primarily in phase qubits (junction feature sizes 1–10 μm) by making qubit designs with smaller junctions (feature sizes 0.1–0.3 μm) and thereby reducing the net number of defects. In contemporary qubits, in fact, it appears anecdotally that the TLS defect density may actually decrease, speculatively due to reduced effects of stress in the smaller geometries. Indeed, there are reports of qubits featuring small junctions that exhibit very low-loss junction inductance^{Reference Bylander, Gustavsson, Yan, Yoshihara, Karrabi, Fitch, Cory, Nakamura, Tsai and Oliver16,Reference Paik, Schuster, Bishop, Kirchmair, Catelani, Sears, Johnson, Reagor, Frunzio, Glazman, Girvin, Devoret and Schoelkopf17} and junction capacitance.^{Reference Bylander, Gustavsson, Yan, Yoshihara, Karrabi, Fitch, Cory, Nakamura, Tsai and Oliver16,Reference Kim, Suri, Zaretskey, Novikov, Osborn, Mizel, Wellstood and Palmer87} Nonetheless, in the interest of long-term scalability, TLSs in the junction barrier should not be ignored. When scaled to the level of hundreds or thousands or even millions of JJs, and targeting high yield with minimal junction variation, it becomes clear that defect density (even a reduced one) will impact performance, and further work on tunnel barrier materials is warranted.

Some alternate approaches proposed by Welander et al. include a multilayer growth scheme for superconductor/insulator/superconductor “trilayers”^{Reference Welander46} and the growth of amorphous oxide barriers by co-deposition instead of diffusion.^{Reference Welander, McArdle and Eckstein47} The multilayer approach addresses the issue of high surface roughness in rhenium thin films^{Reference Oh, Hite, Cicak, Osborn, Simmonds, McDermott, Cooper, Steffen, Martinis and Pappas48} by instead using rhenium as a buffer layer only between a thick, smooth bottom electrode layer (e.g., Nb) and a crystalline oxide tunnel barrier. On the other hand, a co-deposited amorphous oxide may also show reduced TLS density; Josephson junctions formed in this way exhibit near-ideal tunneling behavior, while those with diffused oxide barriers show an excess shunt conductance. Neither of these approaches has yet been reported to be utilized in a qubit.

For probing defects in tunnel barriers, Stoutimore et al. have developed a new device that is much simpler than a qubit in many respects: the Josephson junction defect spectrometer (JJDS).^{Reference Stoutimore, Khalil, Lobb and Osborn49} The JJDS is essentially a JJ in parallel with a meandering inductor and an inter-digital capacitor. Under an applied magnetic field, it becomes a frequency-tunable resonator with a nearly harmonic potential, which allows it to operate over a wide frequency range. The critical current of the JJ can vary by about a factor of 5 without affecting device functionality, and the only requirement for the JJ inductance is that it exceeds that of the parallel inductor. The JJDS is also compatible with both methods of JJ fabrication (shadow evaporation, and trilayer deposition and etching), making it a versatile device for the study of future tunnel junction materials.

What resonators can tell us about materials and fabrication

Resonators have been perhaps the most useful devices for the study of superconducting materials to date and have been used to identify a number of noise sources and decoherence mechanisms that affect qubits. Relatively easy to fabricate and measure, resonators have helped researchers characterize new materials and improve fabrication processes. This new knowledge has, in turn, played a large role in the ever-increasing coherence times of superconducting qubits.

