The phase qubit features a single Josephson junction biased with a current I, as shown in Fig. 1(a). From the resistively and capacitively shunted junction (RCSJ) model^[15] and considering Eq. (1), it is easy to find that the system is governed by 3 4 Here C and R are junction’s capacitance and effective resistance, and m = (Φ₀/2π)²C, E_J = Φ₀ I_c/2π. Equation (3) describes a fictitious particle with mass m moving in a potential U(φ) with damping coefficient m/RC. The Hamiltonian of the system can be written as H = Q²/2C + U(φ) with charge Q = 2en. The Cooper pair number operator n = −i∂_φ is the quantum-mechanical conjugate to φ.^[15]

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. (color online) (a) Current-biased Josephson junction as a phase qubit and (b) its potential and energy levels. (c) and (d) Corresponding ones of the rf-SQUID-type phase qubit, which can also be operated as a flux qubit.

The qubit has a washboard-like potential U(φ), which is depicted in Fig. 1(b) together with the energy levels in a selected well. Experimentally one can tilt the potential by changing I so that the barrier height and the level spacing can be tuned. The bottom two states are used for the qubit states, which can be initialized when the barrier is set sufficiently high and manipulated with microwave pulses. The state readout is accomplished by setting appropriate barrier heights at which tunneling out of the potential well from the ground and excited states can be distinguished. Tunneling results in the switching of the junction from the zero to finite voltage state which is then detected.^[5]

One can also use the magnetic flux Φ as the experimentally controllable parameter instead of I by using an rf-SQUID type configuration shown in Fig. 1(c). In this case, when the flux quantization condition Eq. (2) is taken into account, the system is then governed by 5 6 in which ϕ = Φ/Φ₀ and ϕ_e = Φ_e/Φ₀ are the normalized total and external fluxes, respectively, and , β = 2πLI_c/Φ₀ with the loop inductance L. It can be verified that when β > 1, the two lowest potential wells are symmetric when ϕ_e = 1/2, which correspond to the two circulating currents of opposite directions in the loop. Changing ϕ_e away from 1/2 will tilt the potential wells, and also the barrier height and level spacing. Figure 1(d) shows the case when ϕ_e ≠ 1/2, in which the bottom two states in the left well can be used as the phase qubit states. The state manipulation is similar to the situation discussed above but the state readout is realized by a SQUID detector which monitors the circulating current direction change upon tunneling out from the left to the right potential well. The rf-SQUID type phase qubit has been developed and studied extensively by the UCSB group.^[16–19]

Figure 2 shows the photograph of an rf-SQUID-type phase qubit fabricated using the multilayer process.^[20] The device layout is similar to the design developed in Refs. [18] and [19] but the fabrication process is changed to include the shadow evaporation of the Josephson junctions. The design looks a bit complicated largely due to the effort to minimize the crosstalks among different parts of the circuits. In the design, the SQUID detector contains three Josephson junctions, where a small junction is in parallel with two larger ones connected in series whose critical current is 1.7 times that of the small one. The advantage of the 3-junction dc-SQUID is that it requires no external flux bias for the operation as a magnetometer. One is able to modulate the flux sensitivity and turn on (off) the coupling during qubit measurement (operation).^[18]

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. (color online) Optical microscope image of an rf-SQUID-type phase qubit. The red open square and circles enclose the shadow evaporated junctions of the qubit and detector SQUID, respectively. The bar at the bottom-right corner indicates a length of 100 μm. (Figure reproduced with permission from Ref. [20]).

It can be seen from Fig. 1(d) that the qubit can be operated as a flux type device if the two bottom states |↑ ⟩ and |↓ ⟩ in two different potential wells are used.^[21] The two states correspond to the currents circulating in the qubit loop with different polarity. This rf-SQUID type flux qubit, which differs from the persistent-current flux qubit, is the key component in the quantum annealer built by the D-Wave Systems Inc., which we will discuss in further detail below.

