There are various types of superconducting qubits, including phase qubit, flux qubit, and charge qubit, the latter of which is developed to the transmon and Xmon qubits. The increase of the quantum coherence time is among the key issues in superconducting qubit studies; extensive studies considering various possible mechanisms causing the qubit decoherence have been performed. For example, studies have been performed to reduce the dielectric loss from the qubit surrounding materials, which are always present, such as the substrate, insulation layers, and oxidized tunnel barrier, or metal surface. The dielectric loss at low temperatures, believed to emerge from the two-level states (TLS), can be significantly reduced when the junction size is reduced down to the submicron level. Therefore, smaller junctions combined with shunt capacitors with low-dielectric-loss materials are preferably used. It has been experimentally demonstrated that parallel plate capacitors with a low dielectric loss can provide an improvement of the energy relaxation times by a factor of 20 in the case of phase qubits.

Early experiments of the charge qubits demonstrated only nanosecond-scale coherence times. A significant progress has been achieved with the charge-qubit-derived transmon, which is immune to the 1/f charge noise, thus removing the leading source of dephasing, leading to increased microsecond-scale coherence times. A further important improvement can be achieved with the understanding of a resonator modifying the qubit electromagnetic environment, based on the so-called “Purcell effect”. The qubit spontaneous emission can be effectively suppressed by coupling it to a resonator, which significantly modifies the modes and density of states of the electromagnetic field in the external circuitry. The reduced spontaneous emission effectively enhances the qubit energy relaxation times.

Transmon (or Xmon) and persistent-current flux qubits only need a modest shunt capacitance. The shunt capacitors can be formed from two electrodes with the substrate as the insulation layer between them. This facilitates the reduction of the dielectric loss if a low-loss substrate is used and if the substrate surface is carefully treated. Dielectric losses of the two-dimensional (2D) resonators, together with the qubits as a whole, have been carefully studied in the past decade. A convenient concept, called “participation ratio”, is often used. The participation ratio is defined as the electric field energy stored in the lossy material divided by the total electric field energy stored in the entire device. The participation ratio and dielectric loss decrease with the increase of the device feature size (footprint), which demonstrates that the loss is interface- or surface-limited. Some experiments indicate that the dominant loss originates from the substrate–air and substrate–metal interfaces, while the metal–air interface has a smaller influence. The three-dimensional (3D) transmon is advantageous in this respect as the sizes of the devices are larger, thus reducing the participation ratio and loss from interfaces.

In addition to the advances discussed above, i.e., reduction of the dielectric loss, design of qubits immune to the external noises, and modification of the electromagnetic environment, we mention other studies on the improvement of materials and fabrication techniques, and improvement of filtering and shielding against stray radiation causing quasiparticle dissipation. All of these studies have led to increased qubit coherence times up to the present level of 10–100 μs and beyond.

As discussed above, there are many types of superconducting qubits; in addition, there are also many ways of coupling them together, including capacitive and inductive couplings, and couplings through Josephson junctions and resonators. Single and multi-qubit quantum gate operations, required for the universal gate-based quantum computation, have been experimentally demonstrated with an improving fidelity, starting from the early years after the discovery of the qubit coherence. Single qubit logic operations of different angle rotations about the x, y, and z axes of the Bloch sphere are performed, e.g., in the quantronium, phase, and transmon qubits, which suffice the usual single qubit phase gate, phase-flip gate, NOT gate, and Hadamard gate. Two qubit logic operations, such as controlled-NOT (cNOT) gates, are realized in charge and persistent-current qubits. The square root of the i-SWAP gate is also realized in the phase qubit. The three-qubit Toffoli gate (ccNOT), Toffoli-equivalent controlled controlled phase gate (ccZ), and four-qubit controlled controlled controlled phase gate (cccZ) are successfully demonstrated in transmon qubits. Universal gate sets are proposed, and experimentally implemented and characterized.

