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Superconducting quantum computing is a branch of quantum computing andsolid state physics that implementssuperconductingelectronic circuits asqubits in a quantum processor. These devices are typically microwave-frequency electronic circuits containingJosephson junctions, which are fabricated on solid state chips.

Superconducting circuits are one of many possible physical implementations of qubits, the quantum computer's equivalent of a traditionalbit in a classiccomputer.^[1] Qubits refer to a two-state quantum mechanical system, and have two logic states, theground state and theexcited state, often denoted${\displaystyle |g⟩\text{ and }|e⟩}$ (for ground and excited), or${\displaystyle |0⟩\text{ and }|1⟩}$.^[2] Superconducting quantum computing implementations are categorized as "solid state" quantum computers, where qubits are intrinsically integrated in a solid-state device. Solid state quantum computers also borrow fabrication techniques developed for solid state classical computation.^[3]

Superconducting architecture is the dominant method in the industry for developing quantum processing units, or QPUs. Research in superconducting quantum computing is conducted by companies such asGoogle,^[4]IBM,^[5]IMEC,^[6]BBN Technologies,^[7]Rigetti,^[8] andIntel.^[9] Alternatives to superconducting qubits include trapped ions, and neutral atoms.

Ongoing research in superconducting quantum computing includes device-level improvement, developing error correction methods, and demonstratingquantum advantage by comparing a quantum processor's performance to a classical computer.

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 alogrithm, aquantum algorithm for finding theprime factors of an integer, which could in theory break RSA encryption. Similarly, Lov Grover proposed theGrover search algorithm, which provides an alternative tobinary search that can be done with quadratic speedup.

At the time, superconducting quantum circuits were already being used to construct highly sensitiveSQUID devices, and had also been used to demonstratemacroscopic quantum phenomena, such as quantized energy levels. It became apparent that these superconducting qubits could be used to achieve quantum computation.^[10]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.^[11]

In 1999, a paper^[12] was published byYasunobu Nakamura, demosntrating the first superconducting qubit. It is a form ofCooper pair box, now known as the "charge qubit". Although the design had been proposed in 1997 by the Saclay team (includingDevoret), this paper was the first to show coherent control and readout, in the form ofRabi 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.^[13] Part of the problem was that this initial design maintained coherence for less than a nanosecond, not long enough to do any calculations.

In the following years, several other superconducting qubits were invented, including thephase qubit,flux qubit, quatronium, thetransmon qubit, and the fluxonium. The transmon design, which has reduced sensitivity to charge noise, is now widely and primarily used in superconducting quantum computing.^[10] Further developments in readout and design have allowed superconducting transmon qubits to reach millisecond coherence times.^[14]

Google in 2016, implemented 16 qubits to convey a demonstration of theFermi-Hubbard Model. In another experiment, Google used 17 qubits to optimize theSherrington-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.^[15] The task wasrandom 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 beingIBM Condor, a 1,121-qubit quantum processor.^[16][17] In 2025, Google announced one of the first independently verifiable quantum advantages on hardware using theWillow processor.^[18]

Classicalcomputation models rely on physical implementations consistent with the laws ofclassical mechanics.^[19] Some very small systems, or certain systems under extreme conditions, are instead described by thequantum mechanics, which obey different sets of physcial 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 asGrover Search orShor'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, usequbits (analogous to bits) andquantum gates (analogous to classical gates).

A qubit is any two-level system in quantum mechanics. Much like a classicalbit, it is a system with two possiblestates. However, the difference lies in the fact that because a qubit obeys the laws of quantum mechanics, it is capable of occupying aquantum 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 nonlinearLC circuit with a capacitor and aJosephson junction, a superconducting element with a nonlinear inductance.^[20] 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.^[20] In practice, however, noise in the system can still cause it to leave the computational subspace. In many cases, the higher energy levels of a superconducting qubit need to be considered.^[21] 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 operationtemperatures.

