A quantum engineer's guide to superconducting qubits

Quantum processors harness the intrinsic properties of quantum mechanical systems—such as quantum parallelism and quantum interference—to solve certain problems where classical computers fall short.^1–6 Over the past two decades, rapid developments in the science and engineering of quantum systems have advanced the frontier in quantum computation, from the realm of scientific explorations on single isolated quantum systems toward the creation and manipulation of multiqubit processors.^7,8 In particular, the requirements imposed by larger quantum processors have shifted the mindset within the community, from solely scientific discovery to the development of new, foundational engineering abstractions associated with the design, control, and readout of multiqubit quantum systems. The result is the emergence of a new discipline termed “quantum engineering,” which serves to bridge basic sciences, mathematics, and computer science with fields generally associated with traditional engineering.

One prominent platform for constructing a multiqubit quantum processor involves superconducting qubits, in which information is stored in quantum degrees of freedom (DOFs) of nanofabricated, anharmonic oscillators (AHOs) constructed from superconducting circuit elements. In contrast to other platforms, e.g., electron spins in silicon^9–14 and quantum dots,^15–18 trapped ions,^19–23 ultracold atoms,^24–27 nitrogen-vacancies in diamonds,^28,29 and polarized photons,^30–33 where the quantum information is encoded in natural microscopic quantum systems, superconducting qubits are macroscopic in size and lithographically defined.

One remarkable feature of superconducting qubits is that their energy-level spectra are governed by circuit element parameters and thus are configurable; they can be designed to exhibit “atomlike” energy spectra with the desired properties. Therefore, superconducting qubits are also often referred to as “artificial atoms,” offering a rich parameter space of possible qubit properties and operation regimes, with predictable performance in terms of transition frequencies, anharmonicity, and complexity.

While there are many other excellent reviews on superconducting qubits, see, e.g., Refs. 34–43, this work specifically aims to introduce new quantum engineers (academic and industrial alike) to the terminology and state-of-the-art practices used in the rapidly accelerating field of superconducting quantum computing. The reader is assumed to be familiar with the basic concepts that span classical physics, quantum mechanics, and electrical engineering. In particular, readers will find it useful to have had previous exposure to classical mechanics, the Schrödinger equation, the Bloch sphere representation of qubit states, second quantization, basic concepts of superconductivity, electromagnetism, introductory circuit analysis, classical Boolean logic, linear dynamical systems, analog and digital signal processing, and familiarity with microwave components such as transmission lines and mixers. These topics will be introduced as they arise, but having basic prior knowledge will be helpful.

This review is organized in the following four sections: first, in Sec. II, we explore the parameter space available when designing superconducting circuits. In particular, we look at the promising capacitively shunted planar qubit modalities and how these can be engineered with the desired properties, such as transition frequency, anharmonicity, and reduced susceptibility to various sources of noise. In this section, we also introduce several ways in which interactions between qubits can be engineered, in order to implement two-qubit entangling operations, needed for a universal gate set.

In Sec. III, we discuss systematic and stochastic noise, the concepts of noise strength and qubit noise susceptibility, and the common sources of noise which lead to decoherence in superconducting circuits. We introduce the Bloch-Redfield model of decoherence, characterized by longitudinal and transverse relaxation times T₁ and T₂, and discuss the implications of 1/f noise. We then define the noise power spectral density (PSD), which is commonly used to characterize noise processes, and describe how it drives decoherence. Finally, we close the section with a review of coherent control methods used to mitigate certain types of coherence and reversible noise.

In Sec. IV, we provide a review of how single- and two-qubit operations are typically implemented in superconducing circuits, by using a combination of local magnetic flux control and microwave drives. In particular, we discuss the family of two-qubit gates arising from a capacitive coupling between qubits, and introduce several recent advances that have been demonstrated to achieve high-fidelity gates, as well as applications in quantum information processing that use these gates. The continued development of high-fidelity two-qubit gates in superconducting qubits is a highly active research area. For this reason, we include sufficient technical details that a reader may use this review as a starting point to critically assess the pros and cons of the various gates, as well as develop an appreciation for the types of gate-engineering already implemented in-state-of-the-art superconducting quantum processors.

Finally, in Sec. V, we discuss the physics and engineering associated with the dispersive readout technique, typically used to measure the individual qubit states in modern quantum processors. After a discussion of the theory behind dispersive coupling, we give an introduction to design of Purcell filters and the development of quantum-limited parametric amplifiers (PAs).

In this section, we will demonstrate how quantum systems based on superconducting circuits can be engineered to achieve certain desired properties. Using the most common qubit modalities, we discuss how properties such as the qubit transition frequency, anharmonicity, and noise susceptibility can be tailored by the choice of circuit topology and element parameter values. We also discuss how to engineer the interactions between different quantum systems, in particular, the cases of qubit-qubit and qubit-resonator couplings.

A quantum mechanical system is governed by the time-dependent Schrödinger equation

where $|ψ(t)⟩$ is the state of the quantum system at time t, $ℏ$ is the reduced Planck's constant h/2π, and $Ĥ$ is the “Hamiltonian” that describes the total energy of the system. The “hat” is used to indicate that $Ĥ$ is a quantum operator. As the Schrödinger equation is a first-order linear differential equation, the temporal dynamics of the quantum system may be viewed as a straightforward example of a linear dynamical system with a formal solution

The time-independent Hamiltonian $Ĥ$ governs the time evolution of the system through the operator $e−iĤt/ℏ$⁠. Thus, just as with classical systems, determining the Hamiltonian of a system—whether the classical Hamiltonian H or its quantum counterpart $Ĥ$—is the first step to deriving its dynamical behavior. In Sec. IV, we consider the case when the Hamiltonian is time-dependent in the context of qubit control.

To understand the dynamics of a superconducting qubit circuit, it is natural to start with the classical description of a linear LC resonant circuit [Fig. 1(a)]. In this system, energy oscillates between electrical energy in the capacitor C and magnetic energy in the inductor L. In the following, we will arbitrarily associate the electrical energy with the “kinetic energy” and the magnetic energy with the “potential energy” of the oscillator. The instantaneous, time-dependent energy in each element is derived from its current and voltage

where $V(t′)$ and $I(t′)$ denote the voltage and current of the capacitor or inductor.

To derive the classical Hamiltonian, we follow the standard approach used in classical mechanics: the Lagrange-Hamilton formulation. Here, we represent the circuit elements in terms of one of its generalized circuit coordinates, charge or flux. In the following, we pick flux, defined as the time integral of the voltage

In this example, the voltage at the node is also the branch voltage across the element. In this section, we will simply refer to these as node voltages and fluxes for convenience. For a more detailed discussion of nodes and branches in this context, we refer the reader to Ref. 44.

Note that in the following, we could have exchanged our associations with kinetic energy (momentum coordinate) and potential energy (position coordinate), and instead start with the charge variable Q(t), which is the time integral of the current I(t).

By combining Eqs. (3) and (4), using the relations $V=L dI/dt$ and $I=C dV/dt$⁠, and applying the integration by parts formula, we can write down energy terms for the capacitor and inductor in terms of the node flux

The Lagrangian is defined as the difference between the kinetic and potential energy terms and can thus be expressed in terms of Eqs. (5) and (6)

From the Lagrangian in Eq. (7), we can further derive the Hamiltonian using the Legendre transformation, for which we need to calculate the momentum conjugate to the flux, which in this case, is the charge on the capacitor

The Hamiltonian of the system is now defined as

as one would expect for an electrical LC circuit. Note that this Hamiltonian is analogous to that of a mechanical harmonic oscillator, with mass m = C and resonant frequency $ω=1/LC$⁠, which expressed in position, x, and momentum, p, coordinates takes the form $H=p2/2m+mω2x2/2$⁠.

The Hamiltonian described above is classical. In order to proceed to a quantum-mechanical description of the system, we need to promote the charge and flux coordinates to quantum operators, whereas the classical coordinates satisfy the Poisson bracket

the quantum operators similarly satisfy a “commutation relation”

where the operators are indicated by hats. From this point forward, however, the hats on operators will be omitted for simplicity.

In a simple LC resonant circuit [Fig. 1(a)], both the inductor L and the capacitor C are linear circuit elements. Defining the reduced flux $ϕ≡2πΦ/Φ0$ and the reduced charge n = Q/2e, we can write down the following quantum-mechanical Hamiltonian for the circuit

where E_C = e²/(2C) is the charging energy required to add “each” electron of the Cooper-pair to the island and $EL=(Φ0/2π)2/L$ is the inductive energy, where $Φ0=h/(2e)$ is the superconducting magnetic flux quantum. Moreover, the quantum operator n is the excess number of Cooper-pairs on the island, and ϕ—the reduced flux—is denoted the “gauge-invariant phase” across the inductor. These two operators form a canonical conjugate pair, obeying the commutation relation [ϕ, n] = i. We note that the factor 4 in front of the charging energy E_C is solely a historical artifact, namely, that this energy scale was first defined for single-electron systems and then adopted to two-electron Cooper-pair systems.

The Hamiltonian in Eq. (13) is identical to the one describing a particle in a one-dimensional quadratic potential, a quantum harmonic oscillator (QHO). We can treat ϕ as the generalized position coordinate, so that the first term is the kinetic energy and the second term is the potential energy. We emphasize that the functional form of the potential energy influences the eigensolutions. For example, the fact that this term is quadratic (⁠$UL∝ϕ2$⁠) in Eq. (13) gives rise to the shape of the potential in Fig. 1(b). The solution to this eigenvalue problem gives an infinite series of eigenstates $|k⟩, (k=0,1,2,…)$⁠, whose corresponding eigenenergies E_k are all equidistantly spaced, i.e., $Ek+1−Ek=ℏωr$⁠, where $ωr=8ELEC/ℏ=1/LC$ denotes the resonant frequency of the system, see Fig. 1(b). We may represent these results in a more compact form (second quantization) for the quantum harmonic oscillator (QHO) Hamiltonian

where $a†(a)$ is the creation (annihilation) operator of a single excitation of the resonator. The Hamiltonian in Eq. (14) is written as energy. It is, however, often preferred to divide by $ℏ$ so that the expression has units of radian frequency, since we will later resonantly drive transitions at a particular frequency or reference the rate at which two systems interact with one another. Therefore, from here on, $ℏ$ will be omitted.

