In this Tutorial, we introduce basic conceptual elements to understand and build a gate-based superconducting quantum computing system.

## I. INTRODUCTION

Quantum computing is considered as a next-generation information processing technology. The basic element of a quantum computing system is a quantum bit, often called a qubit. Over the last few decades, considerable progress has been made toward realizing quantum computing systems by physically implementing a qubit in various systems such as ion traps, quantum dots, nuclear spins, and cavity quantum electrodynamics. The scalability of such a qubit is considered to be a prerequisite for a practical quantum computer of the future. In this regard, a solid-state qubit has been considered to be indispensable. Superconducting quantum systems are one of the most promising candidates because, in these systems, qubits are intrinsically integrated in a solid-state device, and their wide range of choice for the qubit parameters is a considerable advantage, which in turn gives flexibility in designing such quantum circuits.

In this Tutorial, we try to provide basic conceptual elements to understand and build a potentially scalable superconducting quantum computing system based on gate operations. The logical flow is roughly from principle to practice. After introducing the qubit and structure of a universal quantum computing system (Sec. II), we explain a superconducting circuit that can be used as a qubit (Secs. III and IV) and how to implement basic functions that are required for quantum computation (Secs. V and VI). Then, we introduce a quantum error-correction scheme, called the surface code, that is believed to be suitable for superconducting qubit systems (Sec. VII). Last, we deal with practical topics, such as how to characterize and control a quantum system (Secs. VIII and IX). The contents of this Tutorial are briefly summarized in Table I.

Section . | . |
---|---|

II | Universal quantum computing system |

This section introduces a quantum bit, quantum gates, and a possible structure of a universal quantum computing system. | |

III | Superconducting qubit |

This section describes elementary circuits that can be used as qubits and their properties in various circuit parameter regimes. | |

IV | Effect of noise |

This section discusses mechanisms of loss of quantum information and several noise-resilient circuit designs. | |

V | Coupling |

This section explains coupling schemes between a qubit and other quantum systems with classical analogies. | |

VI | Implementation of quantum computation |

This section explains how to implement basic functions that are required for quantum computation, such as readout, gate operation, and initialization. | |

VII | Quantum error correction |

This section explains how to construct an error-free logical qubit and how to perform logical gate operations in the context of the surface code. | |

VIII | Characterizing a quantum system |

This section describes standard procedures for the quantum system characterization. | |

IX | Controlling a quantum system |

This section explains several useful techniques for controlling a quantum system and their working principles. |

Section . | . |
---|---|

II | Universal quantum computing system |

This section introduces a quantum bit, quantum gates, and a possible structure of a universal quantum computing system. | |

III | Superconducting qubit |

This section describes elementary circuits that can be used as qubits and their properties in various circuit parameter regimes. | |

IV | Effect of noise |

This section discusses mechanisms of loss of quantum information and several noise-resilient circuit designs. | |

V | Coupling |

This section explains coupling schemes between a qubit and other quantum systems with classical analogies. | |

VI | Implementation of quantum computation |

This section explains how to implement basic functions that are required for quantum computation, such as readout, gate operation, and initialization. | |

VII | Quantum error correction |

This section explains how to construct an error-free logical qubit and how to perform logical gate operations in the context of the surface code. | |

VIII | Characterizing a quantum system |

This section describes standard procedures for the quantum system characterization. | |

IX | Controlling a quantum system |

This section explains several useful techniques for controlling a quantum system and their working principles. |

Since this is a Tutorial, the topics covered here are very selective rather than comprehensive. Hence, we cite references that are more accessible to readers. Another reason for this is that many concepts and experimental techniques for superconducting circuits were originally developed in other branches of science—tracing all historical literature is not meaningful for readers. For comprehensive reviews on this field, see Refs. 1–6.

Regarding the difficulty of this Tutorial, we assume that readers are somewhat familiar with quantum mechanics, especially the Dirac notation and the occupation number representation (second quantization), and elementary statistical mechanics, such as the Boltzmann distribution. Since superconducting quantum computing systems are electrical circuits, knowledge on basic electrical engineering will be helpful, especially the S-parameters. However, readers do not need to be masters of these topics. Reading this Tutorial does not require deep physical insights—it is more like learning a new language.^{7} Once you get used to it, you will enjoy it.

Before entering the main part, we would like to point out that the word “scaling” in quantum engineering is different from that in the semiconductor industry. In the semiconductor industry, scaling means reducing the size of the information processing device used, such as a transistor, and the energy cost per bit, so that we can integrate more and more devices into a chip. In quantum engineering, “scaling” simply means adding more qubits because physical quantities involved in operations of a superconducting quantum computing platform, such as the charge of a Cooper pair and magnetic flux quantum, are already at the quantum limit, and a quantum information processing device is lossless. Thus, the dramatic size reduction as demonstrated in Moore’s law may not be expected for superconducting qubits.

A set of formulas for deriving equations in this Tutorial are summarized in Table II.

$ e A ^ B ^ e \u2212 A ^= B ^+[ A ^, B ^]+ 1 2 ![ A ^,[ A ^, B ^]]+ 1 3 ![ A ^,[ A ^,[ A ^, B ^]]]+ 1 4 ![ A ^,[ A ^,[ A ^,[ A ^, B ^]]]]+\cdots ,$ |

$[ A ^ B ^, C ^]= A ^[ B ^, C ^]+[ A ^, C ^] B ^,[ a ^, a ^ \u2020]=1,[ a ^ \u2020 a ^, a ^ \u2020]= a ^ \u2020,[ a ^ \u2020 a ^, a ^]=\u2212 a ^,$ |

$[ \sigma ^ x, \sigma ^ y]=2 i \sigma ^ z,[ \sigma ^ y, \sigma ^ z]=2 i \sigma ^ x,[ \sigma ^ z, \sigma ^ x]=2 i \sigma ^ y,$ |