A quantum gate is a generalization of alogic gate describing thetransformation of qubits from their initial state to a different state, often a superposition.

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.^[23]

Two qubit gates, such as theiSWAP gate, can be achieved through coherent exchange or parametric coupling between two qubits.^[23] In coherent exchange, the transition frequency of one of the qubits is tuned such that it matches the transition frequency of the second qubit. This method relies on the frequency tunability of the qubit, and does not work in fixed-frequency cases.^[23] Parametric coupling on the other hand is done by changing the coupling constant between two qubits at the sum or difference of their two frequencies.^[24]

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 theDiVincenzo's criteria,^[25] 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.^{[citation needed]}

Superconducting qubits already meet a large number of DiVincenzo's criteria. They are already highly scalable from afabrication standpoint, they can be initialized by thermal relaxation, and single-qubit gates combined with two-qubit gates form auniversal gate set. However, superconducting qubits still struggle with having shortcoherence times, making preservingquantum information a challenge.

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.^[26]

This phenomenon occurs because the basic charge carriers are pairs ofelectrons (known asCooper pairs), rather than single electrons as found in typical conductors.^[27] While single electrons arefermions (with half-integer spin), Cooper pairs of electrons arebosons (with integer spin), and as such they no longer obey thePauli exclusion principle, meaning theseCooper pairs can occupy the same states. Under certain conditions, this allows them to form a state of matter known as aBose–Einstein condensate, where all of the pairs of electrons in the condensate each occupy the same position in space and have equalmomentum. 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 infiniteconductivity and near zeroresistance.

Superconductivity generally only occurs nearabsolute 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 are networks ofelectrical elements described by a single condensatewave function, whereincharge flow is well-defined by somecomplexprobability amplitude. Quantization in the circuit results from complexamplitudecontinuity, since only discrete numbers ofmagnetic flux quanta can penetrate a superconducting loop. Parameters of superconducting circuits are designed by setting (classical) values to the electrical elements composing them, such ascapacitance orinductance.^{[citation needed]}

One distinguishing attribute of superconducting quantum circuits is theJosephson junction, anelectrical element which does not exist innormal conductors. Thejunction is a weak connection between two superconductors on either side of a thin layer ofinsulator material only a fewatoms thick. The resulting Josephson junction device exhibits theJosephson effect, whereby the condensate wave function on the two sides of the junction are weakly correlated. Current flows through the junction due toquantum tunneling.

The Josephson junction exhibits a nonlinear inductance, which allows foranharmonic oscillators for which energy levels are discretized (orquantized) with nonuniform spacing between energy levels, denoted${\displaystyle \mathrm{\Delta }E}$.^[2] In contrast, thequantum harmonic oscillatorcannot be used as a qubit as there is no way to address only two of its states, given that the spacing between every energy level and the next is exactly the same.

For any qubit implementation, the logicalquantum states${\displaystyle \{|0⟩,|1⟩\}}$ aremapped to different states of the physical system (typically to discreteenergy levels or theirquantum superpositions). Different superconducting qubit designs have different ranges of Josephson energy to charging energy ratio. Josephson energy refers to the energy stored in Josephson junctions when current passes through, and charging energy is the energy required for one Cooper pair to charge the junction's total capacitance.^[28] Josephson energy can be written as

${\displaystyle U_j=-\frac{I_0{\mathrm{\Phi }}_0}{2\pi }\cos \delta }$,

where${\displaystyle I_0}$ is the critical current parameter of the Josephson junction,${\displaystyle {\mathrm{\Phi }}_0=\frac{h}{2e}}$ is (superconducting)flux quantum, and${\displaystyle \delta }$ is thephase difference across the junction.^[28] Notice that the term${\displaystyle cos\delta }$ indicates nonlinearity of the Josephson junction.^[28] Charge energy is written as

${\displaystyle E_C=\frac{e^2}{2C}}$,

where${\displaystyle C}$ is the junction's capacitance and${\displaystyle e}$ is electron charge.^[28]

Circuit quantization is a method of obtaining a quantum mechanical description of an electrical circuit. The end result is aHamiltonian 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 thecurrent andvoltage. Then, generalizedKirchhoff's circuit laws are applied at every node of the circuit network to obtain the system'sequations of motion. Finally, these equations of motion must be reformulated toLagrangian mechanics such that aquantum Hamiltonian is derived describing the total energy of the system.^{[citation needed]}

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.

