Gate-based quantum-computing platform built from Josephson-junction-based microwave circuits (circuit QED). Foundational: Josephson 1962 equations for supercurrent tunneling; Devoret-Martinis-Clarke 1985 macroscopic quantum tunneling;…
superconducting-qubits
Transmon: E_J/E_C ~ 50 for exponentially-suppressed charge-noise sensitivity
Koch-Yu-Gambetta-Houck-Schuster-Blais-Devoret-Girvin-Schoelkopf 2007 transmon qubit: a Cooper-pair box shunted by a large capacitor to…
Fluxonium: E_J/E_L ~ 1 — high anharmonicity (~GHz), long T₁ (ms)
Manucharyan-Koch-Glazman-Devoret 2009 fluxonium: a Josephson junction shunted by a large superinductance (typically L ≈ μH, implemented as…
Dispersive regime |Δ| ≫ g: qubit-state-dependent resonator frequency pull 2χ
Wallraff-Schuster-Blais 2004 dispersive-readout scheme for circuit QED. Starting from the Jaynes-Cummings Hamiltonian H_JC = ℏω_q σ_z/2 +…
Josephson energy: E_J cos φ̂ with I_c = 2πE_J/Φ_0; Φ_0 = h/(2e) flux quantum
Josephson 1962 dc relations: the potential energy of a Josephson junction as a function of the gauge-invariant phase difference φ across…
Transmon anharmonicity: α = ω_12 − ω_01 ≈ −E_C (leading order in (E_C/E_J)^(1/2))
Transmon (E_J/E_C ≫ 1) is effectively a weakly-anharmonic oscillator. Duffing-oscillator expansion of the H_trans = 4E_C(n̂−n_g)² − E_J…
Dispersive shift χ = g²/Δ for two-level limit; χ = g² α/(Δ(Δ+α)) for transmon
Blais-Huang-Wallraff-Girvin-Schoelkopf 2004 dispersive shift for circuit-QED. In the two-level (σ_z) truncation of the Jaynes-Cummings…
Josephson (Φ, Q) symplectic phase space: [Φ, Q] = iℏ; ΔΦ·ΔQ ≥ ℏ/2 symplectic-capacity bound
Superconducting-circuit symplectic phase space. Setup: a single superconducting-circuit node has canonical coordinate Φ (flux) and…
LC oscillator Hamiltonian H = Q²/(2C) + Φ²/(2L) as diagonal bilinear form in (Φ, Q)
LC-oscillator Hamiltonian as diagonal bilinear form. Setup: parallel LC circuit with inductance L, capacitance C; the stored energy is H_LC…
Transmon Hamiltonian H = 4E_C(n̂−n_g)² − E_J·cos(φ̂); charge-basis Schur triangulation
Transmon Hamiltonian charge-basis Schur triangulation. Setup: a transmon qubit is a Josephson junction shunted by a large capacitance,…
LC vacuum uncertainty ΔΦ·ΔQ = ℏ/2 (symplectic-capacity saturation by Gaussian vacuum)
LC ground-state Heisenberg uncertainty saturation. From framework F7, [Φ̂, Q̂] = iℏ implies ΔΦ · ΔQ ≥ ℏ/2 via Robertson-Schrödinger. The…
LC plasma frequency ω_LC = 1/√(LC) from bilinear-form diagonalisation; ℏω·(a†a+1/2)
LC plasma frequency from bilinear-form diagonalisation. From framework F8, the LC-Hamiltonian bilinear form H_LC = Φ²/(2L) + Q²/(2C) has…
Transmon dispersive shift χ = g²·α/(Δ·(Δ + α)) (Koch 2007 finite-anharmonicity correction)
Transmon dispersive shift with finite-anharmonicity correction (Koch et al. PRA 76, 042319, 2007). From framework F9 (transmon Hamiltonian…
Transmon (Koch 2007)
J Koch 2007 transmon-qubit: charge-insensitive flux-tunable; modern IBM + Google qubits; T2 > 100 us coherence.
Circuit QED (Blais 2004)
A Blais 2004 + Wallraff 2004 circuit-QED strong-coupling SC-resonator-qubit; modern dispersive-readout + multi-qubit-bus.
Flux qubit (Mooij 1999)
Mooij-Orlando-Levitov-Lloyd 1999 flux-qubit; persistent-currents +/- 1 uA; modern fluxonium 2014 100x improved coherence.
Cooper-pair box (Nakamura 1999)
Y Nakamura 1999 first SC-qubit Rabi-oscillations; modern transmon descendant; basis of all SC-qubit architectures.
Dispersive readout (Blais 2007)
Blais 2007 + Reed 2010 dispersive-readout via cavity-pull; modern parametric-amp + JPA + JTWPA quantum-limited noise.
Logical qubit (Google 2024)
Google Quantum AI 2024 'Quantum error correction below the surface code threshold'; d=7 surface-code logical-qubit; below-threshold…
Transmon (Koch 2007)
J Koch-Yale 2007 transmon-qubit anharmonicity; modern modern foundational text + 2024 fluxonium + heavy-fluxonium + cat-qubit alternatives.
Charge qubit (Nakamura 1999)
Y Nakamura-J S Tsai 1999 first SC-qubit Cooper-pair-box; modern modern foundational + transmon evolution + 2024 dispersive readout.
Circuit QED (Blais 2004)
A Blais-Gambetta-Wallraff 2004 circuit-QED; modern modern foundational text + dispersive-readout + 2024 single-qubit gate >99.99% fidelity.
Surface code Yale (2014)
Schoelkopf-Yale 2014 + Google 2014 surface-code; modern modern foundational + 2024 Google distance-7 below threshold + Quantinuum logical.
DD (Viola 2000)
L Viola 2000 dynamical-decoupling; modern modern foundational text + 2024 ML-pulse-design + DRAG-pulses + Carr-Purcell-Meiboom-Gill scq.
RB (Knill 2008)
E Knill 2008 randomized-benchmarking; modern modern foundational + Magesan 2011 interleaved + 2024 cross-entropy benchmarking Arute scq.