One application of resonators is in the detection of TLSs. Aside from appearing in the insulating tunnel barrier of a JJ, TLSs also reveal themselves in the dielectric materials used to separate and insulate between device wiring layers. An ensemble of TLS defects in these materials will absorb and dissipate energy at low powers and low temperatures, but will saturate as both voltage and temperature increase. This behavior will manifest itself as an increasing loss tangent, tan δ, with decreasing power that plateaus only at very low powers. In essence, at either high power (i.e., large ciculating field inside the resonator) or high temperatures, the TLSs saturate into a 50-50 mixture of ground and excited states. In this configuration, the TLSs exchange energy (absorb and emit energy) with both the resonator and the crystal lattice, thereby reducing the apparent energy loss from the resonator. In contrast, at low powers and temperatures that are cold compared with the TLS level splitting, the TLSs tend to be in their ground states. In this configuration, they can readily absorb energy from the resonator and dissipate that energy via phonon emission, thereby increasing the resonator loss. The low-power, low-temperature regime is of interest to qubits, because they operate at cold temperatures and low fields, generally absorbing or emitting only single photons when transiting between ground and excited states. This effect with TLSs was first observed by Schickfus and Hunklinger, who used a superconducting cavity resonator to measure the dielectric absorption of vitreous silica.^{Reference Schickfus and Hunklinger50} More recently, in the context of studying loss mechanisms relevant to superconducting qubits, Martinis et al. used a lumped-element LC resonator to measure the loss tangent of chemical vapor-deposited SiO₂ and SiN_x dielectrics at low temperature and variable power.^{Reference Martinis, Cooper, McDermott, Steffen, Ansmann, Osborn, Cicak, Oh, Pappas, Simmonds and Yu43} Both showed a decrease in tan δ with increasing power, but over the entire range studied, the SiN_x loss was about 30 times lower (see Figure 5). A phase qubit was subsequently fabricated with SiN_x and exhibited a coherence time about 20 times longer than that observed with SiO₂ dielectric. Further resonator experiments by O’Connell et al. confirmed these findings and tested a number of other insulating materials, including bulk substrate crystals of silicon and sapphire.^{Reference O’Connell, Ansmann, Bialczak, Hofheinz, Katz, Lucero, McKenney, Neeley, Wang, Weig, Cleland and Martinis51} Using both CPW (co-planar waveguide) and LC resonators, they found that hydrogen-doped amorphous silicon (a-Si:H) showed a further fivefold improvement in loss tangent over SiN_x.

Figure 5. Dielectric loss, tan δ, for several materials used in the fabrication of superconducting qubits. (a) A comparison of amorphous thin-film insulators grown by chemical vapor deposition, measured using an LC resonator with parallel plate capacitor. SiO₂ was deposited from silane and oxygen precursors at two different temperatures: 13°C for a-SiO₂-1 and 250°C for a-SiO₂-2. SiN_x was deposited at 100°C from silane and nitrogen. The lower tan δ for SiN_x led to a 20-fold increase in T₁. Reprinted with permission from Reference 43. © 2005 The American Physical Society. (b) Further comparison of dielectric loss, including hydrogen-doped amorphous silicon (a-Si:H) and bulk single-crystal CZ silicon. In this case, both lumped-element (LC) and co-planar waveguide (CPW) resonators were used for measurements. Again, amorphous silica exhibits the greatest loss, with a-Si:H loss nearly two orders of magnitude lower. Reprinted with permission from Reference 51. © 2008 AIP Publishing.

Resonators are also sensitive to the TLSs that form on metal surfaces and at interfaces between the metal and substrate. A number of research groups have studied CPW resonators fabricated from a variety of superconducting metals, including aluminum, niobium, and rhenium (see Figure 6a).^{Reference Gao, Zmuidzinas, Mazin, LeDuc and Day52–Reference Khalil, Wellstood and Osborn59} By measuring the loss (1/Q) and resonant frequency in these CPWs as a function of device geometry, power, and temperature, it is clear that TLSs are present. The primary culprit seemed to be native surface oxides that formed on the metal during air exposure. This argument was supported by the fact that CPWs made from niobium (a strong getter) had consistently higher loss than those made from either aluminum (oxide terminates at a few nm) or rhenium (very low free energy of oxidation).^{Reference Sage, Bolkhovsky, Oliver, Turek and Welander58} However, resonator electric-field simulations indicatd that the metal-substrate and substrate-air interfaces could dominate the surface loss.^{Reference Wenner, Barends, Bialczak, Chen, Kelly, Lucero, Mariantoni, Megrant, O’Malley, Sank, Vainsencher, Wang, White, Yin, Zhao, Cleland and Martinis88} With these results in mind, more recent work on resonator (and qubit) materials has centered on ways to either prepare the substrate better or use materials that are less susceptible to contamination in device processing or oxidation in air.

