Continuous-time evolution via probabilistic angle interpolation and its applications

Abstract

We explore the applicability of a stochastic time-evolution algorithm based on probabilistic angle interpolation. To simplify the pre-processing of the algorithm, we take the continuous-time limit, thereby explicitly eliminating Trotter errors and streamlining the resource analysis. We also introduce a noise-mitigation method tailored to it. As demonstrations, we apply the algorithm to two representative problems: estimating the ground-state energy of the $H_3^+$ molecular Hamiltonian and computing out-of-time-ordered correlators in the sparse Sachdev--Ye--Kitaev model. We evaluate the protocol's performance through numerical simulations and experiments on a trapped-ion quantum computer, Quantinuum Reimei.

Figures (6)

• Figure 1: Fidelities between the exact ground state and adiabatically prepared approximate ground states for the fixed adiabatic time $T=8$. Blue data are computed with TE-PAI using 500 samples with statistical uncertainties represented by the opaque region. The data points correspond to $\Delta=\{\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8},\frac{1}{10},\frac{1}{12}, \frac{1}{14}\}$. Orange data are computed with the statevector simulation of the corresponding Trotter circuit. The data points correspond to the number of Trotter steps $\{1,2,3,4,5,6,7\}$.
• Figure 2: Noiseless simulation of ground state energy estimation with TE-PAI & TETRIS and first-order Trotterization. The adiabatic time is $T=8$, and the Hadamard test angle is $s=10$. Blue (red) data are computed using the former by sampling 500 (1000) circuits and executing 1 shot for each. The data points correspond to $\Delta=\{\frac{1}{2},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\frac{1}{10},\frac{1}{12}\}$. Orange data are computed by executing 500 shots of a Trotter circuit. The data points correspond to the number of Trotter steps $\{1,2,3,4,5,6,7,8\}$.
• Figure 3: The interferometric circuit to compute the real part of OTOC \ref{['eq:def_OTOC']}. The qubit at the top is the single-qubit ancillary register initialized to $\ket{+}$, while the remainder constitutes the system register with the input state $|0^n\rangle$. The unitary circuits $\hat{U}_1$ and $\hat{U}_2$ are sampled independently from TE-PAI circuits. At the end of the circuit, the ancillary qubit is measured in the $X$ basis.
• Figure 4: Noisy simulation of the interferometric circuit for OTOC calculations with ZNE (Fig. \ref{['circ:OTOC']}). All $CX$s and single-qubit gates are subject to depolarizing noise of the parameters $10^{-3}$ and $10^{-5}$, respectively. The blue data are obtained by executing 500 circuits with $\Delta=0.05$, while the orange data are obtained by running 1000 circuits with $\Delta=0.1$. Each circuit is executed with 5 shots. The green data points represent the extrapolated values. The OTOC computed by noiseless simulation of continuous dynamics is shown by the black curve, with the gray region representing its statistical error.
• Figure 5: Energy estimate relative to the exact ground state energy calculated with Reimei against the number of samples. The blue curve shows the mean of $E_{\rm est}-E_{\rm exact}$ with the shaded region representing the statistical error. The dashed line corresponds to $E_{\rm HF}-E_{\rm exact}$, which is the energy of the initial state $\ket{\rm HF}$ relative to the exact ground state energy.