$ \sigma ^ \xb1= 1 2( \sigma ^ x \xb1 i \sigma ^ y), \sigma ^ + \sigma ^ \u2212= 1 2( \sigma ^ z + I ^),[ \sigma ^ z, \sigma ^ \xb1]=\xb12 \sigma ^ \xb1,[ \sigma ^ +, \sigma ^ \u2212]= \sigma ^ z.$ |

$ e A ^ B ^ e \u2212 A ^= B ^+[ A ^, B ^]+ 1 2 ![ A ^,[ A ^, B ^]]+ 1 3 ![ A ^,[ A ^,[ A ^, B ^]]]+ 1 4 ![ A ^,[ A ^,[ A ^,[ A ^, B ^]]]]+\cdots ,$ |

$[ A ^ B ^, C ^]= A ^[ B ^, C ^]+[ A ^, C ^] B ^,[ a ^, a ^ \u2020]=1,[ a ^ \u2020 a ^, a ^ \u2020]= a ^ \u2020,[ a ^ \u2020 a ^, a ^]=\u2212 a ^,$ |

$[ \sigma ^ x, \sigma ^ y]=2 i \sigma ^ z,[ \sigma ^ y, \sigma ^ z]=2 i \sigma ^ x,[ \sigma ^ z, \sigma ^ x]=2 i \sigma ^ y,$ |

$ \sigma ^ \xb1= 1 2( \sigma ^ x \xb1 i \sigma ^ y), \sigma ^ + \sigma ^ \u2212= 1 2( \sigma ^ z + I ^),[ \sigma ^ z, \sigma ^ \xb1]=\xb12 \sigma ^ \xb1,[ \sigma ^ +, \sigma ^ \u2212]= \sigma ^ z.$ |

## II. UNIVERSAL QUANTUM COMPUTING SYSTEM

### A. Essential elements

#### 1. Quantum bit

Note that a qubit and a spin-1/2 system are mathematically identical. This allows us to represent the qubit state conveniently as an arrow, called the Bloch vector, in the Bloch sphere [Fig. 1(a)]. Conventionally, the qubit quantization axis is set as the $z$ axis, and the north and south poles represent $ 0$ and $ 1$, respectively. Hence, the longitudinal component of the Bloch vector corresponds to the polarization of the qubit, and the transverse component corresponds to the coherence between the two basis states.

When we use the Bloch sphere, we are free to choose a frame of reference. In the majority of the literature, including this Tutorial, the dynamics of the qubit state are described in the rotating frame [Fig. 1(b)]. To determine the rotating frame frequency, we have to know the dynamics we want to focus on. Then, we eliminate the trivial evolution by performing a unitary transformation, which changes our frame of reference. Note that this is conceptually and mathematically identical to switching into the interaction picture. Usually, the qubit frequency, the resonator frequency (see Sec. V B), or the external drive frequency (Sec. VI C) is chosen as the rotating frame frequency.

A qubit is often implemented by the two lowest states of a quantum system, such as (artificial or natural) atoms [Fig. 1(a)]. This subspace is called the computational subspace. In general, any Hilbert space whose dimension is truncated into two can be used as a qubit. This generalized definition of a qubit is essential for constructing a logical qubit (Sec. VII).

In this Tutorial, the notations denoting the qubit states, { $ 0$, $ 1$, $ 2$ (higher excitation level)} and { $ g$, $ e$, $ f$}, are used interchangeably to avoid confusion with the photon or the charge number states. In addition, $ \omega q$, which we call the qubit frequency, is the transition frequency between $ 0$ and $ 1$, and $ \omega i - j$ (with a hyphen in the subscript) is the transition frequency between $ i$ and $ j$; $ \omega i j$ (without a hyphen in the subscript) indicates the energy level of the two-qubit state, $ i\u2297 j$ (or $ i j$ in the short form).

#### 2. Quantum gate

A quantum gate is a discrete control acting on qubits inducing the unitary evolution of the quantum states of the qubits. Quantum computation is basically a series of quantum gate operations.

^{8,9}

Any multiqubit gate operation can be decomposed into a set of single-qubit and controlled-NOT (CNOT) gates. Thus, the gate set {single-qubit gates, CNOT} is called a universal quantum gate set. An arbitrary single-qubit gate can be well approximated by the discrete gate set { $H$, $S$, $T$} (Solovay–Kitaev theorem^{10}). Hence, we can rewrite a universal gate set as { $H$, $S$, $T$, CNOT}. The definitions of these gates and other popular gates are summarized in Table III.

Name . | Function . | Symbol . | Matrix . |
---|---|---|---|

Pauli-X (X) | $R^x(\pi )$ | $0110$ | |

Pauli-Y (Y) | $R^y(\pi )$ | $0\u2212ii0$ | |

Pauli-Z (Z) | $R^z(\pi )$ | $100\u22121$ | |

Hadamard (H) | $R^x(\pi )R^x(\pi /2)$ | $12111\u22121$ | |

Phase (S) | $R^z(\pi /2)$ | $100i$ | |

π/8 (T) | $R^z(\pi /4)$ | $100ei\pi /4$ | |

$Controlled-NOT(CNOT)$ | $X^\psi tif\psi c=1$ | $1000010000010010$ |

Name . | Function . | Symbol . | Matrix . |
---|---|---|---|

Pauli-X (X) | $R^x(\pi )$ | $0110$ | |

Pauli-Y (Y) | $R^y(\pi )$ | $0\u2212ii0$ | |

Pauli-Z (Z) | $R^z(\pi )$ | $100\u22121$ | |

Hadamard (H) | $R^x(\pi )R^x(\pi /2)$ | $12111\u22121$ | |

Phase (S) | $R^z(\pi /2)$ | $100i$ | |

π/8 (T) | $R^z(\pi /4)$ | $100ei\pi /4$ | |

$Controlled-NOT(CNOT)$ | $X^\psi tif\psi c=1$ | $1000010000010010$ |

Among these universal quantum gates, the quantum gates generated by the $H$, $S$, and CNOT gates form a group called the Clifford group. This group is important in quantum computation, especially for quantum error correction (Sec. VII) and efficient gate qualification (Sec. IX D). However, it is known that a quantum computer operated by only Clifford gates can be simulated efficiently on a probabilistic classical computer (Gottesman–Knill theorem^{10}). Thus, a non-Clifford gate, such as the $T$ gate, is required to show the advantage of quantum computation.