Superconducting circuits and qubits behave quantum mechanically on a macroscopic scale with conveniently controllable system parameters, which provide an excellent platform for studies on quantum physics, atomic physics, and quantum optics. A very large number of studies have been published related to this field; we mention only a few of them: (1) resonant escapes and bifurcation phenomena of nonlinear systems under a strong driving; (2) macroscopic quantum tunneling in cuprate materials and phase diffusion in the quantum regime; (3) quantum stochastic synchronization in dissipative quantum systems; (4) Landau–Zener–Stückelberg interference and precise control of quantum states in the tripartite system; (5) topological phase diagram and phase transitions in interacting quantum systems; (6) Schrödinger cat living in two boxes; (7) Autler–Townes splitting phenomena; (8) stimulated Raman adiabatic passage and coherent population transfer; (9) electromagnetically induced transparency, a phenomenon similar to the Autler–Townes splitting; however, it is physically distinct; (10) resonance fluorescence and correlated emission lasing.

Quantum simulation, i.e., the use of a controllable quantum system to simulate another quantum system, was proposed for the first time by Richard Feynman, envisioned as the main application of a quantum computer. There are two types of quantum simulations: digital and analog quantum simulations. The former simulates a quantum system by dividing the time evolution of a Hamiltonian, expressed as a sum of local terms, into small intervals, which can be realized by a sequence of qubit gate operations. In the analog quantum simulation, the Hamiltonian of the system to be simulated is directly mapped onto the Hamiltonian of the simulator, which can be experimentally controlled. Quantum simulations are promising for applications in various fields including atomic physics, chemistry, condensed-matter physics, cosmology, and high-energy physics.

In particular, in condensed-matter physics, the implementation of digital quantum simulators for the Heisenberg, frustrated Ising, and Fermi–Hubbard models was proposed, which were later experimentally validated. Digital and analog simulations of the Fermi–Hubbard models were discussed. Quantum emulation of creation of anyonic excitations was also achieved experimentally by dynamically generating the ground and excited states of the toric code model. There are many proposals for analog simulators, which need experimental tests: transverse-field Ising or other spin models; Holstein molecular-crystal model exhibiting electron–phonon-induced small-polaron formation; multi-connected Jaynes–Cummings lattice model with Mott-insulator–superfluid–Mott-insulator phase transition; pairing Hamiltonians developed from nearest-neighbor-interacting qubits; quantum emulation of spin systems with topologically protected ground states.

Adiabatic quantum computing or quantum annealing was proposed as a heuristic technique for a quantum enhanced optimization approximately two decades ago. This architecture, known to be equivalent to the standard gate-based quantum computation, is simpler and provides a more practical approach for applications in the near term. The first superconducting quantum annealing processors were developed by the D-Wave Systems Inc. based on the rf-SQUID type flux qubits and Nb-junction technology. The D-Wave One, D-Wave Two, D-Wave 2X, and D-Wave 2000Q machines have become available in the years of 2011, 2013, 2015, and 2017 with coupled 128, 512, 1,024, and 2,048 qubits, respectively. Although the superconducting quantum annealing has attracted a significant attention, both academically and industrially, a consensus on the physics and its potential to achieve an optimization algorithm has not been reached; further studies are ongoing.

Devoret and Schoelkopf discussed key stages in the development of a practical superconducting quantum computer. Briefly, one needs single and multiple physical qubits, which satisfy the first five DiVincenzo criteria: well-defined quantum two-state system, ability to initialize the state, large (relative) coherence times, single- and two-qubit logic gate operations, and state measurement capability. This is followed by logical qubits and memories, which have significantly larger 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 has attracted a significant attention as it is of crucial importance for the realization of quantum computing. In order to perform error correction, the fidelity of gate operation must be above a certain threshold, which can be measured by the double π-pulse, quantum process tomography, and randomized benchmarking methods. It was shown that, using a 5-qubit array, an average single-qubit gate fidelity of 99.92% and two-qubit gate fidelity of up to 99.4% could be achieved. This places the quantum computing at the fault tolerance threshold for surface code error correction. The protection of classical states from environmental bit-flip errors was also reported and the suppression of these errors with an increased system size on a 9-qubit array was demonstrated. In addition, a quantum error detection protocol on a 2 × 2 planar qubit lattice was reported. The protocol could detect both bit-flip and phase-flip errors, enabling the detection of an arbitrary quantum error. A different approach to implement a full quantum error correction using real-time feedback for encoding, monitoring naturally occurring errors, decoding, and correction was employed. A quantum error correction system was demonstrated, which 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.

