Emulator Noise Model

Users who have direct access to the Quantinuum API have the option of experimenting with the physical noise parameters of the emulator. When deviating from the default emulation model, users should not assume that performance predicted with modified error parameters will match hardware performance. As updates to the quantum computers are made, the emulator noise parameters and the underlying error models are subject to change to accommodate performance improvements, updates in the methodology for measuring device parameters and research into the noise sources themselves.

All the errors are applied even when only certain parameters are specified. Only the parameters specified are overridden. To turn off certain error parameters, explicitly set them to 0.

Physical Noise

The emulator runs with default error parameters that represent a noise environment that closely resembles the respective hardware. These error parameters can be set and used to override the default error parameters and do finer-grain tweaks of the error model. Modification of the error parameters away from default values is an advanced option and not recommended as a starting point for emulations of hardware performance.

Physical Noise API Options

Name	API Option	Description
1-Qubit Fault Probability	p1	Probability of a fault occurring during a 1-qubit gate.
2-Qubit Fault Probability	p2	Probability of a fault occurring during a 2-qubit gate.
Bit Flip Measurement Probability	p_meas	Probability of a bit flip being applied to a measurement. Either a float or a tuple of 2 floats. If it is a single float then that error rate is used to bitflip both 0 and 1 measurement results. If a tuple is supplied, the first element is the probability a bit flip is applied if a 0 result occurs during measurement while the second error rate if a 1 is measured.
Crosstalk Measurement Fault Probability	p_crosstalk_meas	Probability of a crosstalk measurement fault occurring.
Initialization Fault Probability	p_init	Probability of a fault occurring during initialization of a qubit.
Crosstalk Initialization Fault Probability	p_crosstalk_init	Probability of a cross-talk fault occurring during initialization of a qubit.
Ratio of 1-Qubit Spontaneous Emission to p1	p1_emission ratio	Fraction of p1 that is spontaneous emission for a single qubit instead of asymmetric depolarizing noise.
Ratio of 1-Qubit Spontaneous Emission in 2-Qubit Gate to p2	p2_emission_ratio	Fraction of p2 that is spontaneous emission for a 1- qubit in a 2-qubit gate instead of asymmetric depolarizing noise.

The 1- and 2-qubit fault probabilities are largely modeled using asymmetric depolarizing channels; however, there is smaller probability that a spontaneous emission event happens. The probability is about an order of magnitude lower than the corresponding asymmetric depolarizing error rate. The spontaneous emission error rates can be scaled using the scaling parameters given in the Scaling section. If a spontaneous emission event happens then \(\frac{1}{4}\) the time \(Pauli-X\) is applied, \(\frac{1}{4}\) the time Pauli-Y is applied, and \(\frac{1}{2}\) the time leakage is applied[1].

The 2-qubit fault probability corresponds to the asymmetric depolarizing probability of the H-Series fully entangling 2-qubit gate, \(ZZ()\). The probability of asymmetric depolarizing error for the arbitrary angle 2-qubit gate, \(RZZ(\theta)\), depends on the angle \(\theta\). The spontaneous emission error channel is the same for both \(ZZ()\) and \(RZZ(\theta)\).

Dephasing Noise

The noise model includes a memory error for which \(Z\) is applied. This is often called “dephasing” or “memory” noise and depends on the duration for which the qubits are idling or transporting in the trap. We potentially model two types of dephasing noise: one where the probability of applying \(Z\) is quadratically dependent on the duration, \(d\) and another where the probability is linearly dependent on \(d\). Note, we apply both sorts of noise simultaneously. For state vector simulations, the quadratic noise is modeled in the emulator by default as coherent noise. For this coherent quadratic dephasing noise, the \(RZ\) gate is applied with an angle, \(a\), proportional to quadratic dephasing frequency, \(f\) multiplied by the duration, \(d\). The resulting probability of the \(RZ\) gate applying a \(Z\) operation on a plus state is \(sin⁡(0.5fd)^2\), which is why we call this a form of quadratic dephasing.

For the stabilizer simulator, by default this quadratic noise is modeled incoherently by applying Pauli-\(Z\) with probability, \(sin⁡(0.5 f d)^2\), to model more closely the quadratic dependency with frequency, \(f\) and duration, \(d\), as seen in the coherent model. Note, stabilizer simulations can only simulate Clifford and measurement-like gates, so the \(RZ\) gate cannot be applied directly.

For both state vector and stabilizer simulations, linear dephasing is modeled with \(Z\) applied using a probability equal to the linear dephasing rate multiplied by the duration.

Switching between the coherent and incoherent quadratic dephasing model can be accomplished by setting coherent_dephasing either True or False. As mentioned, coherent_dephasing is True by default for the state vector simulations and False by default for stabilizer simulations. If coherent_dephasing is set to False then the frequency for the quadratic error model (quadratic_dephasing_rate) is multiplied by coherent_to_incoherent_factor to attempt to make up for increased noise due to coherent effects; however, how sensitive circuits are to coherent effects depends on the circuit. Therefore, users may want to adjust this factor appropriately.

In addition, a transport dephasing parameter (transport_dephasing) and an idle dephasing parameter (idle_dephasing) are both turned on by default. Both can be toggled off.