Superconducting quantum computing devices are typically designed in theradio-frequency spectrum, cooled indilution refrigerators below 15 mK and addressed with conventional electronic instruments, e.g.frequency synthesizers andspectrum analyzers. Typical dimensions fall on the range of micrometers, with sub-micrometer resolution, allowing for the convenient design of aHamiltonian system with well-establishedintegrated circuit technology.

Manufacturing superconducting qubits follows a process involvinglithography, depositing of metal,etching, and controlledoxidation.^[29] This process is similar, though not the same, as CMOS (Complementary Metal-Oxide-Semiconductor) fabrication used for commercial silicon computer chips.^[30] A major difference is the use ofelectron-beam lithography, as opposed to optical lithographic techniques, which is hard to scale and has low yield.^[31] 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 usuallyaluminum, deposited on asilicon substrate, but can also beniobium ortantalum, both d-band superconductors.^[32]

Improvements in fabrication continue to improve the lifetime of superconducting qubits and have made significant improvements since the early 2000s.^[29]: 4

Cryogenicdilution 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 thermalquasiparticles exist within the circuit, which can interfere with the circuit's superconductivity. These so-called 'equilibrium quasiparticles' are exponentially suppressed at lower temperatures.^[33] 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.

The three primary superconducting qubit archetypes are thephase,charge andflux 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.^[34]

Of the three archetypes, phase qubits allow the most of Cooper pairs to tunnel through the junction, followed by flux qubits, and charge qubits allow the fewest.

The charge qubit, also known as theCooper pair box, possesses a Josephson to charging energy ratio on the order of magnitude. For charge qubits, different energy levels correspond to an integer number ofCooper pairs on a superconducting island (a small superconducting area with a controllable number of charge carriers).^[36] Indeed, the first experimentally realized qubit was the Cooper pair box, achieved in 1999.^[37]

The flux qubit (also known as a persistent-current qubit) possesses a Josephson to charging energy ratio on the order of magnitude${\displaystyle 10}$. For flux qubits, the energy levels correspond to differentinteger numbers of magnetic flux quanta trapped in a superconducting ring.

The phase qubit possesses a Josephson to charge energy ratio on the order of magnitude${\displaystyle {10}^6}$. For phase qubits, energy levels correspond to different quantum charge oscillationamplitudes across a Josephson junction, where charge andphase are analogous to momentum and position respectively as analogous to aquantum harmonic oscillator. Note that in this context phase is the complex argument of the superconducting wave function (also known as the superconductingorder parameter), not the phase between the different states of the qubit.

Superconducting Qubit Archetypes^[38]

Type Aspect	Charge qubit	RF-SQUID qubit (prototype of the Flux Qubit)	Phase qubit
Circuit
Hamiltonian	${\displaystyle H=E_C(N-N_g{)}^2-E_J\cos \varphi }$ In this case${\displaystyle N}$ is the number ofCooper pairs totunnel through the junction,${\displaystyle N_g=C{V}_0/2e}$ is the charge on thecapacitor in units of Cooper pairs number,${\displaystyle E_C=(2e{)}^2/2(C_J+C)}$ is the charging energy associated with both capacitance${\displaystyle C}$ and Josephson junction capacitance${\displaystyle C_J}$.	${\displaystyle H=\frac{q^2}{2C_J}+{\left(\frac{{\mathrm{\Phi }}_0}{2\pi }\right)}^2\frac{{\varphi }^2}{2L}-E_J\cos \left[\varphi -\mathrm{\Phi }\frac{2\pi }{{\mathrm{\Phi }}_0}\right]}$ Note that${\displaystyle \varphi }$ is only allowed to take values greater than${\displaystyle 2\pi }$ and is alternatively defined as the time integral of voltage along inductance${\displaystyle L}$.	${\displaystyle H=\frac{(2e{)}^2}{2C_J}{q}^2-I_0\frac{{\mathrm{\Phi }}_0}{2\pi }\varphi -E_J\cos \varphi }$Here${\displaystyle {\mathrm{\Phi }}_0}$ is magnetic flux quantum.
Potential

