Quantinuum System Model H1
Emulator Product Data Sheet

Version 6.2 February 27, 2023

TABLE OF CONTENTS• INTRODUCTION .............................................................................................. 3• FEATURES ...................................................................................................... 3• USE CASES ..................................................................................................... 3• FUNCTIONAL REQUIREMENTS ..................................................................... 3• EMULATOR ACCESS AND OUTPUT ............................................................. 4• PERFORMANCE.............................................................................................. 4• EMULATION METHOD .................................................................................... 4• NOISE MODEL ................................................................................................ 5
Physical Noise ................................................................................................................................... 6Dephasing Noise ................................................................................................................................ 7
Arbitrary Angle Noise Scaling ............................................................................................................. 8Scaling ............................................................................................................................................... 8• APPENDIX ....................................................................................................... 8© 2023 by Quantinuum. All rights reserved.

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INTRODUCTION
This Product Data Sheet covers all features and characteristics of the Quantinuum System
Model H1 Emulator, which includes the H1-1 and H1-2 emulators.

FEATURESHigh fidelity noise models and parameters closely mimicking System Model H1(hardware) performance. Each emulator uses the same physical noise model, but noiseparameters reflect performance of the device being emulated.
Uses identical API for job submission as System Model H1, enabling seamless
translation from emulator to hardware
Uses identical compiler as System Model H1, containing all the native gates (single-qubit
rotations, two-qubit ZZ gates, arbitrary angle ZZ gates), transport operations andclassical operations used in System Model H1Provides identical output format as System Model H1Allows usage of unique System Model H1 attributes: all-to-all connectivity and qubit reuse
after mid-circuit measurement
Available even while System Model H1 is offline to enable maximized productivity anddevelopment time

USE CASESThe System Model H1 Emulator provides a high-fidelity emulation of System Model H1. Usecases include:
Debugging of quantum code before running on physical hardware
Optimization of quantum code in the presence of noise mechanisms
Exploring new algorithms and techniques for quantum error correction
Introduction to System Model H1 and its unique differentiating capabilities such as qubit
reuse after mid-circuit measurement, all-to-all connectivity, and high-fidelity gates

FUNCTIONAL REQUIREMENTS
The System Model H1 Emulator is meant to be a functional emulation of System Model H1and therefore supports the same functional operations as H1. Specifically, the System Model
H1 Emulator supports:
OPENQASM 2.0 circuits
Quantinuum QASM enhancements, including classical logic, math, and program flow
control
Quantinuum native gate set1 – 𝑅𝑧 (𝜆), 𝑈1𝑞 (𝜃, 𝜑), 𝑍𝑍, 𝑅𝑍𝑍(𝜃)Common compound gates from OPENQASM library, e.g., CX, H
User-defined compound gates
User option of noiseless simulation or inclusion of System Model H1 noise models
Large quantum circuits with a limit of 10,000 on the number of shots
Identical queuing prioritization as System Model H1

1 For definition of native gates, please request a copy of the Quantinuum System Model H1 Product Data Sheet© 2023 by Quantinuum. All rights reserved.

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EMULATOR ACCESS AND OUTPUT
Communication with the System Model H1 Emulator occurs through an API endpoint basedon the OpenQASM 2.0 standard (Cross, Lev, John, & Jay, 2017). Interface details are givenin the Quantinuum Application Programming Interface (API) Specification.
Users can select a System Model H1 Emulator in the machine list API, designated with the“E” suffix machine name. The output of H1 Emulator is a JSON-formatted array, identical tothe output format of System Model H1. Through the Job Submission API, users may selectthe type of emulator used and turning on or off the application of the error model.