The original charge number and phase operators can be expressed as $n=nzpf×i(a−a†)$ and $ϕ=ϕzpf×(a+a†)$⁠, where $nzpf=[EL/(32EC)]1/4$ and $ϕzpf=(2EC/EL)1/4$ are the “zero-point fluctuations” of the charge and phase variables, respectively. Quantum mechanically, the quantum states are represented as wavefunctions that are generally distributed over a range of values of n and ϕ and, consequently, the wavefunctions have nonzero standard deviations. Such wavefunction distributions are referred to as “quantum fluctuations,” and they exist, even in the ground state, where they are called zero-point fluctuations.

The linear characteristics of the QHO have a natural limitation in its applications for processing quantum information. Before the system can be used as a qubit, we need to be able to define a computational subspace consisting of only two energy states (usually the two-lowest energy eigenstates) in between which transitions can be driven without also exciting other levels in the system. Since many gate operations, such as single-qubit gates (Sec. IV), depend on frequency selectivity, the equidistant level-spacing of the QHO, illustrated in Fig. 1(b), poses a practical limitation.⁴²⁷ 

To mitigate the problem of unwanted dynamics involving noncomputational states, we need to add anharmonicity (or nonlinearity) into our system. In short, we require the transition frequencies $ωq0→1$ and $ωq1→2$ be sufficiently different to be individually addressable. In general, the larger the anharmonicity the better it is. In practice, the amount of anharmonicity sets a limit on how short the pulses used to drive the qubit can be. This is discussed in detail in Sec. IV D 3.

To introduce the nonlinearity required to modify the harmonic potential, we use the Josephson junction—a nonlinear, dissipationless circuit element that forms the backbone in superconducting circuits.^46,47 By replacing the linear inductor of the QHO with a Josephson junction, playing the role of a nonlinear inductor, we can modify the functional form of the potential energy. The potential energy of the Josephson junction can be derived from Eq. (3) and the two Josephson relations

resulting in a modified Hamiltonian

where $EC=e2/(2CΣ), CΣ=Cs+CJ$ is the total capacitance, including both shunt capacitance C_s and the self-capacitance of the junction C_J, and $EJ=IcΦ0/2π$ is the Josephson energy, with I_c being the critical current of the junction.⁴²⁸ After introducing the Josephson junction in the circuit, the potential energy no longer takes a manifestly parabolic form (from which the harmonic spectrum originates), but rather features a cosinusoidal form, see the second term in Eq. (16), which makes the energy spectrum nondegenerate. Therefore, the Josephson junction is the key ingredient that makes the oscillator anharmonic and thus allows us to identify a uniquely addressable quantum two-level system, see Fig. 1(d).

Once the nonlinearity has been added, the system dynamics is governed by the dominant energy in Eq. (16), reflected in the E_J/E_C ratio. Over time, the superconducting qubit community has converged toward circuit designs with $EJ≫EC$⁠. In the opposite case when $EJ≤EC$⁠, the qubit becomes highly sensitive to charge noise, which has proven more challenging to mitigate than flux noise, making it very hard to achieve high coherence. Another motivation is that current technologies allow for more flexibility in engineering the inductive (or potential) part of the Hamiltonian. Therefore, working in the $EJ≤EC$ limit, makes the system more sensitive to the change in the potential Hamiltonian. Therefore, we will focus here on the state-of-the-art qubit modalities that fall in the regime $EJ≫EC$⁠. For readers who are interested in the physics in the $EJ≤EC$ regime, such as the earlier Cooper-pair box charge qubit, we refer to Refs. 48–51.

To access the $EJ≫EC$ regime, one preferred approach is to make the charging E_C small by shunting the junction with a large capacitor, $Cs≫CJ$⁠, effectively making the qubit less sensitive to charge noise—a circuit commonly known as the transmon qubit.⁵² In this limit, the superconducting phase ϕ is a good quantum number, i.e., the spread (or quantum fluctuation) of ϕ values represented by the quantum wavefunction is small. The low-energy eigenstates are therefore, to a good approximation, localized states in the potential well, see Fig. 1(d). We may gain more insight by expanding the potential term of Eq. (16) into a power series (since ϕ is small), that is

The leading quadratic term in Eq. (17) alone will result in a QHO, recall Eq. (13). The second term, however, is quartic which modifies the eigensolution and disrupts the otherwise harmonic energy structure. Note that, the negative coefficient of the quartic term indicates that the anharmonicity $α=ωq1→2−ωq0→1$ is negative and its limit in magnitude thus cannot be made arbitrarily large. For the case of the transmon, α = –E_C is usually designed to be 100–300 MHz, as required to maintain a desirable qubit frequency $ωq=(8EJEC−EC)/ℏ=3-6$GHz, while keeping an energy ratio sufficiently large (⁠$EJ/EC≥50$⁠) to suppress charge sensitivity.⁵² Fortunately, the charge sensitivity is exponentially suppressed for an increased E_J/E_C, while the reduction in anharmonicity only scales as a weak power law, leading to a workable device.

Including terms up to fourth order and using the QHO eigenbases, the system Hamiltonian resembles that of a Duffing oscillator

Since $|α|≪ωq$⁠, we can see that the transmon qubit is basically a weakly anharmonic oscillator (AHO). If excitation to higher noncomputational states is suppressed over any gate operations, either due to a large enough $|α|$ or due to robust control techniques such as the derivative reduction by adiabatic gate (DRAG) pulse, see Sec. IV D 3, we may effectively treat the AHO as a quantum two-level system, simplifying the Hamiltonian to

where σ_z is the Pauli-z operator. However, one should always keep in mind that higher levels physically exist.⁵³ Their influence on the system dynamics should be taken into account when designing the system and its control processes. In fact, there are many cases where the higher levels have proven useful to implement more efficient gate operations.⁵⁴ 

In addition to reducing the charge dispersion, the use of a large shunt capacitor also enables us to engineer the electric field distribution of the quantum system, and thus the participation of surface loss mechanisms. In the development of the 3D transmon,⁵⁵ e.g., a 2D transmon coupled to a 3D cavity, it was demonstrated that by making the gap between the two lateral capacitor plates large (compared to the film thickness) the coherence time increases since a smaller portion of the electric field interacts with the lossy interfaces, e.g., metal-substrate and substrate-vacuum interfaces, which has been studied extensively.^56–61 

To implement fast gate operations with high-fidelity, as needed to implement quantum logic, many (though not all⁶³) of the quantum processor architectures implemented today feature tunable qubit frequencies.^64–67 For instance, in some cases, we need to bring two qubits into resonance to exchange (swap) energy, while we also need the capability of separating them during idling periods to minimize their interactions. To do this, we need an external parameter which allows us to access one of the degrees of freedom of the system in a controllable fashion.

One widely used technique is to replace the single Josephson junction with a loop interrupted by two identical junctions—forming a DC superconducting quantum interference device (DC-SQUID).⁶⁸ Due to the interference between the two arms of the SQUID, the effective critical current of the two parallel junctions can be decreased by applying a magnetic flux threading the loop, see Fig. 2(a). Due to the fluxoid quantization condition, the algebraic sum of branch flux of all of the inductive elements along the loop plus the externally applied flux equal an integer number of superconducting flux quanta, that is

where $φe=πΦext/Φ0$⁠. Using this condition, we can eliminate one degree of freedom and treat the SQUID-loop as a single junction, but with the important modification that E_J is tunable (via the SQUID critical current) by means of the external flux Φ_ext. The effective Hamiltonian of the so-called split transmon (ignoring the constant) is

We can see that Eq. (21) is analogous to Eq. (16), with E_J replaced by $EJ′(φe)=2EJ| cos (φe)|$⁠. The magnitude of the net, effective Josephson energy $EJ′$ has a period of Φ₀ in applied flux and spans from 0 to its maximum value 2E_J. Therefore, the qubit frequency can be tuned periodically with Φ_ext, see Fig. 2(b).

While the split transmon enables frequency tunability by the externally applied magnetic field, it also introduces sensitivity to random flux fluctuations, known as flux noise. At any working point, the slope of the qubit spectrum, $∂ωq/∂Φext$⁠, indicates to first order how strongly this flux noise affects the qubit frequency. The sensitivity is generally nonzero, except at multiples of the flux quantum, $Φext=kΦ0$⁠, where k is an integer, where $∂ωq/∂Φext=0$⁠.

One recent development has focused on reducing the qubit sensitivity to flux noise, while maintaining sufficient tunability to operate our quantum gates. The idea is to make the two junctions in the split transmon asymmetric,⁶⁹ see Fig. 2(c). This yields the following Hamiltonian

where $EJΣ=EJ1+EJ2$ and $d=(γ−1)/(γ+1)$ is the junction asymmetry parameter, with $γ=EJ2/EJ1$⁠. Again, we can treat the two junctions as a single-junction transmon, with an effective Josephson energy $EJ′(φe)$⁠. In particular, we can recognize the two special cases; for d = 0, the Hamiltonian in Eq. (22) reduces to the symmetric case with $EJ′(φe)=EJΣ| cos (φe)|$⁠, as in Eq. (21) with $EJΣ=2EJ$⁠. In the other limit, when $|d|→1, EJ′(φe)→EJΣ$ and the flux-tunability of the Josephson energy vanishes, which is equivalent to the single junction case, recall Eq. (16).

From the discussion above we see that going from symmetric to asymmetric transmons does not change the circuit topology. This seemingly trivial modification, however, has a profound impact for practical applications. As we can see from the qubit spectra, Fig. 2(d), the flux sensitivity is suppressed across the entire tunable frequency range. For example, the performance of the cross-resonance gate is optimized with a certain frequency detuning between two qubits.⁷⁰ Therefore, by using an asymmetric transmon, a small frequency-tuning range is introduced that is sufficient to compensate for fabrication variations, without introducing unnecessary large susceptibility to flux noise and thus maintaining high coherence. For another example, a surface code scheme based on the adiabatic controlled phase (CPHASE)-gate requires specific frequency configuration among qubits in order to avoid frequency crowding issues, and asymmetric transmons fit well with its well-defined frequency range.⁷¹ In general, as the quantum processors scale up and fabrication improves, asymmetric transmons are likely to be found in wider applications in the future.

