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

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.

TABLE I.

Brief description of the contents.

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.

TABLE II.

Useful formulas.

e A ^ B ^ e A ^ = B ^ + [ A ^ , B ^ ] + 1 2 ! [ A ^ , [ A ^ , B ^ ] ] + 1 3 ! [ A ^ , [ A ^ , [ A ^ , B ^ ] ] ] + 1 4 ! [ A ^ , [ A ^ , [ A ^ , [ A ^ , B ^ ] ] ] ] + , 
[ A ^ B ^ , C ^ ] = A ^ [ B ^ , C ^ ] + [ A ^ , C ^ ] B ^ , [ a ^ , a ^ ] = 1 , [ a ^ a ^ , a ^ ] = a ^ , [ a ^ a ^ , a ^ ] = a ^ , 
[ σ ^ x , σ ^ y ] = 2 i σ ^ z , [ σ ^ y , σ ^ z ] = 2 i σ ^ x , [ σ ^ z , σ ^ x ] = 2 i σ ^ y , 
σ ^ ± = 1 2 ( σ ^ x ± i σ ^ y ) , σ ^ + σ ^ = 1 2 ( σ ^ z + I ^ ) , [ σ ^ z , σ ^ ± ] = ± 2 σ ^ ± , [ σ ^ + , σ ^ ] = σ ^ z . 
e A ^ B ^ e A ^ = B ^ + [ A ^ , B ^ ] + 1 2 ! [ A ^ , [ A ^ , B ^ ] ] + 1 3 ! [ A ^ , [ A ^ , [ A ^ , B ^ ] ] ] + 1 4 ! [ A ^ , [ A ^ , [ A ^ , [ A ^ , B ^ ] ] ] ] + , 
[ A ^ B ^ , C ^ ] = A ^ [ B ^ , C ^ ] + [ A ^ , C ^ ] B ^ , [ a ^ , a ^ ] = 1 , [ a ^ a ^ , a ^ ] = a ^ , [ a ^ a ^ , a ^ ] = a ^ , 
[ σ ^ x , σ ^ y ] = 2 i σ ^ z , [ σ ^ y , σ ^ z ] = 2 i σ ^ x , [ σ ^ z , σ ^ x ] = 2 i σ ^ y , 
σ ^ ± = 1 2 ( σ ^ x ± i σ ^ y ) , σ ^ + σ ^ = 1 2 ( σ ^ z + I ^ ) , [ σ ^ z , σ ^ ± ] = ± 2 σ ^ ± , [ σ ^ + , σ ^ ] = σ ^ z . 

1. Quantum bit

A qubit is a two-level system whose quantum mechanical state displays phase coherence between two basis states, 0 and 1. The phase coherence between two quantum states can be defined as follows. The quantum state of a qubit, in general, is a linear superposition of the two basis states 0 and 1,
ψ = α 0 + β 1 ,
(1)
where α and β are complex numbers and | α | 2 + | β | 2 = 1. We can define the relative quantum phase φ of the two basis states as
φ arg ( α β ) .
(2)
If φ of a given quantum mechanical state has a definite value, we say that a given quantum mechanical state displays phase coherence (often just called coherence) between the states 0 and 1.
An arbitrary qubit state can also be expressed in the density matrix form:
ρ ^ = ψ ψ = 0 1 0 1 α 2 α β α β β 2
(3)
where kets and bras around the density matrix indicate the basis. Note that the diagonal elements represent the populations of the basis states and the off-diagonal elements represent the phase coherences between these states. Hence, the populations and coherences depend on the chosen basis.

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.

FIG. 1.

(a) Bloch sphere representation of the qubit state. In the energy level diagram, the two lowest states in the dashed boundary are used for computation. This subspace is called the computational subspace. To selectively control these two levels, ω q ω 1 - 2 is required, where ω q is the transition frequency between 0 and 1, and ω 1 - 2 is the transition frequency between 1 and 2. (b) In the rotating frame, “trivial evolution” is eliminated such that we can concentrate on the dynamics we are interested in. The axes with the prime indicate that we are in the rotating frame. As in the majority of the literature, all Bloch spheres in this Tutorial are in the rotating frame.

FIG. 1.

(a) Bloch sphere representation of the qubit state. In the energy level diagram, the two lowest states in the dashed boundary are used for computation. This subspace is called the computational subspace. To selectively control these two levels, ω q ω 1 - 2 is required, where ω q is the transition frequency between 0 and 1, and ω 1 - 2 is the transition frequency between 1 and 2. (b) In the rotating frame, “trivial evolution” is eliminated such that we can concentrate on the dynamics we are interested in. The axes with the prime indicate that we are in the rotating frame. As in the majority of the literature, all Bloch spheres in this Tutorial are in the rotating frame.

Close modal
In spherical coordinates, the Bloch vector s can be written as
s = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) ,
(4)
where θ and ϕ are the polar and azimuthal angles, respectively [Fig. 1(a)]. Conversion from the Bloch vector to the density matrix can be done by using the Pauli matrices,
ρ ^ = 1 2 ( I ^ + s x σ ^ x + s y σ ^ y + s z σ ^ z ) = cos 2 θ 2 e i ϕ cos θ 2 sin θ 2 e i ϕ cos θ 2 sin θ 2 sin 2 θ 2 .
(5)
One possible mapping between Eqs. (3) and (5) is α = cos ( θ / 2 ) and β = e i ϕ sin ( θ / 2 ).
The rotation of the Bloch vector with the angle η about the k axis is done by the rotation operator R ^ k ( η ),
R ^ k ( η ) exp i η σ ^ k 2 = cos η 2 I ^ i sin η 2 σ ^ k .
(6)
Here, we used the formula
exp ( i η A ^ ) = 1 η 2 2 ! + I ^ i η η 3 3 ! + A ^ = cos ( η ) I ^ i sin ( η ) A ^ , if A ^ 2 = I ^ .
(7)

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, ω q, which we call the qubit frequency, is the transition frequency between 0 and 1, and ω i - j (with a hyphen in the subscript) is the transition frequency between i and j; ω i j (without a hyphen in the subscript) indicates the energy level of the two-qubit state, i j (or i j in the short form).

A generic two-qubit state can be written as
ψ = α 00 + β 01 + γ 10 + δ 11 ,
(8)
where α, β, γ, and δ are complex numbers and | α | 2 + | β | 2 + | γ | 2 + | δ | 2 = 1. In the density matrix form,
(9)

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.

Consider a closed quantum system described by the time-independent Hamiltonian
H ^
. The time evolution of such a system is described by a unitary operator U ^ ( t ), which is called the time-evolution operator or the propagator,8,9
ψ ( t ) = U ^ ( t ) ψ ( 0 ) .
(10)
The Schrödinger equation connects U ^ ( t ) and H ^,
U ^ ( t ) = e i H ^ t / .
(11)
Thus, a gate operation is implemented by engineering the system Hamiltonian such that the resulting unitary evolution of the qubits implements the target gate.

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 theorem10). 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.

TABLE III.

Universal quantum gate set. R^k(η) (k = x, y, z) is the rotation operator defined in Eq. (6). ψt(c) indicates the quantum state of the target (control) qubit. X^ is the X gate in the operator form.