PERFORMANCE
The performance of the System Model H1 Emulator is measured in the fidelity to hardware.
With inclusion of accurate and up-to-date noise models, the System H1 Emulator can
provide a high-fidelity representation of System Model H1 output. Fidelity is verified at
Quantinuum by comparison between the emulator and hardware outputs. However, noisemodels cannot fully capture the behavior of System Model H1; users should expect somevariance. In the case of exceptional or unexplained variance, users should contact
Quantinuum technical support at QCsupport@quantinuum.com to discuss the circuit andresults.

EMULATION METHODThe System Model H1 Emulator, accessible via the API, receives instructions directly fromthe same compilers used by the System Model H1 physical quantum hardware. These
compilers translate the submitted quantum program into a set of instructions comprising of
the native gate operations and the transport operations necessary to reconfigure the ion
chain at each step of the program.
Users can choose between either a state-vector or stabilizer emulation method; in bothcases results are performed shot-by-shot. The state-vector emulation method can run any
general quantum circuits, while the stabilizer emulation method is restricted to circuits
involving only quantum unitary gates that are Clifford operations.
The error model for the emulation can be turned on or off, allowing noisy or noise-freeemulations, respectively. The emulated error model includes:
depolarizing gate noiseleakage errors, crosstalk noisedephasing noise due to transport and qubit idling
Except for dephasing, errors on physical qubits are modeled as stochastic processes. Forthe state-vector emulation, dephasing is handled as a coherent Z rotation according to adephasing rate and the duration the qubit spends in transport or while idling while other
qubits are being gated. For the stabilizer emulation, the dephasing noise is treated as astochastic Z error where the probability of a Z error is equal to the Pauli twirledapproximation of the coherent dephasing channel, which is proportional to the square of thedephasing rate multiplied by the duration.

© 2023 by Quantinuum. All rights reserved.

Page 4NOISE MODEL
Users who have direct access to the Quantinuum API have the option of experimenting withthe 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.
All parameters listed in Table 1 are the default settings of the System Model H1 emulators.
As updates to the System Model H1 quantum computers are made, the emulator noise
parameters and the underlying error model are subject to change to accommodate
performance improvements, updates in the methodology for measuring devices parameterand research into the noise sources themselves
All the errors are applied even when only certain parameters are specified. Only theparameters specified are overridden. To turn off certain error parameters, explicitly set themto 0.
For more information on the errors observed, see the following publications: Realization of
Real-Time Fault-Tolerant Quantum Error Correction, Implementing Fault-tolerant Entangling
Gates on the Five-qubit Code and the Color Code.
Table 1 Default Settings of the System Model H1 Emulators

Default Settings

H1-1

H1-2

GeneralQubits

20

12

All-to-all

All-to-all

53Single-Qubit Fault Probability (p1)4.474 × 10−5

3.408 × 10−5

Two-Qubit Fault Probability (p2)2.048 × 10−3

2.5 × 10−3

Bit Flip Measurement Probability (0 outcome) (p_meas)

1.7 × 10−3

1.833 × 10−3

Bit Flip Measurement Probability (1 outcome) (p_meas)
Crosstalk Measurement Fault Probability(p_crosstalk_meas)
Initialization Fault Probability (p_init)

4.5 × 10−3

4.6 × 10−3

6.745 × 10−6

5.273 × 10−5

3.62 × 10−5

5 × 10−6

5.020 × 10−6

2 × 10−5

0.15

0.15

0.3

0.3

Quadratic Dephasing Rate (quadratic_dephasing_rate)

0.1220139

Linear Dephasing Rate (linear_dephasing_rate)

0.0

0.0

Connectivity
Parallel two-qubit operations
Physical Noise

Crosstalk Initialization Probability (p_crosstalk_init)
Ratio of Single-Qubit Spontaneous Emission to p1(p1_emission_ratio)
Ratio of Single-Qubit Spontaneous Emission in TwoQubit Gate to p2 (p2_emission_ratio)
Dephasing Noise© 2023 by Quantinuum. All rights reserved.