We see that split transmon qubits, be it symmetric or not, still share the same topology as the single junction version, yielding a sinusoidal potential. Therefore, the degree to which the properties of these qubits can be engineered has not fundamentally changed. In particular, the limited anharmonicity in transmon-type qubits intrinsically causes significant residual excitation to higher-energy states, undermining the performance of gate operations. To go beyond this, it is necessary to introduce additional complexity into the circuit.

One outstanding development in this regard is the invention of the flux qubit,^72,73 where the qubit loop is interrupted by three (or four) junctions, see Fig. 2(e). On one branch is one smaller junction; on the other branch are two identical junctions, both a factor γ larger in size compared to the small junction. The addition of one more junction as compared to the split transmon is nontrivial, as it changes the circuit topology and reshapes the potential energy profile.

Each junction is associated with a phase variable, and the fluxoid quantization condition again allows us to eliminate one degree of freedom. Consequently, we have a two-dimensional potential landscape, which in comparison to the simpler topology of the transmon, complicates the problem both conceptually and computationally. Fortunately, under the assumed setting that the array junctions are larger in size (γ > 1), it is usually a good approximation to treat the problem as a particle moving in a quasi-1D potential, which also helps us gain more insight and intuition about the system and draw qualitative conclusions. The Hamiltonian under this “quasi-1D approximation” reads

Note that the phase variable in Eq. (23) is the sum of the branch phases across the two array junctions, ϕ = (φ₁ + φ₂)/2, assuming the same current direction across φ₁ and φ₂. The external magnetic flux is denoted $φe=2πΦext/Φ0$⁠. The second term in Eq. (23) is contributed by the small junction with Josephson energy E_J, whereas the third term takes into account the two array junctions, together with Josephson energy 2γE_J. Clearly, the sum of these two terms no longer has the characteristics of a simple cosinusoid, and the final potential profile as well as the corresponding eigenstates depend on both the external flux φ_e and the junction area ratio γ.

The most common working point for this system is when $φe=π+2πk$⁠, where k is an integer—that is when half a superconducting flux quantum threads the qubit loop. At this flux bias point, the qubit spectrum reaches its minimum, and the qubit frequency is first-order insensitive to flux noise, see Fig. 2(f). This point is often referred to as “the flux degeneracy point,” where flux qubits tend to have the optimal coherence time.

At this operation point, the potential energy may assume a single-well (⁠$γ≥2$⁠) or a double-well (γ < 2) profile. The single-well case shares some similarities with the transmon qubit, where the quadratic and quartic terms of the Hamiltonian determines the harmonicity and anharmonicity, respectively. The capacitively shunted flux qubit (CSFQ)^62,74 was explored in this regime, demonstrating long coherence and decently high anharmonicity. Note that as opposed to the transmon qubit, the anharmonicity of the CSFQ is “positive” (α > 0). While the improvement in anharmonicity can be associated with reshaping the energy potential, the improved coherence over the first flux qubits can be attributed to the introduction of the capacitive shunt, similar to the modified Cooper-pair box leading to the transmon qubit.

The double-well case obtained for γ < 2 was demonstrated and investigated much earlier.^72,73 The intuitive picture based on circulating current states—so it gets the name persisting-current flux qubit (PCFQ)—gives a satisfying physical description of the qubit degrees of freedom. However, from the perspective of a quantum engineer, the qubit properties are of more interest, even if sometimes we may lose physical intuition about the system in certain regimes; such as when γ ≈ 2 and there are no clear circulating current states. The most important feature of the PCFQ is that its anharmonicity can be much greater than the transmon and CSFQ and the transition matrix elements $|⟨1|n̂|0⟩|,|⟨1|ϕ̂|0⟩|$ become considerably smaller given equivalent E_J/E_C. Therefore, a longer relaxation time can be expected. These features have been demonstrated even more prominently in its close relative, the fluxonium qubit.⁷⁵ 

The flux qubit is a striking example that illustrates how one dramatically can engineer the qubit properties through the choice of various circuit parameters. The introduction of array junctions and consequent biharmonic profile generates rich dynamics as well as broad applications. An extention of this idea is the fluxonium qubit, which generated substantial interest recently, due partly to its capability of engineering the transition matrix elements to achieve millisecond T₁ time, and due partly to the invention of novel gate schemes applicable to such well-protected qubits.^76,77

Compared to flux qubits, which usually contain two or three array junctions,⁷⁸ the number of array junctions in the fluxonium qubit is dramatically increased,^75,79 in some cases, to the order of 100, see Fig. 2(g). Following the same quasi-1D approximation as for the flux qubit, the last term in Eq. (23) becomes $−NγEJ cos (ϕ/N)$⁠, where N denotes the number of array junctions. For large N, the argument in the cosine term ϕ/N becomes sufficiently small that a second order expansion is a good approximation. This results in the fluxonium Hamiltonian

where E_L = (γ/N)E_J is the inductive energy of the effective inductance contributed by the junction array—often known as superinductance due to its large value.^79–81 Therefore, we can treat the potential energy as a quadratic term modulated by a sinusoidal term, similar to that of an rf-SQUID type flux qubit.⁸² However, the kinetic inductance of the Josephson junction array is in general much larger than the geometric inductance of the wire in an rf-SQUID.

Depending on the relative magnitude of E_J and E_L, the fluxonium system could involve plasmon states (in the same well) and fluxon states (in different wells). There are a variety of schemes to utilize them for quantum information processing. Generally, the spectrum of the transition between the lowest energy states is similar to that of the flux qubit, see Fig. 2(h). Both long coherence and high anharmonicity can be expected at the flux sweet spot.

Lastly, we want to point out a further extension—the 0–π qubit—which has even stronger topological protection from noise.^83,84 However, the strongly suppressed sensitivity to external fluctuations also makes it hard to manipulate.

To generate entanglement between individual quantum systems—it is necessary to engineer an interaction Hamiltonian that connects degrees of freedom in those individual systems. In this section, we discuss the physical coupling mechanism and its representation in the qubit eigenbasis. The use of coupling to form 2-qubit gates is discussed in Sec. IV.

The Hamiltonian of two coupled systems takes a generic form

where H₁ and H₂ denote the Hamiltonians of the individual quantum systems, which could be any combination of the qubit circuits mentioned in Secs. II A and II B. The last term, H_int, is the interaction Hamiltonian, which couples the variables of both systems. In superconducting circuits, the physical form of the coupling energy is either an electric or magnetic field (or a combination thereof).

To achieve capacitive coupling, a capacitor is placed between the voltage nodes of the two participating circuits, yielding an interaction Hamiltonian H_int of the form

where C_g is the coupling capacitance and V₁(V₂) is the voltage operator of the corresponding voltage node being connected. Figure 3(a) illustrates a realistic example of a direct capacitive coupling between the top nodes of two transmon qubits. Circuit quantization in the limit of $Cg≪C1,C2$ yields

where the expressions in brackets are the two Hamiltonians of the individual qubits, [see Eq. (16)], and we take $Vi=(2e/Ci)ni$ in Eq. (26). From Eq. (27), we see that the coupling energy depends on the coupling capacitance as well as the matrix elements of the voltage operators. The dependencies are bilinear in the perturbative limit (⁠$Cg≪C1,C2$⁠).

To implement the coupling capacitance, one only need bring the edges of the capacitor pads into close proximity, as has been demonstrated in-state-of-the-art planar designs.⁸⁵ The coupling capacitance is determined by the planar capacitor geometry as well as the surrounding environment, such as the dielectric constant of the substrate and the ground plane proximity.

In the case of inductive coupling, a mutual inductance shared by two loops is the coupling mechanism, yielding an interaction Hamiltonian of the form

where M₁₂ denotes the mutual inductance and $I1(I2)$ is the current-operator of the loop current. A typical example is two closely positioned (rf-SQUID type) flux qubits, as illustrated in Fig. 3(c). The system Hamiltonian can be expressed as

where the individual qubit Hamiltonians are identical to that of the fluxonium in Eq. (24), and the current operators, $Ii=Ici sin (ϕi)$ with $i∈1,2$⁠, is the familiar DC-Josephson relation for each junction, see Eq. (15). In this case, the strength of the inductive coupling energy depends on the mutual inductance as well as the matrix element of the current operators.

To realize a mutual inductance, two looped circuits are brought into close proximity to one another, or, to make them stronger, overlapping with each other,⁸⁶ and even may share the same wire or Josephson junction inductor.^87–90 In the case of a Josephson junction, and for certain metals, the inductance is dominated by “kinetic inductance” contributions, rather than solely geometric inductance.^91,92 Kinetic inductance arises from the mechanical, inertial mass of the charge carriers, but is only practically witnessed in very high-conductance materials like superconductors. A primary feature of kinetic inductance is that its values can vastly exceed those of conventional geometric inductances, which are generally limited by electromagnetic considerations.⁷⁹ 

Regardless of its physical realization, the effect of a coupling on system dynamics is determined by its form as represented in the eigenbasis of the individual systems. That is, how H_int appears in the representation spanned by the eigenbasis of H₁ ⊗ H₂.

Let us start with the previous example of two capacitively coupled transmon qubits [Fig. 3(a)]. Using second quantization, the system Hamiltonian in Eq. (27) can be expressed as

where the expression within brackets represent the Duffing oscillator Hamiltonian for the qubits and g is the coupling energy. Since we define $V∝n∝i(a−a†)$⁠, and consequently $I∝ϕ∝(a+a†)$⁠, the original $n1n2$-term becomes what is shown in Eq. (30). Such a coupling is called “transverse,” because the coupling Hamiltonian has nonzero matrix elements only at off-diagonal positions with respect to both oscillators, i.e., $i⟨k|ai−ai†|k⟩i=0$ for any integer k and for $i∈1,2$⁠, and in this case, $i⟨k±1|ai−ai†|k⟩i≠0$⁠.

