# Depolarization — scqubits Documentation (2023)

Noise may cause depolarization of the qubit by inducing spontaneous transitions among eigenstates. scqubits uses thestandard perturbative approach (Fermi’s Golden Rule) to approximate the resulting transition rates due to differentnoise 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 atthe 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 ofexcitations (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=Falsewill calculate only a single-directional transition rate from the state indexed i to the state indexed j (both of whichcal 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

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

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

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

(Video) "Next generation superconducting qubits for quantum computing" presented by Jens Koch, Northwestern

To see a detailed signature of this method, see the API description of qubits that support this particular noise channel. These areCos2phi,Fluxonium,FullZeroPi,TunableTransmon,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

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 iand i to j are combined to give total effective depolarizationtime (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

References: [Nguyen2019], [Smith2020]

## 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

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

where $$L_J$$ is the relevant inductance or superinductance, and $$Q_{\rm ind}$$ the corresponding inductivequality 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.

(Video) Seminar Talk - Next-generation superconducting qubits for quantum computing by Dr. Jens Koch

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

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 iand i to j are combined to give total effective depolarizationtime (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

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 areTunableTransmon,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

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 iand i to j are combined to give total effective depolarizationtime (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.

(Video) Jens Koch: Lecture IV

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

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 areTunableTransmon,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

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 iand i to j are combined to give total effective depolarizationtime (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:

(Video) PHYSICS 20170110 22 Emerson 019

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

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 inductivequality 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 areTunableTransmon,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

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 iand i to j are combined to give total effective depolarizationtime (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

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 areFluxonium,FluxQubit,FullZeroPi,Transmon,TunableTransmon,ZeroPi.

## References

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