Superconducting quantum computing

Quantum computers were first proposed by Richard Feynman, who in 1982 proposed using such a computer to simulate and understand the properties of other quantum systems. In the 1990s, two quantum algorithms were published, which further stirred interest in realizing quantum computers. Peter Shor proposed Shor's algorithm, a quantum algorithm for finding the prime factors of an integer, which could in theory break RSA encryption. Similarly, Lov Grover proposed the Grover search algorithm, which provides an alternative to binary search that can be done with quadratic speedup. superconducting quantum processor pictured above in 2023, is based on transmon qubits and is part of one of the first circuit-based commercial quantum computers. At the time, superconducting quantum circuits were already being used to construct highly sensitive SQUID devices, and had also been used to demonstrate macroscopic quantum phenomena, such as quantized energy levels. It became apparent that these superconducting qubits could be used to achieve quantum computation. This was especially true because such "solid state" approaches to quantum computing were seen as far more viable than other approaches at the time, including NMR (nuclear magnetic resonance) quantum computing, due in part to the fact that existing fabrication techniques would apply. In 1999, a paper was published by Yasunobu Nakamura, demonstrating the first superconducting qubit. It is a form of Cooper pair box, now known as the "charge qubit". Although the design had been proposed in 1997 by the Saclay team (including Devoret), this paper was the first to show coherent control and readout, in the form of Rabi oscillations between the ground and excited states of the qubit. However, even after this first result, it was unclear if superconducting qubits would be viable, and some argued that the system was not truly capable of containing quantum information. Part of the problem was that this initial design maintained coherence for less than a nanosecond, not long enough to do any calculations. , the 2025 Nobel laureate in physics, led the team at Google Quantum AI that built the Sycamore processor, which, in 2019, claimed the first evidence of quantum supremacy. In the following years, several other superconducting qubits were invented, including the phase qubit, flux qubit, quatronium, the transmon qubit, and the fluxonium. The transmon design, which has reduced sensitivity to charge noise, is now widely and primarily used in superconducting quantum computing. Google in 2016, implemented 16 qubits to convey a demonstration of the Fermi-Hubbard Model. In another experiment, Google used 17 qubits to optimize the Sherrington-Kirkpatrick model. In 2019, Google produced the Sycamore quantum computer which performed a task in 200 seconds that Google claimed would have taken 10,000 years on a classical computer. The task was random circuit sampling, a common benchmark for claims of "quantum supremacy" or quantum advantage. As of 2025, superconducting quantum processors have exceeded 1,000 qubits, the largest being IBM Condor, a 1,121-qubit quantum processor. In 2025, Google announced one of the first independently verifiable quantum advantages on hardware using the Willow processor. == Background ==

Quantum computing Classical computation models rely on physical implementations consistent with the laws of classical mechanics. Some very small systems, or certain systems under extreme conditions, are instead described by the quantum mechanics, which obey different sets of physical rules. Quantum computation is a method of constructing a quantum system for the purpose of encoding information. Applications of a quantum computer would include simulating quantum phenomena beyond the scope of classical approximation, and speeding up certain calculations, particularly those that involve an "oracle". Certain algorithms designed for quantum computers, such as Grover Search or Shor's algorithm, are believed to be able to do some calculations better than their classical counterparts. Gate-based quantum computing is a method of quantum computing that, much like traditional computing, use qubits (analogous to bits) and quantum gates (analogous to classical gates). Qubits A qubit is any two-level system in quantum mechanics. Much like a classical bit, it is a system with two possible states. However, the difference lies in the fact that because a qubit obeys the laws of quantum mechanics, it is capable of occupying a quantum superposition of both states. The primary requirement for physically constructing a qubit is the ability to be able to individually address the first two states, in this case energy levels, of the system. This is difficult, as most systems contain a