If we can ignore higher energy levels (⁠$k≥2$⁠) either because of sufficient anharmonicity or through careful control protocols that ensure these levels never have influence, we may truncate the Hamiltonian in Eq. (30) to

This is a Hamiltonian of two spins, coupled by an exchange interaction. As we will see in Sec. IV D 1, such a Hamiltonian is most commonly used in contemporary implementations and can generate various types of two-qubit entangling gates. Note that, more often, we see that the interaction term is expressed in $σxσx$ instead of $σyσy$⁠. The choice in the context here is arbitrary and does not change the dynamics. However, when both capacitive and inductive couplings are present in the system, both $σxσx$ and $σyσy$ may be needed. In this case, the voltage operator $V∝i(a−a†)$ (reduced to σ_y after two-level approximation in the lab frame) is transversal to the current operator $I∝(a+a†)$ (reduced to σ_x) and both of them may be transverse to the qubit. A similar example is demonstrated between a qubit and a resonator by Lu et al.⁹³ 

Transverse coupling can be engineered between a qubit and a harmonic oscillator, see Fig. 3(b). In this case, the Hamiltonian becomes

where ω_q and ω_r denote the qubit and resonator frequencies, and $σ+=|0⟩⟨1|$ and $σ−=|1⟩⟨0|$ describes the processes of exciting and de-exciting the qubit, respectively. Here, we have assumed that the coupling is in the dispersive limit, i.e., $g≪ωq,ωr$⁠, hence ignoring the double (de)excitation terms proportional to $σ+a†$ and $σ−a$⁠, which under typical operation regimes oscillate sufficiently fast to average to zero. The Hamiltonian in Eq. (32), is the standard model used for describing how a two-level atom interacts with a resonant cavity that houses it. Such a structure is also known as cavity quantum electrodynamics (cQED), and it is extended to the circuit version here. It has many useful applications in superconducting quantum information architectures, such as high-fidelity readout,⁹⁴ see Sec. V, cavity buses,⁹⁵ quantum memory,^96,97 quantum computation with cat states,^98–100 etc.

Here, we briefly mention the use of a cavity or resonator to mediate coupling between two qubits, which may be physically well-separated (≈1 cm). Since most superconducting resonators are in the GHz frequency range, they can be made much longer than any dimension of a qubit circuit (≈1 mm). One can use such a resonator to mediate coupling between two or more otherwise noninteracting qubits. An example is shown in Fig. 3(b), where two transmon qubits are both capacitively coupled to the center resonator. The two-level system Hamiltonian is:

It can be shown that in the dispersive limit, i.e., $gir≪|ωi−ωr|$⁠, the resonator can—after proper transformation and approximation—be treated as an isolated system, and the composite system simplified to two transversely coupled qubits, see Eq. (31).

We now turn to the previous example of two inductively coupled flux qubits, see Fig. 3(c). Assume that the double-well potential [Fig. 2(g)] has a relatively high interwell barrier, which leads to an exponentially small qubit transition frequency at the energy degeneracy point, (Φ_e = π). Around this degeneracy point, the off-diagonal matrix element of $sin (ϕ)$ is zero, i.e., the ground and excited states are localized in different wells and $⟨1| sin (ϕ)|1⟩−⟨0| sin (ϕ)|0⟩≠0$⁠. We can then rewrite the Hamiltonian in Eq. (29) as

Now, the coupling axis is the same as the qubit quantization axes and therefore termed “longitudinal coupling.” Note, however, that the physical $σxσx$ and $σzσz$ couplings can change in the qubit frame.

Longitudinal coupling is an important type of interaction, because it can generate entanglement without energy exchange. Moreover, it is found a necessary ingredient in the application of quantum annealing, where certain hard combinatorial optimization problems can be modeled by the Ising Hamiltonian in Eq. (34) and finding its ground state would solve this problem.

An intermediate qubit mode may also be used as a coupler in the longitudinal case. In Fig. 3(d), an additional rf-SQUID is used to mediate the coupling. The coupling strength can be tuned by the flux bias of the coupler SQUID.¹⁰¹ Note that a tunable coupler may also be realized in a structure with capacitive couplings.⁶³ A tunable coupler is useful because it provides a wide range of coupling strengths,¹⁰² a high on-off ratio¹⁰³ for reducing gate error-rates, and more ways of achieving high-fidelity entangling gates.^67,104–106 The trade-off is an additional control line.

In addition to the pure transversal and longitudinal qubit-qubit interactions, there are also examples of mixed types of interaction Hamiltonians¹⁰⁷ 

which are longitudinal with respect to a qubit, but transverse with respect to a harmonic oscillator in a qubit-resonator system. Such a model is called longitudinal but one should note that it is only longitudinal to one participating system. It is hard to engineer physically longitudinal coupling with respect to a harmonic oscillator, since either the E-field (V) or the B-field (I) is transverse with respect to the eigen field of the harmonic oscillator. Note, however, that a transversal model such as in Eq. (32) may be transformed into a longitudinal one in certain operating regimes, see Sec. V.

In some applications, such as for quantum annealing, both longitudinal and transverse couplings are desired (σ_zσ_z coupling for mapping the problem and σ_xσ_x coupling for enhancing the annealing performance) and require independent control.

Random, uncontrollable physical processes in the qubit control and measurement equipment or in the local environment surrounding the quantum processor are sources of noise that lead to decoherence and reduce the operational fidelity of the qubits. In this section, we introduce the basics of noise leading to decoherence in superconducting circuits, and we discuss coherent control methods to mitigate certain types of noise.

In a closed system, the dynamical evolution of a qubit state is deterministic. That is, if we know the starting state of the qubit and its Hamiltonian, then we can predict the state of the qubit at any time in the future. However, in open systems, the situation changes. The qubit now interacts with uncontrolled degrees of freedom in its environment, which we refer to as fluctuations or noise. In the presence of noise, as time progresses, the qubit state looks less and less like the state we would have predicted and, eventually, the state is lost. There are many different sources of noise that affect quantum systems, and they can be categorized into two primary types: systematic noise and stochastic noise.

Systematic noise arises from a process that is traceable to a fixed control or readout error. For example, we apply a microwave pulse to the qubit that we believe will impart a 180° rotation. However, the control field is not tuned properly and, rather than rotating the qubit 180°, the pulse slightly over-rotates or under-rotates the qubit by a fixed amount. The underlying error is “systematic,” and it therefore leads to the same rotation error each time it is applied. However, when such erroneous pulses are used in practice in a variety of control sequences, the observed results may appear to be influenced by random noise. This is because the pulse is generally not applied in the same way for each experiment: it could be applied a different number of times, interspersed with different pulses in different orders, and therefore generally differs from experiment to experiment. However, once systematic errors are identified, they can generally be corrected through proper calibration or the use of improved hardware.

The second type of noise is stochastic noise, arising from random fluctuations of parameters that are coupled to our qubit.¹⁰⁸ For example, thermal noise of a 50 Ω resistor in the control lines leading to the qubit will have voltage and current fluctuations—Johnson noise—with a noise power that is proportional to both temperature and bandwidth. Or, the oscillator that provides the carrier for a qubit control pulse may have amplitude or phase fluctuations. Additionally, randomly fluctuating electric and magnetic fields in the local qubit environment—e.g., on the metal surface, on the substrate surface, at the metal-substrate interface, or inside the substrate—can couple to the qubit. This creates unknown and uncontrolled fluctuations of one or more qubit parameters, and this leads to qubit decoherence.

The degree to which a qubit is affected by noise is related to the amount of noise impinging on the qubit, and the qubit's susceptibility to that noise. The former is often a question of materials science and fabrication; that is, can we make devices with lower levels of noise. Or, it may be related to the quality of the control electronics and cryogenic engineering to limit the levels of noise on the control lines that necessarily connect to the qubits to control them. The latter—qubit susceptibility—is a question of qubit design. Qubits can be designed to trade-off sensitivity to one type of noise at the expense of increased sensitivity to other types of noise. Thus, materials science, fabrication engineering, electronics design, cryogenic engineering, and qubit design all play a role in creating devices with high coherence. In general, one should strive to eliminate the sources of noise, and then design qubits that are insensitive to the residual noise.

The qubit response to noise depends on how the noise couples to it—either through a longitudinal or a transverse coupling as referenced to the qubit quantization axis. This can be visualized using a Bloch Sphere picture of the qubit state, as illustrated in Fig. 4 and discussed in detail in Sec. III B.

The “Bloch sphere” is a unit sphere used to represent the quantum state of a two-level system (qubit). Figure 4(a) shows a Bloch sphere with a “Bloch vector” representing the state $|ψ⟩=α |0⟩+β |1⟩$⁠. If we visualize the Bloch sphere as the planet Earth, then by convention, the north pole represents state $|0⟩$ and the south pole state $|1⟩$⁠. For pure quantum states such as $|ψ⟩$⁠, the Bloch vector is of unit length, $|α|2+|β|2=1$⁠, connecting the center of the sphere to any point on its surface.

The z-axis connects the north and south poles. It is called the “longitudinal axis,” since it represents the “qubit quantization axis” for the states $|0⟩$ and $|1⟩$ in the qubit eigenbasis. In turn, the x–y plane is the “transverse plane” with “transverse axes” x and y. In this Cartesian coordinate system, the unit Bloch vector $a→=(sin θ cos ϕ, sin θ sin ϕ,cos θ)$ is represented using the polar angle $0≤θ≤π$ and the azimuthal angle $0≤ϕ<2π$⁠, as illustrated in Fig. 4(a). Following our convention, state $|0⟩$ at the north pole is associated with +1, and state $|1⟩$ (the south pole) with –1. We can similarly represent the quantum state using the angles θ and ϕ,

The Bloch vector is stationary on the Bloch sphere in the “rotating frame picture.” If state $|1⟩$ has a higher energy than state $|0⟩$ (as it generally does in superconducting qubits), then in a stationary frame, the Bloch vector would precess around the z-axis at the qubit frequency $(E1−E0)/ℏ$⁠. Without loss of generality (and much easier to visualize), we instead “choose” to view the Bloch sphere in a reference frame where the x and y-axes also rotate around the z-axis at the qubit frequency. In this “rotating frame,” the Bloch vector appears stationary as written in Eq. (36). The rotating frame will be described in detail in Sec. IV D 1 in the context of single-qubit gates.

For completeness, we note that the density matrix $ρ=|ψ⟩⟨ψ|$ for a pure state $|ψ⟩$ is equivalently

where I is the identity matrix, and $σ→=[σx,σy,σz]$ is a vector of Pauli matrices. If the Bloch vector $a→$ is a unit vector, then ρ represents a pure state ψ and Tr(ρ²) = 1. More generally, the Bloch sphere can be used to represent “mixed states,” for which $|a→|<1$⁠; in this case, the Bloch vector terminates at points “inside” the unit sphere, and $0≤Tr(ρ2)<1$⁠. To summarize, the surface of the unit sphere represents pure states, and its interior represents mixed states.⁶ 

Within the standard Bloch-Redfield^109–111 picture of two-level system dynamics, noise sources weakly coupled to the qubits have short correlation times with respect to the system dynamics. In this case, the relaxation processes are characterized by two rates (see Fig. 4),

which contains the pure dephasing rate Γ_φ. We note that the definition of Γ₂ as a sum of rates presumes that the individual decay functions are exponential, which occurs for Lorentzian noise spectra (centered at ω = 0) such as white noise (short correlation times) with a high-frequency cutoff.