### B. Structure

A gate-operation-based, universal, and scalable superconducting quantum computer will likely have the following structure (Fig. 2):^{11,12}

**Physical resources:**This layer is a collection of physical qubits and necessary circuits for the control and readout of the physical qubits.**Error correction resources:**In this layer, errors acting on quantum information stored in a set of physical qubits are corrected. This operation produces a single error-free logical qubit. For this, high-fidelity controls, such as initialization, gate operation, readout, and feedback, for physical qubits are required.**Logical resources:**Initialization, gate operation, and readout of logical qubits are performed in this layer.**Algorithmic resources:**Quantum algorithms, such as Shor’s factoring and Grover’s search algorithms, are performed in this layer.

## III. SUPERCONDUCTING QUBIT

### A. Design criteria

A superconducting qubit is the two lowest energy eigenstates of an artificial atom made of a superconducting circuit. To be a useful qubit, the circuit must be designed to satisfy the following conditions:

*Proper operating frequency range*: A qubit must have a transition frequency that is significantly higher than the thermal energy of a typical solid-state system to observe quantum nature. The only continuous refrigeration method for solid state devices below 0.3 K is to use a dilution refrigerator, whose base temperature is usually about 10 mK ( $\u223c200$ MHz). This means that the transition frequency of a qubit must be at least a few gigahertz. At the same time, the qubit transition frequency should be sufficiently lower than the superconducting energy gap of the host superconductor so as not to excite quasiparticles. For aluminum, which is the most popular material for superconducting qubit systems, the energy gap is about 100 GHz.*Large anharmonicity*: To be a well-defined two-level system, a qubit should have anharmonicity $\alpha \u2261 \omega 1 - 2\u2212 \omega q$ of at least $\u223c100$ MHz to perform a reasonably fast gate operation (see Sec. IX A 1 for the gate time and frequency selectivity). Recently, it has been found that having a third level in an accessible frequency range can be beneficial, such as for initialization or two-qubit gate operation (see Secs. VI D and VI E 1).*Long coherence time*: The assigned quantum state should last for a long time compared with the time for gate operations.*Ease of coupling*: For readout and (multi)qubit gate operation, a reasonably strong coupling between a qubit and another quantum system, such as a resonator or neighboring qubit, should be achieved easily.*Ease of control*: The quantum state should be brought to a superposition easily and straightforwardly by an external mean.*Ease of fabrication*: A qubit should be easy to fabricate with standard nanotechnology for good reproducibility.

### B. Josephson junction

A superconductor is a macroscopic quantum mechanical system in the sense that it can be described by a single macroscopic wavefunction, i.e., the order parameter $\Psi $. However, this property is not a sufficient condition for being a qubit; we need a confinement potential to have discrete energy eigenstates such as electrons in the Coulomb potential forming an atom. Moreover, to control the two lowest energy eigenstates selectively, the potential must be anharmonic to have distinct energy separation between eigenstates.

The solution for discrete energy eigenstates is to make an electrical circuit. In a superconducting circuit, the quantized energy level emerges from the quantization of the charge and the magnetic flux stored in various electrical components just like the position and the momentum of electrons in a real atom. (Since the charge and the magnetic flux are collective coordinates that represent the cooperative motion of large numbers of electrons, the circuit quantization is essentially phenomenological.^{14})

^{15}

^{,}

^{16}

### C. Elementary circuits

#### 1. Generic Hamiltonian

We can categorize elementary circuits of superconducting qubits into two groups, an island and a loop (Fig. 4). In the early literature, these two kinds of qubits were called a charge qubit and a flux qubit, respectively, on the basis of the spread of the wavefunctions in the number (charge) and phase (flux) bases [typical wavefunctions of a charge qubit are shown in Fig. 5(d)].^{17} However, such a classification is valid only for a certain parameter range; it does not work well for sophisticated qubits whose wavefunctions often show exotic distributions in both the number and the phase bases. Therefore, we simply categorize circuits of superconducting qubits based on the geometry. Then, we will show how the qubit properties change as we tune the circuit parameters. The knowledge acquired in this way can also be used for analyzing more complex qubits.

*relative*number and phase operators between two superconductors, and this number imbalance of electrons is much less than the number of electrons in each superconductor. For details, see Refs. 15 and 19.) The resulting circuit Hamiltonian $ H ^ q$ is given by

#### 2. Island-based qubit

In the small $ E J/ E C$ limit, the $ E C$ term is dominant in Eq. (15); as a result, the wavefunctions are localized in the number basis as shown in Fig. 5(d), suggesting that the number basis will be more convenient to describe the physics in this regime. In Fig. 5(a), the gray lines indicate the $ E C$ term associated with $ N = 0$ and $ \xb1 1$. At $ N ext=0.5$, $ N = 0$ and $ 1$ are energetically degenerated. Here, the $ E J$ term hybridizes these two states via coherent charge tunneling [Fig. 4(a)], resulting in an anticrossing whose size is approximately $ E J$. At zero bias, a similar, but significantly smaller, hybridization occurs between $ N = \u2212 1$ and $ 1$. This results in the first excitation level at $\u2248$4 $ E C$.