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Coherent to Incoherent Factor(coherent_to_incoherent_factor)
Arbitrary Angle Noise Scaling

2.5

2.5

Fit Parameter 1 (przz_a)

1.39

1.39

Fit Parameter 2 (przz_b)

0.014

0.014

Fit Parameter 3 (przz_c)

1.39

1.39

Fit Parameter 4 (przz_d)

0.014

0.014

Physical NoiseThe emulator runs with default error parameters that represent a noise environment thatclosely resembles the respective hardware. These error parameters can be set and used tooverride 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 notrecommended as a starting point for emulations of hardware performance.
Single-Qubit Fault Probability (p1): probability of a fault occuring during a single-qubitgate
Two-Qubit Fault Probability (p2): probability of a fault occuring during a two-qubit gate
Bit Flip Measurement Probability (p_meas): probability of a bit flip being applied to ameasurement. Either a float or a tuple of 2 floats. If it is a single float then that error rateis 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 thesecond error rate if a 1 is measured.
Crosstalk Measurement Fault Probability (p_crosstalk_meas): probability of acrosstalk measurement fault occurring per qubit in the gate zones
Initialization Fault Probability (p_init): probability of a fault occuring during initializationof a qubit per qubit in the gate zones
Crosstalk Initiailization Fault Probability (p_crosstalk_init): probability of a cross-talkfault occurring during initialization of a qubit
Ratio of Single-Qubit Spontaneous Emission to p1 (p1_emission ratio): fraction of
p1 that is spontaneous emission for a single qubit instead of depolarizing noise
Ratio of Single-Qubit Spontaneous Emission in Two-Qubit Gate to p2(p2_emission_ratio): fraction of p2 that is spontaneous emission for a single qubit in atwo-qubit gate instead of depolarizing noise
The single and two-qubit fault probabilities are largely modeled using depolarizing channels;
however, there is smaller probability that a spontaneous emission event happens. Theprobability is about an order of magnitude lower than the corresponding 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 ¼ the time 𝑋 isapplied, ¼ the time 𝑌 is applied, and ½ the time leakage is applied. For more details see:
Realization of Real-Time Fault-Tolerant Quantum Error Correction.
The two-qubit fault probability corresponds to the depolarizing probability of the System
Model H1 fully entangling two-qubit gate, 𝑍𝑍(). The probability of depolarizing error for thearbitrary angle two-qubit gate, 𝑅𝑍𝑍(𝜃), depends on the angle 𝜃. The spontaneous emission
error channel is the same for both 𝑍𝑍() and 𝑅𝑍𝑍(𝜃).
© 2023 by Quantinuum. All rights reserved.

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Dephasing NoiseThe noise model includes a memory error for which 𝑍 is applied. This is often called"dephasing" or "memory" noise. We potentially model two types of dephasing noise: one
where the probability of applying 𝑍 is quadratically dependent on the duration for which
qubits are idling or transporting in the trap, and another where the probability is linearly
depedent on the duration. Note, we apply both sorts of noise simutaneously.
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 anangle propotional to frequency x duration. The probability of the 𝑅𝑍 gate applying a 𝑍
operation on a plus state is sin( frequency x duration⁄2)2, which is why we call this a form ofquadratic dephasing.
For the stabilizer simulatior, by default this quadratic noise is modelled incoherent byapplying Pauli 𝑍 with probability sin( frequency x duration⁄2)2 to model more closely thequadratic depency with frequency and time as seen in the coherent model. Note, stabilizer
simulations can only simulate Clifford and measurement-like gates, so the 𝑅𝑍 gate cannot beapplied directly. For both the state-vector and stabilizer simulations, linear dephasing is alsomodeled with 𝑍 applied with probability rate x duration.
Switching between the coherent and incoherent quadratic dephasing model can beaccomplished 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 thequadratic error model (quadratic_dephasing_rate) is multiplied bycoherent_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) is turned on by default
and an idle dephasing parameter is turned on by defult (idle_dephasing). Both of these canbe toggled off.
Coherent Quadratic Dephasing Model : the gate 𝑅𝑍 (frequency x duration) is applied
during transport and qubit idling, applied by default for the state-vector simulator, wherefrequency is equal to quadratic_dephasing_rate (in usings of 2 pi rads per sec). Thismodel is used if coherent_dephasing is True.
Incoherent Quadratic Dephasing Model: Pauli 𝑍 is applied during transport and qubit
idling according to the probability sin( frequency x duration⁄2)2, applied by default for thestabilizer simulator, where frequency is equal to quadratic_dephasing_rate x
coherent_to_incoherent_factor (all in units of 2𝜋 radians per second). This model is used
if coherent_dephasing is False. This model is mostly used to minimic coherent dephasing
noise for stabilizer simulations.
Linear Dephasing Model: Pauli 𝑍 is applied during transport and qubit idling according
to the probability of linear_dephasing_rate x duration (where linear_dephasing_rate is inunits of per sec. and duration is in units of seconds). This model is used in conjunction
with both the coherent or incoherent quadratic dephasing model.