The impact of noise on the qubit can be visualized on the Bloch sphere in Fig. 4(a). For an initial state (t = 0)

the Bloch-Redfield density matrix ρ_BR for the qubit is written^112,113

There are a few important distinctions between Eqs. (43) and (39), which we list here and then describe in more detail in Secs. III B 2 a–III B 2 c.

• First, we have introduced the “longitudinal decay function” $exp (−Γ1t)$⁠, which accounts for longitudinal relaxation of the qubit.

• Second, we introduced the “transverse decay function” $exp (−Γ2t)$⁠, which accounts for transverse decay of the qubit.

• Third, we have introduced an explicit phase accrual $exp (iδωt)$⁠, where $δω=ωq−ωd$⁠, which generalizes the Bloch sphere picture to account for cases where the qubit frequency ω_q differs from the rotating-frame frequency ω_d, as we will see later when discussing measurements of T₂ using Ramsey interferometry,^114,115 and in Sec. IV D 1, in the context of single-qubit gates.

• Fourth, we have constructed the matrix such that for $t≫(T1, T2)$⁠, the upper-left matrix element will approach a unit value, indicating that all populations relax to the ground state, while the other three matrix elements decay to zero. This is related to the assumption that the environmental temperature is low enough that thermal excitations of the qubit from the ground to the excited state rarely occur.

The longitudinal relaxation rate Γ₁ describes depolarization along the qubit quantization axis, often referred to as “energy decay” or “energy relaxation.” In this language, a qubit with polarization p = 1 is entirely in the ground state $(|0⟩)$ at the north pole, p = –1 is entirely in the excited state $(|1⟩)$ at the south pole, and p = 0 is a completely depolarized mixed state at the center of the Bloch sphere.

As illustrated in Fig. 4(b), longitudinal relaxation is caused by “transverse noise,” via the x- or y-axis, with the intuition that off-diagonal elements of an interaction Hamiltonian are needed to connect and drive transitions between states $|0⟩$ and $|1⟩$⁠.

Depolarization occurs due to energy exchange with an environment, generally leading to both an “up transition rate” $Γ1↑$ (excitation from $|0⟩$ to $|1⟩$⁠), and a “down transition rate” $Γ1↓$ (relaxation from $|1⟩$ to $|0⟩$⁠). Together, these form the longitudinal relaxation rate Γ₁

T₁ is the 1/e decay time in the exponential decay function in Eq. (43), and it is the characteristic time scale over which the qubit population will relax to its steady-state value. For superconducting qubits, this steady-state value is generally the ground state, due to Boltzmann statistics and typical operating conditions. Boltzmann equilibrium statistics lead to the “detailed balance” relationship $Γ1↑=exp (−ℏωq/kBT)Γ1↓$⁠, where T is the temperature and $kB$ is the Boltzmann constant, with an equilibrium qubit polarization approaching $p=tanh(ℏωq/2kBT)$⁠. Typical qubits are designed at frequency $ωq/2π≈5$ GHz and are operated at dilution refrigerator temperatures T ≈ 20 mK. In this limit, the up-rate $Γ1↑$ is exponentially suppressed by the Boltzmann factor $exp (−ℏωq/kBT)$⁠, and so only the down-rate $Γ1↓$ contributes significantly, relaxing the population to the ground state. Thus, qubits generally spontaneously lose energy to their cold environment, but the environment rarely introduces a qubit excitation. As a result, the equilibrium polarization approaches unity [see Eq. (43)].^116,117

Only noise at the qubit frequency mediates qubit transitions, whether absorption or emission, and this noise is generally “well behaved” (short correlation time, many modes weakly coupled to qubit, no divergences) around the qubit frequency for superconducting qubits. The intuition is that qubit-transition linewidths are relatively narrow in frequency, and so the noise generally does not vary much over this narrow frequency range. Although there are a few notable exceptions, for example, qubit decay in the presence of hot quasiparticles,^118–120 which can lead to nonexponential decay functions, longitudinal depolarization measurements generally present exponential decay functions consistent with the Bloch-Redfield picture.

An example of a T₁ measurement is shown in Fig. 5(a). The qubit is prepared in its excited state using an $Xπ$-pulse, and then left to spontaneously decay to the ground state for a time τ, after which the qubit is measured. A single measurement will project the quantum state into either state $|0⟩$ or state $|1⟩$⁠, with probabilities that correspond to the qubit polarization. To make an estimate of this polarization, one needs to identically prepare the qubit and repeat the experiment many times. This is analogous to flipping a coin: any single flip will yield heads or tails, but the probability of obtaining a heads or tails can be estimated by flipping the coin many times and taking the ensemble average. The resulting exponential decay has a characteristic time T₁ = 85 μs.

The “pure dephasing” rate Γ_ϕ describes depolarization in the x–y plane of the Bloch sphere. It is referred to as “pure dephasing,” to distinguish it from other phase-breaking processes such as energy excitation or relaxation.

As illustrated in Fig. 4(c), pure dephasing is caused by “longitudinal noise” that couples to the qubit via the z-axis. Such longitudinal noise causes the qubit frequency ω_q to fluctuate, such that it is no longer equal to the rotating frame frequency ω_d, and causes the Bloch vector to precess forward or backward in the rotating frame. Intuitively, we can imagine identically preparing several instances of the Bloch vector along the x-axis. For each instance, the stochastic fluctuations of qubit frequency will result in a different precession frequency, resulting in a net fanout of the Bloch vector in the x–y plane. This eventually leads to a complete depolarization of the azimuthal angle ϕ. Note that this stochastic effect will be captured in the transverse relaxation rate Γ₂ (Sec. III B 2 c); it is “not” the deterministic term $exp (±iδωt)$ that appears in Eq. (43), which represents intentional detuning of the qubit reference frame.

There are a few important distinctions between pure dephasing and energy relaxation. First, in contrast to energy relaxation, pure dephasing is not a resonant phenomenon; noise at any frequency can modify the qubit frequency and cause dephasing. Thus, qubit dephasing is subject to broadband noise. Second, since pure dephasing is elastic (there is no energy exchange with the environment), it is in principle “reversible.” That is, the dephasing can be “undone”—with quantum information being preserved—through the application of unitary operations, e.g., dynamical decoupling pulses,⁷⁸ see Sec. III D 2.

The degree to which the quantum information can be retained depends on many factors, including the bandwidth of the noise, the rate of dephasing, the rate at which unitary operations can be performed, etc. This should be contrasted with spontaneous energy relaxation, which is an “irreversible” process. Intuitively, once the qubit emits energy to the environment and its myriad uncontrollable modes, the quantum information is essentially lost with no hope for its recovery and reconstitution back into the qubit.

The transverse relaxation rate $Γ2=Γ1/2+Γφ$ describes the loss of coherence of a superposition state, for example $(1/2)(|0⟩+|1⟩)$⁠, pointed along the x-axis on the equator of the Bloch sphere as illustrated in Fig. 4(d). Decoherence is caused in part by longitudinal noise, which fluctuates the qubit frequency and leads to pure dephasing Γ_φ (red). It is also caused by transverse noise, which leads to energy relaxation of the excited-state component of the superposition state at a rate Γ₁ (blue). Such a relaxation event is also a phase-breaking process, because once it occurs, the Bloch vector points to the north pole, $|0⟩$⁠, and there is no longer any knowledge of which direction the Bloch vector had been pointing along the equator; the relative phase of the superposition state is lost.

Transverse relaxation T₂ can be measured using Ramsey interferometry, as shown and described in Fig. 5(b). The protocol positions the Bloch vector on the equator using a $Xπ/2$-pulse. Typically, the carrier frequency of this pulse is slightly detuned from the qubit frequency by an amount δω. As a result, the Bloch vector will precess around the z-axis at a rate δω. This is done for convenience sake, so that the resulting Ramsey measurement will oscillate, making it easier to analyze. After precessing for a time τ, a second $Xπ/2$-pulse projects the Bloch vector back on to the z-axis. Repeated measurements are made to take an ensemble averaged estimate of the qubit polarization, as a function of τ. The resulting oscillations in Fig. 5(b) feature an approximately exponential decay function with time $T2*=98 μs$⁠. The “*” indicates that the Ramsey experiment is sensitive to “inhomogeneous broadening.” That is, it is highly sensitive to quasistatic, low-frequency fluctuations that are constant within one experimental trial, but vary from trial to trial, e.g., due to 1/f-type noise. This sensitivity to quasistatic noise is related to the corresponding N = 0 noise filter function shown in Fig. 5(d) being centered at zero-frequency, as described in more detail in Sec. III D 2.

The Hahn echo shown in Fig. 5(c) is an experiment that is less sensitive to quasistatic noise. By placing a Y _π pulse at the center of a Ramsey interferometry experiment, the quasistatic contributions to dephasing can be “refocused,” leaving an estimate T_2E that is less sensitive to inhomogeneous broadening mechanisms. The pulses are generally chosen to be resonant with the qubit transition for a Hahn echo, since any frequency detuning would be nominally refocused anyway. The resulting decay function in Fig. 5(c) is essentially exponential with time $T2E=120 μs$⁠.

With the known T₁ and T₂ times, one can infer the pure dephasing time T_φ from Eq. (41), provided the decay functions are exponential. In superconducting qubits, however, the broadband dephasing noise (e.g., flux noise, charge noise, critical-current noise, …) tends to exhibit a 1/f-like power spectrum. Such noise is singular near ω = 0, has long correlation times, and generally does not fall within the Bloch-Redfield description. The decay function of the off-diagonal terms in Eq. (43) is generally nonexponential, and for such cases, the simple expression in Eq. (41) is not applicable.