NameFunctionSymbolMatrix
Pauli-X (XR^x(π)  0110 
Pauli-Y (YR^y(π)  0ii0 
Pauli-Z (ZR^z(π)  1001 
Hadamard (HR^x(π)R^x(π/2)  121111 
Phase (SR^z(π/2)  100i 
π/8 (TR^z(π/4)  100eiπ/4 
Controlled-NOT(CNOT) X^ψtifψc=1  1000010000010010 
NameFunctionSymbolMatrix
Pauli-X (XR^x(π)  0110 
Pauli-Y (YR^y(π)  0ii0 
Pauli-Z (ZR^z(π)  1001 
Hadamard (HR^x(π)R^x(π/2)  121111 
Phase (SR^z(π/2)  100i 
π/8 (TR^z(π/4)  100eiπ/4 
Controlled-NOT(CNOT) X^ψtifψc=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 theorem10). Thus, a non-Clifford gate, such as the T gate, is required to show the advantage of quantum computation.

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.

FIG. 2.

Structure of a universal quantum computing system.

FIG. 2.

Structure of a universal quantum computing system.

Close modal

In this Tutorial, the physical resources, the error correction resources, and part of the logical resources are briefly covered. For quantum algorithms, see the standard textbooks on quantum computation, such as Refs. 10 and 13.

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:

  1. 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 ( 200 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.

  2. Large anharmonicity: To be a well-defined two-level system, a qubit should have anharmonicity α ω 1 - 2 ω q of at least 100 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).

  3. Long coherence time: The assigned quantum state should last for a long time compared with the time for gate operations.

  4. 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.

  5. Ease of control: The quantum state should be brought to a superposition easily and straightforwardly by an external mean.

  6. Ease of fabrication: A qubit should be easy to fabricate with standard nanotechnology for good reproducibility.

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 Ψ. 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)

The solution for the anharmonicity is a Josephson junction where a pair of superconductors are weakly coupled [Fig. 3(a)]. In a superconducting circuit, a Josephson junction acts as a nonlinear inductor, resulting in an anharmonic potential. Since a superconductor is a macroscopic quantum mechanical system, only two quantities are required to describe the physics of a Josephson junction: the number imbalance of electrons N and the relative phase φ between the two superconductors. Here, N corresponds to the difference in | Ψ | 2 of the two superconductors. The equations of motion regarding these two quantities, called the Josephson equations, are given by15,
d N ( t ) d t = 2 E J sin φ ( t ) and d φ ( t ) d t = 2 e V ( t ) ,
(12)
where E J is the Josephson energy, which is a measure of the ability of Cooper pairs to tunnel through the junction; is the reduced Planck constant; e is the magnitude of the charge carried by a single electron; and V is the voltage difference maintained across the junction. The popular form of the left equation in Eq. (12) is16 
I s ( t ) = I c sin φ ( t ) ,
(13)
where I s is a zero-voltage supercurrent flow through the junction and I c ( = 2 e E J / ) is the maximum current that can flow through the junction, i.e., the critical current of the junction.
FIG. 3.

(a) Schematic diagram of a Josephson junction where a pair of superconductors are weakly coupled via an oxide tunnel barrier. The phase and the number difference of the macroscopic wavefunctions Ψ 1 and Ψ 2 fully determine the physics of the junction. (b) A DC SQUID can be considered as a variable Josephson junction tuned by an external magnetic flux Φ. The symbol of a cross in a square represents a Josephson junction.

FIG. 3.

(a) Schematic diagram of a Josephson junction where a pair of superconductors are weakly coupled via an oxide tunnel barrier. The phase and the number difference of the macroscopic wavefunctions Ψ 1 and Ψ 2 fully determine the physics of the junction. (b) A DC SQUID can be considered as a variable Josephson junction tuned by an external magnetic flux Φ. The symbol of a cross in a square represents a Josephson junction.

Close modal
Here, we point out that a DC Superconducting Quantum Interference Device (DC SQUID), which consists of two Josephson junctions and a superconducting loop [Fig. 3(b)], can be considered as a variable Josephson junction whose effective Josephson energy E J , eff as a function of the external flux bias Φ is given by
E J , eff ( φ ext ) = E J , 1 2 + E J , 2 2 + 2 E J , 1 E J , 2 cos φ ext ,
(14)
where φ ext( 2 π Φ / Φ 0) is the phase offset due to the external flux bias. This idea is useful for making tunable superconducting devices.

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.

FIG. 4.

(a) and (b) Elementary circuits of a superconducting qubit. The dashed boundary line indicates the charge island. In a certain parameter range, their operations as qubits can be understood as the coherent tunneling of charge or flux (see Secs. III C 2 and III C 3). (c) Parallel circuit composed of an inductor with L, a capacitor with C, and a Josephson junction with the critical current I c as a generic model of a superconducting qubit.

FIG. 4.

(a) and (b) Elementary circuits of a superconducting qubit. The dashed boundary line indicates the charge island. In a certain parameter range, their operations as qubits can be understood as the coherent tunneling of charge or flux (see Secs. III C 2 and III C 3). (c) Parallel circuit composed of an inductor with L, a capacitor with C, and a Josephson junction with the critical current I c as a generic model of a superconducting qubit.

Close modal
FIG. 5.

(a)–(c) Energy levels of an island-based qubit with three different E J / E C ratios. (d) and (e) Wavefunctions of the ground state g and the excited state e in the number and phase bases. (f) If the displacement in the phase basis is reasonably small, which is the case when E J / E C 1, we can approximate the cosine potential U (solid line) as a weakly nonlinear harmonic potential (dashed line). (g) Transition frequency between g and e, ω q, as a function of E J / E C ratio at N ext = 0 (dashed line) and 0.5 (solid line). The inset shows the E J / E C dependence of Δ ω q[ ω q ( N ext = 0 ) ω q ( 0.5 )]. (h) Anharmonicity α( ω e - f ω q, where ω e - f is the transition frequency between e and the higher excitation level f) as a function of E J / E C ratio.

FIG. 5.

(a)–(c) Energy levels of an island-based qubit with three different E J / E C ratios. (d) and (e) Wavefunctions of the ground state g and the excited state e in the number and phase bases. (f) If the displacement in the phase basis is reasonably small, which is the case when E J / E C 1, we can approximate the cosine potential U (solid line) as a weakly nonlinear harmonic potential (dashed line). (g) Transition frequency between g and e, ω q, as a function of E J / E C ratio at N ext = 0 (dashed line) and 0.5 (solid line). The inset shows the E J / E C dependence of Δ ω q[ ω q ( N ext = 0 ) ω q ( 0.5 )]. (h) Anharmonicity α( ω e - f ω q, where ω e - f is the transition frequency between e and the higher excitation level f) as a function of E J / E C ratio.

Close modal
We introduce a Hamiltonian for a parallel circuit composed of a capacitor with the capacitance C including the intrinsic capacitance of a junction, an inductor with the inductance L, and a Josephson junction, as a generic model of a superconducting qubit [Fig. 4(c)]. This particular circuit is easily quantized by treating N and φ as the operators N ^ and φ ^. (The standard introduction to superconducting circuit quantization is Ref. 18.) Here, the number operator N ^ is conjugate to the phase operator φ ^: N ^ = i / φ ^. (They satisfy the relation e i φ ^ N ^ e i φ ^ = N ^ 1. The popular form of this relation is [ φ ^ , N ^ ] = i. However, this form is not mathematically rigorous because the phase operator is not Hermitian. It holds approximately only if N ^ and φ ^ are the 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
H ^ q = 4 E C ( N ^ N ext ) 2 + 1 2 E L φ ^ 2 E J cos ( φ ^ φ ext ) ,
(15)
where E C ( e 2 / 2 C ) is the capacitive energy, which is the energy cost to charge a capacitor with a single electron (the factor of 4 comes from the Cooper pairing), and E L [ ( Φ 0 / 2 π ) 2 / L ] is the inductive energy, which is the energy cost to “charge” an inductor with a single flux quantum Φ 0. The E J term, which represents the energy stored in the junction, was obtained by integrating the electrical work I s V d t with Eqs. (12) and (13). Last, N ext is the charge offset due to the external voltage bias and φ ext( 2 π Φ / Φ 0) is the phase offset due to the external flux bias Φ. Inserting this phase offset in the Josephson junction term is based on the assumption that a magnetic flux penetrates into the loop through the junction, not through the inductor.

