Superconducting Quantum Computer Architecture

(W2-C1, W2-D1, W2-D2)For node flux \(\Phi\), the capacitive network provides the kinetic term, while inductors and Josephson junctions provide the potential term:

\[\mathcal L=\frac12\dot{\Phi}^{T} C\dot{\Phi}-U(\Phi),\qquad Q=\frac{\partial\mathcal L}{\partial\dot{\Phi}}\]

A Legendre transform gives \(H(\Phi, Q)\), after which we impose \([\hat\Phi_i,\hat Q_j]=i\hbar\delta_{ij}\). A linear LC circuit produces only equally spaced energy levels; it is the Josephson term

\[U_J(\varphi)=-E_J\cos\varphi,\qquad E_J=\frac{\Phi_0 I_c}{2\pi}\]

that breaks harmonicity and allows selective control of the two lowest levels. In an advanced course, students need to complete the full path from schematic to Hamiltonian at least once; memorizing the transmon formula is not enough to predict the consequences of parasitic modes, coupling, or layout changes.

Boundary:This week does not go deeply into the microscopic BCS derivation. Here, superconductors primarily provide a macroscopic phase, low-loss wiring, and the Josephson element. Microscopic material loss is deferred to T6.

A quantized LC circuit already has discrete energy levels:

\[ \hat H_{LC}=\frac{\hat Q^2}{2C}+\frac{\hat\Phi^2}{2L} =\hbar\omega\left(\hat a^\dagger\hat a+\frac12\right), \qquad \omega=\frac1{\sqrt{LC}}. \]

First establish that an LC circuit really is a quantum system

Define the characteristic impedance \(Z=\sqrt{L/C}\), so the flux and charge operators can be written as

\[ \hat\Phi=\Phi_{\rm zpf}(\hat a+\hat a^\dagger), \qquad \hat Q=-iQ_{\rm zpf}(\hat a-\hat a^\dagger), \]

\[ \Phi_{\rm zpf}=\sqrt{\frac{\hbar Z}{2}}, \qquad Q_{\rm zpf}=\sqrt{\frac{\hbar}{2Z}}, \qquad \Phi_{\rm zpf}Q_{\rm zpf}=\frac{\hbar}{2}. \]

Even in the ground state, flux and charge retain zero-point fluctuations. The energy eigenstates \(\ket m\) are also genuinely discrete:

\[ E_m=\hbar\omega\left(m+\frac12\right), \qquad m=0,1,2,\ldots \]

The problem, then, is not that the LC oscillator is “insufficiently quantum.” Quite the opposite: it is an overly ideal linear quantum system in which every rung obeys exactly the same rule.

An equally spaced spectrum makes all adjacent transition frequencies coincide

The transition frequency between adjacent levels is

\[ \omega_{m,m+1} =\frac{E_{m+1}-E_m}{\hbar} =\omega, \]

Therefore,

\[ \omega_{01}=\omega_{12}=\omega_{23}=\cdots. \]

A resonant microwave tone sees not only \(\ket0\leftrightarrow\ket1\) but also every higher transition. Frequency selectivity cannot tell the system to use only the two lowest levels.

The drive Hamiltonian shows why leakage is unavoidable

Suppose the oscillator is driven by a capacitively coupled microwave voltage. Ignoring a coupling prefactor that does not affect the main point, the control term can be written as

\[ \hat H_d(t) =\hbar\epsilon(t)(\hat a+\hat a^\dagger). \]

The matrix element between adjacent levels is

\[ \langle m+1|\hat a+\hat a^\dagger|m\rangle =\sqrt{m+1}. \]

The drive initially pushes \(\ket0\) toward \(\ket1\). Once \(\ket1\) acquires population, the same frequency also drives \(\ket1\rightarrow\ket2\) exactly on resonance, with a matrix element larger by a factor of \(\sqrt2\). The same pattern continues upward. Therefore, \(\{\ket0,\ket1\}\) is not a closed invariant subspace under this control.

A stronger statement: a linear drive creates a coherent state, not a pure \(\ket1\) state

Under the rotating-wave approximation, a driven harmonic oscillator can be written as

\[ \hat H_{\rm rot}(t) =\hbar\bigl[\epsilon(t)\hat a^\dagger+\epsilon^*(t)\hat a\bigr]. \]

If the initial state is the vacuum, any such linear pulse can only produce a displacement:

\[ \ket0\longrightarrow\hat D(\alpha)\ket0=\ket\alpha, \qquad P_m=e^{-|\alpha|^2}\frac{|\alpha|^{2m}}{m!}. \]

To maximize \(P_1\) one must choose \(|\alpha|^2=1\), but then \(P_1^{\rm max}=e^{-1}\approx0.368\), and the remaining population is still distributed over \(\ket0,\ket2,\ldots\). In other words, applying a conventional qubit “\(\pi\) pulse” to a bare harmonic oscillator cannot deterministically flip \(\ket0\) to \(\ket1\).

A qubit also requires spectral addressability

Define the anharmonicity

\[ \alpha\equiv\omega_{12}-\omega_{01}. \]

For a harmonic oscillator, \(\alpha=0\). Only when nonlinearity makes \(\alpha\neq0\) can the drive frequency select the computational transition through \(\omega_{01}\) while remaining off-resonant from \(\omega_{12}\). For a simple pulse, we generally want

\[ |\alpha|\gg\Omega_R,\;\kappa,\;\gamma_\phi, \]

where \(\Omega_R\) is the Rabi rate. A faster drive has a broader bandwidth, and the leakage amplitude typically increases on the scale of \(\Omega_R/|\alpha|\); a slower drive instead accumulates more decoherence. This is the speed–leakage trade-off later addressed by pulse shaping and DRAG control.

