Depolarization¶

Noise may cause depolarization of the qubit by inducing spontaneous transitions among eigenstates. scqubits uses the standard perturbative approach (Fermi’s Golden Rule) to approximate the resulting transition rates due to different noise channels.

The rate of a transition from state \(i\) to state \(j\) can be expressed as

\[\Gamma_{ij} = \frac{1}{\hbar^2} |\langle i| B_{\lambda} |j \rangle|^2 S(\omega_{ij}),\]

where \(B_\lambda\) is the noise operator, and \(S(\omega_{ij})\) the spectral density function evaluated at the angular frequency associated with the transition frequeny, \(\omega_{ij} = \omega_{j} - \omega_{i}\). \(\omega_{ij}\) is positive in the case of decay (the qubit emits energy to the bath), and negative in case of excitations (the qubit absorbs energy from the bath).

Unless stated otherwise, it is assumed that the depolarizing noise channels satisfy detailed balanced. This implies

\[\frac{S(-\omega)}{S(\omega)} = \exp{\frac{\hbar \omega}{k_B T}},\]

where \(T\) is the bath temperature, and \(k_B\) Boltzmann’s constant.

Note

By default all \(t_1\) methods estimate the coherence depolarization times from the sum of the upward and downard rates. This behavior is controlled by the arugment total, which can be modified by the user. For example, setting total=False will calculate only a single-directional transition rate from the state indexed i to the state indexed j (both of which cal also be changed by the user through providing their values as arguments)

Capacitive noise¶

Method name	`t1_capacitive`
\(B_\lambda\)	\(2e \hat{n}\)

Capacitive noise corresponds to noise coming from a lossy capacitance. The assumed spectral density reads

\[S(\omega) = \frac{\omega \hbar}{|\omega| C_J Q_{\rm cap}(\omega)} \left(1 + \coth \frac{\hbar |\omega|}{2 k_B T} \right)\]

where \(C_J\) is the relevant capacitance, and \(Q_{\rm cap}\) the corresponding capacitive quality factor. The default value of the frequency-dependent quality factor is assumed to be

\[Q_{\rm cap}(\omega) = 10^{6} \left( \frac{2 \pi \times 6 {\rm GHz} }{ |\omega|} \right)^{0.7}.\]

This method can be customized further through various noise related parameters shown below:

Parameter name	Default value	Description
i	1	Initial state index
j	0	Final state index
Q_cap	\(Q_{\rm cap}(\omega)\)	Capacitive quality factor Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
total	True	Return both up and down rates
get_rate	False	Return rate instead of time

Qubits that support this noise channel include: TunableTransmon, Fluxonium, FullZeroPi, ZeroPi.

References: [Nguyen2019], [Smith2020]

Inductive noise¶

Method name	`t1_inductive`
\(B_\lambda\)	\(\frac{\Phi_0}{2\pi} \hat{\phi}\)

Inductive noise due to lossy inductance. The assumed spectral density reads

\[S(\omega) = \frac{\omega \hbar}{|\omega| L_{J} Q_{\rm ind}(\omega)} \left(1 + \coth \frac{\hbar |\omega|}{2 k_B T} \right)\]

where \(L_J\) is the relevant inductance or superinductance, and \(Q_{\rm ind}\) the corresponding inductive quality factor. The default value of the frequency-dependent quality factor is assumed to be

\[Q_{\rm ind}(\omega) = 500 \times 10^{6} \frac{ K_{0} \left( \frac{h \times 0.5 {\rm GHz}}{2 k_B T} \right) \sinh \left( \frac{\hbar |\omega| }{2 k_B T} \right)}{K_{0} \left( \frac{\hbar |\omega|}{2 k_B T} \right)\ \sinh \left( \frac{\hbar |\omega| }{2 k_B T} \right)},\]

where \(K_0\) is the Bessel function of the second kind.

This method can be customized further through various noise related parameters shown below:

Parameter name	Default value	Description
i	1	Initial state index
j	0	Final state index
Q_ind	\(Q_{\rm ind}(\omega)\)	Inductive quality factor Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
total	True	Return both up and down rates
get_rate	False	Return rate instead of time

Qubits that support this noise channel include: Fluxonium.