Figure 3(a) shows the schematic of a charge qubit. Clearly, the Hamiltonian of the system can be written as H = (Q – C_g V_g)²/2(C + C_g) − E_J cos φ, with Q = 2en being the charge on the island separated by the tunneling barrier and the capacitor C_g, where C_g and V_g are the gate capacitance and gate voltage, respectively. Denoting the charging energy scale E_C = e²/2(C + C_g), the Hamiltonian becomes

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. (color online) (a) The charge qubit schematic. The transmon qubit can be derived by shorting the gate voltage source (Vg = 0). Energy spectra in the cases of EC ≫ EJ (b) and EC ≪ EJ (c). (d) Potential well and energy levels of the transmon qubit.

In the opposite regime where E_C ≪ E_J, the energy spectrum is quite different and becomes almost flat, as can be seen in Fig. 3(c). Koch et al. proposed a new type of qubit operating in this regime, which was termed “transmon”.^[23–25] The transmon qubit has an obvious advantage that it is much less sensitive to the charge noise due to the flat spectrum and therefore has a longer coherence time as compared to the charge qubit. A flat spectrum also means that one does not need to tune n_g as in the case for the charge qubit. The lowest two energy levels are used as the qubit levels as is depicted in Fig. 3(d). Experimentally, the transmon qubit can be built by simply connecting a capacitor in parallel with a Josephson junction, namely by shorting the voltage source V_g in Fig. 3(a). In this case, the transmon qubit is similar to the phase qubit in Fig. 1(a) with identical Hamiltonian when I = 0 and the junction is connected in parallel with a capacitor.

Figure 4(a) shows a transmon qubit imbedded in three-dimensional (3D) cavity resonator.^[26] The enlarged part is the qubit with a shadow evaporated junction at the center and the two large pads being the capacitor electrodes on one hand and making the coupling to the resonator on the other. The resonator coupled with the qubit is used to detect the qubit states and also protect the qubit from the environmental electromagnetic noises (see below). Coherence times as long as T₁ = 60 μs, T₂ = 20 μs, and T_echo = 25 μs were measured in the energy relaxation, Ramsey, and spin-echo experiments, respectively. Figure 4(b) shows a recently reported two-dimensional (2D) transmon qubit,^[27] where the two large horizontal pads are the capacitor electrodes with the junction located at the center. Two smaller pads at the bottom are used for the capacitive couplings to the other qubits, while the small pad at the top is leading to a resonator that is used for both the microwave excitation and qubit state measurement. Coherence times of T₁ and T_echo around 30 μs and 20 μs were demonstrated experimentally.^[28]

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. (color online) Optical microscope images of three-dimensional (3D) (a) and two-dimensional (2D) (b) transmon qubits. Figures reproduced with permission from Refs. [26] and [27], respectively.

A new qubit design named Xmon^[29] was developed by the UCSB group based on the earlier planar transmon qubit.^[23–25] As can be seen in Fig. 5, the Xmon qubit has a cross shape with its four arms connected to XY control, Z control, coupling to other qubit, and the resonator, which is used for the qubit state readout. The extreme flexibility makes the Xmon qubit suitable for various studies of the coupled qubit system. The energy relaxation time T₁ above 40 μs was experimentally demonstrated in these devices.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. (color online) (a), (b) Optical micrographs of an Xmon qubit. (c) The electrical circuit of the qubit. Figure reproduced with permission from Ref. [29].

The qubit state measurement of the transmon and Xmon devices is realized by coupling them to a resonator, e.g., to a 3D resonator as shown in Fig. 4(a) or to a 2D coplanar waveguide (CPW) resonator as shown in Figs. 4(b) and 5(a). This qubit-resonator circuit quantum electrodynamics (QED) architecture enables the quantum nondemolition measurement for the qubit state,^[30–32] which will be discussed in Section 3.