A precise statement: A harmonic oscillator is not incapable of carrying quantum information. With multiple Fock states, cat states, or another bosonic encoding, an oscillator can still serve as a logical qubit. But if only ordinary linear microwave control is allowed, a bare LC oscillator does not naturally provide an independently controllable two-lowest-level qubit.

In-class task: Draw harmonic and anharmonic spectra and mark which transitions are connected by a drive at the same frequency.

Conclusion to Question 3:“Discrete energy levels” only guarantee that the system is quantum; “unequally spaced energy levels” are what make \(\ket0\leftrightarrow\ket1\) frequency-selective and suppress leakage to a controllable level.

2.2 Why Nonlinearity Is Necessary

If a quantum circuit has equally spaced levels, the frequency from \(\ket{0}\) to \(\ket{1}\) is identical to the frequency from \(\ket{1}\) to \(\ket{2}\). Microwave control therefore cannot address only the qubit subspace \(\ket{0}, \ket{1}\) and will readily excite higher levels.

A Josephson junction supplies the nonlinear inductance required by superconducting quantum circuits. Through this nonlinearity, a superconducting circuit can acquire unequally spaced energy levels, allowing its two lowest levels to serve as a qubit.

4.1 From LC Circuit to Quantum LC Oscillator

In a classical circuit, an LC oscillator exchanges energy between its capacitor and inductor. The capacitor stores electric-field energy, while the inductor stores magnetic-field energy. Quantizing the LC oscillator gives equally spaced energy levels:

\[ E_m = \hbar\omega\left(m + \frac{1}{2}\right) \]

Such a harmonic oscillator is unsuitable as a qubit because adjacent levels have the same spacing, so a control pulse cannot select only the \(\ket{0} \leftrightarrow \ket{1}\) transition.

Josephson tunneling provides a nondissipative, phase-controlled nonlinear element:

\[ I=I_c\sin\varphi, \qquad V=\frac{\Phi_0}{2\pi}\dot\varphi, \qquad U_J(\varphi)=-E_J\cos\varphi. \]

The Josephson energy is not a separate assumption: it follows by integrating the two Josephson relations

The supercurrent branch of an ideal junction obeys \(I=I_c\sin\varphi\), while the phase evolves with voltage according to \(V=(\Phi_0/2\pi)\dot\varphi\). The instantaneous power flowing into the element is

\[ P=IV =I_c\sin\varphi\, \frac{\Phi_0}{2\pi}\dot\varphi. \]

If this power is stored in the junction potential, then

\[ \frac{dU_J}{dt} =\frac{dU_J}{d\varphi}\dot\varphi =\frac{\Phi_0 I_c}{2\pi}\sin\varphi\,\dot\varphi. \]

and therefore

\[ \frac{dU_J}{d\varphi}=E_J\sin\varphi, \qquad E_J\equiv\frac{\Phi_0 I_c}{2\pi} =\frac{\hbar I_c}{2e}, \]

Integrating gives

\[ U_J(\varphi)=-E_J\cos\varphi+\text{constant}. \]

Thus, saying that “a Josephson junction provides a cosine potential” does not mean forcing the circuit into a preconceived atomic model. The energy landscape follows directly from the measurable current–phase relation.

It behaves like an inductor, but not a linear one

For a linear inductor, \(I=\Phi/L\), so the potential is always quadratic. For a junction, \(\dot I=I_c\cos\varphi\,\dot\varphi\). Comparing \(V=L_J\dot I\) gives the differential inductance

\[ L_J(\varphi) =\frac{\Phi_0}{2\pi I_c\cos\varphi}. \]

\(L_J\) varies with phase: this is the circuit-language description of nonlinearity. For small amplitudes with \(\varphi\approx0\), \(L_J(0)=\Phi_0/(2\pi I_c)\) and the junction appears approximately linear; at larger amplitudes, the variation of \(\cos\varphi\) can no longer be neglected. At the same time, the ideal Josephson supercurrent is nondissipative, so a junction can provide strong nonlinearity in a low-loss circuit—a combination difficult to achieve with an ordinary resistor or semiconductor diode. Real devices still have junction capacitance, quasiparticle loss, and material defects; “ideally nondissipative” must not be mistaken for completely lossless.

With capacitance included, phase becomes a quantum coordinate moving in a cosine potential

Combine the junction with a total capacitance \(C_\Sigma\) and define the charging energy as

\[ E_C\equiv\frac{e^2}{2C_\Sigma}. \]

The Hamiltonian becomes

\[ \hat H =4E_C(\hat n-n_g)^2-E_J\cos\hat\varphi. \]

In the “quantum particle” picture, \(\varphi\) is position and \(n\) is momentum; \(E_C\) sets the kinetic-energy scale, while \(E_J\) sets the depth of the periodic potential. Changing the junction critical current changes \(E_J\), and changing the shunt capacitance changes \(E_C\). The “mass, confinement strength, and spectrum” of this artificial atom can therefore all be designed through its circuit.

Higher-order terms of the cosine break equal spacing

In the linearized approximation, retain only the quadratic term near a potential minimum:

\[ -E_J\cos\varphi =-E_J+\frac{E_J}{2}\varphi^2 -\frac{E_J}{24}\varphi^4 +\frac{E_J}{720}\varphi^6-\cdots. \]

The quadratic potential of a linear inductor can produce only a harmonic spectrum; the cosine potential of a Josephson junction provides a nonlinear inductance. When the cosine is expanded, the quadratic term establishes the dominant oscillation frequency, while the higher-order terms break the equal spacing and produce an anharmonic spectrum that can be selectively controlled.