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'skinetic andpotential energy components (analogous to a particle in apotential well). For the Hamiltonians denoted,${\displaystyle \varphi }$ is the superconducting wave function phase difference across the junction,${\displaystyle C_J}$ is the capacitance associated with the Josephson junction, and${\displaystyle q}$ 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 aninverse function of the circuit capacitance and that the shape of the potential is governed by regularinductors and Josephson junctions. Schematic wave solutions in the third row of the table show the complexamplitude 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 valueonly where the depicted wave functionoscillates. All three rows are essentially different presentations of the same physical system.

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,^[39]transmon,^[40] Xmon,^[41] and quantronium.^[42]

Transmons are a special type of qubit with ashunted capacitor specifically designed to mitigatenoise. The transmon qubit model a charge-phase hybrid qubit based on the Cooper pair box^[44]. The increased ratio of Josephson to charge energy mitigates noise. The Hamiltonian for the transmon is:

$${\displaystyle \hat{H}=4E_C{\left(\hat{n}-n_g\right)}^2-E_J\cos \hat{\varphi }}$$

where n is the number of Cooper pairs transferred between the island and$\varphi$ is the phase difference across the junction.^[45]

Two transmons can be coupled using acoupling capacitor.^[2] For this 2-qubit system the Hamiltonian is written

${\displaystyle \hat{H}=\frac{\hslash J}{2}({\sigma }_1^x{\sigma }_2^x+{\sigma }_1^y{\sigma }_2^y)}$,

where${\displaystyle J}$ iscurrent density and${\displaystyle \sigma }$ issurface charge density.^[2]

Transmon qubits are the most popular design of modern superconducting qubits, and are implemented in superconducting quantum processors such as Google'sWillow processor, a chip with 105 physical transmon qubits.^[46] Other companies that use transmon qubits includeIBM,Rigetti, andIQM.

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.

Many variations of the transmon design exist and are active areas of research. They aim to improve upon failings of the transmon design.

The Xmon is similar in design to a transmon in that it originated based on the planar transmon model.^[48] 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.^[49]

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 viagate voltage.

The Unimon consists of a single Josephson junction shunted by a linear inductor (possessing an inductance not depending on current) inside a (superconducting)resonator.^[50] Unimons have increased anharmonicity and display faster operation time resulting in lower susceptibility to noise errors.^[50] Unimon qubits also have decreased susceptibility to flux noise and complete insensitivity to dc charge noise.^[35] However, unimon qubits have a limited tunability range.

The unimon qubit was first formulated in 2022 by researchers fromIQM Quantum Computers,Aalto University, andVTT Technical Research Centre of Finland, and is still largely in the research stages of design.^[52]

Fluxonium qubits are a specific type of flux qubit whose Josephson junction is shunted by a linear inductor of${\displaystyle E_J\gg E_L}$ where${\displaystyle E_L=(\hslash /2e{)}^2/L}$.^[53] In practice, the linear inductor is usually implemented by a Josephson junction array that is composed of a large number (can be often${\displaystyle N>100}$) of large-sized Josephson junctions connected in a series. Under this condition, the Hamiltonian of a fluxonium can be written as:

${\displaystyle \hat{H}=4E_C{\hat{n}}^2+\frac{1}{2}{E}_L(\hat{\varphi }-{\varphi }_{\mathrm{e}\mathrm{x}\mathrm{t}}{)}^2-E_J\cos \hat{\varphi }}$.