The persistent-current flux qubit was first proposed by Mooij et al., which had its original structure as shown in Fig. 6(a) in the absence of the shunting capacitor C_S.^[33,34] The qubit is formed with a superconducting loop interrupted by three Josephson junctions, with one junction a factor of α smaller in area than the other two. The junctions contribute much inductance and a small loop with negligible inductance is used, making it less sensitive to the external noise. Considering the energy of the form 1 − cos φ stored in the each junction,^[15] the qubit potential can be written as: 8 U ( φ 1 , φ 2 ) = E J ( 2 − cos φ 1 − cos φ 2 ) + α E J [ 1 − cos ( 2 π f + φ 1 − φ 2 ) ] , in which φ_1,2 are the phase differences across the two larger junctions, and f is the magnetic flux frustration defined by φ₁ − φ₂ + φ₃ = −2π f with φ₃ being the phase difference of the smaller junction.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. (color online) (a) A persistent-current flux qubit schematic. (b) A contour plot of the potential in Eq. (8). (c) The potential along the direction indicated by a red line in panel (b) when Φe ≠ Φ0/2. States in the two potential wells correspond to the currents circulating in the qubit loop with opposite polarities.

The general feature of U(φ₁, φ₂) is plotted in Fig. 6(b). A double-well potential can be seen more clearly in panel (c), which results from a cut along the red line in panel (b) and appears asymmetric as the external flux bias Φ_e ≠ Φ₀/2. The barrier height separating the two wells can be adjusted by changing α. Usually, α is in the range of 0.5–0.7 or below so that quantum tunneling occurs leading to the hybridization of the |↑ ⟩ and |↓ ⟩ states near their degeneracy point of Φ_e = Φ₀/2. The |↑ ⟩ and |↓ ⟩ states correspond to the currents circulating in the qubit loop with opposite directions and the two hybridized states are used as the qubit states.

The probability of observing |↑ ⟩ or |↓ ⟩ state is different in the qubit ground and excited states when it is not biased at degeneracy point. Hence one may detect the qubit states using a dc-SQUID that monitors the loop flux difference between the two states.^[6] At the degeneracy point, the probability of observing the two basis states is the same and there is no net current difference between the ground and excited states. In this case, it is possible to determine the qubit state by applying a fast flux bias pulse to adiabatically shift the qubit to a non-degeneracy point. It is also possible to detect the quantum state using the inductive method similar to that used for transmon and Xmon.^[35–37] This method, which can work at the degeneracy point, has the advantage of avoiding the voltage state of readout dc-SQUID so its back action to the qubit is minimized.

One can replace the small junction with a dc-SQUID, called α loop, which allows the modulation of the critical current and the qubit level spacing (or gap) by external flux. This type of gap tunable flux qubit has been demonstrated by several groups.^[38–40] Using this design, Fedorov et al. observed macroscopic quantum coherence oscillations^[41] and Zhu et al. coherently coupled a flux qubit to the nitrogen–vacancy (NV) centers.^[42]

Compared with the phase and transmon (Xmon) qubits, the flux qubit is advantageous in that it has a much larger anharmonicity. On the other hand, You et al.^[43] proposed a design to increase the qubit coherence time by shunting the small junction with a capacitor C_S to suppress the charge noise, as is shown in Fig. 6(a). Systematic works by the MIT and UC Berkeley groups showed a significant increase in the coherence times, with T₁ reaching 40 μs–50 μs and T₂ approaching 2T₁ ∼ 80 μs.^[44] The devices also showed strong anharmonicity of 500 MHz–700 MHz and the improved sample reproducibility.

One important feature of the flux qubits reported in Ref. [44] is the reduced α value. It is shown that for α greater than 0.5 the coherence time is much shorter. When α is decreased below 0.5, the double well feature of the qubit potential weakens with decreasing barrier height and the flux qubit becomes similar to the transmon qubit. It is reported that the flux qubits with the modified design eliminating the quadratic term show the T₁ values around 80 μs (see the device in Fig. 7).^[45] Recent work also shows that by implementing a pumping sequence to reduce the number of unpaired quasiparticles in close proximity to the device, a threefold enhancement in qubit relaxation times and a comparable reduction in coherence variability can be achieved.^[46]

Fig. 7.

