Superconducting Qubits

Superconducting qubits are the workhorse of the noisy intermediate-scale quantum era. Google's Sycamore used 53 transmons to claim a sampling advantage in 2019, and the same lineage now powers processors of hundreds of qubits from Google, IBM, and others, with the field pivoting from qubit count toward error correction. In 2023 a Google device ran the surface code below threshold for the first time, showing that adding physical qubits reduced the logical error rate — the crucial demonstration that superconducting hardware can, in principle, be scaled to fault tolerance. Everything in this article is load-bearing there: gate speed and leakage are set by the anharmonicity $\alpha$, readout fidelity by the dispersive shift $\chi$, and the whole logical-error budget by $T_1$ and $T_2$, which materials and fabrication improvements have pushed from nanoseconds in 1999 to hundreds of microseconds today. The open frontiers are concrete — coherence limited by two-level-system defects in oxides and interfaces, crosstalk and frequency crowding as chips grow, the wiring and cooling bottleneck of routing thousands of microwave lines into a dilution refrigerator, and novel encodings (fluxonium, cat and bosonic codes) that trade circuit complexity for intrinsic noise protection. The platform that proved a circuit can be quantum is now being asked to stay quantum at scale.

Picture a marble rolling in a well. A perfectly parabolic well — the LC oscillator — gives quantum levels spaced by exactly one quantum $\hbar\omega_p$, evenly, forever. That even spacing is fatal: any microwave tone resonant with the bottom transition is equally resonant with every rung above it, so you can never trap the dynamics in just two levels. The Josephson junction replaces the parabola with a $-E_J\cos\varphi$ valley. Near the bottom it still looks parabolic, but it flattens toward the rim, and a flatter well binds its upper states more weakly, so the top rungs of the ladder crowd together. That crowding — the anharmonicity — is what lets a single resonant pulse address only the $0\leftrightarrow1$ transition and leave the rest alone. The first figure puts the two wells and their ladders side by side and attaches the real qubit numbers.

The circuit's two dynamical variables are the Cooper-pair number $\hat n$ transferred across the junction and the superconducting phase difference $\hat\varphi$; they are conjugate, $[\hat\varphi,\hat n]=i$, so localizing one delocalizes the other. Which one is sharp — whether the qubit is "charge-like" or "flux-like" — is set entirely by the dimensionless ratio $E_J/E_C$. Everything downstream is microwave engineering: the qubit frequency lands in the few-gigahertz band, so a superconducting quantum computer is really a refrigerated microwave laboratory, with arbitrary single-qubit rotations shaped from pulses on a carrier resonant with $\omega_{01}$ and the rotation axis set by the pulse phase. Measurement is indirect — you never touch the qubit, you watch a resonator coupled to it. The second figure shows that read: the resonator line splits in two, and which peak you see tells you the qubit's state.

Start with the circuit that fails. A capacitor and a linear inductor give a harmonic Hamiltonian in the charge and phase operators; quantizing it is textbook, and the result is an evenly spaced oscillator with no preferred pair of levels.

where: $E_C=e^2/2C$ is the charging energy ($C$ the total shunt capacitance, $e$ the electron charge); $E_L=(\Phi_0/2\pi)^2/L$ is the inductive energy ($\Phi_0=h/2e$ the flux quantum, $L$ the linear inductance); $\hat n$ counts Cooper pairs transferred; $\hat\varphi$ is the phase difference, with $[\hat\varphi,\hat n]=i$. Reading it: this is a harmonic oscillator with plasma frequency $\hbar\omega_p=\sqrt{8E_CE_L}$, so $E_{k+1}-E_k=\hbar\omega_p$ for every $k$ — the fatal degeneracy. Now replace the linear inductor with a Josephson junction. Its energy is not quadratic in the phase but cosinusoidal, and that single change is the origin of everything that makes the circuit a qubit.

where: $E_J=\dfrac{\Phi_0 I_c}{2\pi}=\dfrac{\hbar I_c}{2e}$ is the Josephson energy ($I_c$ the junction critical current); the term $-E_J\cos\hat\varphi$ is the nonlinear inductive potential that replaces the parabola. Reading it: expanded to $\tfrac12 E_J\hat\varphi^2-\tfrac{1}{24}E_J\hat\varphi^4+\cdots$, the quartic term is the leading nonlinearity — the source of the anharmonicity. In the transmon limit $E_J\gg E_C$ the phase barely fluctuates, so the quartic term is a small perturbation and the circuit is a weakly anharmonic (Duffing) oscillator. It is convenient to write it directly in the number basis of the underlying oscillator.

