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Workings of the hybrid approach combining Clifford and fullstate simulation. This includes, on the one hand, the efficient simulation of Pauli gates using Pauli frame tracking and the emulation of non-Clifford gates by utilizing the Pauli frame as a lookup table, which allows for representing multi-qubit Pauli rotations as single-qubit Pauli rotations with a modified rotation axis. The updated element is highlighted in green, respectively.

Demonstration of the sequential updates applied to the respective Pauli frame as defined in Eq.~\eqref{eq:framemat} under the gate-by-gate action of the Cliffords. Thus, by tracking updates to the Pauli frame, we can efficiently simulate this Clifford circuit.

Illustrating the commutation behavior as described in Eq.~\eqref{eq:single_to_multirot}, we can commute the Clifford unitary $U$ (here given by a single CNOT gate as an example) past the rotation, translating the single qubit rotation to a multi-qubit one.

The unitary $e^{-i \frac{\theta}{2} P}$ generated by the Pauli $P=\sigma_x \otimes \sigma_y \otimes \sigma_z \otimes \sigma_x$ acting on 4 qubits can be decomposed in the presented CNOT staircase~\cite{trotter1959product, whitfield2011simulation} for implementation with one and two qubit gates.

Comparison of the simulation time (runtime [s]) for random Hamiltonians with varying locality~$L$, 100 terms and 24 qubits, comparing results based on CFHS and IQS with and without MPI in this exemplary case. We observe a significant reduction in simulation time with our new CFHS method in both cases, with and without MPI. The linear increase of IQS simulation time in terms of the locality $L$, in contrast to the constant scaling of CFHS in locality~$L$, is nicely visible for results obtained with and without MPI.

Demonstrating the impact when enabling MPI for IQS and CFHS using 128 MPI ranks. We present the ratio of the runtimes for random Hamiltonians with 100 terms each, with the presented mean and standard deviation of Hamiltonians for different localities $L$ and specific numbers of qubits $n_{\text{qubits}}$.

: Simulation time [s] for selected chemistry Hamiltonians without MPI parallelization.

: Simulation time [s] for selected chemistry Hamiltonians with MPI parallelization.

Impact when utilizing MPI for IQS and CFHS using 128 MPI ranks as before. Here, we show the results presenting the ration of runtime of using no MPI and MPI for selected chemistry Hamiltonians from Hamlib as given in Table~\ref{tab:params_rh}. We show the mean and standard deviation taken over the Hamiltonians with the same number of qubits $n_{\text{qubits}}$ but different numbers of terms and localities, respectively.

: Histogram of the compilation time ratio distribution for CFHS/IQS.

: Mean and standard deviation of the compilation time ratio distribution for CFHS/IQS.

Increase of the compilation time [s] for random Hamiltonians each with 100 terms, averaged over the respective locality of the Hamiltonian for given numbers of qubits. We compare both the compilation time with and without MPI parallelization, as well as the compilation time results obtained via IQS and the CFHS simulator. We showcase the mean and standard deviation as error bars.

: Rescaled simulation time [s] without using MPI for random Hamiltonians. For better visibility, we present the results for a locality $L > 14$ in detail.

: Rescaled simulation time [s] using MPI for random Hamiltonians. For better visibility, we present the results for a locality $L > 16$ in detail.

: Histogram of the compilation time ratio distribution for CFHS/IQS.

: Mean and standard deviation of the compilation time ratio distribution for CFHS/IQS.

: Rescaled runtime as in Eq.~\eqref{eqn:workload} without MPI usage.

: Rescaled runtime as in Eq.~\eqref{eqn:workload} with MPI parallelization.