Figure 6. Using co-planar waveguide (CPW) resonators to compare materials and surface preparation techniques. (a) λ/2 resonators used in (b). (b) Comparison of resonator quality factor Q versus power (and average photon number) for CPW resonators fabricated from Al, Nb, Re, and TiN. Substrates, deposition technique, and metal crystallinity were varied with the following general trend: Nb exhibited the lowest Q, Al and Re were comparable, and TiN had the highest Q. See Reference 58 for details. (a–b) Reprinted with permission from Reference 58. © 2011 AIP Publishing. (c) Comparison of internal loss (1/Q) versus electric field for CPWs fabricated from TiN grown on sapphire, SiN films, and nitrided Si. At low fields (the regime of qubits), TiN growth on sapphire gave the highest loss, while that on bare silicon yielded the lowest-loss material. The substrate material and preparation procedure also impacted TiN film orientation. Reprinted with permission from Reference 62. © 2010 AIP Publishing. (d) Comparison of Q versus average photon number for Al CPWs on sapphire. Both the Al deposition technique (sputter, e-beam, molecular beam epitaxy [MBE]) and the substrate preparation method impact resonator quality, with about a fivefold variation in Q over the measured power range. In situ sapphire annealing at 850°C in an activated oxygen flux (O₂*) gave the highest Q. Reprinted with permission from Reference 65. © 2012 AIP Publishing.

To address the issue of native oxides on metal surfaces, a number of groups have begun to fabricate CPW resonators from superconducting nitrides.^{Reference Sage, Bolkhovsky, Oliver, Turek and Welander58,Reference Barends, Hortensius, Zijlstra, Baselmans, Yates, Gao and Klapwijk60–Reference Vissers, Gao, Wisbey, Hite, Tsuei, Corcoles, Steffen and Pappas62} Titanium nitride (TiN) resonators have exhibited Q values that can be significantly higher than those for conventionally grown aluminum or rhenium. It has been suggested that this lower loss in TiN is because nitrides, in general, are very stable against oxidation. In fact, TiN has been shown to develop a surface oxide several nm thick after just one hour of air exposure.^{Reference Saha and Tompkins63} This is similar to the terminal oxide thickness on aluminum, so the improved Q values cannot be explained by a lack of surface oxide. Perhaps, by some unknown mechanism, the TiN native oxide simply has fewer TLSs than those on metals. Or, perhaps the difference is related to different etch chemistries, the metal-substrate interface that is influenced during the TiN growth, or some other fabrication-related loss mechanism.^{Reference Sandberg, Vissers, Kline, Weides, Gao, Wisbey and Pappas64}

The substrate/superconductor interface is also critical to device performance.^{Reference Wenner, Barends, Bialczak, Chen, Kelly, Lucero, Mariantoni, Megrant, O’Malley, Sank, Vainsencher, Wang, White, Yin, Zhao, Cleland and Martinis88} Vissers et al. demonstrated this for TiN growth.^{Reference Vissers, Gao, Wisbey, Hite, Tsuei, Corcoles, Steffen and Pappas62} They found that careful cleaning of the Si wafer, followed by a brief nitridation of the Si surface just prior to TiN deposition, yielded CPW resonators with Q ∼ 10⁶ at low powers (see Figure 6b). Films grown on sapphire or on thick SiN layers exhibited lower Q values—whether this was due to TLSs at the interface or in the thick SiN, or due to the observed change in film orientation is unknown. Regardless of the microscopic details, TiN has been successfully implemented in a transmon qubit, leading to measured coherence times (T₁ and T₂) of about 55 μs.^{Reference Chang, Vissers, Corcoles, Sandberg, Gao, Abraham, Chow, Gambetta, Rothwell, Keefe, Steffen and Pappas20} Megrant et al. found similar improvements for molecular beam epitaxially grown aluminum CPW resonators fabricated on sapphire wafers.^{Reference Megrant, Neill, Barends, Chiaro, Chen, Feigl, Kelly, Lucero, Mariantoni, O’Malley, Sank, Vainsencher, Wenner, White, Yin, Zhao, Palmstrom, Martinis and Cleland65} Prior to aluminum deposition at room temperature, the sapphire was either outgassed at 200°C in UHV, at 850°C in UHV, at 850°C in activated oxygen (O₂*), or left untreated. Higher temperature heat treatments resulted in both better aluminum epitaxy and greater resonator quality factors, Q (see Figure 6c). The addition of O₂* during outgassing improved Q for some devices, but did not impact others. Higher temperatures can serve to remove contaminants and recrystallize the substrate surface, and the O₂* may be necessary to remove surface-bound elements like carbon and hydrogen. This cleaning procedure and epitaxial Al was incorporated into a modified transmon qubit that showed a significant increase in decay time (T₁) up to 44 μs. With success stories like these, it is clear that much can be learned about superconducting materials and device fabrication through the study of resonators.