near-infinite number of energy levels. In superconducting quantum computers, these qubits are constructed using superconducting resonant circuits. Each superconducting qubit is essentially a nonlinear LC circuit with a capacitor and a Josephson junction, a superconducting element with a nonlinear inductance. Because the circuit is non-linear, there is unequal spacing between its energy levels, allowing the first two states to be individually addressable. In theory, due to its nonlinearity, the qubit is affected only by photons with the energy difference required to jump from the ground state to the excited state. This is especially true in transmons, which have weak anharmonicity by design. Because the circuit is superconducting, it has zero resistance, and dissipates almost no energy. However, this comes at the price of extremely low operation temperatures. Quantum gates A quantum gate is a generalization of a logic gate describing the transformation of qubits from their initial state to a different state, often a superposition. (CCNOT) is implemented using a combination of single and two-qubit gates. Toffoli gates have been experimentally implemented using three superconducting transmon qubits coupled to a microwave resonator. In superconducting qubits, quantum gates are implemented as microscopic pulses applied to the circuit using microwave resonators. Pulses are sent through resonators capacatively coupled to the qubit, which are harmonic oscillators that are detuned from the qubit itself. By applying an external drive to the qubit, the normal unitary evolution of the system implements a single qubit gate after a certain length of time has passed. Two qubit gates, such as the iSWAP gate, can be achieved through coherent exchange or parametric coupling between two qubits. Criteria for a viable quantum processor There are many possible physical implementations of qubits, with superconducting circuits being one of them. In order for a given implementation to be considered viable for constructing a quantum computer, one set of criterion is the DiVincenzo's criteria, a set of criteria for the physical implementation of superconducting quantum computing. The initial five criteria ensure that the quantum computer is in line with the postulates of quantum mechanics and the remaining two pertaining to the relaying of this information over a network. Superconducting qubits already meet a large number of DiVincenzo's criteria. They are already highly scalable from a fabrication standpoint, they can be initialized by thermal relaxation, and single-qubit gates combined with two-qubit gates form a universal gate set. However, superconducting qubits still struggle with having short coherence times, making preserving quantum information a challenge. Superconductors Superconducting qubits are circuits made from superconducting metal material. Superconductivity is a phenomenon that occurs in some metals at low temperatures where electrical current experiences zero resistance in a material, allowing the current to flow without loss of energy, and be nearly dissipation-less. This phenomenon occurs because the basic charge carriers are pairs of electrons (known as Cooper pairs), rather than single electrons as found in typical conductors. While single electrons are fermions (with half-integer spin), Cooper pairs of electrons are bosons (with integer spin), and as such they no longer obey the Pauli exclusion principle, meaning these Cooper pairs can occupy the same states. Under certain conditions, this allows them to form a state of matter known as a Bose–Einstein condensate, where all of the pairs of electrons in the condensate each occupy the same position in space and have equal momentum. In this way, there is nothing distinguishing the pairs from each other, and they occupy the same state. As a result, the electron pairs move coherently as a single wave, bypassing the disturbances in the lattice that usually cause resistance. Thus, superconductors possess near infinite conductivity and near zero resistance. Superconductivity generally only occurs near absolute zero, since that is when it is more energetically favorable for electrons to pair up than repel each other. This is one of the primary reasons why superconducting qubits must be cooled to ultra cold temperatures. Superconducting electrical circuits Superconducting electrical circuits are networks of electrical elements described by a single condensate wave function, wherein charge flow is well-defined by some complex probability amplitude. Quantization in the circuit results from complex amplitude continuity, since only discrete numbers of magnetic flux quanta can penetrate a superconducting loop. Parameters of superconducting circuits are designed by setting (classical) values