where: $\omega_{01}$ is the $0\to1$ transition frequency; $\alpha=\omega_{12}-\omega_{01}$ is the anharmonicity, negative for a transmon. Reading it: the second term vanishes in the two-level subspace but shifts every higher transition, so $\omega_{12}=\omega_{01}+\alpha$ and a pulse tuned to $\omega_{01}$ is detuned from $\omega_{12}$ by $|\alpha|$. The explicit spectrum falls out of the quartic perturbation theory (Derivation 1) and is the pair of numbers a device physicist actually quotes.

where: $\sqrt{8E_JE_C}=\hbar\omega_p$ is the plasma energy; $E_C$ is the charging energy. Reading it: the qubit frequency is essentially the plasma frequency shifted down by one charging energy, and the anharmonicity is just $-E_C$ — so a transmon with $E_C/h\approx 250\,\mathrm{MHz}$ has $\alpha/2\pi\approx-250\,\mathrm{MHz}$. Cooling is not optional, and one line says why: the transition energy must dwarf the thermal energy or the qubit will not stay in $|0\rangle$.

where: $k_B$ is Boltzmann's constant; $\bar n_{th}$ is the thermal excited-state population. Reading it: at $T=15\,\mathrm{mK}$ the ratio $\hbar\omega_{01}/k_BT\approx16$, so $\bar n_{th}\approx e^{-16}\approx10^{-7}$ — the qubit initializes to its ground state by passive cooling (Derivation 3). To measure the qubit you couple it to a microwave resonator; near resonance the pair is a Jaynes–Cummings system, the same Hamiltonian that governs a single atom in a cavity.

where: $\omega_r,\omega_q$ are the resonator and qubit frequencies; $g$ is the qubit–resonator coupling (vacuum Rabi rate); $\hat\sigma_\pm,\hat\sigma_z$ are the qubit ladder and inversion operators. Reading it: on resonance the qubit and resonator swap a photon at rate $g$; far off resonance they cannot, and the coupling survives only as a shift — the dispersive regime. Detune strongly, $|\Delta|=|\omega_q-\omega_r|\gg g$, and a Schrieffer–Wolff rotation (Derivation 2) turns the swap term into a state-dependent pull on the resonator.

where: $\Delta=\omega_q-\omega_r$ is the detuning; $\chi$ is the dispersive shift. Reading it: the resonator frequency becomes $\omega_r\pm\chi$ depending on $\hat\sigma_z$, and because $\chi\hat a^\dagger\hat a\hat\sigma_z$ commutes with $\hat\sigma_z$, reading the resonator measures the qubit without flipping it — a quantum non-demolition readout. Finally, two device relations complete the working toolkit: a two-junction (split) transmon tunes its frequency with flux, and coherence bookkeeping ties the decay and dephasing times together.

where: $\Phi_{ext}$ is the flux threading the SQUID loop; $E_{J\Sigma}$ is the summed junction energy; $T_1$ is the energy-relaxation time, $T_\varphi$ the pure-dephasing time, and $T_2$ the transverse coherence time. Reading it: flux tunes $\omega_{01}(\Phi_{ext})\approx\sqrt{8E_J^{\mathrm{eff}}E_C}-E_C$ for two-qubit gates, while $T_2\le 2T_1$ always, saturating when dephasing is negligible.

The behaviour of a single Josephson circuit is governed almost entirely by the dimensionless ratio $E_J/E_C$, where $E_J=\Phi_0 I_c/2\pi$ is the Josephson energy and $E_C=e^2/2C$ is the single-electron charging energy.

1 · Designing a transmon from E_J and E_C (fully worked)

A junction is fabricated with Josephson energy $E_J/h=15\,\mathrm{GHz}$ and shunted so that $E_C/h=0.25\,\mathrm{GHz}$ (thus $E_J/E_C=60$). Find the qubit frequency, the anharmonicity, and the thermal excited-state population at $15\,\mathrm{mK}$.

Step 1. Plasma frequency: $\hbar\omega_p=\sqrt{8E_JE_C}$, so $\omega_p/2\pi=\sqrt{8\cdot15\cdot0.25}\,\mathrm{GHz}=\sqrt{30}=5.477\,\mathrm{GHz}$. This is the harmonic part — the frequency the circuit would have with no anharmonicity.

Step 2. Qubit frequency: $\hbar\omega_{01}=\hbar\omega_p-E_C$, so $\omega_{01}/2\pi=5.477-0.250=\mathbf{5.23\ GHz}$, squarely in the standard 4–6 GHz band. The junction pulls the transition down by exactly one charging energy.

Step 3. Anharmonicity: $\hbar\alpha\approx-E_C$, so $\alpha/2\pi\approx\mathbf{-250\ MHz}$. The relative anharmonicity is $\alpha/\omega_{01}\approx-4.8\%$ — small, but the $|2\rangle$ level is still 250 MHz off resonance, enough to run 20–40 ns gates with DRAG shaping. This is the transmon compromise: weak anharmonicity bought in exchange for charge-noise immunity.