If we assume that the qubit is coupled to many independent fluctuators, then, regardless of their individual statistics, they will in concert generate noise with a Gaussian distribution due to the central limit theorem. We therefore say that the longitudinal fluctuations exhibit Gaussian-distributed 1/f noise. For 1/f noise spectra, the phase decay function is itself a Gaussian $exp [−(t/Tφ,Gt)2]$⁠, where we write $Tφ,G$ to distinguish it from T_φ used in Eq. (41). Furthermore, this function is separable from the T₁-type exponential decay, because the T₁-noise remains regular at the qubit frequency. The density matrix in Eq. (43) becomes, following Refs. 78 and 112,

where the decay function $⟨ exp (−χN(t))⟩$ contains the “coherence function” $χN(t)$⁠, which generalizes pure dephasing to include nonexponential decay functions. As we shall see later, the subscript N labeling the decay function refers to the number of π-pulses used to refocus the low-frequency noise, which impacts the form of the decay function. Because the function is no longer purely exponential, we cannot formally write the transverse relaxation decay function as $exp (−t/T2)$⁠. However, an exponential decay remains a practically reasonable approximation for $Tφ≳T1$⁠. We also note that the energy decay component of the transverse relaxation is $exp (−t/2T1)$⁠, and so T₂ can never be larger than 2T₁. In the absence of pure dephasing, the maximum T₂ = 2T₁ is reached.

As an example, consider the Ramsey interferometry data in Fig. 5(b). Since the dephasing is relatively weak, the transverse relaxation function as $exp (−t/T2)$ is a reasonable fit and yields T₂ = 95 μs. However, using the value T₁ = 85 μs from Fig. 5(a) and dividing out $exp (−t/2T1)$ from the data in Fig. 5(b), the remaining pure dephasing decay function is shown in Fig. 5(d) and assumes a Gaussian envelope $⟨ exp (−χN(t))⟩=exp [−(t/Tφ,Gt)2]$⁠, with $Tφ,G=98 μs$⁠. The Hahn echo data in Fig. 5(c) may be treated similarly.

For completeness, in addition to 1/f dephasing mechanisms, we note that there are also “white” pure dephasing mechanisms, which give rise to an exponential decay function for the dephasing component of T₂. One common example is dephasing due to the shot noise of residual photons in the readout resonator coupled to superconducting qubits, as we discuss in Sec. III C 3.

The frequency distribution of the noise power for a stationary noise source λ is characterized by its PSD $Sλ(ω)$

The Wiener-Khintchine theorem states that the PSD is the Fourier transform of the autocorrelation function $cλ(τ)=⟨λ(τ)λ(0)⟩$ of the noise source λ. Since the integration limits are $(−∞,∞)$⁠, this is the bilateral PSD. Symmetrizing the PSD allows one to consider only positive frequencies, which is termed a unilateral PSD. Both unilateral and bilateral PSDs are used, often with the same notation, and so one needs to know how the PSD is defined, to keep track of the factors 2 and π, and also be aware of the implications for quantum systems.

For classical systems, the noise power spectral density is symmetric. This is because the autocorrelation function of real signals is itself a real function, and the Fourier transform of a real temporal function is symmetric in the frequency domain. Dephasing noise is caused by real, fluctuating fields, and so its PSD is generally symmetric. Examples of such classical noise include thermal (Johnson) noise and 1/f noise¹²² (see Fig. 6).

In turn, the inverse Fourier transform of the PSD will yield the autocorrelation function

This implies that integrating the noise power spectral density with τ = 0 yields the second moment of the noise, or, for zero-mean fluctuations, the variance.

However, the autocorrelation function for a quantum system may be complex-valued due to the fact that quantum operators generally do not commute at different times. This means that time-ordering of the operators matters, and the PSD need not be symmetric in frequency. This is generally the case for transverse noise causing longitudinal energy relaxation. Noise at a positive frequency $S(ωq)$ corresponds to energy transfer from the qubit to the environment, including both stimulated and spontaneous emission, associated with the down-rate $Γ1↓$⁠. Noise at a negative frequency $S(−ωq)$ corresponds to energy transfer to the qubit from the environment, associated with the up-rate $Γ1↑$⁠. This energy transfer becomes exponentially suppressed when the qubit frequency is larger than thermal energy (k_B T), as shown in Fig. 6. For a detailed discussion, see Refs. 123 and 124. Spontaneous emission to a cold environment or electromagnetic vacuum, represented by Nyquist noise in Fig. 6, is an example of an asymmetric noise PSD.¹²¹ 

In general, making a connection between $Sλ(ω)$ and the measured qubit decay functions is the basis for noise spectroscopy up to second-order statistics.^78,125–128 The search for higher-order spectra related to non-Gaussian noise is a current topic of active research.¹²⁹ 

There are many sources of stochastic noise in superconducting qubits, and we refer the reader to Ref. 40 for a review. Here, we briefly present several of the most common types of noise, their effect on coherence, and refer the reader to the references for a more detailed discussion.

“Charge noise” is ubiquitous in solid-state devices. It arises from charged fluctuators present in the defects or charge traps that reside in interfacial dielectrics, the junction tunnel barrier, and in the substrate itself. These are often modeled as an ensemble of fluctuating two-level systems or as bulk dielectric loss.^130,131 For example, in the case of a transmon qubit, the electric field between the capacitor plates traverses and couples to dielectric defects residing on the metal surfaces of the plates (for lateral-plate-type capacitors) or the capacitor dielectric between the plates (for parallel-plate-type capacitors). The electric field variable is transverse with respect to the quantization axis of the transmon qubit, which means that this noise is mainly responsible for energy relaxation (T₁). Additionally, if the E_J/E_C ratio of the transmon is not made sufficiently large (smaller than around 60), the qubit frequency itself will also be sensitive to broadband charge fluctuations. In this case, low-frequency charge noise couples longitudinally to the transmon and causes pure dephasing (T_φ).

Charge noise is modeled primarily as a combination of inverse-frequency noise and Nyquist noise, also referred to as “ohmic” noise. At lower frequencies, the spectral density takes the form

with quasiuniversal values $AQ2=(10−3e)2/Hz$ at 1 Hz, and $γQ≈1$⁠. In addition to large 1/f fluctuations, early charge qubits often witnessed discrete, charge offsets reminiscent of random telegraph noise. Together, these two mechanisms severely limited the utility of charge qubits, and served as a strong motivation to move to capacitively shunted charge qubits (transmons), which greatly reduced the qubit longitudinal sensitivity to charge noise. At higher frequencies, the power spectrum takes the form $SQ(ω)=BQ2[ω/(2π×1 Hz)]$⁠, where the noise strength $BQ2$ at 1 Hz can assume a range of values depending on the level of dissipation in the system. Likewise, the cross-over from 1/f-like behavior to f-like behavior generally occurs at around 1 GHz, but will vary higher or lower between samples depending on the degree of dissipation.^62,132

Another commonly observed noise in solid-state devices is magnetic “flux noise.” The origin of this noise is understood to arise from the stochastic flipping of spins (magnetic dipoles) that reside on the surfaces of the superconducting metals comprising the qubit,¹³³ resulting in random fluctuations of the effective magnetic field that biases flux-tunable qubits.

For example, in the case of the split transmon, the external magnetic field threading the loop couples longitudinally to the qubit and modulates the transition frequency via the Josephson energy E_J (except at $φe=0$⁠, where the qubit is first-order insensitive to magnetic-field fluctuations). Because the flux noise is longitudinal to the transmon, it contributes to pure dephasing (T_φ). However, in the case of the flux qubit, and depending on the flux-bias point, the flux noise may be either longitudinal—causing dephasing T_φ—or it may couple transversely and thus contribute to T₁ relaxation.^62,78 The noise power spectrum of these fluctuations generally exhibits a “quasiuniversal” dependence

with $γΦ≈0.8−1.0$ and $AΦ2≈(1 μΦ0)2/Hz$⁠, and has been shown to extend from less than millihertz to beyond gigahertz frequencies.^{78,127,128,134,135}

The large, low-frequency weighting of the 1/f power distribution enables the use of engineered error mitigation techniques—such as dynamical decoupling—to achieve better coherence^78,136,137 and for improving single and two-qubit gate fidelity.¹³⁸ It was recently demonstrated that 1/f flux noise is also a T₁-mechanism when extended out to the qubit frequency,⁶² and one similarly expects a crossover to ohmic flux noise at high enough frequencies.¹³⁹ 

Although much is known about the statistics and number of the defects presumed responsible for flux noise, their precise physical manifestation remains uncertain.^133,140 The fact that the 1/f noise is quasiuniversal and largely independent of device, strongly suggests a common origin for the noise. Recent studies suggest that adsorbed molecular oxygen may be responsible for flux-noise.^140,141

In the circuit QED architecture, resonator “photon number fluctuation” is another major decoherence source.¹⁴² Residual microwave fields in the cavity have photon-number fluctuations that in the dispersive regime impact the qubit through an interaction term $χσzn$⁠, see Sec. II C 2, leading to a frequency shift $ΔStark=2ηχn¯$⁠, where $n¯$ is the average photon number, and $η=κ2/(κ2+4χ2)$ effectively scales the photon population seen by the qubit due to the interplay between the qubit-induced dispersive shift of the resonator frequency (χ) and the resonator decay rate (κ).

In the dispersive limit, the noise is longitudinally coupled to the qubit and leads to pure dephasing at a rate

The fluctuations originate from residual photons in the resonator, typically due to radiation from higher temperature stages in the dilution refrigerator.^106,143 The corresponding noise spectral density is of Lorentzian type

which exhibits an essentially white noise spectrum up to a 3 dB cutoff frequency ω = κ set by the resonator decay rate κ, see Ref. 62.

“Quasiparticles,” i.e., unpaired electrons, are another important noise source for superconducting devices.¹¹⁹ The tunneling of quasiparticles through a qubit junction may lead to both T₁ relaxation and pure dephasing T_φ, depending on the type of qubit, the bias point, and the junction through which the tunneling event occurs.^118,120

Quasiparticles are naturally excited due to thermodynamics, and the quasiparticle density in equilibrium superconductors should be exponentially suppressed as the temperature decreases. However, below about 150 mK, the quasiparticle density observed in superconducting devices—generally in the range 10^–8–10^–6 per Cooper pair—is much higher than what the BCS theory would predict for a superconductor in equilibrium with its cryogenic environment at 10 mK. The reason for this excess quasiparticle population is unclear, but it is very likely related to the presence of additional, nonthermal mechanisms that increase the generation rates, “bottleneck effects” that occur at millikelvin temperatures to reduce recombination rates, or a combination of both.