More systematic plots regarding the two observations, (i) the flattening of the energy band and (ii) the suppression of the anharmonicity in the large $ E J/ E C$ limit, are given in Figs. 5(g) and 5(h), respectively. Note that the difference between the transition frequencies at $ N ext=0$ and 0.5, denoted by $\Delta \omega q$, decreases exponentially as shown in the inset of Fig. 5(g). This indicates that the energy levels are completely flat if $ E J/ E C\u227350$.

The anharmonicity at $ N ext=0$ and 0.5 also collapses into a single curve because of the flattening of the energy band [Fig. 5(h)]. The crucial observation is that, although $\alpha $ is also approaching zero, its slope is algebraic rather than exponential. This suggests that we can use the circuit in the large $ E J/ E C$ limit as a charge-insensitive qubit, which is called a transmon (see Sec. IV B for the implementation of a transmon).^{20}

#### 3. Loop-based qubit

A loop-based qubit is not as simple as an island-based qubit because we have to consider all terms in Eq. (15). We start with the effect of $ E L$. Since $ E L$ is a function of $\phi $, it is convenient to take the phase basis, and consequently, to treat $ E J$ and $ E L$ terms as the potential. We first consider the regime in which $ E J/ E L\u226b1$. In this regime, the periodic shape is prominent in the potential as shown in Figs. 6(a)–6(d). When $ E J/ E C\u226a1$ [Fig. 6(e)], the energy level diagram is almost independent of $\Phi $, and $ \omega q\u2248 8 E L E C/\u210f$. The reason is that the oscillating potential is averaged out owing to the large kinetic energy [Fig. 6(a)], and consequently, only the harmonic terms are effective in Eq. (15). In this regime, $ N ^$ is localized. Thus, $ N ^$ corresponds to the position in the spring-block oscillator analogy, $L$ corresponds to the mass, and $ C \u2212 1$ corresponds to the spring constant.

For $ E J/ E C>1$, the physics of a loop-based qubit can be understood in a similar way to that of an island-based qubit. In Figs. 6(f) and 6(g), the gray lines show the $ E L$ term in Eq. (15) associated with $ \phi = 0$ and $ \xb1 2 \pi $, which means that the numbers of trapped fluxes in the loop are 0 and $\xb11$, respectively. Note that, at $\Phi / \Phi 0=0.5$ ( $ \phi ext=\pi $), the potential has a double-well shape, resulting in energy degeneracy between $ \phi \u2248 + \pi $ and $ \phi \u2248 \u2212 \pi $. These degenerated states correspond to two superposed currents circulating in opposite directions [Fig. 6(b) and its inset]. Similarly to the degeneracy point in an island-based qubit, the hybridization mediated by the kinetic energy ( $ E C$ term) breaks the degeneracy, resulting in an anticrossing. This process can be understood as coherent flux tunneling between the flux island (loop) and the flux reservoir [Fig. 4(b)]. On the basis of this explanation, it is easy to understand that $ \omega q$ at $\Phi / \Phi 0=0.5$ decreases monotonically as a function of $ E J/ E C$ [Fig. 6(m), dashed lines].

At zero flux bias, the first excitation level is formed through the hybridization of states $ \phi \u2248 \xb1 2 \pi $, as shown in Fig. 6(c). Since this hybridization requires the tunneling of two potential barriers, the energy gap is significantly smaller than that at $\Phi / \Phi 0=0.5$. Hence, $ \omega q$ at zero bias is approximately $2 \pi 2 E L$[ $= E L ( \xb1 2 \pi ) 2/2$] and weakly depends on $ E J/ E C$. This explains why $ \omega q$ at zero bias shows a plateau in Fig. 6(m).

As $ E C$ decreases further [Fig. 6(h)], the ground and excited states at zero bias become bound states within a well of the periodic potential. In this case, we can approximate the potential as a weakly nonlinear harmonic potential [Fig. 6(d)] as we did in Sec. III C 2. Hence, $ \omega q\u2248 8 E J E C/\u210f$. This explanation suggests that the physics of a loop-based qubit in this regime is actually close to that of two island-based qubits connected by an inductor, i.e., inductively shunted junction [the inset of Fig. 6(d)]. The reason is that $L$ is very large in the regime $ E J/ E L\u226b1$ such that the reactance at $ \omega q$ is significant, whereas the circuit is electrically shorted at the low-frequency limit.

In the regime $ E J/ E L\u223c1$, the harmonic contribution to the potential is substantial; thus, it is difficult to separate the contributions from the periodic and harmonic potentials to the energy levels. One consequence is that the minimum of the potential becomes almost flat at $\Phi / \Phi 0=0.5$ as shown in Fig. 6(j). The other consequence is that, in Figs. 6(k) and 6(l), the first excitation level near zero bias already has a parabolic shape rather than a cross shape because the first excitation level at zero bias is mostly governed by the physics shown in Fig. 6(d), rather than the hybridization shown in Fig. 6(c). This explains why $ \omega q$ at $\Phi / \Phi 0=0$ in this regime decreases monotonically with increasing $ E J/ E C$ without any plateau in Fig. 6(m).

The experimentally accessible range of $ E J/ E C$ is typically from $\u223c0.1$ to $\u223c100$. In this range, $ \omega q$ of a loop-based qubit with $ E J/ E L\u226b1$ at $\Phi / \Phi 0=0.5$ is often too low to satisfy condition 1 in Sec. III A, while $ \omega q$ of a qubit with $ E J/ E L\u223c1$ at zero bias is too high. Regarding the anharmonicity, a loop-based qubit with $ E J/ E L\u226b1$ is more advantageous than that with $ E J/ E L\u223c1$ as shown in Fig. 6(n). If $ E L$ increases even further such that $ E J/ E L\u226a1$, the potential becomes almost harmonic, and as a result, the circuit does not show enough anharmonicity to be a qubit.