First treat the quadratic part as a harmonic oscillator:

\[ \hat H_0 =4E_C\hat n^2+\frac{E_J}{2}\hat\varphi^2-E_J, \qquad \hbar\omega_p=\sqrt{8E_JE_C}. \]

The corresponding zero-point amplitudes may be chosen as

\[ \hat\varphi=\varphi_{\rm zpf}(\hat a+\hat a^\dagger), \qquad \hat n=i n_{\rm zpf}(\hat a^\dagger-\hat a), \]

\[ \varphi_{\rm zpf} =\left(\frac{2E_C}{E_J}\right)^{1/4}, \qquad n_{\rm zpf} =\left(\frac{E_J}{32E_C}\right)^{1/4}, \qquad \varphi_{\rm zpf}n_{\rm zpf}=\frac12. \]

Now treat the quartic term \(-E_J\hat\varphi^4/24\) as a first-order perturbation. Because

\[ \langle m|(\hat a+\hat a^\dagger)^4|m\rangle =6m^2+6m+3, \]

we obtain the low-energy approximation

\[ E_m\approx -E_J+\sqrt{8E_JE_C}\left(m+\frac12\right) -\frac{E_C}{12}(6m^2+6m+3). \]

Thus adjacent transitions no longer have the same frequency:

\[ \hbar\omega_{m,m+1} \approx\sqrt{8E_JE_C}-E_C(m+1), \]

\[ \hbar\omega_{01}\approx\sqrt{8E_JE_C}-E_C, \qquad \hbar\omega_{12}\approx\sqrt{8E_JE_C}-2E_C, \]

\[ \alpha\equiv\omega_{12}-\omega_{01} \approx-\frac{E_C}{\hbar}. \]

The minus sign means that the spacing between adjacent levels decreases at higher energy. This result comes from the negative quartic correction of the cosine; it is precisely what detunes the \(\omega_{01}\) drive from \(\omega_{12}\) and makes selective control of the two lowest levels possible.

The same cosine provides two complementary intuitions in two different bases

• phase basis: \(-E_J\cos\varphi\) forms periodic wells; as \(E_J/E_C\) increases, the low-energy wavefunction becomes more localized near one minimum.
• charge basis: \(-E_J\cos\hat\varphi\) couples \(\ket n\) and \(\ket{n\pm1}\), turning crossings of the bare charge-state energies into avoided crossings.

Conclusion to Question 4: A Josephson junction does not simply “add another inductor”; it replaces a quadratic potential with a cosine potential designable through \(E_J\). Its quadratic part sets the dominant frequency, while its quartic and higher-order terms create anharmonicity. This is how an artificial atom acquires a controllable spectrum.

3.3 Josephson Junction

A Josephson junction consists of two superconductors separated by an ultrathin insulating layer. It is commonly called an SIS junction: superconductor–insulator–superconductor. Although the middle layer is insulating, Cooper pairs can still cross the junction by quantum tunneling and form a Josephson current.

Josephson Relations

The current through a Josephson junction depends nonlinearly on the phase difference:

\[ I = I_c \sin\varphi \]

When a voltage \(V\) exists across the junction, the phase difference evolves in time:

\[ V = \frac{\Phi_0}{2\pi}\frac{d\varphi}{dt} \]

where \(\Phi_0 = h/2e\) is the superconducting flux quantum. The Josephson energy can be written as:

\[ E_J = \frac{\Phi_0 I_c}{2\pi} \]

3.4 Josephson Junction as a Nonlinear Inductor

The key value of a Josephson junction is that it provides nonlinear inductance. The energy of an ordinary linear inductor is approximately quadratic in phase; the Josephson-junction energy is instead:

\[ U_J(\varphi) = -E_J \cos\varphi \]

This cosine potential makes the circuit's energy levels unequally spaced, allowing the two lowest levels of the quantum oscillator to be selected as a qubit.

Nonlinearity enables precise qubit control

Quantum description of a Josephson junction

▲ A Josephson junction whose intrinsic material properties are described by \(H_0\). A voltage \(V\) is applied across the junction, and the interaction between the superconducting wavefunctions is represented by \(K\), with equation of motion: \[ i\hbar \frac{\partial}{\partial t} \begin{pmatrix} \sqrt{n_A} e^{i\theta_A} \\ \sqrt{n_B} e^{i\theta_B} \end{pmatrix} = \begin{pmatrix} H_0 - \frac{qV}{2} & K \\ K & H_0 + \frac{qV}{2} \end{pmatrix} \begin{pmatrix} \sqrt{n_A} e^{i\theta_A} \\ \sqrt{n_B} e^{i\theta_B} \end{pmatrix}. \] For \(q = -2e\) (a Cooper pair), the equation becomes: \[ i\hbar \frac{\partial}{\partial t} \begin{pmatrix} \sqrt{n_A} e^{i\theta_A} \\ \sqrt{n_B} e^{i\theta_B} \end{pmatrix} = \begin{pmatrix} H_0 + eV & K \\ K & H_0 - eV \end{pmatrix} \begin{pmatrix} \sqrt{n_A} e^{i\theta_A} \\ \sqrt{n_B} e^{i\theta_B} \end{pmatrix}. \]

Complete derivation of \(\theta_A, n_A \)

The first equation is: \[ i\hbar \frac{\partial}{\partial t} (\sqrt{n_A} e^{i\theta_A}) = \frac{i}{2} \frac{e^{i\theta_A}}{\sqrt{n_A}} \frac{\partial n_A}{\partial t} - \sqrt{n_A} e^{i\theta_A} \frac{\partial \theta_A}{\partial t} = (H_0 + eV) \sqrt{n_A} e^{i\theta_A} + K\sqrt{n_B} e^{i\theta_B}. \] Separating real and imaginary parts gives: \[ \frac{i}{2}\hbar \frac{\partial n_A}{\partial t} - n_A \hbar \frac{\partial \theta_A}{\partial t} = (H_0 + eV)n_A + K\sqrt{n_A n_B} e^{i(\theta_B - \theta_A)}. \] Its complex-conjugate equation is: \[ -\frac{i}{2}\hbar \frac{\partial n_A}{\partial t} - n_A \hbar \frac{\partial \theta_A}{\partial t} = (H_0 + eV)n_A + K\sqrt{n_A n_B} e^{-i(\theta_B - \theta_A)}. \]