One important property of the fluxonium qubit is the longerqubit lifetime at the half flux sweet spot, which can exceed 1 millisecond.^[53][54] Another crucial advantage of the fluxonium qubit when biased at the sweet spot is its largeanharmonicity. 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.^[55][56]

The 0-π qubit is a protected qubit design where logical states are protected by circuit symmetry.^[57] 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.^[58]

TheGHz energy gap between energy levels of a superconducting qubit is designed to be compatible with available electronic equipment, due to theterahertz gap (lack of equipment in the higherfrequency band). Thesuperconductor 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 impliesenergy fluctuations of 20 GHz. Temperatures of tens of millikelvins are achieved indilution refrigerators and allow qubit operation at a ~5 GHz energy level separation. Qubit energy level separation is frequently adjusted by controlling a dedicatedbias current line, providing a "knob" to fine tune the qubit parameters.

A single qubit gate is achieved by rotation in theBloch sphere. Rotations between different energy levels of a single qubit are induced bymicrowave pulses sent to anantenna ortransmission line coupled to the qubit with afrequency resonant with the energy separation between levels. Individual qubits may be addressed by a dedicatedtransmission line or by a shared one if the other qubits are offresonance. Theaxis of rotation is set byquadrature amplitude modulation of microwave pulse, while pulse length determines theangle of rotation.^[59]

More formally (following the notation of^[59]) for a driving signal

${\displaystyle \mathcal{E}(t)={\mathcal{E}}^x(t)\cos ({\omega }_{d}t)+{\mathcal{E}}^y(t)\sin ({\omega }_{d}t)}$

of frequency${\displaystyle {\omega }_d}$, a driven qubit Hamiltonian in arotating wave approximation is

${\displaystyle H^R/\hslash =(\omega -{\omega }_d)|1⟩⟨1|+\frac{{\mathcal{E}}^x(t)}{2}{\sigma }_x+\frac{{\mathcal{E}}^y(t)}{2}{\sigma }_y}$,

where${\displaystyle \omega }$ is the qubit resonance and${\displaystyle {\sigma }_x,{\sigma }_y}$ arePauli matrices.

To implement a rotation about the${\displaystyle X}$ axis, one can set${\displaystyle {\mathcal{E}}^y(t)=0}$ and apply a microwave pulse at frequency${\displaystyle {\omega }_d=\omega }$ for time${\displaystyle t_g}$. The resulting transformation is

${\displaystyle U_x=\exp \left\{-\frac{i}{\hslash }{\int }_0^{t_g}{H}^{R}dt\right\}=\exp \left\{-i{\int }_0^{t_g}{\mathcal{E}}^x(t)dt\cdot {\sigma }_x/2\right\}}$.

This is exactly therotation operator${\displaystyle R_X(\theta )}$ by angle${\displaystyle \theta ={\int }_0^{t_g}{\mathcal{E}}^x(t)dt}$ about the${\displaystyle X}$ axis in the Bloch sphere. A rotation about the${\displaystyle Y}$ axis can be implemented in a similar way. Showing the two rotation operators is sufficient for satisfyinguniversality as every single qubit unitary operator${\displaystyle U}$ may be presented as${\displaystyle U=R_X({\theta }_1)R_Y({\theta }_2)R_X({\theta }_3)}$ (up to a globalphase which is physically inconsequential) by a procedure known as the${\displaystyle X-Y}$ decomposition.^[60] Setting${\displaystyle {\int }_0^{t_g}{\mathcal{E}}^x(t)dt=\pi }$ results in the transformation

${\displaystyle U_x=\exp \left\{-i{\int }_0^{t_g}{\mathcal{E}}^x(t)dt\cdot {\sigma }_x/2\right\}=e^{-i\pi {\sigma }_x/2}=-i{\sigma }_x}$

up to the global phase${\displaystyle -i}$ and is known as theNOT gate.