## IV. EFFECT OF NOISE

### A. Relaxation

#### 1. Concept

*independently*of its direction.

^{10,13}) $ T 2$ is the time constant for the decay of the transverse component of the Bloch vector to zero. Note that there are two contributions to $ T 2$ in Fig. 7(a): one is the shortening of arrows and the other is the spreading of arrows. The shortening of arrows is due to the growth of the longitudinal component, whereas the spreading is due to the loss of the phase coherence of the qubit, called dephasing. As shown in Figs. 7(b) and 7(c), dephasing is caused by the temporal fluctuation in qubit transition frequency. Hence, both thermalization and dephasing contribute to $ T 2$, while $ T 1$ is entirely determined by thermalization. This explanation can be written as

^{21}

#### 2. Thermalization

^{21}

^{24–26}

Equations (19)–(22) suggest that thermalization due to various noise processes acting on a qubit is determined by the circuit parameters and the off-diagonal matrix elements of $ N ^$ and $ \phi ^$, i.e., the overlap between wavefunctions in the circuit variable space.

Currently, there are three approaches to suppressing thermalization:

*Clean environment*: This approach eliminates the noise source by removing any unnecessary quantum systems, such as defects, which could possibly couple to the qubit. Naturally, this approach requires much knowledge and engineering regarding materials, such as host superconductors, substrates, and oxide layers.^{27–29}For example, it is known that a qubit on a silicon substrate usually shows a shorter $ T 1$ than that on a sapphire substrate, partly because of a lossy amorphous silicon oxide layer.^{30}For a comprehensive review for this approach, see Ref. 31.*Reducing participation ratio*: This approach reduces losses in dielectric media, such as oxides or organics at the surface of a qubit, by minimizing the participation ratio that is defined as the fraction of the electric field energy stored within the volume of each dielectric medium.^{32,33}Since an electric field in a planar device is highly concentrated near the edges, a qubit made of large superconducting pads with a simple design shows good performance in general.^{30,32}*Reducing wavefunction overlap*: We can engineer the potential by choosing the geometry and parameters of the circuit to minimize the effective dipole moment, i.e., the wavefunction overlap in the circuit variable space, as shown in Fig. 7(e). This is the strategy that the so-called protected qubit takes.^{34–39}However, reducing the effective dipole moment inevitably makes the qubit difficult to control [compare Eqs. (22) and (34)].

#### 3. Dephasing

^{21}

^{20,21}

On the basis of what we have learned thus far, we explain two approaches to suppressing dephasing.

*Geometry*: We can select a circuit geometry that is insensitive to a certain type of noise. A fixed-frequency island-based qubit is insensitive to flux noise simply because there is no loop that can contain a flux [Fig. 4(a)]. For a loop-based qubit, the sensitivity to flux noise depends on the circuit parameters. If the qubit is in a circuit parameter range in which the qubit states are the circulating current states shown in Fig. 4(b) and the inset of Fig. 6(b), the qubit is insensitive to charge noise. The reason is that a continuously flowing DC supercurrent does not allow any charge offset within the current path, i.e., the circuit is electrically shorted in the low-frequency limit. However, such a state is sensitive to flux noise. If the circuit parameters are chosen such that the qubit states are similar to the island-like qubit shown in the inset of Fig. 6(d), then the qubit states are sensitive to charge noise but less sensitive to flux noise.*Bias dependence*: Since $ \Gamma \phi $ is proportional to $ \u2202 \lambda \omega q$ [Eq. (25)], we can choose the circuit parameters that give minimal bias dependence as shown in Fig. 7(f). In this regard, operating a qubit at a bias at which $ \u2202 \lambda \omega q=0$, called a sweet spot [red circles in Fig. 7(f)], is necessary because the qubit is first-order insensitive to noise at this particular bias [ $ \Gamma \phi =0$ in Eq. (24)].

Note that the energy of a qubit is conserved during dephasing, in contrast to thermalization. This allows us to recover the phase coherence by applying pulses that can revert the direction of the time evolution. Such a technique is called refocusing and will be discussed in Sec. IX C.

In Sec. IV B, we briefly explore several noise-resilient qubit designs and discuss how to improve the robustness of the qubit by tuning the circuit parameters.

### B. Noise-resilient designs

#### 1. Island-based qubit

The most successful noise-resilient design of an island-based qubit is a transmon. As mentioned in Sec. IV, the dephasing rate of an island-based qubit in Fig. 8(a) is insensitive to flux noise because of the absence of a loop. To suppress the effect of charge noise, the transmon design pushes strategy 2 in Sec. IV A 3 to the limit: eliminating the $ N ext$ dependence by choosing $ E J/ E C=50$–100 (Fig. 5).

This limit can be achieved by adopting a shunt capacitor [red capacitor in Fig. 8(a)]. The shunt capacitor takes the majority of the effect of the charge noise and thus minimizes this effect on the junction. The physics of this idea is the same as adding a heavy mass to reduce the sensitivity to mechanical noise.

As mentioned in Sec. III C 2, the tradeoff is the reduced anharmonicity: in the large $ E J/ E C$ limit, the qubit wavefunctions are localized in the phase space; hence, a transmon is a weakly nonlinear harmonic oscillator. From this reasoning, we can easily imagine that, if we treat the qubit wavefunction as a rolling glass bead in a potential well, the bead sees more anharmonicity as the kinetic energy ( $ E C$) increases. Indeed, the anharmonicity of a transmon is roughly given by $\u2212 E C$ (see Sec. V B 1). $ E C$ is usually chosen 100–500 MHz to satisfy condition 2 in Sec. III A. Then, $ E J$ must be 10–30 GHz to satisfy condition 1 in Sec. III A. The resulting circuit parameters are summarized in Table IV.