Adding the two equations gives: \[ -2n_A \hbar \frac{\partial \theta_A}{\partial t} = 2(H_0 + eV)n_A + 2K\sqrt{n_A n_B} \cos(\theta_B - \theta_A), \] Rearranging yields the equation for \(\theta_A\): \[ \hbar \frac{\partial \theta_A}{\partial t} = -(H_0 + eV) - K\sqrt{\frac{n_B}{n_A}} \cos(\theta_B - \theta_A). \]
Subtracting the two equations gives: \[ i\hbar \frac{\partial n_A}{\partial t} = 2iK\sqrt{n_A n_B} \sin(\theta_B - \theta_A), \] That is: \[ \hbar \frac{\partial n_A}{\partial t} = 2K\sqrt{n_A n_B} \sin(\theta_B - \theta_A). \]

Complete derivation of \(\theta_B, n_B \)

The second equation is: \[ i\hbar \frac{\partial}{\partial t} (\sqrt{n_B} e^{i\theta_B}) = \frac{i}{2} \frac{e^{i\theta_B}}{\sqrt{n_B}} \frac{\partial n_B}{\partial t} - \sqrt{n_B} e^{i\theta_B} \frac{\partial \theta_B}{\partial t} = K\sqrt{n_A} e^{i\theta_A} + (H_0 - eV)\sqrt{n_B} e^{i\theta_B}. \] Splitting it gives: \[ \frac{i}{2}\hbar \frac{\partial n_B}{\partial t} - n_B \hbar \frac{\partial \theta_B}{\partial t} = K\sqrt{n_A n_B} e^{-i(\theta_B - \theta_A)} + (H_0 - eV)n_B. \] Its complex-conjugate equation is: \[ -\frac{i}{2}\hbar \frac{\partial n_B}{\partial t} - n_B \hbar \frac{\partial \theta_B}{\partial t} = K\sqrt{n_A n_B} e^{i(\theta_B - \theta_A)} + (H_0 - eV)n_B. \]

Adding the two equations gives: \[ -2n_B \hbar \frac{\partial \theta_B}{\partial t} = 2K\sqrt{n_A n_B} \cos(\theta_B - \theta_A) + 2(H_0 - eV)n_B, \] Rearranging yields the equation for \(\theta_B\): \[ \hbar \frac{\partial \theta_B}{\partial t} = -(H_0 - eV) - K\sqrt{\frac{n_A}{n_B}} \cos(\theta_B - \theta_A). \]
Subtracting the two equations gives: \[ i\hbar \frac{\partial n_B}{\partial t} = -2iK\sqrt{n_A n_B} \sin(\theta_B - \theta_A), \] That is: \[ \hbar \frac{\partial n_B}{\partial t} = -2K\sqrt{n_A n_B} \sin(\theta_B - \theta_A). \]

Rearrangement gives the following equations:

• For \(\theta_A\): \[ \hbar \frac{\partial \theta_A}{\partial t} = -(H_0 + eV) - K\sqrt{\frac{n_B}{n_A}} \cos(\theta_B - \theta_A). \]
• For \(\theta_B\): \[ \hbar \frac{\partial \theta_B}{\partial t} = -(H_0 - eV) - K\sqrt{\frac{n_A}{n_B}} \cos(\theta_B - \theta_A). \]
• For \(n_A\): \[ \hbar \frac{\partial n_A}{\partial t} = 2K\sqrt{n_A n_B} \sin(\theta_B - \theta_A). \]
• For \(n_B\): \[ \hbar \frac{\partial n_B}{\partial t} = -2K\sqrt{n_A n_B} \sin(\theta_B - \theta_A). \]

Define the phase difference \(\Delta \theta = \theta_B - \theta_A\) and subtract the first two equations to obtain: \[ \hbar \frac{\partial \Delta \theta}{\partial t} = 2eV + \frac{K}{\hbar} \left( \frac{n_B - n_A}{\sqrt{n_A n_B}} \right) \cos\Delta \theta. \] Combine the latter two equations and define the current \(I\) as: \[ -\frac{\partial n_A}{\partial t} = \frac{\partial n_B}{\partial t} \equiv \frac{I}{-2e}. \] Rearranging gives: \[ I = \frac{4eK}{\hbar} \sqrt{n_A n_B} \sin\Delta \theta \equiv I_c \sin\Delta \theta, \] where \(I_c\) is the critical current of the Josephson junction.

Define the nonlinear inductance of the Josephson junction: \[ L_J(\Delta \theta) = \left(\frac{2e}{\hbar I_c \cos\Delta \theta}\right). \] The energy stored in an inductor is: \[ E = \int IV dt = \int I_0 \sin\Delta \theta \cdot \frac{\hbar}{q} \frac{\partial \Delta \theta}{\partial t} dt = -\frac{I_0 \hbar}{2e} \cos\Delta \theta \equiv -\frac{I_0 \Phi_0}{2\pi} \cos\Delta \theta, \] where the flux quantum is \(\Phi_0 = \frac{2\pi\hbar}{2e}\), and define: \[ E_J = \frac{I_0 \Phi_0}{2\pi}. \]

▲ A linear \(LC\) circuit becomes a nonlinear \(LC\) circuit by incorporating the nonlinear inductance of a Josephson junction. \(L_J\) is the junction inductance.

4.2 Nonlinear LC Circuit

A transmon can be understood as a nonlinear LC oscillator. It replaces an ordinary linear inductor with a Josephson junction and adds a large shunt capacitance. The Josephson junction supplies nonlinearity, while the shunt capacitor lowers the charging energy, making the qubit insensitive to offset-charge noise.