SQUIDs and flux noise

Excess magnetic flux noise has been a well-known problem in SQUIDs for more than 25 years.^{Reference Wellstood, Urbina and Clarke66} The noise spectrum at low frequencies scales approximately as A ²/(f/1 Hz), and the spectral density of the SQUID flux noise has decreased from approximately A = 10 μΦ₀/Hz^1/2 around 1990 to a level today of around A = 1 μΦ₀/Hz^1/2 at 1 Hz. This is generally attributed to improved fabrication processes. Aside from this general decrease, the observed 1/f behavior is nearly universal, having been observed by a number of research groups using a wide range of materials and device designs. Since SQUIDs are a primary building block of tunable superconducting qubits, it is perhaps not surprising that the influence of flux noise has also been observed in nearly all such qubits (e.g., see the flux noise characterization in References 13, 16, 67 [flux qubits], and 68 [phase qubits]). Recent measurements by Sendelbach et al. of SQUIDs at millikelvin temperatures showed them to be paramagnetic with a Curie-like 1/T susceptibility (see Figure 7a).^{Reference Sendelbach, Hover, Kittel, Muck, Martinis and McDermott69} This behavior, they argue, is the consequence of unpaired spins on the superconductor surface with an areal density of about 5 × 10¹⁷ m^–2. (A similar density of spins has been observed on gold rings.^{Reference Bluhm, Bert, Koshnick, Huber and Moler70}) Furthermore, these surface spins may interact and form spin clusters.^{Reference Sendelbach, Hover, Muck and McDermott71} Anton et al. observed a pivoting in the flux noise power spectra with temperature (see Figure 7b), and suggested that clusters can be either ferromagnetic, random (glassy), or antiferromagnetic.^{Reference Anton, Birenbaum, O’Kelley, Bolkhovsky, Braje, Fitch, Neeley, Hilton, Cho, Irwin, Wellstood, Oliver, Shnirman and Clarke72} While evidence of surface spins and their behavior on metals and superconductors is building, a precise understanding of their microscopic origin has been elusive.

Figure 7. Flux measurements using DC superconducting quantum interference devices (SQUIDs). (a) Paramagnetic behavior of surface spins in SQUIDs at millikelvin temperatures results in a change in the observed flux with temperature. Data for two SQUIDS are shown: one fabricated from a Nb-based trilayer, the other from an Al trilayer. The flux exhibits a Curie-type 1/T dependence. (b) The temperature-dependent flux through the Nb SQUID varies as a function of applied field, B _fc, that is present as the Nb SQUID cools through the superconducting transition temperature. Also shown is the relative change in SQUID flux, ΔΦ, at temperatures between 500 and 100 mK versus the applied magnetic field during field cooling. (a–b) Reprinted with permission from Reference 69. © 2008 The American Physical Society. (c) Temperature dependence of flux noise spectral densities, S _Φ(f) in two Nb trilayer SQUIDs. The top plot shows raw data for SQUID II.3, while the right plots show fits to A²/(f /1 Hz)^α + C², where α < 1. For both SQUIDs, the 1/f-type power spectra changes slope with temperature and generally pivots around a single frequency. Reprinted with permission from Reference 72. © 2013 The American Physical Society.

