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

Noise operator	t1_capacitive
\(B_\lambda\)	\(2e \hat{n}\)

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

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

where \(C\) 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

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

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are Cos2phi, Fluxonium, TunableTransmon.

The parameters that determine what transitions are taken into account during the calculation of \(T_1\) due to this channel, are i, j and total. Their properties are described below.

Parameter name	Default value	Description
i	1	Index of the first state involved in the transition
j	0	Index of the second state involved in the transition
total	True	Determines how the \(T_1\) time (or rate) is calculated. If total=False then a transition from state i to state j is assumed. Depending on whether \(i>j\) or \(i<j\), the resulting \(T_1\) time (or rate) corresponds to a relaxation or excitation process, respectively. If total=True then both transition rates from j to i and i to j are combined to give total effective depolarization time (or rate).

Warning

By default, total=True is used when calculating the \(T_1\) coherence time for this channel. This means that both the excitation and relaxation rates are combined to give an effective \(T_1\) depolarization time (or rate). See table above for details.

Other parameters that could be used for further customization are:

Parameter name	Default value	Description
Q_cap	\(Q_{\rm cap}(\omega)\)	Capacitive quality factor Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
get_rate	False	Return rate instead of time

References: [Smith2020] (and see [Nguyen2019] for a related discussion)

Inductive noise

Noise operator	t1_inductive
\(B_\lambda\)	\(\frac{\Phi_0}{2\pi} \hat{\phi}\)

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

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

where \(L\) 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

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

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

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are: Cos2phi, Fluxonium.

The parameters that determine what transitions are taken into account during the calculation of \(T_1\) due to this channel, are i, j and total. Their properties are described below.

Parameter name	Default value	Description
i	1	Index of the first state involved in the transition
j	0	Index of the second state involved in the transition
total	True	Determines how the \(T_1\) time (or rate) is calculated. If total=False then a transition from state i to state j is assumed. Depending on whether \(i>j\) or \(i<j\), the resulting \(T_1\) time (or rate) corresponds to a relaxation or excitation process, respectively. If total=True then both transition rates from j to i and i to j are combined to give total effective depolarization time (or rate).

Warning

By default, total=True is used when calculating the \(T_1\) coherence time for this channel. This means that both the excitation and relaxation rates are combined to give an effective \(T_1\) depolarization time (or rate). See table above for details.

Other parameters that could be used for further customization are:

Parameter name	Default value	Description
Q_ind	\(Q_{\rm ind}(\omega)\)	Inductive quality factor Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
get_rate	False	Return rate instead of time

References: [Nguyen2019], [Smith2020]

Charge-coupled impedance noise

Noise operator	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

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

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are TunableTransmon, Fluxonium, FullZeroPi.

The parameters that determine what transitions are taken into account during the calculation of \(T_1\) due to this channel, are i, j and total. Their properties are described below.

Parameter name	Default value	Description
i	1	Index of the first state involved in the transition
j	0	Index of the second state involved in the transition
total	True	Determines how the \(T_1\) time (or rate) is calculated. If total=False then a transition from state i to state j is assumed. Depending on whether \(i>j\) or \(i<j\), the resulting \(T_1\) time (or rate) corresponds to a relaxation or excitation process, respectively. If total=True then both transition rates from j to i and i to j are combined to give total effective depolarization time (or rate).

Warning

By default, total=True is used when calculating the \(T_1\) coherence time for this channel. This means that both the excitation and relaxation rates are combined to give an effective \(T_1\) depolarization time (or rate). See table above for details.

Other parameters that could be used for further customization are:

Parameter name	Default value	Description
Z	50	Complex Impedance of coupled line (\(\Omega\)) Can be function of \(\omega\), or a number
T	0.015	Temperature (in K)
get_rate	False	Return rate instead of time

References: [Schoelkopf2003], [Ithier2005]

Flux-bias line noise

Noise operator	t1_flux_bias_line
\(B_\lambda\)	\(\frac{\partial \hat{H}}{\partial \Phi_x}\)

Noise due to current fluctuations in the flux-bias line. 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.

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are TunableTransmon, Fluxonium, FullZeroPi, ZeroPi.

The parameters that determine what transitions are taken into account during the calculation of \(T_1\) due to this channel, are i, j and total. Their properties are described below.

Parameter name	Default value	Description
i	1	Index of the first state involved in the transition
j	0	Index of the second state involved in the transition
total	True	Determines how the \(T_1\) time (or rate) is calculated. If total=False then a transition from state i to state j is assumed. Depending on whether \(i>j\) or \(i<j\), the resulting \(T_1\) time (or rate) corresponds to a relaxation or excitation process, respectively. If total=True then both transition rates from j to i and i to j are combined to give total effective depolarization time (or rate).

Warning

By default, total=True is used when calculating the \(T_1\) coherence time for this channel. This means that both the excitation and relaxation rates are combined to give an effective \(T_1\) depolarization time (or rate). See table above for details.

Other parameters that could be used for further customization are:

Parameter name	Default value	Description
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)
get_rate	False	Return rate instead of time

References: [Koch2007], [Groszkowski2018],

Quasiparticle-tunneling noise

Noise operator	t1_quasiparticle_tunneling
\(B_\lambda\)	\(\sin(\hat{\phi}/2)\) (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

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

** This form assumes that the external flux is grouped with the inductive term of the Hamiltonian. In qubits where the flux is grouped with the Josephson term, the noise operator is appropriately transformed.

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are TunableTransmon, Fluxonium, FullZeroPi, ZeroPi.

The parameters that determine what transitions are taken into account during the calculation of \(T_1\) due to this channel, are i, j and total. Their properties are described below.

Parameter name	Default value	Description
i	1	Index of the first state involved in the transition
j	0	Index of the second state involved in the transition
total	True	Determines how the \(T_1\) time (or rate) is calculated. If total=False then a transition from state i to state j is assumed. Depending on whether \(i>j\) or \(i<j\), the resulting \(T_1\) time (or rate) corresponds to a relaxation or excitation process, respectively. If total=True then both transition rates from j to i and i to j are combined to give total effective depolarization time (or rate).

Warning

By default, total=True is used when calculating the \(T_1\) coherence time for this channel. This means that both the excitation and relaxation rates are combined to give an effective \(T_1\) depolarization time (or rate). See table above for details.

Other parameters that could be used for further customization are:

Parameter name	Default value	Description
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)
get_rate	False	Return rate instead of time

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

User-defined noise

Noise operator	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. To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These are Fluxonium, FluxQubit, FullZeroPi, Transmon, TunableTransmon, ZeroPi.