to the electrical elements composing them, such as capacitance or inductance. Josephson junctions One distinguishing attribute of superconducting quantum circuits is the Josephson junction, an electrical element which does not exist in normal conductors. The junction is a weak connection between two superconductors on either side of a thin layer of insulator material only a few atoms thick. The resulting Josephson junction device exhibits the Josephson effect, whereby the condensate wave function on the two sides of the junction are weakly correlated. Current flows through the junction due to quantum tunneling. The Josephson junction exhibits a nonlinear inductance, which allows for anharmonic oscillators for which energy levels are discretized (or quantized) with nonuniform spacing between energy levels, denoted \Delta E. Josephson energy can be written as : U_j = - \frac{I_0 \Phi_0}{2 \pi} \cos \delta, where I_0 is the critical current parameter of the Josephson junction, \textstyle \Phi_0 = \frac{h}{2e} is (superconducting) flux quantum, and \delta is the phase difference across the junction. Notice that the term cos \delta indicates nonlinearity of the Josephson junction. Charge energy is written as : E_C = \frac{e^2}{2C}, where C is the junction's capacitance and e is electron charge. Circuit quantization Circuit quantization is a method of obtaining a quantum mechanical description of an electrical circuit. The end result is a Hamiltonian describing the energy of the system, from which other properties such as the ground and excited state can be derived. In circuit quantization, all electrical elements in the circuit are rewritten in terms of the condensate wave function's amplitude and phase, as opposed to the current and voltage. Then, generalized Kirchhoff's circuit laws are applied at every node of the circuit network to obtain the system's equations of motion. Finally, these equations of motion must be reformulated to Lagrangian mechanics such that a quantum Hamiltonian is derived describing the total energy of the system. Circuit quantum electrodynamics Properties of superconducting electrical circuits coupled to a resonator are described by the framework of circuit quantum electrodynamics, or cQED. Superconducting qubits generally need to be connected to a resonator in order to protect them from environmental noise, and to allow them to be coupled to each other. The cQED framework is similar to cavity QED and uses largely the same techniques. In physical implementations, the resonator is usually an on-chip coplanar waveguide readout resonator, a superconducting LC resonator, or a high purity cavity. == Hardware and technology ==

Superconducting quantum computing devices are typically designed in the radio-frequency spectrum, cooled in dilution refrigerators below 15 mK and addressed with conventional electronic instruments, e.g. frequency synthesizers and spectrum analyzers. Typical dimensions fall on the range of micrometers, with sub-micrometer resolution, allowing for the convenient design of a Hamiltonian system with well-established integrated circuit technology. Manufacturing Manufacturing superconducting qubits follows a process involving lithography, depositing of metal, etching, and controlled oxidation. This process is similar, though not the same, as CMOS (Complementary Metal-Oxide-Semiconductor) fabrication used for commercial silicon computer chips. A major difference is the use of electron-beam lithography, as opposed to optical lithographic techniques, which is hard to scale and has low yield. However, electron beams allow for a much sharper resolution, which is often necessary for certain device designs. The superconductor used to make superconducting circuits is usually aluminum, deposited on a silicon substrate, but can also be niobium or tantalum, both d-band superconductors. Improvements in fabrication continue to improve the lifetime of superconducting qubits and have made significant improvements since the early 2000s. Refrigeration Cryogenic dilution refrigerators are used to keep the superconducting circuits cold. They are cooled to temperatures below 15 mK. Although superconductivity itself onsets before this temperature, a large population of thermal quasiparticles exist within the circuit, which can interfere with the circuit's superconductivity. These so-called 'equilibrium quasiparticles' are exponentially suppressed at lower temperatures. Therefore, it is favorable to cool the circuit to as low of a temperature as possible. Inside of the dilution fridge, the superconducting circuits are connected to various filters and amplifiers that enable the qubit to be read out from observers outside of the dilution fridge. == Qubit types ==