The ability to couple qubits is essential for implementing 2-qubitgates. 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 theDC-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.^[61] 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 connectedunit cell of 8 qubits in Chimera graph configuration.Quantum algorithms typically require coupling between arbitrary qubits. Consequently, multipleswap operations are necessary, limiting the length of quantum computation possible before processordecoherence.

TheHeisenberg model of interactions, written as

${\displaystyle {\hat{\mathcal{H}}}_{\mathrm{X}\mathrm{X}\mathrm{Z}}/\hslash =\sum _{i,j}{J}_{\mathrm{X}\mathrm{Y}}({\hat{\sigma }}_{\text{x}}^i{\hat{\sigma }}_{\text{x}}^j+{\hat{\sigma }}_{\text{y}}^i{\hat{\sigma }}_{\text{y}}^j)+J_{\mathrm{Z}\mathrm{Z}}{\hat{\sigma }}_{\text{z}}^i{\hat{\sigma }}_{\text{z}}^j}$,

serves as the basis for analog quantum simulation of spin systems and the primitive for an expressive set of quantum gates, sometimes referred to asfermionic simulation (orfSim) gates. In superconducting circuits, this interaction model has been implemented using flux-tunable qubits with flux-tunable coupling,^[62] allowing the demonstration of quantum supremacy.^[63] In addition, it can also be realized in fixed-frequency qubits with fixed-coupling using microwave drives.^[64] The fSim gate family encompasses arbitrary XY and ZZ two-qubit unitaries, including the iSWAP, the CZ, and the SWAP gates (seeQuantum logic gate).

Another method of coupling two or more qubits is by way of aquantum bus, by pairing qubits to this intermediate. A quantum bus is often implemented as amicrowave cavity modeled by a quantum harmonic oscillator. Coupled qubits may be brought in and out of resonance with the bus and with each other, eliminating the nearest-neighbor limitation. Formalism describing coupling iscavity quantum electrodynamics. In cavity quantum electrodynamics, qubits are analogous to atoms interacting with anoptical photon cavity with a difference of GHz (rather than the THz regime of electromagnetic radiation). Resonant excitation exchange among these artificial atoms is potentially useful for direct implementation of multi-qubit gates.^[65] Following the dark statemanifold, the Khazali-Mølmer scheme^[65] performs complex multi-qubit operations in a single step, providing a substantial shortcut to the conventional circuit model.

One popular gating mechanism uses two qubits and a bus, each tuned to different energy level separations. Applying microwave excitation to the first qubit, with a frequency resonant with the second qubit, causes a${\displaystyle {\sigma }_x}$ rotation of the second qubit. Rotation direction depends on the state of the first qubit, allowing acontrolled phase gate construction.^[66]

Following the notation of,^[66] the drive Hamiltonian describing the excited system through the first qubit driving line is formally written

${\displaystyle H_D/\hslash =A(t)\cos ({\tilde{\omega }}_{2}t)\left({\sigma }_x\otimes I-\frac{J}{{\mathrm{\Delta }}_{12}}{\sigma }_z\otimes {\sigma }_x+m_{12}I\otimes {\sigma }_x\right)}$,

where${\displaystyle A(t)}$ is the shape of the microwave pulse in time,${\displaystyle {\tilde{\omega }}_2}$ is resonance frequency of the second qubit,${\displaystyle \{I,{\sigma }_x,{\sigma }_y,{\sigma }_z\}}$ are thePauli matrices,${\displaystyle J}$ is the coupling coefficient between the two qubits via the resonator,${\displaystyle {\mathrm{\Delta }}_{12}\equiv {\omega }_1-{\omega }_2}$ is qubit detuning,${\displaystyle m_{12}}$ is stray (unwanted) coupling between qubits, and${\displaystyle \hslash }$ is thereduced Planck constant. The timeintegral over${\displaystyle A(t)}$ determines the angle of rotation. Unwanted rotations from the first and third terms of the Hamiltonian can be compensated for with single qubit operations. The remaining component, combined with single qubit rotations, forms a basis for thesu(4)Lie algebra.