Equation (15) suggests that characteristics of a superconducting qubit can be engineered by three circuit parameters, E J, E C, and E L. In Secs. III C 2 and III C 3, we explore how these circuit parameters determine the basic properties of a qubit.

2. Island-based qubit

The Hamiltonian of an island-based qubit is Eq. (15) in the E L 0 and φ ext 0 limits,
H ^ q = 4 E C ( N ^ N ext ) 2 E J cos φ ^ .
(16)
Therefore, the properties of an island-based qubit are mainly determined by the ratio E J / E C.

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 ± 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 = 1 and 1. This results in the first excitation level at 4 E C.

As E J / E C increases, the contribution from the anticrossing dominates [Fig. 5(b)]; eventually, in the large E J / E C limit, the energy levels become flat, i.e., insensitive to N ext [Fig. 5(c)]. In this regime, the wavefunctions are localized in the phase basis as shown in Fig. 5(e); hence, it is reasonable to treat the E J term in Eq. (15) as the periodic potential and the E C term as the kinetic term. In addition, since the kinetic term is much less than the potential term ( E J / E C 1), the displacement in the phase basis during the evolution of the qubit state is small. Thus, as depicted in Fig. 5(f), we can approximate the periodic potential (solid line) as a weakly nonlinear harmonic potential (dashed line). Then, the qubit frequency can be obtained by approximating Eq. (16) as a quantum harmonic oscillator (QHO) ( cos φ ^ 1 φ ^ 2 / 2),
H ^ q 4 E C N ^ 2 + 1 2 E J φ ^ 2 ,
(17)
where N ext is ignored because the energy levels are insensitive to N ext. By comparing Eq. (17) with a standard spring-block harmonic oscillator with the spring constant k and the mass m, we obtain ω q 8 E J E C / from ω 0 = k / m, where ω 0 is the resonance frequency of the spring-block oscillator. Since we are considering the regime where φ ^ is localized, C corresponds to m, and L J 1 corresponds to k, where L J[ ( Φ 0 / 2 π ) 2 / E J] is the effective inductance of the junction. Moreover, in this large E J / E C regime, the anharmonicity will decrease with an increase in E J / E C because the Hamiltonian [Eq. (16)] becomes closer to that of a quantum harmonic oscillator [Eq. (17)] in this direction.

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 Δ ω 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 50.

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 α 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 φ, 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 1. In this regime, the periodic shape is prominent in the potential as shown in Figs. 6(a)6(d). When E J / E C 1 [Fig. 6(e)], the energy level diagram is almost independent of Φ, and ω q 8 E L E C / . 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 1 corresponds to the spring constant.

FIG. 6.

(a)–(h) Potential U [(a)–(d)] and energy level diagrams [(e)–(h)] of a loop-based qubit when E J / E L 1. The potential-energy level correspondence is [(a) and (e)], [(b), (c) and (f), (g)], and [(d) and (h)]. Φ is the external flux bias and Φ 0 is the flux quantum. The inset in (b) shows the circulating currents in the circuit; the inset in (d) shows a circuit describing a loop-based qubit as two charge islands (dashed lines) that are connected by an inductor. Gray lines in (f)–(h) indicate how the ground and first excited levels appear from Eq. (15). (i)–(l) Similar diagrams for the potential and the energy levels when E J / E L 1. (m) Transition frequencies ω q at Φ / Φ 0 = 0 (solid lines in the upper panel) and 0.5 (dashed lines), and their difference Δ ω q (lower panel) as a function of E J / E C. Arrows in the upper panel indicate the E J / E L and E J / E C ratios employed in (e)–(h). (n) Anharmonicity as a function of E J / E C. At Φ / Φ 0 = 0.5, α is positive, while it is negative at zero bias. In (m) and (n), the numbers near solid lines indicate the corresponding E J / E L ratios.

FIG. 6.

(a)–(h) Potential U [(a)–(d)] and energy level diagrams [(e)–(h)] of a loop-based qubit when E J / E L 1. The potential-energy level correspondence is [(a) and (e)], [(b), (c) and (f), (g)], and [(d) and (h)]. Φ is the external flux bias and Φ 0 is the flux quantum. The inset in (b) shows the circulating currents in the circuit; the inset in (d) shows a circuit describing a loop-based qubit as two charge islands (dashed lines) that are connected by an inductor. Gray lines in (f)–(h) indicate how the ground and first excited levels appear from Eq. (15). (i)–(l) Similar diagrams for the potential and the energy levels when E J / E L 1. (m) Transition frequencies ω q at Φ / Φ 0 = 0 (solid lines in the upper panel) and 0.5 (dashed lines), and their difference Δ ω q (lower panel) as a function of E J / E C. Arrows in the upper panel indicate the E J / E L and E J / E C ratios employed in (e)–(h). (n) Anharmonicity as a function of E J / E C. At Φ / Φ 0 = 0.5, α is positive, while it is negative at zero bias. In (m) and (n), the numbers near solid lines indicate the corresponding E J / E L ratios.

Close modal

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 φ = 0 and ± 2 π, which means that the numbers of trapped fluxes in the loop are 0 and ± 1, respectively. Note that, at Φ / Φ 0 = 0.5 ( φ ext = π), the potential has a double-well shape, resulting in energy degeneracy between φ + π and φ π. 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 ω q at Φ / Φ 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 φ ± 2 π, 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 Φ / Φ 0 = 0.5. Hence, ω q at zero bias is approximately 2 π 2 E L[ = E L ( ± 2 π ) 2 / 2] and weakly depends on E J / E C. This explains why ω 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, ω q 8 E J E C / . 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 1 such that the reactance at ω q is significant, whereas the circuit is electrically shorted at the low-frequency limit.

In the regime E J / E L 1, 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 Φ / Φ 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 ω q at Φ / Φ 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 0.1 to 100. In this range, ω q of a loop-based qubit with E J / E L 1 at Φ / Φ 0 = 0.5 is often too low to satisfy condition 1 in Sec. III A, while ω q of a qubit with E J / E L 1 at zero bias is too high. Regarding the anharmonicity, a loop-based qubit with E J / E L 1 is more advantageous than that with E J / E L 1 as shown in Fig. 6(n). If E L increases even further such that E J / E L 1, the potential becomes almost harmonic, and as a result, the circuit does not show enough anharmonicity to be a qubit.

1. Concept

In general, the states of a qubit cannot last forever; after some time, they relax back to the ground state because of the interaction between the qubit and the surrounding environment. This is the reason for having condition 3 in Sec. III A. We can define two experimentally measurable time scales that characterize the relaxation of a quantum state [Fig. 7(a)]: one is the longitudinal relaxation time ( T 1) and the other is the transverse relaxation time ( T 2). As the name implies, T 1 is the time constant for recovering the longitudinal component of the Bloch vector to its thermal equilibrium value. Thus, the physical process responsible for T 1 is thermalization of the qubit. (We note that a considerable number of papers use the term “depolarizing” to describe the physical process responsible for T 1. We do not use this term to avoid confusion with the depolarizing channel, which contracts the Bloch vector 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 as21 
1 T 1 = Γ , 1 T 2 = Γ 2 + Γ φ ,
(18)
where Γ is the decay rate of the excited state population and Γ φ is the dephasing rate. The reason for the factor of 2 in Γ / 2 will be given in Sec. VI A. The measurement procedures for T 1 and T 2 are described in Sec. VIII B.
FIG. 7.