	Figure Option ViewDownloadNew Window
	Fig. 7. (color online) (a), (b) Microscope images of a persistent-current qubit. The horizontal λ/2 CPW resonator at the bottom in panel (a) couples to the qubit for the qubit state measurement. Figure reproduced with permission from Ref. [45].

Superconducting qubits have many types as discussed previously and there are also many ways of coupling them together, including capacitive and inductive couplings, and couplings via Josephson junctions and resonators.^[9,10] Single- and multi-qubit quantum gate operations, required for the universal gate-based quantum computation, have been experimentally demonstrated with improving fidelity, starting from the early years after finding the qubit coherence. Single qubit logic operations of different angle rotations about the x, y, and z axes of the Bloch sphere are performed, for instance, in the quantronium,^[74] phase,^[75] and transmon^[76] qubits, which suffice the usual single qubit phase gate, phase-slip gate, NOT gate, and Hadamard gate. Two qubit logic operations such as controlled-NOT (cNOT) gates are realized in charge^[77] and persistent-current qubit.^[77,78] The square root of i-SWAP gate is also realized in the phase qubit.^[79] The three qubit Toffoli gates (ccNOT) are successfully demonstrated in transmon qubits.^[80,81] Universal gate sets are proposed^[82] and are experimentally implemented and characterized.^[83]

Superconducting circuits and qubits behave quantum mechanically on a macroscopic scale with the system parameters conveniently controllable, which provide an excellent tool for the studies of quantum physics, atomic physics, and quantum optics. There are vast published works in the literature, out of which we only mention a few: (i) Resonant escapes and bifurcation phenomena of nonlinear systems under strong driving.^[84,85] (ii) Macroscopic quantum tunneling in cuprate materials and phase diffusion in the quantum regime.^[86,87] (iii) Quantum stochastic synchronization in dissipative quantum systems.^[88] (iv) Landau–Zener–Stückelberg interference and the precise control of quantum states in the tripartite system.^[89,90] (v) Topological phase diagram and phase transitions in interacting quantum systems.^[91] (vi) A Schröinger cat living in two boxes.^[92] (vii) The Autler–Townes splitting phenomena.^[93–97] (viii) The stimulated Raman adiabatic passage and coherent population transfer.^[98,99] (ix) The electromagnetically induced transparency, a phenomenon similar to Autler–Townes splitting but physically distinct.^[100,101] (x) Resonance fluorescence and correlated emission lasing.^[102,103]

Adiabatic quantum computing or quantum annealing was proposed as a heuristic technique for quantum enhanced optimization nearly two decades ago.^[104–107] This architecture, known to be equivalent to the standard gate-based quantum computation,^[108,109] is simpler and provides a more practical approach for applications in the near term. The first superconducting quantum annealing processors were built by the D-Wave Systems Inc.^[110,111] based on the rf-SQUID type flux qubits^[21] and the Nb-junction technology.^[112,113] The D-Wave One, D-Wave Two, D-Wave 2X, and D-Wave 2000Q machines have become available in the years 2011, 2013, 2015, and 2017 with coupled 128, 512, 1024, and 2048 qubits, respectively. Although superconducting quantum annealing has received much attention both academically and industrially, a consensus on some physics and its potential for achieving optimization algorithm has not been reached. For more recent studies the readers are referred to the works in Refs. [114]–[117] by Google Inc. and D-Wave Systems Inc., for instance, and references therein.