© 2023 by Quantinuum. All rights reserved.

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Arbitrary Angle Noise Scaling
The System Model H1 systems have a native arbitrary-angle 𝑍𝑍 gate, 𝑅𝑍𝑍(𝜃). Forimplementation of this gate in the System Model H1 Emulator, certain parameters relate to
the strength of the depolarizing noise. These parameters depend on the angle 𝜃. This is𝜋normalized so that 𝜃 = 2 gives the two-qubit fault probability (p2).
The parameters for depolarizing noise are fit parameters that fit the noise estimated as theangle 𝜃 changes per this equation:(𝑝𝑟𝑧𝑧𝑎 √|𝜃|⁄𝜋 + 𝑝𝑟𝑧𝑧𝑏 ) ∗ 𝑝2 for 𝜃 < 0(𝑝𝑟𝑧𝑧𝑐 √|𝜃|⁄𝜋 + 𝑝𝑟𝑧𝑧𝑑 ) ∗ 𝑝2for 𝜃 > 00for 𝜃 = 0

Fit Parameter 1 (przz_a)
Fit Parameter 2 (przz_b)
Fit Parameter 3 (przz_c)
Fit Parameter 4 (przz_d)

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 theerror model, which include:
Scaling (scale): scale all error rates in the model linearly
P1 Scaling (p1_scale): scale the probability of single-qubit gates having a fault
P2 Scaling (p2_scale): scale the probability of two-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 afault
Emission Scaling (emission_scale): scale the probability that a spontaneous emission
event happens during a single or two-qubit gate
Cross-talk Scaling (crosstalk_scale): scale the probability that measurement or
intialization crosstalk events get applied to qubits, during mid-circuit measurement andreset (initialization), "crosstalk" noise can occur that effectively measures other qubits inthe trap or cause them to leak.
Leakage Scaling (leakage_scale): scale the probability that a leakage even happens
during single or two-qubit gates as well as during initalization or crosstalk; on the device
half the time, spontaneous emission leads to a leakage event

APPENDIXA H-System Quantum Credit (HQC) is defined as:© 2023 by Quantinuum. All rights reserved.

Page 8𝐻𝑄𝐶 = 5 +𝑁1𝑞 + 10 𝑁2𝑞 + 5 𝑁𝑚
𝐶5000where 𝑁1𝑞 is the number of single-qubit gates, 𝑁2𝑞 is the number of native two-qubit gates,
𝑁𝑚 is the number of state preparation and measurement operations in a circuit, including theinitial implicit state preparation and any intermediate and final measurements and state
resets, and 𝐶 is the shot count. When a circuit is submitted (whether to the Syntax Checker,
System Model H1, or System Model H1 Emulator) the cost in HQCs is returned with theresults.

© 2023 by Quantinuum. All rights reserved.

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