Type . | E_{J} (GHz)
. | E_{J}/E_{C}
. | E_{J}/E_{L}
. | β
. | N_{J}
. | Reference . |
---|---|---|---|---|---|---|

Transmon | 10–30 | 50–100 | 40 and 41 | |||

Flux qubit | 10–100 | 10–100 | ∼1 | ≈2 | 2–3 | 23 |

Fluxonium | 1–10 | 1–10 | 3–10 | 2–5 | 10–100 | 42 |

A DC SQUID is employed to tune the qubit frequency as explained in Sec. III B [rightmost figure in Fig. 8(a)]. However, in this case, the transmon is exposed to the flux noise. Therefore, we need to design a DC SQUID with minimal flux dependence based on Eq. (14).^{43} In addition, we have to operate the tunable transmon at the flux bias sweet spot.

#### 2. Loop-based qubit

The main difficulty in implementing a loop-based qubit is designing an inductor with sufficiently large inductance because the inductance of a superconducting loop made of aluminum or niobium is usually very small such that $ E L> E J$. Consequently, the resulting anharmonicity is too small to satisfy condition 2 in Sec. III A as explained in Sec. III C 3.

A popular strategy is to add multiple Josephson junctions, where the Josephson energy for each junction is $\beta E J$, as an effective inductor. Here, we still want to keep the current flowing in the loop dominated by the main junction [black junction in Fig. 8(b)]. (In the literature, the main junction is often called the “ $\alpha $ junction,” where $\alpha = \beta \u2212 1$, for historical reasons.) Since the flux tunneling rate through a Josephson junction is roughly proportional to $exp\u2061 (\u2212 E J / E C )$ (Ref. 44), $\beta $ must be larger than 1.

^{45,46}

First, we consider the case when $ N J$ is 2 or 3 and $\beta \u22482$ [upper figures in Fig. 8(b)]. The circuit with these parameters, which roughly corresponds to a loop-based qubit with $ E J/ E L\u223c1$, is called a flux qubit. Although the resulting energy level structure from Eq. (26) is not the same as that from Eq. (15), the overall dependence of the energy levels on the circuit parameters is qualitatively similar to that in Figs. 6(k)–6(n).

For a noise-resilient qubit, we need to select $ E J$ to minimize the $ \phi ext$ dependence as mentioned in Sec. IV. At the same time, we also need to satisfy condition 1 in Sec. III A. It was found that $ E J\u223c10$–100 GHz and $\beta \u22482$ balance these two.^{23} However, there is a tradeoff: the qubit becomes sensitive to charge noise because the circulating currents are close to zero even at $\Phi / \Phi 0=0.5$. To circumvent this, a shunt capacitance is added to the main junction as we did for the transmon; thus, we have $ E C=0.1\u22121$ GHz. The final circuit shown in Fig. 8(b) is called a capacitively shunted (C-shunt) flux qubit [upper rightmost figure in Fig. 8(b)].^{47}

One might ask, “can a flux-tunable transmon be considered as a kind of C-shunt flux qubit?” Our answer is yes, but the working flux bias at which criterion 1 in Sec. III A is satisfied is different: a transmon is usually operated at zero flux bias, while a C-shunt flux qubit is operated at $\Phi / \Phi 0=0.5$.

With a sufficiently large $ N J$ ( $\u223c10$–100), Eq. (26) can be treated as a linear inductor with $ E L\u2248\beta E J/ N J$ [lower figures in Fig. 8(b)], if the self-resonance frequency of the junction array is sufficiently higher than that of each junction, $ 8 E J E C/h$.^{48} This condition can be satisfied by limiting $ N J$ to $ N J\u2272 C J / C g$, where $ C J$ is the capacitance across each junction and $ C g$ is the capacitance between the junction array and the ground. By tuning $\beta $ and $ N J$, we can satisfy $ E L\u226a E J$. A superconducting qubit in this regime is called a fluxonium or an RF SQUID qubit. In this case, it is easy for $ \omega q$ at zero bias to satisfy condition 1 in Sec. III A; at $\Phi / \Phi 0=0.5$, $ \omega q$ might be too low. This drawback can be resolved by employing active qubit initialization protocols (see Sec. VI E). According to Fig. 6(h), the anisotropy is significantly larger than that of a flux qubit. Capacitive shunting has also been applied to a fluxonium [lower rightmost figure in Fig. 8(b)], resulting in improved $ T 2$.^{42}

Last, we would like to point out that a Josephson junction array as a linear inductor itself is an interesting system. The reason is that it is difficult, although not impossible,^{49} to make a geometric inductor whose reactance exceeds the superconducting resistance quantum $ R Q=h/ ( 2 e ) 2\u22486.5$ k $\Omega $ because of stray capacitance and radiation to vacuum, whose impedance is about 377 $\Omega $. Such a linear inductor whose impedance is similar to or larger than $ R Q$ is often called a superinductor. Thus, implementations of superinductors have usually been based on kinetic inductance,^{48,50,51} i.e., an inductive contribution to the impedance that arises from kinetic energy of the charge carrier, instead of geometric inductance (for further discussion about kinetic inductance, see Sec. VI B 2). Very recently, a qubit made of Josephson junction arrays with extremely high inductance ( $ E L<100$ MHz) succeeded in implementing the regime shown in Figs. 6(a) and 6(e).^{52}

## V. COUPLING

Thus far, we have explained how to make a qubit out of superconductors. To perform actual computation, a qubit must be coupled to other systems so that the qubit state can be controlled or read. The most commonly used physics for these operations is the cavity quantum electrodynamics.^{53} It provides an integrated control/readout scheme via the interaction between an atom and a cavity. The same physics can be applied to a superconducting circuit as the interaction between a qubit and a resonator. This circuit version of the cavity quantum electrodynamics is called the circuit Quantum ElectroDynamics (cQED).^{6,54,55} In addition, for multiqubit gate operation, qubit–qubit coupling is required. In this section, we discuss how to couple a qubit to other systems.