For a multiple qubit circuit, the demonstration of quantum entanglement and quantum algorithms is an important step towards quantum computing. Among the various studies, the production and tomography of genuinely entangled Greenberger–Horne–Zeilinger (GHZ) states with up to 10 qubits, which connect to a common bus resonator in a superconducting circuit, were demonstrated. Grover search and Deutsch–Jozsa quantum algorithms as well as solving linear equations were also demonstrated. Multi-qubit devices have been developed for quantum simulations of condensed-matter-physics problems. Furthermore, the quantum advantage in implementation of machine learning algorithms using a 5-qubit superconducting processor was demonstrated.

While most present studies are performed on devices, whose components are arranged on a chip in a planar setting, studies have been performed to fabricate devices with components distributed on different chips stacked vertically in order to accommodate more qubits and reduce interference and cross-talk between different qubits. 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 investigations. It is particularly concerning that quantum coherence could be seriously suppressed by the requisite processing steps; the progress in the coming years on this issue will be crucial for the development of chips containing approximately 100 qubits (or more).

Topology is a branch of mathematics, focused on studies of the overall properties of the spatial geometry. These overall topological properties do not change under a local continuous deformation. For example, if we consider a knotted rope, a local bending of the rope does not affect the number and type of knots. In quantum physics, can we use topologically protected physical states to store quantum information? The answer is yes.

There are indeed topological quantum states in physics, such as integer and fractional quantum Hall states, symmetry protected topological insulators, and topological superconductors. Under local perturbations weaker than the topological energy gap, the overall properties of these states do not change. If these topological quantum states are used to encode quantum information, it is possible to obtain fault-tolerant topological quantum computers. X. G. Wen proposed the concept of topological states in the 1990 s when he studied fractional quantum Hall states. In 1997, Kitaev further proposed a spin grid model. In this model, the quantum information is encoded in the degenerate ground state of the topological states. Quantum computing can be achieved by exchanging (braiding) non-Abelian excitations. The operation of the braiding process is related only to the topological nature of the path, and has no relation to the specific details of the path; therefore, there is no need to consider effects of errors introduced during the operation. Therefore, the TQC has a high fault tolerance. M. Freedman and others further discussed the mathematical architecture of the TQC.

In order to perform a TQC, at the current stage, non-Abelian quasiparticles are proposed as a promising approach. Platform candidates include the novel 5/2 fractional quantum Hall state and topological superconductors (such as p-wave superconductor), as they can carry Majorana zero mode (MZM).

What is MZM? In 1928, Dirac proposed a relativistic equation, i.e., the Dirac equation, to describe the motion of spin 1/2 particles (such as electrons), and predicted the existence of an electron antiparticle, positron, which was confirmed by experiments two years later. In 1937, Majorana rewrote the Dirac equation and revealed a real solution, predicting the presence of Majorana fermion. Majorana fermion, as its own antiparticle, is not charged, and follows the Fermi–Dirac statistics. In the field of elementary particles, electrically neutral neutrinos are thought to be Majorana fermions; however, no conclusion has been reached. In the form of an operator, the Majorana operator is a Fermi operator that conforms to γ = γ^†. The electrons can be regarded as a pair of Majorana fermions localized at the same point in space. The creation and annihilation operators of an electron j can be expressed as

From the anti-commutation relation, we obtained: {γ_iα, γ_jβ} = 2δ_ijδ_αβ, . , which is the Majorana operator. The Majorana fermion can be regarded as a “half” electron and always appears in pairs. Certainly, a highly entangled pair of Majorana fermions at the same point in space has no practical significance; the interest of study is focused on spatially separated Majorana fermions.

Although there has been no confirmation of the presence of Majorana fermion in the fields of elementary particles, recently, some traces of it could be observed in condensed matter. In condensed matter, various low-energy quasi-particle excitations can be induced by electromagnetic interactions, possibly producing Majorana fermions. However, collective excitations in condensed matter are mostly collective effects of charged electrons. How can we produce uncharged Majorana fermions? This requires the introduction of superconductivity. The description of quasiparticle excitations in superconductors is provided by the Bogoliubov–de-Gennes equation, corresponding to Bogoliubov quasiparticles as follows:

For the one-dimensional case, Kitaev proposed a p-wave superconductivity model with spinless fermions, referred to as the Kitaev model

The parity of the system is related to the ground state occupations. When the ground state is not occupied, n = 0, P = 1, leading to an even parity; when it is occupied n = 1, P = −1, leading to an odd parity. Owing to the existence of degenerate ground states, together with the extension of effective dimensions (e.g., experimentally, using one-dimensional nanowires to develop “T” structures), the Majorana bound state no longer obeys the Fermi-Dirac statistics, but the non-Abelian statistics.