(a) Relaxation of qubit states in a set of identical measurements represented in the Bloch sphere. Each arrow represents the qubit state for each measurement. Primes ( ) in the axes indicate the rotating frame with the average qubit transition frequency. The numbers in circles indicate the time instant during a single measurement. The thick arrow growing along the z axis represents the longitudinal relaxation, while spreading and shortening arrows in the x y -plane represent the transverse relaxation. (b) In spectroscopic measurements, the qubit transition frequency varies with time because of noises from the surrounding environment. In general, a large deviation from the center is unlikely to occur as shown in the histogram (left figure). Such a fluctuation broadens the qubit spectrum (right figure). (The measurement procedure is described in Sec. VIII A 2.) This phenomenon, called dephasing, corresponds to the spreading of arrows in (a). (c) The temporal fluctuation in qubit transition frequency induces the loss of phase coherence between the signals obtained in each measurement (left figure). The averaged signal is a decaying signal with the time constant T 2 (right figure). (The measurement procedure is described in Sec. VIII B 3.) Note that the Fourier transform connects the decay in time-domain measurement and the spread in spectroscopic measurement; hence, the width of the qubit spectrum is about 1 / π T 2 in Hz as shown in (b). In (b) and (c), T 1 is assumed to be much longer than T 2. (d) Relaxation mechanisms. Thermalization is due to incoherent energy exchange between the qubit and the environment. Dephasing is due to fluctuations in the transition frequency of the qubit, δ ω q. (e) Thermalization can be suppressed by reducing the overlap between the ground-state and excited-state wavefunctions in the circuit variable space, such as N ^ and φ ^ in Eq. (15). In this figure, the circuit variables are denoted by x 1 and x 2 for generality. (f) Dephasing can be suppressed by designing the qubit to be less sensitive to the external bias and operating the qubit at its sweet spot. The figure shows the schematic external bias dependence of the qubit transition frequency ( ω q). Red circles indicate sweet spots.

FIG. 7.

(a) Relaxation of qubit states in a set of identical measurements represented in the Bloch sphere. Each arrow represents the qubit state for each measurement. Primes ( ) in the axes indicate the rotating frame with the average qubit transition frequency. The numbers in circles indicate the time instant during a single measurement. The thick arrow growing along the z axis represents the longitudinal relaxation, while spreading and shortening arrows in the x y -plane represent the transverse relaxation. (b) In spectroscopic measurements, the qubit transition frequency varies with time because of noises from the surrounding environment. In general, a large deviation from the center is unlikely to occur as shown in the histogram (left figure). Such a fluctuation broadens the qubit spectrum (right figure). (The measurement procedure is described in Sec. VIII A 2.) This phenomenon, called dephasing, corresponds to the spreading of arrows in (a). (c) The temporal fluctuation in qubit transition frequency induces the loss of phase coherence between the signals obtained in each measurement (left figure). The averaged signal is a decaying signal with the time constant T 2 (right figure). (The measurement procedure is described in Sec. VIII B 3.) Note that the Fourier transform connects the decay in time-domain measurement and the spread in spectroscopic measurement; hence, the width of the qubit spectrum is about 1 / π T 2 in Hz as shown in (b). In (b) and (c), T 1 is assumed to be much longer than T 2. (d) Relaxation mechanisms. Thermalization is due to incoherent energy exchange between the qubit and the environment. Dephasing is due to fluctuations in the transition frequency of the qubit, δ ω q. (e) Thermalization can be suppressed by reducing the overlap between the ground-state and excited-state wavefunctions in the circuit variable space, such as N ^ and φ ^ in Eq. (15). In this figure, the circuit variables are denoted by x 1 and x 2 for generality. (f) Dephasing can be suppressed by designing the qubit to be less sensitive to the external bias and operating the qubit at its sweet spot. The figure shows the schematic external bias dependence of the qubit transition frequency ( ω q). Red circles indicate sweet spots.

Close modal

The interaction with the surrounding environment is usually treated as various noise processes. The effects of noises are explained further in Secs. IV A 2 and IV A 3.

2. Thermalization

Thermalization of a qubit occurs via incoherent energy exchange between the qubit and the environment [Fig. 7(d)]. The effect of such an interaction is usually modeled as fluctuations in qubit Hamiltonian. In the qubit Hamiltonian, there are physical quantities that mediate the interaction between the qubit and the environment, such as external charge and flux biases. If we denote such a physical quantity as λ, the susceptibility of the qubit Hamiltonian to the fluctuation in λ, denoted by X ^ λ, is given by the derivative of the qubit Hamiltonian H ^ q with respect to λ. For example, the noise caused by fluctuating charges nearby, called charge noise, is coupled to the qubit through the external charge bias; hence, λ = N ext. For the noise caused by fluctuating spins, called flux noise, λ = φ ext. Similarly, the effect of the fluctuation in critical current can be estimated by λ = E J (or I c). Then, for H ^ q in Eq. (15), X ^ λ for these quantities are given by
X ^ N = H ^ q N ext = 8 E C N ^ ,
(19)
X ^ Φ = H ^ q φ ext = E J sin ( φ ^ φ ext ) ,
(20)
X ^ E J = H ^ q E J E J = E J cos ( φ ^ φ ext ) .
(21)
Here, we insert E J in Eq. (21) to set the dimension of X ^ E J identical to the other two equations for fair comparison.
To thermalize a qubit, the environment must be able to dissipate an electromagnetic energy near ω q. Thus, the thermalization process is governed by the noise whose frequency is near ω q. Using Fermi’s golden rule, Γ can be written as21 
Γ = 1 2 λ | 1 | X ^ λ | 0 | 2 | S λ ( ω q ) + S λ ( ω q ) | ,
(22)
where S λ is the noise power spectral density associated with the fluctuation in λ. On the basis of Eq. (22), the thermalization process is often interpreted as the dipole transition associated with the effective dipole moment X ^ λ.
In Eq. (22), S λ ( ω q ) and S λ ( ω q ) represent emission (from 1 to 0) and absorption (from 0 to 1), respectively. When the qubit frequency is much greater than the temperature of the environment T, i.e., ω q k B T, we can safely ignore the contribution from the absorption process. Then, we have
S λ ( ω q ) + S λ ( ω q ) A λ 2 2 π × 1 [ Hz ] ω q μ ,
(23)
where A λ is the noise magnitude, and μ 1 for 1 / f noise and μ 1 for Ohmic noise. It has been reported that Ohmic noise is chiefly responsible for Γ (Refs. 22 and 23) and 1 / f noise is responsible for Γ φ.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 φ ^, i.e., the overlap between wavefunctions in the circuit variable space.

Currently, there are three approaches to suppressing thermalization:

  1. 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.

  2. 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

  3. 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

Dephasing is due to the temporal fluctuation in the transition frequency δ ω q [Fig. 7(d)], which can be expressed as21 
δ ω q | 1 | X ^ λ | 1 0 | X ^ λ | 0 | .
(24)
Equation (24) suggests that the dephasing is determined by the diagonal matrix elements of N ^ and φ ^ [Eqs. (19)–(21)]. In the Bloch sphere [Fig. 7(a)], if ω q is the same as the frequency of the rotating frame, the transverse component will lie along the y axis. However, owing to the fluctuation in ω q, the transverse component rotates around the z axis with amount of rotation differing from measurement to another measurement, resulting in the spreading of arrows [Fig. 7(a)]. As a result, the qubit loses the phase coherence and the averaged transverse component in the Bloch vector decays in time as shown in Fig. 7(c). To yield such a decay, the time scale of the fluctuation must be much slower than the qubit transition (thus, adiabatic) and should be a similar order of magnitude to the measurement time scale. Therefore, the dephasing rate Γ φ is mainly determined by low-frequency noise.
To estimate Γ φ, we have to perform an integration with respect to the frequency of the noise. For this, we set the low-frequency ω low and high-frequency ω high cutoffs. If our time scale of interest τ satisfies ω low τ 1 and ω high τ 1, Γ φ is roughly given by20,21
Γ φ A λ ω q λ ,
(25)
where A λ is the noise magnitude defined in Eq. (23).