### A. Two coupled classical oscillators

Note that, although the coupling spring is always present, its effect on the dynamics strongly depends on the system parameters. When $\kappa $ is the static parameter $ \kappa 0$, the two oscillators exchange their energy only when they are on-resonance [Fig. 9(a)]. Even if the oscillators are off-resonance, we can force them to exchange their energy by modulating $\kappa $ with the frequency difference between the two oscillators, $| f 1\u2212 f 2|$, where $2\pi f i= ( k i + \kappa 0 ) / m i$ [Fig. 9(b)]. These two phenomena can be seen in both classical and quantum systems regardless of whether statistics is fermionic (qubit) or bosonic (resonator).

Next, we inject energy into the system by two methods. One is to modulate the coupling constant with $ f m= f 1+ f 2$. In this case, as one can see in Fig. 9(b), both $ x 1$ and $ x 2$ increase exponentially with time. This is parametric amplification, which is important for realizing noiseless amplification. The concept and applications of parametric amplification will be discussed further in Sec. VI B 2. The other is to drive oscillator 1 with the frequency $ f d$. When $ f d= f 2$, $ x 2$ increases linearly with time.

When we apply the physics learned from these energy injection processes, we need to consider the quantum statistics. If two coupled quantum systems are bosonic, we can simply interpret the displacement of the blocks as the population. However, if one or both of the systems are fermionic, we will see an oscillation in the population, instead of the linear increase that we saw in Fig. 9(c). Such an oscillation is called the Rabi oscillation, which will be discussed further in Sec. VI C.

### B. From circuit to atom

#### 1. Qubit, resonator, and somewhere between them

Although Eq. (32) models several important properties of a weakly anharmonic/nonlinear system successfully, we still need a unified description of superconducting circuits in a wide range of nonlinearity for complicated circuits. Moreover, off-resonant resonator modes have been known to contribute substantially to the relaxation times of a qubit via the Purcell effect (see Sec. VI B 1).^{56} To remedy these issues, semiclassical superconducting circuit quantization methods have been proposed and showed a good agreement with experimental data. Interested readers should see Refs. 33 and 57–60.

For the rest of this section, we model a superconducting qubit as an ideal two-level system because this provides a qualitatively satisfactory picture to understand the physics of various couplings associated with a qubit at the level of this Tutorial.

#### 2. Qubit–resonator coupling

The transverse coupling mediates the energy exchange between the qubit and the resonator [Fig. 10(b)]. Thus, the transverse coupling is effective when the coupled system has a mode whose frequency is close to $ \omega q$ as we saw in Fig. 9(a). The longitudinal coupling changes the qubit frequency. It is effective when $ \omega q$ varies considerably with the external bias [Fig. 10(b)]. Note that the physics of the relaxation processes in Sec. IV can be understood within this framework; the transverse and longitudinal couplings are actually the mechanisms for thermalization and dephasing, respectively.

Equations (34) and (36) suggest that if $ \omega q$ does not depend on the physical parameter that is coupled to the effective dipole moment, the voltage in this case, there is no longitudinal coupling. Note that, because of this, the dominant coupling associated with a qubit at its sweet spot is the transverse coupling. One consequence is that the only possible coupling associated with a capacitively coupled transmon is the transverse coupling because $ \omega q$ of a transmon is insensitive to the external voltage fluctuation, i.e., a transmon is always at its charge bias sweet spot. To implement the longitudinal coupling, a transmon needs a flux-tunable element, such as a DC SQUID, and should be coupled to the target system inductively.^{61}

#### 3. Qubit–qubit coupling

In Eq. (42), the $XX$ interaction corresponds to the transverse interaction. Regarding the longitudinal interaction, there is ambiguity in its definition. If we follow the convention in the qubit–resonator interaction consistently, only the $XZ$ and $ZX$ interactions must be called the longitudinal interactions. However, a considerable number of papers designate all non-transverse interactions, which includes the $ZZ$ interaction, as the longitudinal interactions. In this Tutorial, we use the term “longitudinal interaction” for the qubit–resonator interaction only. For the qubit–qubit interaction, we call the type of interaction explicitly, such as the $XZ$ interaction, for clarity.

If $J$ is static and $| \omega q 1\u2212 \omega q 2|\u226bJ$, it is clear that the coupling term will be averaged out and consequently cannot be used for two-qubit gate operation unless one of the following actions is taken: (i) tuning $ \omega q 1$ or $ \omega q 2$ so that $ \omega q 1\u2248 \omega q 2$; (ii) modulating $J$ with the frequency $| \omega q 1\xb1 \omega q 2|$ to cancel out oscillating factors; or (iii) adding an additional drive term. These strategies are based on the lessons learned in Sec. V A and will be the basis of two-qubit gates in Sec. VI D.

It is often necessary to couple two qubits separated by a macroscopic distance. In this case, a resonator or even a qubit is employed as a coupler—such a scheme is called indirect coupling (Fig. 11). Here, we need to be careful not to excite the coupler itself; otherwise, the information will leak to the Hilbert space of the coupler. Hence, the resonance frequency of the coupler must be significantly far from the transition frequency of the qubits such that $ | \omega r\u2212 \omega q i |\u226b g ( i )$, where $ g ( i )$ is the transverse coupling constant associated with the resonator and qubit $i$. The coupler mediates the exchange of virtual photons between the two qubits. Such a system can also be modeled as Eq. (44).^{66,67}

### C. Strong (transverse) coupling

In this subsection, we consider how to quantify the strength of the transverse coupling because the current standard qubit control and readout methods are based on the transverse coupling. (There are many studies on the potential use of the longitudinal qubit–resonator coupling for quantum computation. Interested readers should see Refs. 61 and 63–65.) For efficient qubit control and readout, we need a reasonably strong qubit–resonator coupling; otherwise, the signal will be too small and the control will be too slow. Similarly, we also need a strong qubit–qubit interaction for efficient two-qubit gate operation (see Secs. VI B 1 and VI D for further explanation). Then, what are the criteria that must be satisfied to be called a strong coupling?