Dephasing Noise Emulator API Options

Name	API Option	Description
Coherent Dephasing	coherent_dephasing	A boolean value determining whether quadratic dephasing is applied (default: True)
Coherent Quadratic Dephasing Model		the gate RZ (frequency x duration) is applied during transport and qubit idling where frequency is equal to quadratic_dephasing_rate (units of \(2\pi\) radians per second). This model is used if coherent_dephasing is True. Applied by default for the state vector simulator.
Quadratic Dephasing Rate	quadratic_dephasing_rate	The frequency, \(f\), in applying \(RZ(fd)\) during transport and idling.
Incoherent Quadratic Dephasing Model		Pauli-\(Z\) is applied during transport and qubit idling according to the probability, \(sin(\frac{fd}{2})^2\), where frequency is equal to quadratic_dephasing_rate multiplied by the coherent_to_incoherent_factor (all in units of \(2\pi\) radians per second). This model is used if coherent_dephasing is False. This model is mostly used to mimic coherent dephasing noise for stabilizer simulations and is applied by default for the stabilizer simulator.
Incoherence Multiplier	coherent_to_incoherent_factor	A multiplier on the quadratic term when running stabilizer simulations to attempt to account for increases in error due to coherent effects in the circuit.
Linear Dephasing Model		Pauli \(Z\) is applied during transport and qubit idling according to the probability of linear_dephasing_rate multiplied by duration (\(fd\)) where linear_dephasing_rate is per second (\(s^{-1}\)), and duration is in units of seconds. This model is used in conjunction with either the coherent or incoherent quadratic dephasing model.
Linear Dephasing Rate	linear_dephasing_rate	The probability of applying \(Z\) with \(p=rd\) where \(r\) is rate and \(d\) is duration. This models the memory error. Note both the quadratic and linear term can be applied in the same simulation.
Transport Dephasing	transport_dephasing	A boolean affecting whether memory noise is applied during transport.
Idle Dephasing	idle_dephasing	A boolean affecting if memory noise is applied due to qubit idling.

Arbitrary Angle Scaling

The H-Series systems have a native arbitrary-angle \(ZZ\) gate, \(RZZ(\theta)\). For implementation of this gate in the H-Series emulators, certain parameters relate to the strength of the asymmetric depolarizing noise. These parameters depend on the angle \(\theta\). This is normalized so that \(\theta=\frac{\pi}{2}\) gives the 2-qubit fault probability (p2). The parameters for asymmetric depolarizing noise are fit parameters that fit the noise estimated as the angle \(\theta\) changes per this equation:

\[\begin{split} \left({przz}_a \times (\frac{|\theta|}{\pi})^{{przz}_{power}} + {przz}_b\right) & \times p_2; \quad \theta < 0\\ \left({przz}_c \times (\frac{|\theta|}{\pi})^{{przz}_{power}} + {przz}_d\right) & \times p_2; \quad \theta < 0\\ \left({przz}_b + {przz}_d \right) & \times 0.5; \quad \theta = 0 \end{split}\]

Arbitrary Angle Scaling API Options

Name	API Option
Fit Parameter 1	\(przz_a\)
Fit Parameter 2	\(przz_b\)
Fit Parameter 3	\(przz_c\)
Fit Parameter 4	\(przz_d\)
Polynomial	\(przz_{power}\)

Scaling

A scaling factor can be applied that multiplies all the default or supplied error parameters by the scaling rate. In this case, a 1 does not change the error rates while 0 makes all the errors have a probability of 0. Other aspects of the noise model can scale specific error rates in the error model, as specified in the table below.

Scaling Noise API Options

Name	API Option	Description
Scaling	scale	Scale all error rates in the model linearly.
P1 Scaling	p1_scale	Scale the probability of 1-qubit gates having a fault.
P2 Scaling	p2_scale	Scale the probability of 2-qubit gates having a fault.
Measurement Scaling	meas_scale	Scale the probability of measurement having a fault.
Initialization Scaling	init_scale	Scale the probability of initialization having a fault.
Memory Scaling	memory_scale	Linearly scale the probability of dephasing causing a fault.
Emission Scaling	emission_scale	Scale the probability that a spontaneous emission event happens during a 1- or 2-qubit gate.
Cross-talk Scaling	crosstalk_scale	Scale the probability that measurement or initialization crosstalk events get applied to qubits, during mid-circuit measurement and reset (initialization), crosstalk noise can occur that effectively measures other qubits in the trap or cause them to leak.
Leakage Scaling	leakage_scale	Scale the probability that a leakage even happens during 1- or 2-qubit gates as well as during initialization or crosstalk; on the device half the time, spontaneous emission leads to a leakage event.

References

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[1]

C. Ryan-Anderson, J. G. Bohnet, K. Lee, D. Gresh, A. Hankin, J. P. Gaebler, D. Francois, A. Chernoguzov, D. Lucchetti, N. C. Brown, T. M. Gatterman, S. K. Halit, K. Gilmore, J. A. Gerber, B. Neyenhuis, D. Hayes, and R. P. Stutz. Realization of real-time fault-tolerant quantum error correction. Phys. Rev. X, 11:041058, Dec 2021. URL: https://link.aps.org/doi/10.1103/PhysRevX.11.041058, doi:10.1103/PhysRevX.11.041058.