Phase, flux, and charge qubits The three primary superconducting qubit archetypes are the phase, charge and flux qubit. These archetypes correspond to limits of the underlying Josephson hamiltonian. Depending on what limit the hamiltonian is in, a different aspect of the qubit will be well defined. The choice of qubit archetype impacts the qubit's transition frequency, anharmonicity (or nonlinearity), and susceptibility to noise. numbers of magnetic flux quanta trapped in a superconducting ring. Phase qubit The phase qubit possesses a Josephson to charge energy ratio on the order of magnitude 10^6. For phase qubits, energy levels correspond to different quantum charge oscillation amplitudes across a Josephson junction, where charge and phase are analogous to momentum and position respectively as analogous to a quantum harmonic oscillator. Note that in this context phase is the complex argument of the superconducting wave function (also known as the superconducting order parameter), not the phase between the different states of the qubit. In the table above, the three superconducting qubit archetypes are reviewed. In the first row, the qubit's electrical circuit diagram is presented. The second row depicts a quantum Hamiltonian derived from the circuit. Generally, the Hamiltonian is the sum of the system's kinetic and potential energy components (analogous to a particle in a potential well). For the Hamiltonians denoted, \phi is the superconducting wave function phase difference across the junction, C_J is the capacitance associated with the Josephson junction, and q is the charge on the junction capacitance. For each potential depicted, only solid wave functions are used for computation. The qubit potential is indicated by a thick red line, and schematic wave function solutions are depicted by thin lines, lifted to their appropriate energy level for clarity. Note that particle mass corresponds to an inverse function of the circuit capacitance and that the shape of the potential is governed by regular inductors and Josephson junctions. Schematic wave solutions in the third row of the table show the complex amplitude of the phase variable. Specifically, if a qubit's phase is measured while the qubit occupies a particular state, there is a non-zero probability of measuring a specific value only where the depicted wave function oscillates. All three rows are essentially different presentations of the same physical system. Hybridizations While the three core forms of superconducting qubits (phase, charge, and flux) are historically how superconducting qubits were categorized, most modern superconducting qubits are a hybridization of these archetypes. Many hybridizations of these archetypes exist including the fluxonium, transmon, Xmon, and quantronium. qubits, four quantum buses, and four readout resonators fabricated by IBM and published in npj Quantum Information in January 2017|alt= Transmon Transmons are a special type of qubit with a shunted capacitor specifically designed to mitigate noise. The transmon qubit model a charge-phase hybrid qubit based on the Cooper pair box. The increased ratio of Josephson to charge energy mitigates noise. The Hamiltonian for the transmon is: \hat{H} = 4 E_C \left( \hat{n} - n_g \right)^2 - E_J \cos \hat{\phi} where n is the number of Cooper pairs transferred between the island and \phi is the phase difference across the junction. Two transmons can be coupled using a coupling capacitor. Other companies that use transmon qubits include IBM, Rigetti, and IQM. The physical design of a transmon qubit can vary depending on the implementation. Common transmon designs include the "transmon cross" which is shaped like an X or cross, and the pad or "paddle transmon", which contains two paddles next to each other. Transmon-like qubits Many variations of the transmon design exist and are active areas of research. They aim to improve upon failings of the transmon design. Xmon The Xmon is similar in design to a transmon in that it originated based on the planar transmon model. An Xmon is essentially a tunable transmon. The major difference between transmon and Xmon qubits is the Xmon qubit is grounded with one of its capacitor pads. Gatemon Another variation of the transmon qubit is the Gatemon. Like the Xmon, the Gatemon is a tunable variation of the transmon. The Gatemon is tunable via gate voltage. Unimon The Unimon consists of a single Josephson junction shunted by a linear inductor (possessing an inductance not depending on current) inside a (superconducting) resonator. Unimons have increased anharmonicity and display faster operation time resulting in lower susceptibility to noise errors. The top right image depicts fluxonium circuit components, and the bottom right image depicts a smaller area Josephson junction. Fluxonium Fluxonium qubits are