The basic transmon circuit contains:

• One Josephson junction or SQUID loop.
• A large shunt capacitor.
• Capacitive coupling to microwave control lines, a readout resonator, or a coupler.

\[ \hat H_{\mathrm T}=4E_C(\hat n-n_g)^2-E_J\cos\hat\varphi. \]

A charge qubit is highly sensitive to offset charge \(n_g\). By increasing the shunt capacitance, a transmon lowers \(E_C\) and raises \(E_J/E_C\), exponentially suppressing the charge dispersion of the energy with respect to \(n_g\). The cost is a smaller magnitude of anharmonicity, requiring a new trade-off between control selectivity and gate speed.

First identify the problem with the original charge qubit

In the charge basis, first set \(E_J=0\). Each integer Cooper-pair number \(n\) then corresponds to a charging parabola:

\[ E_n(n_g)=4E_C(n-n_g)^2. \]

\(n_g\) is set by the gate voltage, background trapped charge, and environmental offset-charge fluctuations. Near \(n_g=1/2\), retaining only \(\ket{n=0}\) and \(\ket{n=1}\), Josephson tunneling gives the approximate matrix

\[ \hat H_{\{0,1\}} \approx \begin{pmatrix} 4E_Cn_g^2 & -E_J/2\\ -E_J/2 & 4E_C(1-n_g)^2 \end{pmatrix}. \]

so the two-level splitting is approximately

\[ \hbar\omega_{01} \approx\sqrt{\bigl[4E_C(1-2n_g)\bigr]^2+E_J^2}. \]

At exactly \(n_g=1/2\) there is a charge sweet spot, \(\partial\omega_{01}/\partial n_g=0\); but once the operating point drifts away, the frequency changes rapidly. If the background charge jumps, the qubit frequency jumps with it. Precise biasing at one point alone is not robust hardware protection.

The transmon's hardware move is simple: make the capacitance large

For total capacitance \(C_\Sigma\),

\[ E_C=\frac{e^2}{2C_\Sigma}. \]

Adding a large shunt capacitor increases \(C_\Sigma\) and lowers \(E_C\), while the junction critical current can still set \(E_J\) independently. Thus \(E_J/E_C\) can be shifted from the charge-qubit regime toward the transmon regime of \(E_J/E_C\gg1\). This does not introduce a different Hamiltonian; it moves the same Hamiltonian into another parameter regime.

In the phase picture, larger \(E_J/E_C\) localizes low-energy wavefunctions near a cosine minimum and reduces the quantum phase-slip amplitude between neighboring wells. Offset charge \(n_g\) plays the role of a boundary phase for the periodic potential; the dependence of energy on \(n_g\) comes precisely from interwell tunneling. As the relative barrier height grows, the energy band flattens with respect to \(n_g\).

The key is not merely that it becomes smaller: charge dispersion is exponentially suppressed

In the asymptotic regime \(E_J/E_C\gg1\), the dominant dependence of an energy level on offset charge can be written as

\[ E_m(n_g)\approx\bar E_m -\frac{\epsilon_m}{2}\cos(2\pi n_g), \]

where the leading scale of the charge-dispersion amplitude is

\[ \epsilon_m \propto E_C\, \exp\!\left[-\sqrt{8E_J/E_C}\right] \times \text{algebraic prefactor}. \]

As \(E_J/E_C\) is increased, \(\partial\omega_{01}/\partial n_g\) does not merely fall slowly as \(1/(E_J/E_C)\); it receives exponential suppression. The transmon's value lies in this asymmetric exchange: a relatively modest parameter change buys very strong suppression of charge-noise dephasing.

But the nonlinearity is not preserved for free

Question 4 gave

\[ \hbar\omega_{01}\approx\sqrt{8E_JE_C}-E_C, \qquad \alpha\equiv\omega_{12}-\omega_{01} \approx-\frac{E_C}{\hbar}. \]

The shunt capacitance lowers \(E_C\), so \(|\alpha|\) also decreases approximately linearly. Fortunately, charge dispersion is exponentially suppressed, whereas anharmonicity decreases only linearly with \(E_C\). A practical parameter window therefore exists in which charge noise is already weak but \(\omega_{01}\) and \(\omega_{12}\) remain distinguishable.

For a target transition frequency and anharmonicity, these two approximations can also be inverted for a first-pass design:

\[ \frac{E_C}{h}\approx\frac{|\alpha|}{2\pi}, \qquad \frac{E_J}{E_C} \approx\frac18 \left(\frac{hf_{01}+E_C}{E_C}\right)^2. \]

A real design still requires numerical diagonalization, a junction-fabrication model, and electromagnetic simulation. Nevertheless, these two equations already show the directional relations among pad capacitance, junction critical current, frequency, and leakage.

Protection does not turn off control along with noise

A capacitive microwave drive couples primarily through \(\hat n\). In the harmonic approximation,

\[ |\langle0|\hat n|1\rangle| \approx \left(\frac{E_J}{32E_C}\right)^{1/4}\neq0. \]

Thus, increasing \(E_J/E_C\) suppresses charge dispersion without eliminating the \(0\leftrightarrow1\) electric-dipole matrix element. The transmon therefore retains effective microwave drive and dispersive readout. The converse is that the same nonzero matrix element means a transmon is not automatically immune to all high-frequency electric noise. It primarily addresses low-frequency offset-charge dephasing, not relaxation in general.

Fixed-frequency and tunable transmons introduce the next trade-off

A single junction gives a fixed \(E_J\), with a simpler structure and no additional flux-noise channel; the cost is that fabrication variation directly becomes frequency spread and may cause frequency collisions. If a symmetric dc SQUID replaces the single junction, then

\[ E_{J,\mathrm{eff}}(\Phi_{\rm ext}) =2E_{J0} \left|\cos\!\left(\pi\frac{\Phi_{\rm ext}}{\Phi_0}\right)\right|, \]

The qubit frequency becomes tunable, allowing collisions to be avoided or flux-mediated gates to be performed. But away from the flux sweet spot, \(\partial\omega_{01}/\partial\Phi_{\rm ext}\neq0\); flux noise, control-line crosstalk, and calibration overhead all increase.