In a 2D p_x + ip_y topological superconductor, there is a zero-energy Majorana bound state at a defect (e.g., flux core). When there is an odd number of magnetic fluxes, a one-dimensional Majorana edge mode propagates at the edges. At the core of the magnetic flux, the superconducting energy gap is closed, and the continuous single-particle energy band becomes discrete energy levels owing to the quantum confinement effect. In s-wave superconductors, zero-point motion drives single-particle excitations away from the zero energy. However, for spinless 2D p_x + i p_y topological superconductors, fermions have a Berry phase of π after a circular motion, which causes the discrete energy level to pass through zero, resulting in the appearance of the Majorana bound state in the magnetic flux core.

As the Majorana bound state exists in topological superconductors, we need to explain the systems hosting topological superconductivity. There are two types of candidate systems: intrinsic topological superconductors and artificial composite structures that can induce topological superconductivity.

It is theoretically considered that the 5/2 fractional quantum Hall state may be an intrinsic topological superconductor resulting from the p-wave pairing of non-Abelian anyons with an effective charge of e/4. There is already some evidence to support the existence of this topological quantum state; however, further experimental studies are required. In addition, Sr₂RuO₄ is considered as a p_x + ip_y wave superconductor with spin triplet pairing. There is a series of experimental evidences in this respect; however, one of the important evidences, i.e., the edge magnetic moment, has not been observed. The phase A of superfluid ³He is also considered to be p-wave pairing. In general, intrinsic topological superconductors have not yet been fully experimentally confirmed.

Topological superconductivity may also be induced in an artificial composite structure. As mentioned above, in order to construct spinless topological superconductors, it is necessary to “freeze” spin degrees of freedom to form spinless fermions. The studies on topological insulators since 2005 have just solved this problem. At the edges of 2D topological insulators, surfaces of 3D topological insulators, and one-dimensional semiconductor nanowires with strong spin–orbit coupling (SOC), spin-momentum-locked helical electrons exist, satisfying the spinless condition. In 2008, Fu and Kane proposed that conventional s-wave superconductors could be used to introduce superconductivity into the helical surface state of a 3D topological insulator through superconducting proximity effect to form a similar 2D p_x + ip_y superconductor, so that the Majorana zero-energy bound state can be generated at the magnetic flux core or in the Josephson junction. Theoretical approaches have also provided further solutions on the implementation of the one-dimensional p-wave superconductivity (Kitaev chain), suggesting that one-dimensional semiconductor nanowires with strong SOC can be used. A magnetic field perpendicular to the SOC opens a Zeeman gap of E_Z at k = 0. When the chemical potential μ is in the Zeeman energy gap, there is only one spin–orbit-locked Fermi surface, thus forming a helical electron state. A superconducting energy gap Δ is then introduced by s-wave superconductivity. When , the nanowire is in the topological superconductor state, and a Majorana bound state appears at both ends. The replacement of the nanowire with ferromagnetic chains also enables the construction of one-dimensional topological superconductor and Majorana bound state at the end-point.

Two types of experiments are mainly employed to detect Majorana bound state: detection of the zero-bias conductance peak (ZBCP) and probing fractional (4π) Josephson effect. As a Majorana bound state appears at the Fermi energy, a ZBCP can be observed in the tunneling spectrum. In addition, the Majorana bound state in a Josephson junction enables a single-electron transport; the effective charge carrying the supercurrent is changed from the original Cooper pair (2e) to a single electron (e), resulting in a 4π-period current–phase relationship and fractional Josephson effect.

In 2012, Leo Kouwenhoven’s group from the Netherlands observed signatures of Majorana bound state in a composite structure containing InSb nanowires with strong SOC and s-wave superconductors. When the parallel magnetic field B increases to ∼ 100 mT, the nanowire enters the topological superconductor phase, inducing a ZBCP in the tunneling spectrum. It is just the ZBCP that points to Majorana bound state. Subsequently, similar experiments on nanowires were repeated by many research groups. In 2018, Kouwenhoven’s group further observed a ZBCP with a quantized conductance value.