a specific type of flux qubit whose Josephson junction is shunted by a linear inductor of E_{J} \gg E_{L} where E_L = (\hbar/2e)^2 / L . In practice, the linear inductor is usually implemented by a Josephson junction array that is composed of a large number (can be often N > 100 ) of large-sized Josephson junctions connected in a series. Under this condition, the Hamiltonian of a fluxonium can be written as: : \hat{H} = 4 E_C \hat{n}^2 + \frac{1}{2} E_L (\hat{\phi}- \phi_\mathrm{ext})^2 - E_J \cos \hat{\phi} . One important property of the fluxonium qubit is the longer qubit lifetime at the half flux sweet spot, which can exceed 1 millisecond. Another crucial advantage of the fluxonium qubit when biased at the sweet spot is its large anharmonicity. In this context, anharmonicity refers to the unequal spacing of energy levels in a superconducting circuit. Large anharmonicity is beneficial because it allows fast local microwave control and mitigates spectral crowding problems, leading to better scalability. 0-π qubit The 0-π qubit is a protected qubit design where logical states are protected by circuit symmetry. The logical states of the qubit are exponentially protected against relaxation and exponentially (first-order) protected to first order against dephasing due to charge (flux) noise. This ideal behavior, however, is not always realistic because it requires that parameter dispersion among nominally identical circuit elements vanishes. == Operation and readout ==

Single qubits The GHz energy gap between energy levels of a superconducting qubit is designed to be compatible with available electronic equipment, due to the terahertz gap (lack of equipment in the higher frequency band). The superconductor energy gap implies a top limit of operation below ~1THz beyond which Cooper pairs break, so energy level separation cannot be too high. On the other hand, energy level separation cannot be too small due to cooling considerations: a temperature of 1 K implies energy fluctuations of 20 GHz. Temperatures of tens of millikelvins are achieved in dilution refrigerators and allow qubit operation at a ~5 GHz energy level separation. Qubit energy level separation is frequently adjusted by controlling a dedicated bias current line, providing a "knob" to fine tune the qubit parameters. Single qubit gates A single qubit gate is achieved by rotation in the Bloch sphere. Rotations between different energy levels of a single qubit are induced by microwave pulses sent to an antenna or transmission line coupled to the qubit with a frequency resonant with the energy separation between levels. Individual qubits may be addressed by a dedicated transmission line or by a shared one if the other qubits are off resonance. The axis of rotation is set by quadrature amplitude modulation of microwave pulse, while pulse length determines the angle of rotation. More formally (following the notation of Setting \int_0^{t_g}\mathcal{E}^x(t) dt=\pi results in the transformation : U_x=\exp\left\{-i\int_0^{t_g}\mathcal{E}^x(t) dt\cdot\sigma_x/2\right\}=e^{-i\pi\sigma_x/2}=-i\sigma_x up to the global phase -i and is known as the NOT gate. Multiple qubits The ability to couple qubits is essential for implementing 2-qubit gates. Coupling two qubits can be achieved by connecting both to an intermediate electrical coupling circuit. The circuit may be either a fixed element (such as a capacitor) or be controllable (like the DC-SQUID). In the first case, decoupling qubits during the time the gate is switched off is achieved by tuning qubits out of resonance one from another, making the energy gaps between their computational states different. This approach is inherently limited to nearest-neighbor coupling since a physical electrical circuit must be laid out between connected qubits. Notably, D-Wave Systems' nearest-neighbor coupling achieves a highly connected unit cell of 8 qubits in Chimera graph configuration. Quantum algorithms typically require coupling between arbitrary qubits. Consequently, multiple swap operations are necessary, limiting the length of quantum computation possible before processor decoherence. Heisenberg interactions The Heisenberg model of interactions, written as \hat{\mathcal{H}}_\mathrm{XXZ}/\hbar =\sum_{ i,j} J_\mathrm{XY}(\hat{\sigma}_\text{x}^{i}\hat{\sigma}_\text{x}^{j} + \hat{\sigma}_\text{y}^{i}\hat{\sigma}_\text{y}^{j}) + J_\mathrm{ZZ}\hat{\sigma}_\text{z}^{i}\hat{\sigma}_\text{z}^{j}, serves as the basis for analog quantum simulation of spin systems and the primitive for an expressive set of quantum gates, sometimes referred to as fermionic simulation (or fSim) gates. In superconducting circuits, this interaction model has been implemented using flux-tunable qubits with flux-tunable coupling, allowing the demonstration of quantum