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

  1. 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.

  2. Bias dependence: Since Γ φ is proportional to λ ω 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 λ ω 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 [ Γ φ = 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.

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).

FIG. 8.

Conceptual evolution of noise-resilient qubit designs from an island-based qubit (a) and a loop-based qubit (b). Dashed boundaries indicate islands. JJ stands for Josephson junction.

FIG. 8.

Conceptual evolution of noise-resilient qubit designs from an island-based qubit (a) and a loop-based qubit (b). Dashed boundaries indicate islands. JJ stands for Josephson junction.

Close modal

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 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.

TABLE IV.

Circuit parameters of several noise-resilient qubit designs. For the flux qubit and fluxonium, which have multiple junctions, EJ is for the smallest junction [the black junctions in Fig. 8(b)]. In addition, the flux qubit and fluxonium considered in this table are capacitively shunted ones.

TypeEJ (GHz)EJ/ECEJ/ELβNJReference
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  
TypeEJ (GHz)EJ/ECEJ/ELβNJReference
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 β 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 “ α junction,” where α = β 1, for historical reasons.) Since the flux tunneling rate through a Josephson junction is roughly proportional to exp ( E J / E C ) (Ref. 44), β must be larger than 1.

The resulting potential term U ^ is given by
U ^ E J cos φ ^ β E J i = 1 N J cos φ ^ i ,
(26)
where N J is the number of additional Josephson junctions and φ ^ i is the phase difference across additional junction i. Note that the loop inductance does not appear in Eq. (26). The reason is that the phase variable associated with the loop inductance is nearly zero because of the large E L; thus, its contribution to the qubit dynamics is small compared with that from the additional junctions. On the other hand, in the phase dimensions associated with the additional junctions, the potential has periodic modulations that can support coherent flux tunneling.45,46

First, we consider the case when N J is 2 or 3 and β 2 [upper figures in Fig. 8(b)]. The circuit with these parameters, which roughly corresponds to a loop-based qubit with E J / E L 1, 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 φ 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 10–100 GHz and β 2 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 Φ / Φ 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 1 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 Φ / Φ 0 = 0.5.

With a sufficiently large N J ( 10–100), Eq. (26) can be treated as a linear inductor with E L β 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 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 β and N J, we can satisfy E L E J. A superconducting qubit in this regime is called a fluxonium or an RF SQUID qubit. In this case, it is easy for ω q at zero bias to satisfy condition 1 in Sec. III A; at Φ / Φ 0 = 0.5, ω 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 6.5 k Ω because of stray capacitance and radiation to vacuum, whose impedance is about 377  Ω. 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 

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.

Before exploring a quantum system, considering a similar classical system is often helpful to understand the physics in the quantum regime. As we will see soon, the physics behind various couplings associated with qubits can be captured using two simple classical harmonic oscillators. Figure 9 shows a schematic diagram of our model system: it is composed of two classical simple harmonic oscillators, each made of a spring and a block. The two oscillators interact via the coupling spring, whose spring constant can be either static [Figs. 9(a) and 9(c)] or time-varying [Fig. 9(b)]. In addition, oscillator 1 may be driven by an external force [Fig. 9(c)]. The equations of motion of this system are given by
m 1 x ¨ 1 = ( k 1 + κ ) x 1 + κ x 2 + A d cos ( 2 π f d t ) , m 2 x ¨ 2 = ( k 2 + κ ) x 2 + κ x 1 ,
(27)
where m i is the mass of oscillator i, where i = 1 , 2; k i is the spring constant of oscillator i; x i is the position of the center of block i; κ is the spring constant of the coupling spring, which can be decomposed into two parts, namely, the static κ 0 and the time-varying κ m cos ( 2 π f m t ); and A d and f d are the amplitude and the frequency of the drive, respectively.
FIG. 9.

Two coupled classical harmonic oscillators without damping. Each oscillator is composed of a spring and a block. k i is the spring constant of oscillator i; f i is the resonance frequency; and x i is the position. Two oscillators are coupled via a coupling spring whose spring constant is κ. The graphs show the solutions of Eq. (27) for various conditions. (a) Evolution of the system when the coupling is static: κ = κ 0. The two oscillators do not interact with each other if f 1 f 2; however, if f 1 = f 2, the oscillators exchange their energy at a rate of 2 δ f κ 0, whose quantity is determined by κ 0. As shown by the Fourier-transformed solution, the energy exchange can be interpreted as a splitting of the resonance frequency with 2 δ f κ 0. (b) Evolution of the system with a time-varying coupling constant κ 0 + κ m cos ( 2 π f m t ), where f m is the modulation frequency. When f m = | f 1 f 2 |, the two oscillators exchange their energy even if f 1 f 2. The Fourier transform shows that the resonance frequency of each oscillator is split by 2 δ f κ m whose quantity is determined by κ m. (c) Evolution of the system with the static coupling and an external drive acting on oscillator 1. Here, oscillator 1 is the control oscillator and oscillator 2 is the target oscillator. The amplitude and the frequency of the drive are denoted by A d and f d, respectively. The parameter sets are given as follows: { m 1 = 10 / ( 2 π ) 2, m 2 = 2.5 / ( 2 π ) 2, k 1 = 10, k 2 = 40, κ 0 = 1}, if not specified. In (a), k 2 = 10 for f 1 = f 2. In (b), { κ m = 3, f m = | f 1 f 2 |} and { κ m = 0.75, f m = f 1 + f 2}. In (c), A d = 30, and f d = f 2 + 1 or f 2. The initial conditions are as follows: for (a) and (b), { x 1 ( t = 0 ) = 1, x ˙ 1 ( 0 ) = 0, x 2 ( 0 ) = 0, x ˙ 2 ( 0 ) = 0}; and for (c), {0, 0, 0, 0}. All numbers are unitless. (d) Correspondence between classical analogies in this figure and required operations for quantum computation covered in this Tutorial. The left column indicates the analogies in this figure and actual quantum oscillators; the right column indicates the target operation. For example, the last row means that “we can understand the physics of the single-qubit gate operation by replacing oscillators 1 and 2 in (c) with a Quantum Harmonic Oscillator (QHO) and a qubit, respectively.”

FIG. 9.