It has been shown that the observed T₁ and excess excited-state population measured in today's state-of-the-art high-coherence transmon are self-consistent with excess “hot” nonequilibrium quasiparticles at the quasiuniversal density of around 10^–7–10^–6 per Cooper pair.^144,145 Although this quasiparticle generation mechanism is not yet well understood, it has been shown that quasiparticles can be transiently pumped away, improving T₁ times and reducing T₁ temporal variation.¹²⁰ 

Similar to the way two qubits are coupled, a qubit may couple and interact with uncontrolled degrees of freedom (DOF) in its environment (the noise sources). The interaction Hamiltonian between the qubit DOF (⁠$Ôq$⁠) and those of the noise source (⁠$λ̂$⁠) may be expressed in a general form

where ν denotes the coupling strength—which is related to the sensitivity of the qubit to environmental fluctuations $∂Ĥq/∂λ$—and we assume that $Ôq$ is a qubit operator within the qubit Hamiltonian $Ĥq$⁠. The noisy environment represented by the operator $λ̂$ produces fluctuations δλ. Note that we retained the hats in this section to remind us that these are quantum operators.

If the coupling is transverse to the qubit, e.g., $Ôq$ is of the type σ_x or $(a+a†)$—see the related case of qubit-qubit coupling treated in Sec. II C—then noise at the qubit frequency can cause transitions between the qubit eigenstates. Since this is a stochastic process, the ensemble-average manifests itself as a decay (usually exponential) of the qubit population toward a certain equilibrium value (usually the qubit ground state $|0⟩$ for $kBT≪ℏωq$⁠). Again, this process is equivalently referred to as “T₁ relaxation,” “energy relaxation,” or “longitudinal relaxation.” As stated above, T₁ is the characteristic time scale of the decay. Its inverse, Γ₁ = 1/T₁ is called the relaxation rate and depends on the power spectral density of the noise S(ω) at the transition frequency of the qubit ω = ω_q

where $∂Ĥq/∂λ$ is the qubit transverse susceptibility to fluctuations δλ, such that $|δλ|2$ is the ensemble average value of the environmental noise sources as seen by the qubit. Equation (53) is equivalent to Fermi's Golden Rule, in which the qubit's transverse susceptibility to noise is driven by the noise power spectral density. The qubit transverse susceptibility can be used to calculate the prefactors; for example, for fluctuations δλ = δn, the relevant term in the transmon Hamiltonian in Eq. (16) is $4EC(n̂−ng)2$⁠, where we allow for an offset charge n_g, and the susceptibility is given by $8ECn̂$⁠. We refer the reader to Refs. 40, 146, and 147 for more details.

If the coupling to the qubit is instead longitudinal, e.g., $Ĥq$ is of the type σ_z or $a†a$⁠, the noise will stochastically modulate the transition frequency of the qubit and thereby introduce a stochastic phase evolution of a qubit superposition state. This gradually leads to a loss of phase information, and it is therefore called pure dephasing (time constant T_φ). Unlike T₁ relaxation, which is generally an irreversible (incoherent) error, pure dephasing T_φ is in principle reversible (a coherent error). The degree of pure dephasing depends on the control pulse sequence applied while the qubit is subject to the noise process.

Consider the relative phase φ of a superposition state undergoing free evolution in the presence of noise. The superposition state's accumulated phase

diffuses due to adiabatic fluctuations of the transition frequency,

where $∂ωq/∂λ=(1/ℏ)|⟨∂Ĥq/∂λ⟩|$ is the qubit's longitudinal sensitivity to λ-noise. For noise generated by a large number of fluctuators that are weakly coupled to the qubit, its statistics are Gaussian. Ensemble averaging over all realizations of the Gaussian-distributed stochastic process $δλ(t)$⁠, the dephasing is

leading to a coherence decay function,

where $g(ω,τ)$ is a dimensionless weighting function.

The function $gN(ω,τ)$ can be viewed as a frequency-domain filter of the noise $Sλ(ω)$ [see Fig. 7(a)]. In general, its filter properties depend on the number N and distribution of applied pulses. For example, considering sequences of π-pulses,^78,148–152

where $δj∈[0,1]$ is the normalized position of the center of the jth π-pulse between the two π/2-pulses, τ is the total free-induction time, and τ_π is the length of each π-pulse,^151,152 yielding a total sequence length $τ+Nτπ$⁠. As the number of pulses increases for a fixed τ, the filter function's peak shifts to higher frequencies, leading to a reduction in the net integrated noise for 1/f^α-type noise spectra with α > 0. Similarly, for a fixed N, the filter function will shift in frequency with τ. Additionally, for a fixed time separation $τ′=τ/N$ (valid for $N≥1$⁠), the filter sharpens and asymptotically peaks at $ω′/2π=1/2τ′$ as more pulses are added. $gN(ω,τ)$ is thus called the “filter function,”^78,150 and it depends on the pulse sequences being applied. From Eq. (57), the pure dephasing decay arises from a noise spectral density that is “shaped” or “filtered” by the sequence-specific filter function. By choosing the number of pulses, their rotation axes, and their arrangement in time, we can design filter functions that minimize the net noise power for a given noise spectral density within the experimental constraints of the experiment (e.g., pulse-modulation bandwidth of the electronics used to control the qubits).

To give a standard example, we compare the coherence integral for two cases: a Ramsey pulse sequence and a Hahn echo pulse sequence. Both sequences involve two π/2 pulses separated by a time τ, during which free evolution of the qubit occurs in the presence of low-frequency dephasing noise. The distinction is that the Hahn echo will place a single π pulse (N = 1) in the middle of the free-evolution period, whereas the Ramsey does not use any additional pulses (N = 0). The resulting filter functions are

where the subscripts N = 0 and N = 1 indicate the number of π-pulses applied for the Ramsey and Hahn echo experiments, respectively. The filter function $g0(ω,τ)$ for the Ramsey case is a sinc-function centered at ω = 0. For noise that decreases with frequency, e.g., 1/f flux noise in superconducting qubits, the Ramsey experiment windows through the noise in S(ω) where it has its highest value. This is the worst choice of filter function for 1/f noise. In contrast, the Hahn echo filter function has a centroid that is peaked at a higher frequency, away from ω = 0. In fact, it has zero value at ω = 0. For noise that decreases with frequency, such as 1/f noise, this is advantageous. This concept extends to larger numbers N of π pulses, and is called a Carr-Purcell-Meiboom-Gill (CPMG) sequence.^153,154 In Fig. 7(b), the T₂ time of a qubit under the influence of strong dephasing noise is increased toward the 2T₁ limit using a CPMG dynamical error-suppression pulse sequence with an increasing number of pulses, N. We refer the reader to Refs. 78, 155, and 156, where these experiments were performed with superconducting qubits.

The qubit is highly sensitive to its noisy environment, and this feature can be used to map out the noise power spectral density. In general, one can map the noise PSD during “free evolution”—periods of time for which no control is applied to the qubit, except for very short dynamical decoupling pulses—and during “driven evolution”—periods of time during which the control fields are applied to the qubit. Both free-evolution and driven-evolution noise is important to characterize, as the noise PSD may differ for these two types of evolution, and both are utilized in the context of universal quantum computation. We refer the reader to Ref. 128 for a summary of noise spectroscopy during both types of evolution.

The Ramsey frequency itself is sensitive to longitudinal noise, and monitoring its fluctuations is one means to map out the noise spectral density over the submillihertz to ∼100 Hz range.^127,157

At higher frequencies, the CPMG dynamical decoupling sequence can be used to create narrow-band filters that “sample” the noise at different frequencies as a function of the free-evolution time τ and the number of pulses N. This has been used to map out the noise PSD in the range 0.1–300 MHz.⁷⁸ One must be careful of the additional small peaks at higher-frequencies, which all contribute to the dephasing used to perform the noise spectroscopy.¹⁵⁸ 

In fact, using pulse envelopes such as Slepians¹⁵⁹—which are designed to have a concentrated frequency response—to perform noise spectroscopy is one means to reduce such errors.¹⁵¹ 

At even higher frequencies, measurements of T₁ can be used in conjunction with Fermi's golden rule to map out the transverse noise spectrum above 1 GHz.^62,78,160

The aforementioned are all examples of noise spectroscopy during free evolution. Noise spectroscopy during driven evolution was also demonstrated using a “spin-locking” technique, where a strong drive along x or y axes defines a new qubit quantization axis, whose Rabi frequency is the new qubit frequency in the spin-locking frame. The spin-locking frame is then used to infer the noise spectrum while the qubit is continually subject to a driving field. For more information, we refer the reader to Ref. 128.

Here, we briefly review a few examples of techniques that have been developed to reduce noise or reduce its impact on decoherence (sensitivity). We stress that improving gate fidelity is a comprehensive optimization task, one that is full of trade-offs. It is thus important to identify what the limiting factors are, what price we have to pay to diminish these limiting factors, and what advantage we can achieve until reaching a better trade-off. These all require an accurate understanding the limitations on the gate fidelity, the sources of decoherence, the properties of the noise, and how it affects the system performance.

Numerous efforts have been undertaken to reduce noise-induced defects due to materials and fabrication.^40,161 In the case of charge noise, significant efforts have been made to reduce the number of defects, such as substrate cleaning,^59,162 substrate annealing,¹⁶³ and trenching.^41,61 In the case of flux noise, several groups have performed experiments to characterize the behavior and properties of magnetic-flux defects.^133,164,165 More recently, a number of groups have tried optical surface treatments to remove these defects.¹⁴⁰ 

In the context of residual quasiparticles, it has been shown that adding quasiparticle traps to the circuit design can reduce the quasiparticle number, particularly in devices that create excess quasiparticles, such as classical digital logic or operation in the presence of thermal radiation¹⁶⁶ 

Another strategy is to reduce qubit sensitivity to the noise by design. A qubit can only lose energy to defects if it couples to them. It has been demonstrated that altering the capacitor geometry to increase the electric-field mode volume reduces the electric field density in the thin dielectric regions that cause loss. This effectively reduces the “participation” of the defects and makes the qubits less sensitive to these noise sources.^55,62,130

In another example, the split transmons built using asymmetric junctions have lower sensitivity to flux noise than their symmetric counterparts at the expense of decreased frequency tunability.⁶⁹ This is a good trade-off to make, because generally one is interested in tuning the qubit frequency over a somewhat restricted range (typically around 1 GHz) about the qubit frequency. When such asymmetric transmons are used in a gate scheme such as the adiabatic $CPHASE$-gate,⁶⁵ (see Sec. IV F) the qubit is less sensitive to flux noise, has a lower dephasing rate, and this should improve the gate fidelity in general.

