High-fidelity Trotter formulas for digital quantum simulation

Quantum simulation promises to address many challenges in fields ranging from quantum chemistry to material science and high-energy physics, and could be implemented in noisy intermediate-scale quantum devices. A challenge in building good digital quantum simulators is the fidelity of the engineered dynamics given a finite set of elementary operations. Here we present a framework for optimizing the order of operations based on a geometric picture, thus abstracting from the operation details and achieving computational efficiency. Based on this geometric framework, we provide two alternative second-order Trotter expansions: one with optimal fidelity at a short timescale, and the second robust at a long timescale. Thanks to the improved fidelity at different timescales, the two expansions we introduce can form the basis for experimental-constrained digital quantum simulation.

Figure 1

Geometric method to calculate the Trotter error. (a) Geometric picture for Trotter expansion of $e^{4A+3B}$ [vector (4,3)] by products of $e^A$'s and $e^B$'s. Unit vector (1,0) represents $e^A$ and unit vector (0,1) represents $e^B$. Any path starting from (0,0) that ends at (4,3) is accurate to first order. The second-order error is the total area enclosed by the diagonal and the path, where any region below (above) the diagonal has an area density $+1$ ($-1$). The third-order error is given by the moments, where, e.g., the infinitesimal moment about $x_2$ axis is the area of the dark-brown square times the distance to the $x_2$ axis. (b) Examples of three different second-order paths. The brown line is the conventional 2T expansion ($e^{2A}{e}^{3B}{e}^{2A}$). The dark-blue dashed path has a global minimum third-order error (2O, $e^B{e}^{3A}{e}^{2B}{e}^A$). The light-blue path has a minimal total distance from each node to the diagonal (2D, $e^A{e}^B{e}^A{e}^B{e}^A$), represented by the sum of absolute areas. All three paths enclose zero total area.

Figure 2

Fidelity comparison for the three Trotter sequences. (a) Geometric illustration of three different second-order approximations to $e^{12A+8B}$. In (a) and (b) dark-blue lines are for the 2O path, light-blue for 2D, and brown for 2T. (b) Numerical evaluation of log fidelity. Here $A=-i{H}_{1}t/n, B=-i{H}_{2}t/n$, where $H_1=\frac{1}{2}({\sigma }_z^1+{\sigma }_z^2), H_2={\sigma }_x^1{\sigma }_x^2$, and $n=1$. At short $t$, 2O shows the best fidelity since it has the smallest third-order error. At very small $t$ we fit the simulated data to $F_l=-{log}_{10}(1-F)=-a{log}_{10}t+b$, and the slopes of 2O, 2D, and 2T are 6.07, 5.99, and 5.99, respectively, close to the expected result, 6, for second-order expansions. 2D starts to outperform 2O around $t=0.13/\parallel H_1\parallel$, where $\parallel ·\parallel$ is the Frobenius norm. Going beyond the time of the simulation, it is possible that 2O outperforms 2D again. However, at these long times the fidelity is low and not useful for quantum simulations. (c) Histogram of crossover point (where 2O and 2D intersect) for 1000 pairs of random $4\times 4$ Hermitian matrices $H_{1,2}$. The black solid curve is the smoothed histogram and the orange curve is a Gaussian fit to the histogram. The mean and variance of the Gaussian distribution are 0.14 and 0.02, respectively.