High-order splitting of non-unitary operators on quantum computers

We show how complex-coefficient product formulas can decompose dissipative dynamics into a sequence of simple Hamiltonian evolutions in real and imaginary time with high-order accuracy. This requires a careful choice of scheme. The unitary substages use positive real coefficients, so the evolutions remain unitary, as using imaginary coefficients here would introduce new non-unitary evolutions. The dissipative substages use complex coefficients with positive real parts, where the positive real parts preserve the dissipative evolution in real time, and the imaginary parts become additional unitary evolutions that are straightforward to implement. There is no general formula for producing such coefficients, which must be instead found numerically, with suitable coefficients up to order 6 having presently been identified [2].

We demonstrate the approach by simulating the classical problem of lossy mechanical wave propagation [3] described by the damped wave equation on a trapped-ion quantum processor (IonQ Forte 1). The problem is separated into the non-commuting contributions from the wave equation and linear damping, which we individually simulate using a novel pseudo-spectral algorithm, and recombine with high-order accuracy using the proposed approach. A step of order 4 achieves greater accuracy than the steps with low orders 1 and 2, despite the increased circuit depth on noisy hardware. The results suggest that high-order operator splitting is an accurate and practical approach for simulating dissipative dynamics on near-term quantum processors.

Fundamentally, we extend the most common method for simulating unitary dynamics with high-order accuracy into one for simulating dissipative dynamics. Therefore, dissipative dynamics can be efficiently simulated with high accuracy when an efficient decomposition of the dynamics into smaller parts that can be simulated can be identified. 

In the presented example of the damped wave equation simulated on a trapped-ion quantum processor, identifying an efficient means of simulating the exact operator is challenging. However,  after separating the problem into its unitary and non-unitary parts, this becomes a Hamiltonian evolution in real-time followed by imaginary-time, which are well-studied problems in quantum computing. The identification of efficient quantum circuits for these simpler problems is much more straightforward, as we demonstrate. Our quantum circuits for simulating the damped wave equation to order 6 accuracy requires 30d log2 N + 60d + 32 CNOT gates per step, where N is the number of grid points per spatial dimension, and d is the number of spatial dimensions. For example, a three-dimensional simulation on a (16384)^3 grid requires 1472 CNOT gates per step. Given current quantum hardware can execute around 1500 gates in its coherence window, such large-scale simulations feasible in the near term.