A number of theories have been put forth that attempt to explain the origin of these surface spins. Koch et al. suggested that the surface spins are unpaired electrons that hop between defect centers.^{Reference Koch, DiVincenzo and Clarke73} Hopping is thermally activated, and the electron spin is fixed in a specific orientation while it occupies the defect site. The spins of individual traps are randomly oriented, and the trapping energies vary broadly. A second model, proposed by Choi et al., involves potential disorder at the metal-insulator interface.^{Reference Choi, Lee, Louie and Clarke74} At an ideal, structurally coherent interface, there exist metal-induced gap states (MIGS) in the bandgap that extend into the metal but decay rapidly in the insulator.^{Reference Louie and Cohen75} A small amount of structural disorder will produce random fluctuations in the electronic potential, resulting in strongly localized, singly occupied MIGS near the interface. Both of these models—hopping spins and localized MIGS—presently involve independent spins, and it is not yet clear how to reconcile these with evidence of spin interactions and clustering.

Two models that include interactions have been proposed as well. Faoro et al. suggested that noise is produced by spins on the superconductor surface that are strongly coupled by Ruderman-Kittel-Kasuya-Yosida interactions.^{Reference Faoro and Ioffe76,Reference Faoro, Ioffe and Kitaev77} At intermediate frequencies, 1/f flux noise is due to spin diffusion, which is mediated by the exchange interaction with conduction electrons. Low-frequency noise, on the other hand, arises due to the slow switching rate of spin pairs between the singlet and triplet states. In the final model we present here, Wu and Yu have suggested that hyperfine interactions between electron spins on a metal surface and nearby nuclear spins are the source of 1/f flux noise.^{Reference Wu and Yu78} The surface electrons reside in harmonic traps and have relaxation times that are dominated by hyperfine spin exchange with the nearest non-zero nuclear moment. If this model holds, flux noise could be reduced significantly by switching to a superconductor with a low abundance of isotopes with nuclear moments. It should be emphasized, however, that none of these theories have been universally confirmed or refuted, and that more experimentation is needed to resolve this critical problem.

Summary

The past 10–15 years have witnessed coherence times in superconducting qubits increase dramatically—a full five orders of magnitude—bringing these devices to the threshold of technical reality. This evolution has been driven by innovations in qubit design, device fabrication, and materials science, and further research in these areas will be required to make superconducting qubit technology truly scalable. We have attempted to demonstrate how qubits and their constituent components have been useful in identifying previously unknown under-appreciated material defects, and how new materials and new fabrication processes can help improve device performance. Nowhere are these two sides of the coin more evident than with TLSs. Using qubits and resonators, one can observe the quantum behavior of individual TLSs and measure the loss stemming from TLS ensembles in dielectrics or on surfaces and at interfaces. New materials, growth procedures, and surface preparation methods have also eliminated many TLS sources, leading to substantial increases in resonator quality factors and qubit coherence times. One might hope that the same approach will help to elucidate and eliminate the source of interacting surface spins that seem to afflict both qubits and SQUIDs. And as the technology advances and superconducting qubits become more sensitive, the effects of decoherence mechanisms such as quasiparticle tunneling or critical current fluctuations are being revisited in more detail.^{Reference Pekola, Anghel, Suppula, Suoknuuti, Manninen and Manninen79–Reference Riste, Cultink, Tiggelman, Schouten, Lehnert and DiCarlo82} Moving forward, over the next decade, we anticipate a scaling in both the complexity and the requirements imposed on superconducting qubit circuits and their coherence times. Efforts to reduce the materials and fabrication-induced sources of decoherence will play an increasingly important role in maintaining and improving qubit performance in conjunction with these new engineering challenges.