As introduced in Sec. III D 2, it is advantageous to leverage the 1/ω distribution of flux noise, wherein a considerable amount of the noise power resides at low frequencies, and so the noise is “quasistatic.” The spin-echo technique,¹¹⁵ which disrupts the free evolution by a π-pulse, is extremely effective in mitigating the pure dephasing by refocusing the coherent phase dispersion due to low-frequency noise. The more advanced versions, such as the CPMG-sequence, use multiple π-pulses to interrupt the system more frequently, pushing the filter band to even higher frequencies—a technique known as “dynamical decoupling.”⁷⁸ 

Returning to excess quasiparticles, it has been shown that quasiparticles can be stochastically pumped away from the qubit region, resulting in longer, and more stable T₁ times.¹²⁰ Although the pumping technique uses a series of π-pulses, this technique differs from dynamical error suppression of coherent errors in that pulses are stochastically applied, and that it addresses incoherent errors (T₁).

In the case of photon shot-noise, in addition to applying dynamical decoupling techniques, there have been several recent studies aimed at reducing the thermal photon flux that reaches the device. This include optimizing the attenuation of the cryogenic setup,^106,144,167 remaking the cryogenic attenuators with more efficient heat sinking,¹⁴³ adding absorptive “black” material to absorb stray thermal photons,^168,169 and adding additional cavity filters for thermalization.¹⁷⁰ 

In this section, we will introduce how superconducting qubits are manipulated to implement quantum algorithms. Since the transmonlike variety of superconducting qubits has so far been the most widely deployed modality for implementing quantum programs, the discussion throughout this section will be focused on modern techniques for transmons. Nonetheless, the techniques introduced here are applicable to all types of superconducting qubits.

We start with a brief review of the gates used in classical computing as well as quantum computing, and the concept of universality. Subsequently we discuss the most common technique of driving single qubit gates via a capacitive coupling of a microwave line, coupled to the qubit. We introduce the notion of “virtual $Z$ gates” and “DRAG” pulsing. In the latter part of this section, we review some of the most common implementations of two-qubit gates in both tunable and fixed-frequency transmon qubits. The single-qubit and two-qubit operations together form the basis of many of the medium-scale superconducting quantum processors that exist today.

Throughout this section, we write everything in the computational basis ${|0⟩,|1⟩}$ where $|0⟩$ is the + 1 eigenstate of σ_z and $|1⟩$ is the –1 eigenstate. We use capitalized serif-fonts to indicate the rotation operator of a qubit state, e.g., rotations around the x-axis by an angle θ is written as

and we use the shorthand notation “$X$” for a full π rotation about the x axis (and similarly for $Y:=Yπ$ and $Z:=Zπ$⁠).

Universal Boolean logic can be implemented on classical computers using a small set of single-bit and two-bit gates. Several common classical logic gates are shown in Fig. 8 along with their truth tables. In classical Boolean logic, bits can take on one of two values: state 0 or state 1. The state 0 represents logical $FALSE$⁠, and state 1 represents logical $TRUE$⁠.

Beyond the trivial “identity operation,” which simply passes a Boolean bit unchanged, the only other possible single-bit Boolean logic gate is the $NOT$ gate. As shown in Fig. 8, the $NOT$ gate flips the bit: $0→1$ and $1→0$⁠. This gate is reversible, because it is trivial to determine the input bit value given the output bit values. As we will see, for two-bit gates, this is not the case.

There are several two-bit gates shown in Fig. 8. A two-bit gate takes two bits as inputs, and it passes as an output the result of a Boolean operation. One common example is the $AND$ gate, for which the output is 1 if and only if both inputs are 1; otherwise, the output is 0. The $AND$ gate, and the other two-bit gates shown in Fig. 8, are all examples of irreversible gates; that is, the input bit values cannot be inferred from the output values. For example, for the $AND$ gate, an output of logical 1 uniquely identifies the input 11, but an output of 0 could be associated with 00, 01, or 10. Once the operation is performed, in general, it cannot be “undone” and the input information is lost. There are several variants of two-bit gates, including,

• $AND$ and $OR$⁠;

• $NAND$ (a combination of $NOT$ and $AND$⁠) and $NOR$ (a combination of $NOT$ and $OR$⁠);

• $XOR$ (exclusive $OR$⁠) and $NXOR$ (⁠$NOT$ $XOR$⁠).

The $XOR$ gate is interesting, because it is a “parity” gate. That is, it returns a logical 0 if the two inputs are the same values (i.e., they have the same parity), and it returns a logical 1 if the two inputs have different values (i.e., different parity). Still, the $XOR$ and $NXOR$ gates are not reversible, because knowledge of the output does not allow one to uniquely identify the input bit values.

The concept of “universality” refers to the ability to perform any Boolean logic algorithm using a small set of single-bit and two-bit gates. A universal gate set can in principle transform any state to any other state in the state space represented by the classical bits. The set of gates which enable universal computation is not unique, and may be represented by a small set of gates. For example, the $NOT$ gate and the $AND$ gate together form a universal gate set. Similarly, the $NAND$ gate itself is universal, as is the $NOR$ gate. The efficiency with which one can implement arbitrary Boolean logic, of course, depends on the choice of the gate set.

Quantum logic can similarly be performed by a small set of single-qubit and two-qubit gates. Qubits can of course assume the classical states $|0⟩$ and $|1⟩$⁠, at the north pole and south pole of the Bloch sphere, but they can also assume arbitrary superpositions $α|0⟩+β|1⟩$⁠, corresponding to any other position on the sphere.

Single-qubit operations translate an arbitrary quantum state from one point on the Bloch sphere to another point by rotating the Bloch vector (spin) a certain angle about a particular axis. As shown in Fig. 9, there are several single-qubit operations, each represented by a matrix that describes the quantum operation in the computational basis represented by the eigenvectors of the σ_z operator, i.e., $|0⟩≡[1 0]T$ and $|1⟩≡[0 1]T$⁠.

For example, the “identity gate” performs no rotation on the state of the qubit. This is represented by a two-by-two identity matrix. The $X$-gate performs a π rotation about the x axis. Similarly, the $Y$-gate and $Z$-gate perform a π rotation about the y axis and z axis, respectively. The $S$-gate performs a π/2 rotation about the z axis, and the T-gate performs a rotation of π/4 about the z axis. The Hadamard gate $H$ is also a common single-qubit gate that performs a π rotation about an axis diagonal in the x–z plane, see Fig. 9.

Two-qubit quantum-logic gates are generally “conditional” gates and take two qubits as inputs. Typically, the first qubit is the “control” qubit, and the second is the “target” qubit. A unitary operator is applied to the target qubit, dependent on the state of the control qubit. The two common examples shown in Fig. 10 are the controlled NOT (⁠$CNOT$-gate) and controlled phase (⁠$CZ$ or $CPHASE$ gate). The $CNOT$-gate flips the state of the target qubit conditioned on the control qubit being in-state $|1⟩$⁠. The $CPHASE$-gate applies a $Z$ gate σ_z to the target qubit, conditioned on the control qubit being in-state $|1⟩$⁠. As we will show later, the $iSWAP$ gate—another two-qubit gate—can be built from the $CNOT$-gate and single-qubit gates. The unitary operator of the $CNOT$ gate can be written in a useful way, highlighting that it applies an $X$-gate (a σ_x operator) $X$ depending on the state of the control qubit

and similarly for the $CPHASE$ gate

Comparing the last equality above with the unitary for the $CNOT$ [Eq. (62)], it is clear that the two gates are closely related. Indeed, a $CNOT$ can be generated from a $CPHASE$ by applying two Hadamard gates

since $HZH=X$⁠. Due to the form of Eq. (63). The $CPHASE$ gate is also denoted the CZ gate, since it implements a controlled $Z$ gate (a controlled-σ_z operation), by analogy with CNOT (a controlled application of the $X$-gate, i.e., the σ_x operation). Inspection of the definition of $CPHASE$ in Fig. 10 makes no distinction between which qubit acts as the target and which as the control and, consequently, the circuit-diagram is sometimes drawn in a symmetric fashion

The $CNOT$ (with qubit 1 as control and qubit 2 as target) can be realized in terms of the $CPHASE$ operation and single-qubit Hadamard gates,

Some two-qubit gates such as $CNOT$ and $CPHASE$ are also called “entangling gates,” because they can take product states as inputs and output entangled states. They are thus an indispensable component of a universal gate set for quantum logic. For example, consider two qubits A and B in the following state:

If we perform a $CNOT$ gate, $UCNOT$⁠, on this state, with qubit A the control qubit, and qubit B the target qubit, the resulting state is (see the truth table in Fig. 10)

which is a state that cannot be factored into an isolated qubit-A component and a qubit-B component. This is one of the two-qubit entangled “Bell states,” a manifestly quantum mechanical state.

A universal set of single-qubit and two-qubit gates is sufficient to implement an arbitrary quantum logic. This means that this gate set can in principle reach “any” state in the multiqubit state-space. How efficiently this is done depends on the choice of quantum gates that comprise the gate set. We also note that each of the single-qubit and two-qubit gates is reversible, that is, given the output state, one can uniquely determine the input state. As we discuss further, this distinction between classical and quantum gates arises, because quantum gates are based on “unitary” operations U. If a unitary operation U is a particular gate applied to a qubit, then its Hermitian conjugate $U†$ can be applied to recover the original state, since $U†U=I$ resolves an identity operation.

The gate-sequences used to represent quantum algorithms have certain similarities to those used in classical computing, with a few striking differences. As an example, we consider first the classical $NOT$ gate (discussed previously), and the related quantum circuit version, shown in Fig. 11.

While the classic bit-flip gate inverts any input state, the quantum bit-flip does not in general produce the antipodal state (when viewed on the Bloch sphere), but rather exchange the prefactors of the wavefunction written in the computational basis. The $X$ operator is sometimes referred to as “the quantum $NOT$” (or “quantum bit-flip”), but we note that X only acts similar to the classical $NOT$ gate in the case of classical data stored in the quantum bit, i.e., $X|g⟩=|g¯⟩$ for $g∈{0,1}$⁠.