Two coupled classical harmonic oscillators without damping. Each oscillator is composed of a spring and a block. k i is the spring constant of oscillator i; f i is the resonance frequency; and x i is the position. Two oscillators are coupled via a coupling spring whose spring constant is κ. The graphs show the solutions of Eq. (27) for various conditions. (a) Evolution of the system when the coupling is static: κ = κ 0. The two oscillators do not interact with each other if f 1 f 2; however, if f 1 = f 2, the oscillators exchange their energy at a rate of 2 δ f κ 0, whose quantity is determined by κ 0. As shown by the Fourier-transformed solution, the energy exchange can be interpreted as a splitting of the resonance frequency with 2 δ f κ 0. (b) Evolution of the system with a time-varying coupling constant κ 0 + κ m cos ( 2 π f m t ), where f m is the modulation frequency. When f m = | f 1 f 2 |, the two oscillators exchange their energy even if f 1 f 2. The Fourier transform shows that the resonance frequency of each oscillator is split by 2 δ f κ m whose quantity is determined by κ m. (c) Evolution of the system with the static coupling and an external drive acting on oscillator 1. Here, oscillator 1 is the control oscillator and oscillator 2 is the target oscillator. The amplitude and the frequency of the drive are denoted by A d and f d, respectively. The parameter sets are given as follows: { m 1 = 10 / ( 2 π ) 2, m 2 = 2.5 / ( 2 π ) 2, k 1 = 10, k 2 = 40, κ 0 = 1}, if not specified. In (a), k 2 = 10 for f 1 = f 2. In (b), { κ m = 3, f m = | f 1 f 2 |} and { κ m = 0.75, f m = f 1 + f 2}. In (c), A d = 30, and f d = f 2 + 1 or f 2. The initial conditions are as follows: for (a) and (b), { x 1 ( t = 0 ) = 1, x ˙ 1 ( 0 ) = 0, x 2 ( 0 ) = 0, x ˙ 2 ( 0 ) = 0}; and for (c), {0, 0, 0, 0}. All numbers are unitless. (d) Correspondence between classical analogies in this figure and required operations for quantum computation covered in this Tutorial. The left column indicates the analogies in this figure and actual quantum oscillators; the right column indicates the target operation. For example, the last row means that “we can understand the physics of the single-qubit gate operation by replacing oscillators 1 and 2 in (c) with a Quantum Harmonic Oscillator (QHO) and a qubit, respectively.”

Close modal

Note that, although the coupling spring is always present, its effect on the dynamics strongly depends on the system parameters. When κ is the static parameter κ 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 κ with the frequency difference between the two oscillators, | f 1 f 2 |, where 2 π f i = ( k i + κ 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.

1. Qubit, resonator, and somewhere between them

Now that we are well equipped with the necessary physics, it is time to move back to quantum. In cQED, the circuit Hamiltonians for a qubit and a resonator are mapped to spin-1/2 fermionic and bosonic Hamiltonians, respectively,
H ^ q ω q σ ^ z 2 , H ^ r ω r a ^ a ^ ,
(28)
where σ ^ z is the Pauli z operator and a ^ ( a ^) is the bosonic creation (annihilation) operator.
Note that this mapping assumes an ideal two-level system and a single-mode resonator. As we will see in Secs. VI D and VI E 1, however, higher excitation levels of a qubit have to be considered in many situations, especially for a transmon whose anharmonicity is weak. On the other hand, the resonator may show a small nonlinearity that has to be considered for high-fidelity control. Hence, we sketch how to model a transmon in the second quantization formalism as an example of a weakly anharmonic/nonlinear system. By expanding Eq. (16), we have
H ^ q 4 E C N ^ 2 + 1 2 E J φ ^ 2 1 24 E J φ ^ 4 .
(29)
In Eq. (29), the first two terms, i.e., harmonic terms, can be diagonalized by defining
N ^ = i N 0 ( b ^ b ^ ) , φ ^ = φ 0 ( b ^ + b ^ ) ,
(30)
where N 0 2 = E J / 32 E C and φ 0 2 = 2 E C / E J are the zero-point fluctuations, and b ^ ( b ^) is the bosonic creation (annihilation) operator for a transmon. Note that the zero-point fluctuations are determined by the E J / E C ratio. After normal ordering, we obtain
H ^ q 8 E J E C E C b ^ b ^ E C 2 b ^ b ^ b ^ b ^ .
(31)
Here, the terms whose mean excitation number is nonzero, such as b ^ b ^ and b ^ b ^ , are ignored because the dynamics induced by these terms will be averaged out at the time scale we are interested in (this is the rotating wave approximation, which will be introduced formally in Sec. V B 2). Thus, a transmon can be mapped into a harmonic oscillator with the Kerr-type nonlinearity,
H ^ q ω q b ^ b ^ + K 2 b ^ b ^ b ^ b ^ ,
(32)
where K is the Kerr coefficient. Equations (31) and (32) suggest that ω q is about 8 E J E C / (because E J / E C 1) and the anharmonicity, i.e., K, is E C. These results are consistent with Secs. III C 2 and IV B 1.

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

Consider a single qubit capacitively coupled to a single-mode resonator without an external Drive. (Usually capacitive coupling is easier to design because, for inductive coupling, we need to consider not only geometric inductance but also kinetic inductance, which is harder to simulate or estimate than the capacitance.) One example is shown in Fig. 10(a). In this case, the physical process of the coupling is the zero-point voltage fluctuation of the resonator acting on the net charge 2 e N ^ ( 2 e is the charge of a Cooper pair) via the coupling capacitor between the qubit and the resonator [the capacitor labeled g in Fig. 10(a)]. Then, 2 e N ^ can be considered as the effective dipole moment of this artificial atom [see Eq. (19)]. The qubit–resonator coupling Hamiltonian H ^ qr can be written as
H ^ qr = 2 e N ^ β V r , 0 ( a ^ + a ^ ) .
(33)
Here, V r , 0( = ω r / 2 C r, where C r is the capacitance of the resonator) is the root mean square voltage of the zero-point fluctuation in the resonator; β is the ratio between the coupling and stored energies, which is the same as the ratio of the coupling capacitance to the total capacitance of the qubit; and ( a ^ + a ^ ) represents the process of populating/depopulating the resonator. Defining the coupling constant,
g i j = 2 e β V r , 0 i N ^ j ,
(34)
where i and j ( i , j = 0 , 1) are the eigenstates of the bare qubit, yields
H ^ qr = i , j g i j | i j | ( a ^ + a ^ ) = ( g x σ ^ x + g z σ ^ z ) ( a ^ + a ^ ) ,
(35)
where g x and g z are defined by
g x g 01 ( = g 10 ) , g z g 00 g 11 2 .
(36)
The g x term is called the transverse coupling because the axis for the Pauli operator is perpendicular to the qubit quantization axis; the g z term is called the longitudinal coupling. Here, a term associated with y is omitted because the choice of x or y is just a matter of convention.
FIG. 10.

(a) In a superconducting circuit, the atom–cavity interaction can be implemented by the qubit–resonator coupling. Here, the resonator can be either a planar resonator or a 3D cavity; in either case, it is usually modeled as an L C circuit. The circuit shows the capacitive coupling between a quarter-wavelength ( λ / 4) resonator and an island-based qubit. Here, g represents the strength of the transverse coupling between the resonator and the qubit; κ represents the loss rate of photons from the resonator; and γ represents the transverse relaxation rate of the qubit. (b) Mechanisms for the transverse and longitudinal coupling between a qubit and a resonator. For the longitudinal coupling, the change in bias shifts the qubit transition frequency ω q at the bias shown by the green square, resulting in a strong longitudinal coupling; at the bias shown by the red circle, the longitudinal coupling is zero.

FIG. 10.

(a) In a superconducting circuit, the atom–cavity interaction can be implemented by the qubit–resonator coupling. Here, the resonator can be either a planar resonator or a 3D cavity; in either case, it is usually modeled as an L C circuit. The circuit shows the capacitive coupling between a quarter-wavelength ( λ / 4) resonator and an island-based qubit. Here, g represents the strength of the transverse coupling between the resonator and the qubit; κ represents the loss rate of photons from the resonator; and γ represents the transverse relaxation rate of the qubit. (b) Mechanisms for the transverse and longitudinal coupling between a qubit and a resonator. For the longitudinal coupling, the change in bias shifts the qubit transition frequency ω q at the bias shown by the green square, resulting in a strong longitudinal coupling; at the bias shown by the red circle, the longitudinal coupling is zero.