Quantum simulation, namely using a controllable quantum system to simulate another quantum system, was first proposed by Richard Feynman and was envisioned as the main application of a quantum computer.^[118,119] Quantum simulation has two types: The digital quantum simulation and the analog quantum simulation. The former simulates a quantum system by dividing the time evolution of a Hamiltonian, written as a sum of local terms, into small intervals which can be realized by a sequence of qubit gate operations.^[120] In the analog quantum simulation, the Hamiltonian of the system to be simulated is directly mapped onto the Hamiltonian of the simulator that can be experimentally controlled. Quantum simulation may find a variety of applications ranging from atomic physics, chemistry, condensed-matter physics, cosmology, and high-energy physics.^[13,121,122]

In condensed-matter physics, in particular, Las Heras et al. proposed the implementation of digital quantum simulators for the Heisenberg and frustrated Ising models^[123] and Fermi–Hubbard models,^[124] which were experimentally validated in Refs. [125] and [126], respectively. Digital and analog simulations of the Fermi–Hubbard models were discussed in Refs. [127]–[129]. Quantum emulation of creating anyonic excitations was also achieved experimentally by dynamically generating the ground and excited states of the toric code model.^[130] There are still many proposals for the analog simulators put forward which need experimental tests: Transverse-field Ising or other spin models;^[131–133] Holstein molecular-crystal model exhibiting electron–phonon-induced small-polaron formation;^[134,135] multi-connected Jaynes–Cummings lattice model with Mott insulator-superfluid-Mott insulator phase transition;^[136,137] pairing Hamiltonians built from nearest-neighbor-interacting qubits;^[138] quantum emulation of spin systems with topologically protected ground states.^[139,140]

Devoret and Schoelkopf discussed some key stages in the development of a practical superconducting quantum computer.^[12] Briefly, one needs the single and multiple physical qubits which satisfy the first five DiVincenzo criteria: A well-defined quantum two-state system, the ability to initialize the state, long (relative) coherence times, single-and two-qubit logic gate operations, and the state measurement capability.^[141] This is followed by moving to the logical qubits and memories, which have much longer coherence times, by means of quantum error correction. Finally, gate operations on single and multiple logical qubits are realized to reach the ultimate goal of fault-tolerant quantum computation.

Quantum error correction is of crucial importance for the realization of quantum computing and has received much attention.^{[28,81,142,143]} In order to perform error correction, the fidelity of gate operation must be above certain threshold, which can be measured by the double π-pulse, quantum process tomography, and randomized benchmarking methods.^[76] Barends et al. showed that, using a 5-qubit array an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity of up to 99.4% could be achieved.^[144] This places the quantum computing at the fault tolerance threshold for surface code error correction. The same authors also reported the protection of classical states from environmental bit-flip errors and demonstrated the suppression of these errors with increasing system size on a 9-qubit array.^[142] Meanwhile, Chow et al. reported a quantum error detection protocol on a two-by-two planar qubit lattice.^[27] The protocol could detect both bit-flip error and phase-flip error, making it possible to detect an arbitrary quantum error.^[28] Ofek et al. adopted a different approach to implement a full quantum error correction by using real-time feedback for encoding, monitoring naturally occurring errors, decoding, and correcting. They demonstrated a quantum error correction system that reached the break-even point (i.e., enhanced lifetime of the encoded information) by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of coherent states of a superconducting resonator.^[143]

For a multiple qubit circuit, the demonstration of quantum entanglement and quantum algorithms is an important step toward quantum computing. Among many other works, Song et al. demonstrated the production and tomography of genuinely entangled Greenberger–Horne–Zeilinger (GHZ) states with up to 10 qubits that connect to a common bus resonator in a superconducting circuit.^[145] Grover search and Deutsch–Jozsa quantum algorithms^[146] as well as solving linear equations^[147] were also demonstrated. Furthermore, Riste et al. showed the quantum advantage in performing a machine learning algorithms using a 5-qubit superconducting processor.^[148]

While most present studies are performed on the devices of which all components are arranged on chip in a planar fashion, efforts have been made to make devices that have components distributed on different chips stacked vertically in order to accommodate more qubits and also reduce interference and cross talk between different qubits.^[149–151] Similar 3D integration is well developed for the semiconductor integrated circuits. However, the application of the technology to the superconducting circuits and qubits is not straightforward and requires further investigation. Of particular concern, quantum coherence could be seriously suppressed by the requisite processing steps and the progress in the coming years on this issue will be crucial for building chips containing around 100 qubits and above.