In addition to the above nanowire experiments, many exploratory experiments have been performed based on 2D and 3D topological insulators: (1) ZBCP in a composite structure composed of Bi₂Se₃ and Sn superconductor; (2) complete Andreev reflection in the edge state of a 2D topological insulator InAs/GaSb; (3) ZBCP with spin selectivity in the flux core in epitaxially grown Bi₂Se₃ thin films on NbSe₂; (4) disappearance of odd-numbered Shapiro steps in Josephson junctions based on semiconductor InSb/AlInSb, 2D topological insulator HgTe/HgCdTe, and 3D topological insulator HgTe, which was considered to be fractional Josephson effect, i.e., an evidence of Majorana bound state. It is worth noting that when the time-reversal symmetry is conserved, theoretically, the fractional Josephson effect should not occur; therefore, the relevant phenomena need to be further analyzed. When the time-reversal symmetry was broken by magnetic doping, the quantum anomalous Hall effect in Cr-doped (Bi,Sb)₂Te₃ was observed. Using the proximity effect to introduce superconductivity into similar materials, the Hall plateau of 1/2 quantum conductance was also detected, which is thought to be derived from a chiral Majorana edge mode. However, there are also other possible explanations for this experimental phenomenon.

Majorana bound state, as an Ising anyon, is expected to be used for TQCs and quantum information storage. There are theoretical proposals on the development of topological quantum computers using one-dimensional nanowires or topological insulators.

The development of a topological qubit requires at least four non-Abelian anyons. Using two nanowires proximitized to superconductors, a magnetic field drives them to a topological superconducting state, thereby producing four Majorana bound states γ₁, γ₂, γ₃, and γ₄. Their degenerate ground states can be divided into four types, according to the different parities of each nanowire: |ee⟩, |oo⟩, |eo⟩, and |oe⟩. When the total parity is conserved, we need to consider only the qubits formed by two degenerate ground states of the same total parity. The state of the qubit, i.e., the specific case of electron occupancy or non-occupancy, can be measured with a single-electron transistor (SET). Using such a qubit, one can also verify Ising anyon fusion rules. First, γ₂ and γ₃ can be fused by the phase change induced by a gate electrode or external magnetic field; γ₁ and γ₄ are then fused, and finally the results are read out through a SET. Currently, studies are ongoing to construct a topological qubit; experiments have successfully produced a Majorana island based on a single nanowire.

However, for the applications of logic gates, the nanowire approach faces natural difficulties, i.e., there is no way to braid Majorana bound state in one-dimensional space. The use of a “T”-shaped nanowire structure or networks has been proposed to achieve this operation. The phase transition of the topological superconductor is related to the chemical potential of the nanowires; therefore, the gate voltage can be locally adjusted to control the boundary of the topological superconductor, thereby moving the Majorana bound state. TQC based on nanowires is one of the main research fields nowadays.

Majorana bound state can also exist in a heterostructure of 3D topological insulators and superconductors. Superconductivity similar to the spinless p_x + ip_y can be induced on the 2D surface of 3D topological insulators through the superconducting proximity effect, hence Majorana bound state can be present in its magnetic flux core or Josephson device. The previously mentioned ZBCP attributed to the Majorana bound state in the magnetic flux core has been observed. In terms of Josephson devices based on 3D topological insulators, some evidence of the existence of Majorana bound state has been observed.

It is obviously easier to braid Majorana bound states in the 2D plane than in the one-dimensional nanowire system. By exchange or braiding Majorana bound states, a series of logic gates can be realized, including the Hadamard and Z gates. However, in order to form a universal gate set, the magic. π/8 phase gate is lacking. It can be obtained by bringing two Majorana bound states close together to induce finite coupling and then separating them far apart. However, this process is not topologically protected and errors can be induced. In addition, interference was proposed as a clever approach to achieve the phase gate; however, it is also not protected. Furthermore, a geometric multistep protocol with an exponential accuracy in the number of steps has been studied to reduce the phase gate error. It was also pointed out that using the hexagonal or square Josephson junction arrays to encode and braid the Majorana bound state (although only Ising anyons), universal topological quantum computation can be realized. During topological quantum computation, the quantum states are measured through fermion parity. An intuitive approach is to employ a SET adjacent to the Majorana qubit, so that the fermion parity can be read out with a high sensitivity. Another more advanced approach is to combine a superconducting qubit in an optical cavity. All of these ideas, among the others, are currently pursued in physical laboratories.