Reassessing the transmon with four design criteria

• Dephasing protection: \(\partial\omega_{01}/\partial n_g\) receives exponential suppression from charge dispersion.
• Relaxation protection: There is no generic \(\langle0|\hat O_{\rm noise}|1\rangle\approx0\); material loss, Purcell loss, and high-frequency electric noise must still be engineered separately.
• Controllability: \(\langle0|\hat n|1\rangle\neq0\), preserving effective microwave control and readout coupling.
• Leakage suppression: \(\Delta_{\rm leakage}\sim\hbar|\alpha|\) is finite; gate speed, pulse shaping, and residual anharmonicity must be balanced.

• Benefits: lower charge dispersion, mature microwave control, and a designable frequency range.
• Costs: weaker anharmonicity, possible leakage, and different noise and calibration burdens for fixed and tunable designs.
• Design principle: the transmon is not the “best artificial atom,” but a successful parameter choice balancing noise sensitivity, control, and manufacturability.

Conclusion for this week: Discrete energy levels alone do not make a qubit. Josephson nonlinearity must create a selectively controllable spectrum, after which \(E_J/E_C\) tunes the trade-off between charge sensitivity and anharmonicity. The transmon became mainstream not because it eliminates every source of noise, but because exponential-versus-linear scaling places it at an especially effective Pareto point that remains controllable and manufacturable.

The current mainstream superconducting qubit

Ref: DOI:10.48550/arXiv.2203.04164

4.3 Transmon Hamiltonian

The transmon Hamiltonian can be written as:

\[ \hat{H}_{\mathrm{transmon}} = 4E_C(\hat{n} - n_g)^2 - E_J\cos\hat{\varphi} \]

where:

• \(\hat{n}\): Cooper pair number operator.
• \(\hat{\varphi}\): the superconducting phase difference across the Josephson junction.
• \(n_g\): offset charge, representing environmental charge noise or gate-induced charge.
• \(E_C = e^2 / 2C_\Sigma\): the charging energy, set by the total capacitance \(C_\Sigma\).
• \(E_J = \Phi_0 I_c / 2\pi\): the Josephson energy, set by the junction critical current.

An important transmon design condition is:

\[ \frac{E_J}{E_C} \gg 1 \]

In this regime, the sensitivity of the energy levels to offset charge \(n_g\) is strongly suppressed, while enough anharmonicity remains for the two lowest levels to serve as a qubit.

4.4 Energy Spectrum and Anharmonicity

Under the approximation \(E_J/E_C \gg 1\), the lowest transmon levels form an approximate weakly anharmonic oscillator. Its \(\ket{0} \rightarrow \ket{1}\) transition frequency is approximately:

\[ \omega_{01} \approx \frac{\sqrt{8E_JE_C} - E_C}{\hbar} \]

The transmon anharmonicity is approximately:

\[ \alpha = \omega_{12} - \omega_{01} \approx -\frac{E_C}{\hbar} \]

Anharmonicity is an important parameter for quantum-gate control. If \(|\alpha|\) is too small, microwave pulses readily cause leakage into \(\ket{2}\); if \(|\alpha|\) is too large, it usually implies larger \(E_C\) and potentially greater charge-noise sensitivity. A transmon must therefore balance charge-noise protection against control selectivity.

4.5 Design of \(E_C\) and \(E_J\)

The core of transmon design is choosing suitable \(E_C\), \(E_J\), and \(E_J/E_C\). These parameters are set respectively by chip geometry, materials, and the Josephson-junction process.

4.6 Fixed-Frequency and Tunable Transmon

Transmons may be divided into fixed-frequency and tunable types. A fixed-frequency transmon uses a single Josephson junction, so its frequency is largely fixed after fabrication. A tunable transmon replaces the single junction with a SQUID loop, allowing the effective Josephson energy to be controlled by an applied magnetic flux.

\[ E_J(\Phi_{\mathrm{ext}}) \approx E_{J,\mathrm{max}} \left| \cos\left(\pi\frac{\Phi_{\mathrm{ext}}}{\Phi_0}\right) \right| \]

A tunable transmon offers greater control flexibility for frequency tuning, two-qubit gates, and avoiding frequency collisions, but it also increases sensitivity to flux noise.

Nonlinear inductive properties of a two-junction SQUID

Frequency-tunable qubits

Combine two Josephson junctions \(J_1\) and \(J_2\) to form a closed loop threaded by flux \( \Phi_B \) and subject to voltage \( V \). From the previous section, the loop integral of the superconducting phase must satisfy: \[ \oint \nabla \theta(r) \cdot d\vec{l} - \frac{q}{\hbar} \Phi_B = 2\pi N \] Since the discontinuities at the junctions contribute \( \Delta\theta_1 \) and \( \Delta\theta_2 \), with \( q = -2e \), substitution gives: \[ \Delta\theta_2 - \Delta\theta_1 + 2\pi \frac{\Phi_B}{\Phi_0} = 2\pi N \] \[ \Delta\theta_2 = \Delta\theta_1 - 2\pi \frac{\Phi_B}{\Phi_0} + 2\pi N \]