Close modal

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 ω q as we saw in Fig. 9(a). The longitudinal coupling changes the qubit frequency. It is effective when ω 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 ω 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 ω 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 

Although Eq. (35) captures all the physics regarding the capacitive coupling, solving Eq. (35) with Eq. (28) is easy. We ignore the g z term because the g x term is the dominant term at the sweet spot as mentioned above. Then, we have
H ^ 1 q = H ^ q + H ^ r + H ^ qr = ω q σ ^ z 2 + ω r a ^ a ^ + g σ ^ x ( a ^ + a ^ ) .
(37)
Here, we omit the subscript x in g for simplicity.
Now, we move to the rotating frame to focus on the dynamics induced by the coupling term. The Hamiltonian defining this frame is
H ^ 0 = ω q σ ^ z 2 + ω r a ^ a ^ .
(38)
The final single-qubit Hamiltonian in the rotating frame H ^ 1 q rot can be obtained by the unitary transformation,
H ^ 1 q rot = e i H ^ 0 t / ( H ^ 1 q H ^ 0 ) e i H ^ 0 t / = g σ ^ x cos ( ω q t ) i σ ^ y sin ( ω q t ) a ^ e i ω r t + a ^ e i ω r t = g [ σ ^ + a ^ e i ( ω q ω r ) t + σ ^ a ^ e i ( ω q ω r ) t + σ ^ + a ^ e i ( ω q + ω r ) t + σ ^ a ^ e i ( ω q + ω r ) t ] ,
(39)
where σ ^ ± = ( σ ^ x ± i σ ^ y ) / 2. In the above equation, if g is a constant, all terms will be averaged out at the time scale we are interested in unless ω q and ω r are reasonably close. In this context, the transverse coupling is told to be effective only if ω q ω r. Moreover, we can safely ignore fast-oscillating terms, σ ^ + a ^ and σ ^ a ^, in most situations. Such an approximation is called the rotating wave approximation (RWA).
After applying the RWA, H ^ qr becomes (now we move back to the inertial frame)
H ^ qr g ( σ ^ + a ^ + σ ^ a ^ ) .
(40)
The physical meaning of g is the exchange of energy between a quantized electromagnetic field and a qubit at a rate of g / 2 π. Such an energy exchange with a well-defined period and phase is called coherent exchange; this will be useful for two-qubit gate operation (Sec. VI D). Equation (40), together with Eq. (28), is called the Jaynes–Cummings Hamiltonian H ^ JC (Refs. 53 and 62),
H ^ JC = ω q σ ^ z 2 + ω r a ^ a ^ + g ( σ ^ + a ^ + σ ^ a ^ ) .
(41)
Equation (41) will be the central equation in Sec. VI B 1.

3. Qubit–qubit coupling

The physics of qubit–qubit coupling is similar to that of qubit–resonator coupling. The Hamiltonian describing the qubit–qubit coupling can be written as
H ^ qq = ( J x x σ ^ x ( 1 ) σ ^ x ( 2 ) + J z x σ ^ z ( 1 ) σ ^ x ( 2 ) + J x z σ ^ x ( 1 ) σ ^ z ( 2 ) + J z z σ ^ z ( 1 ) σ ^ z ( 2 ) ) ,
(42)
where σ i ( j ) ( i = x , z) represents the Pauli operators for qubit j and J k l is the qubit–qubit coupling constant. Note that we have four terms in the qubit–qubit coupling because both systems are fermions (see the last paragraph of Sec. V B 1). For better visibility, we call each term with its subscripts, for example, the J z z term as the Z Z term or the Z Z interaction.

In Eq. (42), the X X 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 X Z and Z X interactions must be called the longitudinal interactions. However, a considerable number of papers designate all non-transverse interactions, which includes the Z Z 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 X Z interaction, for clarity.

We consider transmons coupled directly and capacitively (Fig. 11). In this case, the transverse ( X X) interaction is the dominant interaction as discussed in Sec. V B 2. Hence, we consider the X X term only and omit the subscript x x for simplicity. Similarly to the capacitive qubit–resonator coupling, J is determined by the coupling capacitance C 12 and the voltage fluctuations of the ground states,
J = C 12 V q , 0 ( 1 ) V q , 0 ( 2 ) 2 C 12 C q 1 C q 2 ω q 1 ω q 2 ,
(43)
where V q , 0 ( i ) ( i = 1 , 2) is the root mean square voltage of the ground state of qubit i; ω q i and C q i are the transition frequency and total capacitance of qubit i, respectively. Since a transmon is a weakly nonlinear harmonic oscillator (Sec. IV B 1), V q , 0 ( i ) ω q i / 2 C q i if C 12 C q i. Note that J depends on the transition frequency. Hence, for a coupling associated with a frequency-tunable qubit, J is also tunable.
FIG. 11.

Implementations of qubit–qubit coupling. Two qubits can be coupled either directly or indirectly via a coupling resonator. Here, a half-wavelength ( λ / 2) resonator and a tunable coupler are shown as examples. Solid lines labeled B and E represent magnetic and electric field profiles in the resonators, respectively. For fixed-frequency island-based qubits, the capacitive coupling is the only available coupling scheme. However, for loop-based qubits, not only inductive coupling but also capacitive coupling is possible because a loop-based qubit can also be understood as two superconducting islands as shown in the inset of Fig. 6(d). Note that the circuit for the tunable coupler is the same as that of the neighboring qubits. The qubit–qubit coupling constant is tuned by the external flux bias Φ.

FIG. 11.

Implementations of qubit–qubit coupling. Two qubits can be coupled either directly or indirectly via a coupling resonator. Here, a half-wavelength ( λ / 2) resonator and a tunable coupler are shown as examples. Solid lines labeled B and E represent magnetic and electric field profiles in the resonators, respectively. For fixed-frequency island-based qubits, the capacitive coupling is the only available coupling scheme. However, for loop-based qubits, not only inductive coupling but also capacitive coupling is possible because a loop-based qubit can also be understood as two superconducting islands as shown in the inset of Fig. 6(d). Note that the circuit for the tunable coupler is the same as that of the neighboring qubits. The qubit–qubit coupling constant is tuned by the external flux bias Φ.

Close modal
The resulting two-qubit Hamiltonian is
H ^ 2 q = ω q 1 σ ^ z ( 1 ) 2 + ω q 2 σ ^ z ( 2 ) 2 + J σ ^ x ( 1 ) σ ^ x ( 2 ) .
(44)
We move to the rotating frame defined by
H ^ 0 = ω q 1 σ ^ z ( 1 ) 2 + ω q 2 σ ^ z ( 2 ) 2 .
(45)
Then, we have an equation similar to Eq. (39),
H ^ 2 q rot = e i H ^ 0 t / ( H ^ 2 q H ^ 0 ) e i H ^ 0 t / = J [ σ ^ + ( 1 ) σ ^ ( 2 ) e i ( ω q 1 ω q 2 ) t + σ ^ ( 1 ) σ ^ + ( 2 ) e i ( ω q 1 ω q 2 ) t + σ ^ + ( 1 ) σ ^ + ( 2 ) e i ( ω q 1 + ω q 2 ) t + σ ^ ( 1 ) σ ^ ( 2 ) e i ( ω q 1 + ω q 2 ) t ] .
(46)

If J is static and | ω q 1 ω q 2 | J, 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 ω q 1 or ω q 2 so that ω q 1 ω q 2; (ii) modulating J with the frequency | ω q 1 ± ω 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 | ω r ω q i | 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

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, κ, and γ [Fig. 10(a)]. Here, g / 2 π is the transverse coupling strength in Hz, κ / 2 π is the loss rate of photons from the resonator, i.e., the spectral linewidth of the resonator, in Hz ( κ = ω r / Q, where Q is the quality factor of the resonator), and γ / 2 π( = 1 / π T 2) is the transverse relaxation rate, i.e., the spectral linewidth, of the qubit in Hz. When the system satisfies g > κ / 2 , γ / 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 ith state of the bare qubit and the jth 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.

