System Operation¶
Native Gate Set¶
HSeries utilizes the following native gate set.
Gate 
Expression 
TKET 
Native 1qubit gates 

\(U_{1q} (\theta, \phi)\) 
\(e^{ \frac{i \theta}{2} \left(\cos(\phi \hat{X}) + \sin(\phi \hat{Y})\right) }\) 

\(R_{z}(\lambda)\) 
\(e^{i \frac{\lambda}{2} \hat{Z}}\) 

Fully entangling 2qubit gates 

\(ZZ()\) 
\(e^{i \frac{\pi}{4} \hat{Z} \bigotimes \hat{Z}}\) 

Arbitrary angle 2qubit gates 

\(RZZ(\theta)\) 
\(e^{i \frac{\theta}{2} \hat{Z} \bigotimes \hat{Z} }\) 

General SU(4) entangler 

\(Rxxyyzz (\alpha, \beta, \gamma)\) 
\(e^{\frac{i}{2} (\alpha \hat{X} \bigotimes \hat{X} + \beta \hat{Y} \bigotimes \hat{Y} + \gamma \hat{Z} \bigotimes \hat{Z})}\) 

\(\hat{X}\), \(\hat{Y}\) and \(\hat{Z}\) are the standard Pauli operators, and the twoqubit matrix is written in the \(0,0 \rangle\), \(0,1 \rangle\), \(1,0 \rangle\), \(1,1 \rangle\) basis.
Note that the arbitrary rotation around the \(z\)axis, \(R_z\) (\(\lambda\)), is performed virtually within the software. All other physical gates are constructed from this set.
By default, quantum circuits submitted to the hardware are rebased to the fully entangling \(ZZ\) gate and the arbitrary angle \(RZZ\) gate. Circuits are rebased to the General \(SU(4)\) Entangler only if users specify this option at job submission.
Note
Please note that our native \(U_1q(\theta, \phi)\) gate is NOT IBM’s standard \(U(\theta , \phi, \lambda)\)[1].
\(U_{1q} (\theta, \phi) = U \left( \theta, \phi  \frac{\pi}{2}, \frac{\pi}{2}  \phi \right)\).
The minimum gate angles for the \(RZZ\) gate angle is \(1 \times 10^{4}\) and the minimum \(U_{1q}\) gate angle is \(3 \times 10^{4}\).
Rebasing Quantum Circuits¶
Quantum circuits are rebased to the Quantinuum native gate set as described below.
Gate 
Rebase 
Pauli gate: bitflip 
\(\sigma_x = U_{1q} (\pi,0)\) 
Pauli gate: bit and phase flip 
\(\sigma_y = U_{1q} (\pi,\pi/2)\) 
Pauli gate: phase flip 
\(\sigma_z = R_z(\pi)\) 
Clifford gate: Hadamard 
\(H = U_{1q} (\pi/2,\pi/2)\) 
\(R_z(\pi)\) 

Clifford gate: CNOT 
\(CX^{(c,t)} = U_{1q}^{(t)}(\pi/2,\pi/2)\) 
\(ZZ\) 

\(R_{z}^{(c)}(\pi/2)\) 

\(U_{1q}^{(t)}(\pi/2,\pi)\) 

\(R_{z}^{(t)}(\pi/2)\) 

Pauli interaction: Z basis 
\(R_{zz}(\pi/4) = Rzz(\pi/4)\) 
Pauli interaction: X basis 
\(R_{xx}(\pi/4) = U_{1q}^{(c)}(\pi/2,\pi/2)\) 
\(U_{1q}^{(t)}(\pi/2,\pi/2)\) 

\(R_{zz}(\pi/4)\) 

\(U_{1q}^{(c)}(\pi/2,\pi/2)\) 

\(U_{1q}^{(t)}(\pi/2,\pi/2)\) 
Midcircuit Measurement and Conditional Operations¶
Due to the internal level structure of trappedion qubits, a midcircuit measurement may leave the qubit in a noncomputational state. All midcircuit measurements should be followed by initialization if the qubit is to be used again in that circuit. The qubit may be prepared in the measured state by calling for a measurement followed by initialization and a measurement dependent spinflip.
When a subset of qubits is measured in the middle of the circuit, the classical information from these measurements can be used to condition future elements of the circuit. Although the laser pulses that implement both 1 and 2qubit gates are conditional, the transport operations used to rearrange the physical location of the qubits are not. The qubits will be reconfigured to allow for all gates in all branches irrespective of the midcircuit measurement outcome. In the context of memory error and run time, the effective depth of a circuit with measurement conditioned branching includes all branches.
Intercircuit Performance Validation¶
Jobs submitted with large shot counts are automatically divided into appropriately sized chunks of smaller shot counts in a method called chunking. Chunking ensures that system state checks and dynamic calibrations happen at the appropriate frequency. The number of shots in a chunk is dynamically chosen by the compiler and will vary with the complexity of the circuit. A series of system checks are performed before and after each chunk. If an error is detected, any suspect results are rejected, and the failed chunk shots are rerun at no additional cost. For jobs consisting of multiple chunks, the time between the start date and result date will include all of the system checks and calibrations that happened in the middle of that job and possibly chunks from other jobs in the queue.
Dynamic Calibrations¶
The system automatically schedules and executes calibration routines. There are two types of automated calibrations: those that are executed on a predetermined time interval and those that are triggered when a drift tolerance is exceeded. Because the latter does not follow a predetermined schedule, the circuit throughput and execution time will vary due to these calibrations.
Depth1 Circuit Time¶
We define the depth1 circuit time as the time it takes to arbitrarily permute all qubits and perform 1qubit and 2qubit gates on all \(\left[(N/2)\right]\) pairs, as shown in Figure 1, where \(P\) represents the random permutation of all qubits. Many circuits cannot be fully parallelized into a maximally dense circuit where each qubit participates in a 2qubit gate at each step of the circuit and may have shorter depth1 circuit times as a result.
Estimating Circuit Time¶
The run time for a given circuit depends on many factors, including factors in the circuit structure, such as if the circuit is highly connected or not, as well as factors in the system, such as dynamic calibrations. Run time can be reasonably predicted using the same formula that defines HSystem Quantum Credits (HQCs). A HQC is defined as:
\(HQC = 5 + \frac{N_{1q} + 10 N_{2q} + 5 N_m}{5000} * C\),
where \(N_1q\) is the number of 1qubit gates, \(N_2q\) is the number of native 2qubit gates, \(N_m\) is the number of state preparation and measurement operations in a circuit, including the initial implicit state preparation and any intermediate and final measurements and resets, and \(C\) is the shot count. When a circuit is submitted, the cost in HQCs is returned with the results. For circuits using conditional logic, the charged HQCs include all the gates and measurements across all conditional branches regardless which are executed in the circuit.
Arbitrary Angle ZZ Gates¶
Although an arbitrary angle 2qubit entangling gate can be constructed using two fixedangle 2qubit entangling gates, a direct implementation will lower the error rate in the circuit. Not only is the number of entangling gates reduced, but the error of an \(RZZ(\theta)\) gate also scales with the angle \(\theta\). The error on \(RZZ(\frac{\pi}{2})\) is equal to the error of \(ZZ()\).