Useful qubit equations

Basics (top)

\( \omega = 2 \pi f = \sqrt{\frac{1}{LC}} \)

\( v = \lambda f \)

Planck's constant: \( \hslash = \frac{h}{2\pi} \approx 1.054 \times 10^{-34} J \cdot s \)

Boltzmann's constant: \( k_B \approx 1.380 \times 10^{-23} J K^{-1} \)

Flux quanta: \(\Phi_0 = \frac{h}{2e} \approx 2.067 \times 10^{-15} Wb \)

Qubit Parameters (top)

\( E_C = \frac{e^2}{2C} \)

\( E_L = \frac{\phi_0^2}{2L} \)

\( E_J = \frac{c \sigma}{4e} = \frac{\hslash I_C}{2e} = \frac{\hslash A J_C}{2e} = \frac{1}{\phi_0 J_C A} \)

Useful qubit values (top)

Qubit temperature: Using the Boltzmann formula, we find \(T = \frac{(E_i - E_f) h}{- \ln(p_i / p_j) k_B} \)

Dispersive approximation: \( \Delta \gg |g| \sqrt{n+1} \)

Chi shift: \( \chi \approx \frac{g^2}{\Delta} \)

Quality factor: \(Q = \frac{\omega}{\kappa} = \frac{1}{\tan(\delta)} \)

Note for design: to optimize, \( 2 \chi = \kappa \)

Purcell: \( \Gamma = \frac{\kappa g^2}{\Delta^2} \)

Transmon approximation: \( f_{01} \approx \sqrt{8 E_J E_C} \)

Hamiltonians (top)

Transmon: \( H_t = - 4E_C \frac{d^2}{d\phi^2} - E_J \cos(\phi) \)

Fluxonium: \( H_f = - 4E_C \frac{d^2}{d\phi^2} - E_J \cos(\phi - 2 \pi \frac{\Phi_{ext}}{\Phi_0}) + \frac{1}{2} E_L \phi^2 \)

James-Cummings Hamiltonian: \( H = H_q + H_r + H_{int} = \frac{1}{2} \hslash \omega_q \sigma_z + \hslash \omega_r a^\dagger a + \frac{1}{2} \hslash \Omega (a \sigma_{+} + a^\dagger \sigma_{-} ) \)

Fast flux gates: \( H/h = A \delta \Phi_{ext} \sigma_x + \frac{\omega_q}{2} \sigma_z \)

FF (inductively coupled): \( H \approx H_a + H_b + \frac{L_c}{L_a} E_L \phi_a \phi_b \approx 4 \pi^2 \frac{L_c}{L_a} E_L \sigma_{x,a} \sigma_{x,b} \)

Different qubit regimes (top)

Cooper Pair Box: \(E_J / E_C \approx 1 \)

Transmon: \(E_J / E_C \geq 50 \)

Fluxonium: \(E_J / E_L \geq 10 \)

Heavy Fluxonium: \( E_J / E_C \approx 10 \)

Coherences (top)

\( \Gamma_{\mathrm{diel}} \propto \frac{\omega_q}{E_C Q_{\mathrm{diel}}} \coth{\frac{\hslash \omega_q}{2 k_B T}} | \langle g | \phi | e \rangle|^2 \)

\( T_1 = \)

\( T_2 \leq 2 T_1 \)

Design(top)

\( L \propto \mathrm{wire length}, 1mm \approx 1nH \)

\( C \propto \mathrm{pad area}, 50fF \approx 1\mu m^2\)

\( v_{\mathrm{eff}} = \frac{c}{\sqrt{\epsilon_{\mathrm{eff}}}} \)

\( g = \frac{2 \beta e V_0}{h}; \quad \beta = \frac{C_{qr}}{C_{qr} + C_{gr}} ; V_0 = \sqrt{\frac{\hslash \omega_r}{2C_r}} \)

\( E_J: 12.5GHz \approx 0.1 \mu m^2 \)

\( E_C: 2GHz \approx 10 fF \)

Critical current: \( J_c = \frac{4 \pi e E_J}{A} \)

Materials(top)

Aluminum: \( T_C = 1.2K \)

Tantalum: \( T_C = 4.47K \)

Copper:

Gold:

Stainless Steel:

Benchmarking (top)

Interleaved randomized benchmarking:

\( F(i) = A \alpha^i + B \)

\( r = 1 - \alpha - \frac{1-\alpha}{d}; d = 2^n \)

\( r_C = \frac{(d-1)(1 - \alpha_C / \alpha)}{d} \)

where:
\( i \) = number of gates being played
\( n \) = number of qubits
\(\alpha\)= fitted decay from RB over Clifford group
\( r \) = avg. infidelity across Clifford group
\( \alpha_C \) = fitted fidelity decay from interleaving target gate between random Clifford group
\( r_C \) = infidelity of target gate

Quantum State Tomography purity/fidelity:

Quantum Process Tomography fidelity:

Useful Gates (top)

iSWAP:

bSWAP:

Marki Microwave Power, dBm chart

Marki Microwave Return Loss to VSWR chart

Three Plus One: On Gates

Convenient Conversion Factors