FIG. 12.

Anticrossing due to strong qubit–resonator coupling for the circuit shown in Fig. 10(a). The splitting when ω q = ω r is 2 g.

FIG. 12.

Anticrossing due to strong qubit–resonator coupling for the circuit shown in Fig. 10(a). The splitting when ω q = ω r is 2 g.

Close modal
The physics of the vacuum Rabi splitting/oscillation can be understood using our classical oscillators in Sec. V A. When the coupling is static and there is no external drive, Eq. (27) can be written in the Hamiltonian form
H = p 1 2 2 m 1 + k 1 + κ 0 2 x 1 2 H osc 1 + p 2 2 2 m 2 + k 2 + κ 0 2 x 2 2 H osc 2 κ 0 x 1 x 2 H c .
(47)
The energy exchange and frequency splitting shown in Fig. 9(a) is caused by the coupling term H c. Thus, this coupling term corresponds to the transverse coupling. Although our classical oscillators show essentially the same physics, the vacuum Rabi splitting/oscillation is a highly quantum phenomenon because it is a consequence of coupling between the qubit and the 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).

To implement functions required for quantum computation, we need to know how our qubits evolve during various operations. For a closed quantum system, the evolution of a density matrix ρ ^ is fully described by the von Neumann equation,
d ρ ^ d t = 1 i [ H ^ , ρ ^ ] ,
(48)
where H ^ is the Hamiltonian of the system that the density matrix represents. Note that Eq. (48) is in the Schrödinger picture; in the Heisenberg picture, the density matrix is not time-dependent. If H ^ is time-independent, the solution of Eq. (48) is given by
ρ ^ ( t ) = e i H ^ t / ρ ^ ( 0 ) e i H ^ t / = U ^ ( t ) ρ ^ ( 0 ) U ^ ( t ) ,
(49)
which is the density matrix version of Eq. (10).
However, a qubit is actually an open quantum system—it is always interacting with the environment, a readout resonator, and other control lines, resulting in the relaxation of a quantum state as discussed in Sec. IV. Thus, this relaxation phenomenon has to be included in the equation of motion for precise control. To simplify the situation, we introduce three assumptions. The first one is the Born approximation, which assumes that the interaction between the qubit and the environment is reasonably weak such that the environment is practically unaffected by the system. The second one is the Markovian approximation, which assumes that the noise process acting on the system is memoryless. In other words, the internal dynamics of the environment hides any entanglement with the system as quickly as it arises. The last one is that the initial states of the system and environment are not entangled, i.e., ρ ^ ( t = 0 ) = ρ ^ sys ρ ^ env. With these approximations, the dynamics of the qubit can be well described by the Lindblad master equation, which is given by53,69
d ρ ^ d t = 1 i [ H ^ , ρ ^ ] + k L ^ k ρ ^ L ^ k 1 2 L ^ k L ^ k , ρ ^ .
(50)
To describe the dynamics of the qubit properly, we need to choose the Lindblad operator L ^ k based on the model we have. For example, the effects of an environment on a single qubit can be modeled by
L ^ 1 = Γ σ ^ , L ^ 2 = Γ φ 2 σ ^ z .
(51)
Here, L ^ 1 describes the thermalization process, i.e., the transition from 1 to 0 ( σ ^ ), and L ^ 2 describes the dephasing process. Other effects can also be considered by introducing additional Lindblad operators. The unit of L ^ k is [s 1 / 2].
Let us solve the equation with L ^ 1 only for simplicity. Using the identity L ^ 1 L ^ 1 = Γ | 1 1 | in Eq. (50), we obtain
d d t ρ 00 ρ 01 ρ 10 ρ 11 = Γ ρ 11 ρ 01 / 2 ρ 10 / 2 ρ 11 .
(52)
Solving this is straightforward,
ρ 00 ( t ) ρ 01 ( t ) ρ 10 ( t ) ρ 11 ( t ) = 1 ρ 11 ( 0 ) e Γ t ρ 01 ( 0 ) e Γ t / 2 ρ 10 ( 0 ) e Γ t / 2 ρ 11 ( 0 ) e Γ t .
(53)
Note that the diagonal elements decay with the time constant Γ , whereas the off-diagonal elements decay with Γ / 2. This explains Eq. (18).
The Lindblad master equation can also be applied to a resonator. In this case,
L ^ = κ 2 π a ^ .
(54)

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.

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)].

FIG. 13.

(a) Qubit-state-dependent shift in resonator frequency. This frequency shift, called the dispersive shift, allows us to detect the qubit state by monitoring the S-parameters of the circuit. For the circuit shown in Fig. 10(a), the qubit state can be inferred by measuring the transmission of the circuit at ω r. If the measured phase is A, then the qubit state is g; if the phase is B, the qubit state is e. (b) Resonator-state-dependent shift in qubit frequency. In the strong coupling regime, the qubit frequency can be split by the photon-state-dependent frequency shift. The resonator state is assumed to be a coherence state whose average photon number is n ¯. This figure was obtained by solving the Lindblad equation [Eq. (50)] with Eqs. (51), (54), and (57). For the solution, QuTiP was used.73,74

FIG. 13.

(a) Qubit-state-dependent shift in resonator frequency. This frequency shift, called the dispersive shift, allows us to detect the qubit state by monitoring the S-parameters of the circuit. For the circuit shown in Fig. 10(a), the qubit state can be inferred by measuring the transmission of the circuit at ω r. If the measured phase is A, then the qubit state is g; if the phase is B, the qubit state is e. (b) Resonator-state-dependent shift in qubit frequency. In the strong coupling regime, the qubit frequency can be split by the photon-state-dependent frequency shift. The resonator state is assumed to be a coherence state whose average photon number is n ¯. This figure was obtained by solving the Lindblad equation [Eq. (50)] with Eqs. (51), (54), and (57). For the solution, QuTiP was used.73,74

Close modal

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 ( ω q ω 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 ω q such that the qubit–resonator detuning Δ qr ( ω r ω q ) is much greater than g and κ. 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.

To see the physics in the dispersive limit more clearly, we consider the Jaynes–Cummings Hamiltonian H ^ JC [Eq. (41)]. Here, we treat the qubit–resonator interaction H ^ qr [Eq. (40)] as the perturbation to the uncoupled qubit–resonator Hamiltonian H ^ 0 [Eq. (38)]. Then, we take a unitary transformation that diagonalizes H ^ JC perturbatively to first order in H ^ qr. Such a transformation is called the Schrieffer–Wolff transformation75,76 (the same results can be obtained using the standard perturbation theory53,62). A unitary operator U ^ disp = e S ^ for the Schrieffer–Wolff transformation is defined such that S ^ = S ^ and [ S ^ , H ^ 0 ] = H ^ qr. Then, we have (use the formulas in Table II)
H ^ JC disp = U ^ disp H ^ JC U ^ disp = H ^ 0 + H ^ qr + [ S ^ , H ^ 0 ] + [ S ^ , H ^ qr ] + 1 2 [ S ^ , [ S ^ , H ^ 0 ] ] + 1 2 [ S ^ , [ S ^ , H ^ qr ] ] + H ^ 0 + 1 2 [ S ^ , H ^ qr ] .
(55)
In our case, U ^ disp is given by
U ^ disp = exp g Δ qr ( σ ^ a ^ σ ^ + a ^ ) .
(56)
From Eq. (55), we obtain
H ^ JC disp ( ω q χ ) σ ^ z 2 + ( ω r χ σ ^ z ) a ^