The strength of the qubit–resonator coupling is usually characterized by three quantities: $g$, $\kappa $, and $\gamma $ [Fig. 10(a)]. Here, $g/2\pi $ is the transverse coupling strength in Hz, $\kappa /2\pi $ is the loss rate of photons from the resonator, i.e., the spectral linewidth of the resonator, in Hz ( $\kappa = \omega r/Q$, where $Q$ is the quality factor of the resonator), and $\gamma /2\pi $( $=1/\pi T 2$) is the transverse relaxation rate, i.e., the spectral linewidth, of the qubit in Hz. When the system satisfies $g>\kappa /2,\gamma /2$, the coupling is regarded as a strong coupling. The physical meaning is clear: to ensure a strong qubit–resonator interaction, the photon must stay in the resonator and the qubit needs to keep its coherence while the two systems exchange their energy.

The experimental signature of a strong qubit–resonator or qubit–qubit coupling is an anticrossing called the vacuum Rabi splitting (Fig. 12). Such a situation is well described by the Jaynes–Cummings Hamiltonian [Eq. (41)]. In the Jaynes–Cummings Hamiltonian, when the qubit and the resonator are far off-resonance, the ground state of the entire system is roughly given by $ g 0$ (biases a and c in Fig. 12), where $ i j$ denotes the quantum state where the $i$th state of the bare qubit and the $j$th state of the bare resonator are occupied. At on-resonance, the ground state becomes $( g 1+ e 0)/ 2$ because of the hybridization between the qubit state and the resonator state (bias b in Fig. 12). In the time domain, the population of the two systems oscillates out-of-phase. This oscillation is called the vacuum Rabi oscillation.

*vacuum*mode of the resonator. (There is no longitudinal coupling in our classical oscillators because this system is harmonic. The state of a harmonic system, i.e., boson, cannot be represented in the Bloch sphere because there is no well-defined geometrical quantization axis. However, bosons can couple to each other and exchange their energy; we just call this coupling transverse to be consistent with that for fermionic systems.)

The strong transverse qubit–qubit coupling also yields a similar anticrossing. However, the transition probability, i.e., the strength of the signal, near the anticrossing is more complex than that of the qubit–resonator coupling. The reason is that there are two types of symmetry, triplet and singlet, associated with the quantum states of two entangled qubits, and transitions between different symmetries are forbidden.^{66,68}

Note that, compared with other quantum systems, superconducting planar circuit is particularly convenient system for realizing a strong coupling because the low-dimensional nature of this system results in a strongly concentrated electromagnetic field profile and consequently produces a large $ V r , 0$ in Eq. (34).

## VI. IMPLEMENTATION OF QUANTUM COMPUTATION

### A. Equation of motion

^{53,69}

Although the Lindblad master equation is an appropriate tool to describe the dynamics of a quantum system induced by uncontrolled interactions with the environment, we need another formalism that describes the interaction between the system and a “controlled” environment, such as traveling electromagnetic fields through transmission lines, to model an actual experiment. Input–output theory is a theory for this. Here, “input” refers to the field that drives the system and “output” refers to the field that propagates away from the system. Interested readers should see Refs. 70–72.

### B. Readout

#### 1. Dispersive readout

Readout of a qubit state means to transfer the information of the qubit state to a change in a physical quantity of a classical device. At the time of writing, the standard method of detecting the superconducting qubit state is dispersive readout, i.e., detecting the qubit state by observing the shift in the resonance frequency of a readout resonator interacting with the qubit [Fig. 13(a)].

Advantages of dispersive readout are that (i) it does not rely on the dominant degree of freedom of a qubit, such as charge or flux, and (ii) its nondestructive nature. Before dispersive readout, a single-electron transistor was employed for island-based qubit readout and a DC SQUID was used for loop-based qubit readout because of their excellent sensitivity to charge and flux, respectively. The problem was that if the eigenstates of the qubit show significant spread or superposition in the number or phase basis [Figs. 5(d) and 5(e)], which happens in all noise-resilient qubits mentioned in Sec. IV B, these quantity-specific detection methods are not effective and often suffer from a strong backaction that disturbs the subsequent evolution of the measured observable. As a result, the qubit state becomes uncertain after the readout. This prevents any feedback scheme based on the measurement outcome.

In the dispersive readout scheme, a qubit state is detected and controlled by a resonator via a strong qubit–resonator interaction. However, near on-resonance ( $ \omega q\u2248 \omega r$), we cannot selectively detect or control the qubit state because, in this regime, the strong qubit–resonator interaction hybridizes the qubit and resonator states (see Sec. V C). Hence, we detune $ \omega q$ such that the qubit–resonator detuning $ \Delta qr(\u2261 \omega r\u2212 \omega q)$ is much greater than $g$ and $\kappa $. This limit is called the dispersive limit. In this off-resonant regime, a qubit transition induced by photon exchange with the resonator is negligible. However, the qubit shows small but easily measurable frequency shifts that depend on the resonator state; at the same time, the resonator also shows a small frequency shift that depends on the qubit state. The qubit state is detected by measuring this frequency shift of the resonator.

^{75,76}(the same results can be obtained using the standard perturbation theory

^{53,62}). A unitary operator $ U ^ disp= e S ^$ for the Schrieffer–Wolff transformation is defined such that $ S ^ \u2020=\u2212 S ^$ and $[ S ^, H ^ 0]=\u2212 H ^ qr$. Then, we have (use the formulas in Table II)