Higher levels (outside of the computational subspace) of a pair of coupled superconducting circuits can be used to induce a geometric phase on one of the computational states of the qubits. This leads to an entangling conditional phase shift of the relevant qubit states. This effect has been implemented by flux-tuning the qubit spectra^[67] and by using selective microwave driving.^[68] Off-resonant driving can be used to induce differential ac-Stark shift, allowing the implementation of all-microwave controlled-phase gates.^[69]

Architecture-specific readout, ormeasurement, mechanisms exist. Readout of a phase qubit is explained in thequbit archetypes table above. A flux qubit state is often read using an adjustable DC-SQUIDmagnetometer. States may also be measured using anelectrometer.^[2] A more general readout scheme includes a coupling to a microwaveresonator, where resonance frequency of the resonator is dispersively shifted by the qubit state.^[70][71] Multi-level systems (qudits) can be readout using electron shelving.^[72]

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, andreadout in the 3rd dimension with additionallithography 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.

1. A scalable physical system with well characterized qubits. "Well characterized" implies that theHamiltonian 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.^[25] As superconducting qubits are fabricated on a chip, the many-qubit system is readily scalable. Qubits are allocated on the 2D surface of the chip. The demand for well characterized qubits is fulfilled with (a) qubit non-linearity (accessing only two of the available energy levels) and (b) accessing a single qubit at a time (rather than the entire many-qubit system) by way of per-qubit dedicated control lines and/or frequency separation, or tuning out, of different qubits.
2. Ability to initialize the state of qubits to a simple fiducial state.^[73] 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.
3. Long relevant decoherence times.^[73] Decoherence of superconducting qubits is affected by multiple factors. Most decoherence is attributed to the quality of the Josephson junction and imperfections in the chip substrate. Due to their mesoscopic scale, superconducting qubits are relatively short lived. Nevertheless, thousands of gate operations have been demonstrated in these many-qubit systems.^[74] Recent strategies to improve device coherence include purifying the circuit materials and designing qubits with decreased sensitivity to noise sources.^[53]
4. A "universal" set of quantum gates.^[73] Superconducting qubits allow arbitrary rotations in the Bloch sphere with pulsed microwave signals, implementing single qubit gates.${\displaystyle {\sigma }_z{\sigma }_z}$ and${\displaystyle {\sigma }_x{\sigma }_x}$ couplings are shown for most implementations and for complementing the universal gate set.^[75][76][64] This criterion may also be satisfied by coupling two transmons with a coupling capacitor.^[2]
5. Qubit-specific measurement ability.^[73] In general, single superconducting qubits are used for control or for measurement.
6. Interconvertibility of stationary and flying qubits.^[73] While stationary qubits are used to store information or perform calculations, flying qubits transmit information macroscopically. Qubits should be capable of converting from being a stationary qubit to being a flying qubit and vice versa.
7. Reliable transmission of flying qubits between specified locations.^[73]

The final two criteria have been experimentally proven by research performed byETH Zurich university with two superconducting qubits connected by acoaxial cable.^[77]

Many current challenges faced by superconducting quantum computing lie in the field of microwave engineering.^[70] Some challenges in superconducting qubit design are shaping the potential well and choosing particle mass such that energy separation between two specific energy levels is unique, differing from all other interlevel energy separation in the system, since these two levels are used as logical states of the qubit. Other challenges are mitigating sources of noise in the system. Finally, even more challenges occur a result of scaling to larger and larger device sizes.