Step 4. Charge-noise check: charge dispersion is suppressed as $e^{-\sqrt{8E_J/E_C}}=e^{-\sqrt{480}}=e^{-21.9}\approx3\times10^{-10}$ in relative frequency terms. Offset-charge fluctuations shift the qubit frequency by parts in $10^{9}$ — utterly negligible, which is the entire point of running at $E_J/E_C=60$.

Step 5. Thermal population: $\hbar\omega_{01}/k_B=0.24\,\mathrm{K}\times(5.23/5)=0.251\,\mathrm{K}$, so $\hbar\omega_{01}/k_BT=0.251/0.015=16.7$ and $\bar n_{th}\approx e^{-16.7}\approx\mathbf{6\times10^{-8}}$. The qubit initializes to $|0\rangle$ by passive cooling. Result: $\omega_{01}/2\pi=5.23$ GHz, $\alpha/2\pi=-250$ MHz, $\bar n_{th}\approx6\times10^{-8}$.

2 · Dispersive readout and the coherence budget

The transmon above ($\omega_q/2\pi=5.23\,\mathrm{GHz}$, rounded to $5\,\mathrm{GHz}$ for the estimate) is coupled to a readout resonator at $\omega_r/2\pi=7\,\mathrm{GHz}$ with coupling $g/2\pi=100\,\mathrm{MHz}$. Find the dispersive shift and the state-dependent splitting, and compute $T_2$ given $T_1=100\,\mu\mathrm{s}$ and $T_\varphi=200\,\mu\mathrm{s}$.

Step 1. Detuning: $\Delta=\omega_q-\omega_r$, so $\Delta/2\pi=5-7=-2\,\mathrm{GHz}$. Check the dispersive condition: $g/|\Delta|=0.1/2=0.05\ll1$, so the expansion is valid. We are safely in the regime where the qubit and resonator only virtually exchange photons.

Step 2. Dispersive shift: $\chi=g^2/\Delta$, so $\chi/2\pi=(0.1)^2/(-2)=\mathbf{-5\ MHz}$. The resonator sits at $\omega_r+\chi$ for $|0\rangle$ and $\omega_r-\chi$ for $|1\rangle$. A 5 MHz pull is the lever the measurement uses.

Step 3. State-dependent splitting: $2|\chi|/2\pi=\mathbf{10\ MHz}$. For the two peaks to be resolved the resonator linewidth $\kappa$ must satisfy $\kappa\lesssim2\chi$; a typical $\kappa/2\pi\sim1$–$5\,\mathrm{MHz}$ gives clean single-shot discrimination in a few hundred nanoseconds. This is the number plotted in the second figure.

Step 4. Coherence budget: $1/T_2=1/(2T_1)+1/T_\varphi=1/200+1/200=1/100\,\mu\mathrm{s}^{-1}$, so $T_2=\mathbf{100\ \mu s}$. Even with finite pure dephasing, $T_2$ is capped at $2T_1=200\,\mu\mathrm{s}$; here dephasing pulls it down to $100\,\mu\mathrm{s}$. Result: $\chi/2\pi=-5$ MHz, splitting 10 MHz, $T_2=100\,\mu$s.

The superconducting qubit is where a decades-old question — can a macroscopic electrical variable be quantum? — was answered yes in the laboratory, and then engineered into a computer.

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, §7.8 (physical implementations) · Preskill, Lecture Notes for Physics 219, Ch. 4–5 (open systems and implementations) · Krantz et al., "A Quantum Engineer's Guide to Superconducting Qubits", Appl. Phys. Rev. 6, 021318 (2019) · Koch et al., "Charge-insensitive qubit design derived from the Cooper pair box", Phys. Rev. A 76, 042319 (2007) · Devoret & Schoelkopf, "Superconducting Circuits for Quantum Information: An Outlook", Science 339, 1169 (2013) · Blais et al., "Circuit quantum electrodynamics", Rev. Mod. Phys. 93, 025005 (2021). All quoted device figures ($\omega_{01}\sim4$–$6\,\mathrm{GHz}$, $\alpha\sim-250\,\mathrm{MHz}$, $T_1,T_2\sim10$–$100\,\mu\mathrm{s}$, $\sim10$–$20\,\mathrm{mK}$) are standard representative values from the cited reviews, not fabricated precision figures.

Prerequisites. The Josephson Junction · The DiVincenzo Criteria · The Transmon Qubit. Leads to. Circuit Quantum Electrodynamics · Dispersive Readout · Trapped-Ion Qubits.

Ask the tutor: Want to walk through how the anharmonicity $\alpha=-E_C$ turns a nearby $|2\rangle$ level from a liability for single-qubit gates into the resource that powers the CZ gate?