supremacy. In addition, it can also be realized in fixed-frequency qubits with fixed-coupling using microwave drives. Following the dark state manifold, the Khazali-Mølmer scheme Following the notation of, and by using selective microwave driving. Off-resonant driving can be used to induce differential ac-Stark shift, allowing the implementation of all-microwave controlled-phase gates. Qubit readout Architecture-specific readout, or measurement, mechanisms exist. Readout of a phase qubit is explained in the qubit archetypes table above. A flux qubit state is often read using an adjustable DC-SQUID magnetometer. States may also be measured using an electrometer. Multi-level systems (qudits) can be readout using electron shelving. == Performance ==

Criteria for quantum computation DiVincenzo's criteria DiVincenzo's criteria is a list describing the requirements for a physical system to be capable of implementing a logical qubit. DiVincenzo's criteria is satisfied by superconducting quantum computing implementation. Much of the current development effort in superconducting quantum computing aims to achieve interconnect, control, and readout in the 3rd dimension with additional lithography layers. The list of DiVincenzo's criteria for a physical system to implement a logical qubit is satisfied by the implementation of superconducting qubits. Although DiVincenzo's criteria as originally proposed consists of five criteria required for physically implementing a quantum computer, the more complete list consists of seven criteria as it takes into account communication over a computer network capable of transmitting quantum information between computers, known as the "quantum internet". Therefore, the first five criteria ensure successful quantum computing, while the final two criteria allow for quantum communication. • A scalable physical system with well characterized qubits. "Well characterized" implies that the Hamiltonian function must be well-defined (i.e. the energy eigenstates of the qubit should be able to be quantified). A "scalable system" indicates that this ability to regulate a qubit should be augmentable for multiple more qubits. However, as more qubits are implemented, it leads to an exponential increase in cost and other physical implementations which pale in comparison to the enhanced speed it may offer. A fiducial state is one that is easily and consistently replicable and is useful in quantum computing as it may be used to guarantee the initial state of qubits. One simple way to initialize a superconducting qubit is to wait long enough for the qubits to relax to the ground state. Controlling qubit potential with tuning knobs allows faster initialization mechanisms. • Long relevant decoherence times. Recent strategies to improve device coherence include purifying the circuit materials and designing qubits with decreased sensitivity to noise sources. This criterion may also be satisfied by coupling two transmons with a coupling capacitor. == Challenges ==

Many current challenges faced by superconducting quantum computing lie in the field of microwave engineering. TLS act as resonant, two-level absorbers which drain energy from the qubit, significantly reducing coherence times. They are thought to be caused by deformities during the fabrication process, and surface amorphous oxides that form on or near Josephson junctions. Additionally, coherent TLS defects fluctuate in time, and at the moment, mitigating them requires full recalibration of quantum processors containing 100 qubits around once per day. Quasiparticles Quasiparticles are single-electron excitations that occur when Cooper pairs break. They consist of a superposition of an electron and an 'electron hole'. They occur when a Cooper pair is hit by a photon, causing it to break. There are two kinds of particles; thermally-generated equilibrium quasiparticles, and non-equilibrium quasiparticles which get excited due to other effects in the system. While equilibrium quasiparticles can be suppressed exponentially by operating at low temperatures, but non-equilibrium quasiparticles, due to radiation such as gammas and cosmic ray muons, cannot. Attempts to mitigate quasiparticle generation include increasing shielding against radiation, quasiparticle trapping, gap engineering, or otherwise removing them from the system. Scaling As superconducting quantum computing approaches larger scale devices, researchers face difficulties in qubit coherence, scalable calibration software, efficient determination of fidelity of quantum states across an entire chip, and qubit and gate fidelity. Moreover, superconducting quantum computing devices must be reliably reproducible at increasingly large scales such that they are compatible with these improvements. == References ==