Superconducting quantum computing must mitigatequantum noise (disruptions of the system caused by its interaction with an environment) as well asleakage (information being lost to the surrounding environment). One way to reduce leakage is withparity measurements.^[29] Another strategy is to use qubits with large anharmonicity.^[55][56]

Two-level system (TLS) effects are a dominant source of noise in superconducting qubits.^[78][79] TLS act as resonant, two-level absorbers which drain energy from the qubit, significantly reducingcoherence times. They are thought to be caused by deformities during the fabrication process, and surface amorphous oxides that form on or nearJosephson 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.^[78] As the TLS density increases, it becomes harder to protect a system from TLS effects.^[78]

Attempts to mitigate TLS effects include developing newfabrication techniques, and experimenting with new materials such asniobium andtantalum.^[80]

Quasiparticles are single-electron excitations that occur whenCooper 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.^[81] 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 toradiation such as gammas and cosmic ray muons, cannot.^[81][82]

Attempts to mitigate quasiparticle generation include increasing shielding against radiation,^[83] quasiparticle trapping, gap engineering,^[84] or otherwise removing them from the system.^[85]

As superconducting quantum computing approaches larger scale devices, researchers face difficulties inqubit coherence, scalablecalibration software, efficient determination offidelity of quantum states across an entire chip, and qubit and gate fidelity.^[29] Moreover, superconducting quantum computing devices must be reliably reproducible at increasingly large scales such that they are compatible with these improvements.^[29]

• Stancil, Daniel D.; Byrd, Gregory T. (2022).Principles of Superconducting Quantum Computers (1st ed.). Hoboken, New Jersey: John Wiley & Sons.ISBN 978-1-119-75072-7.OCLC 1302334194. 978-1-119-75074-1 (ebook).
• Salari, Alan (2024).Microwave Techniques in Superconducting Quantum Computers (Unabridged edition). Boston, Massachusetts: Artech House.ISBN 978-1-63081-987-3.OCLC 1405187817. 978-1-63081-988-0 (ebook).
• Krylov, Gleb; Jabbari, Tahereh; Friedman, Eby G. (2024).Single Flux Quantum Integrated Circuit Design (Second Edition). Cham: Springer.doi:10.1007/978-3-031-47475-0.ISBN 978-3-031-47474-3.OCLC 1430662174.
• Absar, Rubaya; Elgabra, Hazem; Ma, Dylan; Zhao, Yiju; Wei, Lan (2024). "Cryogenic CMOS for Quantum Computing". In Liu, Weiqiang; Han, Jie; Lombardi, Fabrizio (eds.).Design and Applications of Emerging Computer Systems (1st ed.). Cham: Springer. pp. 591–621.doi:10.1007/978-3-031-42478-6_22.ISBN 978-3-031-42478-6.OCLC 1418721165.
• Awan, Shakil; Kibble, Bryan P.; Schurr, Jürgen (2011).Coaxial Electrical Circuits for Interference-Free Measurements. Electrical Measurement Series 13. Stevenage: The Institution of Engineering and Technology (IET).doi:10.1049/PBEL013E.ISBN 978-1-84919-069-5.OCLC 761013886.
• Jordan, Andrew N.; Siddiqi, Irfan A. (2024).Quantum Measurement: Theory and Practice (1st ed.). Cambridge University Press.ISBN 978-1-009-10006-9.OCLC 1435711501.
• Zagoskin, Alexandre M. (2011).Quantum Engineering: Theory and Design of Quantum Coherent Structures. Cambridge University Press.ISBN 978-0-521-11369-4.OCLC 768771210.
• Van Duzer, Theodore; Turner, Charles W. (1999).Principles of Superconductive Devices and Circuits (2nd ed.). Prentice Hall.ISBN 978-0-13-262742-9.OCLC 40251418.
• Solymar, Lazlo (1972).Superconductive Tunnelling and Applications. London, England: Chapman and Hall.ISBN 0-412-10210-2.OCLC 488903.

• IBM Quantum offers access to over 20 quantum computer systems.
• TheIBM Quantum Experience offers free access to writing quantum algorithms and executing them on 5 qubit quantum computers.
• IBM's roadmap for quantum computing shows 65 qubit systems available in 2020 and 127 qubits to be available sometime in 2021.

• Quantum information science
• Quantum electronics
• Superconductivity

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