When current \( I = I_1 + I_2 \) is sent through the entire SQUID, use the Josephson current relation from the previous section. For a symmetric SQUID, the two Josephson junctions have identical properties \( I_c \). We obtain: \[ I = I_1 + I_2 = I_0 (\sin\Delta\theta_1 + \sin\Delta\theta_2) \] \[ I = I_0 \left[\sin\Delta\theta_1 + \sin\left(\Delta\theta_1 - 2\pi \frac{\Phi_B}{\Phi_0} + 2\pi N\right)\right] \] \[ I = I_0 \left[\sin\Delta\theta_1 + \sin\left(\Delta\theta_1 - 2\pi \frac{\Phi_B}{\Phi_0}\right)\right] \] Applying a trigonometric identity gives: \[ I = I_0 \left[2 \sin\left(\frac{\Delta\theta_1 + \Delta\theta_1 - 2\pi \frac{\Phi_B}{\Phi_0}}{2}\right) \cos\left(\frac{\Delta\theta_1 - \Delta\theta_1 + 2\pi \frac{\Phi_B}{\Phi_0}}{2}\right)\right] \] \[ I = 2I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \sin\left(\Delta\theta_1 - \frac{\Phi_B}{\Phi_0} \pi\right) \]

The previous section showed that a Josephson junction behaves inductively; the inductive behavior of a SQUID can be analyzed similarly. Differentiate the current with respect to time: \[ \dot{I} = 2I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \cos\left(\Delta\theta_1 - \frac{\Phi_B}{\Phi_0} \pi\right) \dot{\Delta\theta_1} \] The relation among \( \dot{\Delta\theta_1} \) and voltage \( V \) is: \[ V = \frac{\hbar}{2e} \dot{\Delta\theta_1} = \frac{\Phi_0}{2\pi} \dot{\Delta\theta_1} \] Substitution gives the inductance: \[ L_{\text{SQUID}}(\Delta\theta_1, \Phi_B) = \left[\frac{2\pi}{\Phi_0} \cdot 2I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \cos\left(\Delta\theta_1 - \frac{\Phi_B}{\Phi_0} \pi\right)\right] \]

Notice that \( L_{\text{SQUID}} \) depends on the applied flux \( \Phi_B \). This is the key to frequency tuning in a tunable transmon. Define the variable \( \Phi \): \[ \Phi \equiv \frac{\Delta\theta_1}{\pi} \Phi_0 - \Phi_B = \frac{\Phi_0}{\pi} \left(\Delta\theta_1 - \frac{\Phi}{\Phi_0} \pi\right) \] with: \[ \dot{\Phi} = \frac{\Phi_0}{\pi} \dot{\Delta\theta_1} \] The voltage \( V \) and current \( I \) are respectively: \[ V = \frac{1}{2} \dot{\Phi}, \quad I = 2I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \sin\left(\frac{\Phi}{\Phi_0} \pi\right) \]

At fixed external field \( \Phi_B \), the energy stored in a SQUID is: \[ E = \int IV \, dt = \int \left[2I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \sin\left(\frac{\Phi}{\Phi_0} \pi\right) \cdot \frac{1}{2} \dot{\Phi}\right] dt \] Simplifying gives: \[ E = I_0 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \int \sin\left(\frac{\Phi}{\Phi_0} \pi\right) d\Phi \] \[ E = -\frac{I_0 \Phi_0}{\pi} \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \cos\left(\frac{\pi}{\Phi_0} \Phi\right) \]

Tunable Transmon

Frequency-tunable transmon

▲ A \(LC\) circuit composed of two Josephson junctions, with tunable inductance.

As discussed above, a Josephson junction has inductive properties, and a transmon is an LC circuit whose inductor is replaced by a Josephson junction. Here we consider a tunable transmon directly, using a SQUID as the inductor. In practice, fabrication nonuniformity gives different transmons different properties. With tunable transmons, small adjustments of magnetic flux \( \Phi_B \) can bring qubit properties into alignment. In addition, when quantizing the LC circuit above, \([ \Phi, Q ] = i\hbar\) formed a canonical pair. The SQUID derivation naturally relates magnetic flux \( \Phi \) to phase difference \( \Delta\theta \): \[ \Phi \equiv \frac{\Delta\theta_1}{\pi} \Phi_0 - \Phi_B \]

We now write the Hamiltonian: \[ \mathcal{H} = \frac{Q^2}{2C} + \frac{\Phi_\#^2}{2L} = \frac{Q^2}{2C} + \frac{(I_0 \pi)}{2\Phi_0} \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \Phi^2 - \frac{I_0}{24} \left(\frac{\pi}{\Phi_0}\right)^3 \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right) \Phi^4 + \cdots \] \[ = \frac{Q^2}{2C} + \frac{\Phi^2}{2L} - \frac{1}{24L} \left(\frac{\pi}{\Phi_0}\right)^2 \Phi^4 + \cdots \] where: \[ L = \left(\frac{I_0 \pi}{\Phi_0} \cos\left(\frac{\Phi_B}{\Phi_0} \pi\right)\right)^{-1} \]

We now examine how nonlinearity changes the energy levels. Above, we chose new variables \( a \) and \( a^\dagger \): \[ a = \sqrt{\frac{1}{2\hbar} \sqrt{ \frac{L}{C}}} \left(\sqrt{\frac{C}{L}} \Phi + iQ\right), \quad a^\dagger = \sqrt{\frac{1}{2\hbar} \sqrt{ \frac{L}{C}}} \left(\sqrt{\frac{C}{L}} \Phi - iQ\right) \] Conversely: \[ \Phi = \sqrt{\frac{\hbar}{2 }\sqrt{ \frac{L}{C}}} (a^\dagger + a), \quad Q = i\sqrt{\frac{\hbar}{2} \sqrt{ \frac{C}{L}}} (a^\dagger - a) \]

Expand the Hamiltonian as: \[ \mathcal{H} = \frac{\hbar}{\sqrt{LC}} \left[a^\dagger a + \frac{1}{2}\right] - \frac{1}{24L} \left(\frac{\pi}{\Phi_0}\right)^2 \left[i \sqrt{\frac{\hbar}{2\sqrt{L/C}}} (a^\dagger + a)\right]^4 \] After simplification: \[ \mathcal{H} = \frac{\hbar}{\sqrt{LC}} \left[a^\dagger a + \frac{1}{2}\right] - \frac{\hbar^2}{96C} \left(\frac{\pi}{\Phi_0}\right)^2 (a^\dagger + a)^4 \] This expression contains linear and nonlinear terms. The linear term describes the levels of a quantum harmonic oscillator, while the nonlinear term is the Josephson-junction correction to those levels.