References: [Nguyen2019], [Smith2020]

Charge-coupled impedance noise¶

Method name	`t1_charge_impedance`
\(B_\lambda\)	\(2e \hat{n}\)

Noise from a charge coupling to an impedance \(Z(\omega)\). The assumed spectral density reads

\[S(\omega) = \frac{\hbar \omega}{{\rm Re} Z(\omega)} \left(1 + \coth \frac{\hbar |\omega|}{2 k_B T} \right).\]

By default we assume the qubit couples to a infinite transmission line, which leads to

\[{\rm Re} Z(\omega) = 50 \Omega.\]

This method can be customized further through various noise related parameters shown below:

Parameter name	Default value	Description
i	1	Initial state index
j	0	Final state index
Z	50	Complex Impedance of coupled line (\(\Omega\)) Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
total	True	Return both up and down rates
get_rate	False	Return rate instead of time

Qubits that support this noise channel include: TunableTransmon, Fluxonium, FullZeroPi,

References: [Schoelkopf2003], [Ithier2005]

Flux-bias line noise¶

Method name	`t1_flux_bias_line`
\(B_\lambda\)	\(\frac{\partial \hat{H}}{\partial \Phi_x}\)

Noise due to current noisy biasing current coupled to the qubit via flux. The assumed spectral density reads

\[S(\omega) = \frac{M^{2} \omega \hbar}{R} \left(1 + \coth \frac{\hbar |\omega|}{2 k_B T} \right),\]

where \(M\) is the mutual inductance between qubit and the flux line.

This method can be customized further through various noise related parameters shown below:

Parameter name	Default value	Description
i	1	Initial state index
j	0	Final state index
M	400	Mutual inductance between qubit and flux line (in \(\Phi_0/A\))
Z	50	Complex impedance of bias flux line (\(\Omega\)) Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
total	True	Return both up and down rates
get_rate	False	Return rate instead of time

Qubits that support this noise channel include: TunableTransmon, Fluxonium, FullZeroPi, ZeroPi.

References: [Koch2007], [Groszkowski2018],

Quasiparticle-tunneling noise¶

Method name	`t1_quasiparticle_tunneling`
\(B_\lambda\)	\(\cos(\hat{\phi}/2)\) (see note ** below)

Noise due to quasiparticle tunelling. The assumed spectral density reads

\[S(\omega) = \hbar \omega {\rm Re} Y_{\rm qp}(\omega) \left(1 + \coth \frac{\hbar |\omega|}{2 k_B T} \right)\]

where \(L_J\) (with \(E_J = \phi_0^2/L_J\) ) is the relevant inductance or superinductance, and \(Q_{\rm ind}\) the corresponding inductive quality factor. The default value of the frequency-dependent quality factor is assumed to be

The default real part of admittance is assumed to be

\[{\rm Re} Y_{\rm qp}(\omega) = \sqrt{\frac{2}{\pi}} \frac{8 E_J}{R_k \Delta} \ \left(\frac{2 \Delta}{\hbar \omega} \right)^{3/2} x_{\rm qp} \ K_{0} \left( \frac{\hbar |\omega|}{2 k_B T} \right) \sinh \left( \frac{\hbar \omega }{2 k_B T} \right).\]

** In many publications, the operator \(\sin(\hat{\phi}/2)\) is used. This is due to grouping of flux with the quadratic and not the \(\cos\) term of the Hamiltonian.

This method can be customized further through various noise related parameters shown below:

Parameter name	Default value	Description
i	1	Initial state index
j	0	Final state index
Y_qp	\(Y_{\rm qp}\)	Complex admittance (\(\Omega\)) Can be function of \(\omega\), or a number
x_qp	\(3 \times 10^{-6}\)	Quasiparticle density
T	0.015	Temperature (in K)
Delta	\(3.4 \times 10^{-4}\) (for Al)	Superconducting gap (eV)
total	True	Return both up and down rates
get_rate	False	Return rate instead of time

Qubits that support this noise channel include: TunableTransmon, Fluxonium, FullZeroPi, ZeroPi.

References: [Catelani2011], [Nguyen2019], [Pop2014], [Smith2020]

User-defined noise¶

Method name	`t1`
\(B_\lambda\)	user defined

All qubits support user defined noise, where both the noise operator as well as an arbitrary spectral density can be provided.

Qubits that support this noise channel include: Fluxonium, FluxQubit, FullZeroPi, Transmon, TunableTransmon, ZeroPi.