Using perturbation theory, when \( \mathcal{H} = H_0 + \Delta H \), \( \ket{E_0^m} \to \ket{E^m} \), and \( E_0^m \to E^m = E_0^m + \delta E^m \), the correction \( \delta E^m \) to level \( E_0^m \) is:

\[ \delta E^m = \langle E_0^m | \Delta H | E_0^m \rangle = \langle E_0^m | \left[-\frac{\hbar^2}{96C} \left(\frac{\pi}{\Phi_0}\right)^2 (a^\dagger + a)^4\right] | E_0^m \rangle \]

The calculation proceeds as follows:

\[ \langle E_0^m | (a^\dagger + a)^4 | E_0^m \rangle = \langle E_0^m | (aa + aa^\dagger + a^\dagger a + a^\dagger a^\dagger)^2 | E_0^m \rangle \]

Expanding gives:

\[ \langle E_0^m | aa(aa + aa^\dagger + a^\dagger a + a^\dagger a^\dagger) + aa^\dagger(aa + aa^\dagger + a^\dagger a + a^\dagger a^\dagger) + a^\dagger a(aa + aa^\dagger + a^\dagger a + a^\dagger a^\dagger) + a^\dagger a^\dagger(aa + aa^\dagger + a^\dagger a + a^\dagger a^\dagger) \ket{ E_0^m } \]

Because we take an expectation value, only terms with equal numbers of \( a, a^\dagger \) remain:

\[ \langle E_0^m | aaa^\dagger a^\dagger + aa^\dagger aa^\dagger + aa^\dagger a^\dagger a + a^\dagger aaa^\dagger + a^\dagger aa^\dagger a + a^\dagger a^\dagger aa | E_0^m \rangle \]

Continuing to simplify:

\[ \langle E_0^m | a(a^\dagger a + 1)a^\dagger + aa^\dagger aa^\dagger + (a^\dagger a + 1)a^\dagger a + a^\dagger a(a^\dagger a + 1) + a^\dagger aa^\dagger a + a^\dagger (aa^\dagger - 1)a | E_0^m \rangle \]

Rearranging again:

\[ \langle E_0^m | aa^\dagger aa^\dagger + aa^\dagger + aa^\dagger aa^\dagger + a^\dagger aa^\dagger a + a^\dagger a + a^\dagger aa^\dagger a + a^\dagger a + a^\dagger aa^\dagger a + a^\dagger aa^\dagger a - a^\dagger a | E_0^m \rangle \]

Simplifying further:

\[ \langle E_0^m | 2aa^\dagger aa^\dagger + aa^\dagger + 4a^\dagger aa^\dagger a + a^\dagger a | E_0^m \rangle \] \[ = \langle E_0^m | 2(a^\dagger a + 1)(a^\dagger a + 1) + (a^\dagger a + 1) + 4a^\dagger aa^\dagger a + a^\dagger a | E_0^m \rangle \]

Finally, we obtain:

\[ \langle E_0^m | 6a^\dagger aa^\dagger a + 6a^\dagger a + 3 | E_0^m \rangle = 6m^2 + 6m + 3 \]

Define the energy-level spacing:

\[ \Delta E^m_{m-1} = \frac{\hbar}{\sqrt{LC}} - \frac{\hbar^2}{96C} \left(\frac{\pi}{\Phi_0}\right)^2 \left\{6[m^2 - (m-1)^2] + 6\right\} \]

\[ = \frac{\hbar}{\sqrt{LC}} - \frac{\hbar^2 \pi^2}{96C} \frac{e^2}{\pi^2 \hbar^2} \left[6(2m - 1) + 6\right] \]

Finally, we obtain:

\[ \Delta E^m_{m-1} = \frac{\hbar}{\sqrt{LC}} - m\frac{e^2}{8C} \]

Calculate \( \Delta E^{21} - \Delta E^{10} \):

\[ \Delta E^{21} - \Delta E^{10} = \left(\frac{\hbar}{\sqrt{LC}} - 2\frac{e^2}{8C}\right) - \left(\frac{\hbar}{\sqrt{LC}} - \frac{e^2}{8C}\right) \]

\[ = -\frac{e^2}{8C} \equiv -E_C \]

\( E_C \) represents its nonlinear effect.

Ref: DOI:10.1103/RevModPhys.93.025005

2.3 Types of Superconducting Qubits

Superconducting qubits can be classified into several types according to their dominant energy scales and circuit designs. Common examples include the charge qubit, flux qubit, phase qubit, transmon, fluxonium, and 0–π qubit. In modern gate-based superconducting quantum computing, the transmon is among the most widely used designs; fluxonium has also attracted substantial attention for its long coherence, low relaxation rate, and distinctive level structure.

Assignment question: Given a local image of a superconducting quantum chip, can you infer the Hamiltonian it might implement from the geometry?

1. Identify the structures: Mark the qubit pads, Josephson junction, ground, drive line, and readout resonator; explicitly label any uncertain identification.
2. geometry → parameter: Explain which circuit parameters are affected by pad geometry, junction area, spacing, and line length.
3. parameter → Hamiltonian: Identify which structures in the image primarily control \(E_C\), \(E_J\), and the coupling term.
4. Hamiltonian → observable: Predict how \(f_{01}\), anharmonicity, charge dispersion, or coupling changes when the pads grow larger, the junction becomes smaller, or the resonator is moved closer.
5. Limits of inference: List the information that cannot be known from an image alone and must come from process data, EM simulation, or experimental measurement.