High-Order Splitting of Non-Unitary Operators on Quantum Computers

TL;DR

This work tackles the challenge of simulating non-unitary, open-system dynamics on quantum computers by introducing high-order operator splitting with complex coefficients that have positive real parts, enabling stable, accurate integration of $M=H_1+iH_2$ via sequential real- and imaginary-time evolutions. The authors develop and compare splitting schemes up to sixth order, including Castella's fourth-order and Bernier 2023's sixth-order, ensuring $ ext{Re}(a_i)>0$ and $b_i>0$ to avoid instability from dissipative terms. They apply the framework to the damped-wave equation, constructing spectrally accurate quantum circuits using quantum spectral transforms, and demonstrate a six_order scheme on a 9-qubit, 128-grid-point test with 1,208 CNOTs, illustrating practical feasibility within coherence times. The approach yields near-linear time scaling with high-order accuracy and significantly reduced circuit depth for diagonalizable problems, offering a viable route for industrially relevant PDE simulations on near-term quantum hardware.

Abstract

We present a high-order splitting method for simulating non-unitary dynamics by sequential real- and imaginary-time Hamiltonian evolutions. Complex-coefficient splitting methods with positive real parts are chosen for stable integration in a quantum circuit, avoiding the unstable, norm-amplifying negative steps that arise from real-coefficient splitting at high orders. The method is most beneficial for dynamics that naturally separate into unitary and dissipative components, with broad applications across science and engineering. These systems frequently admit compact spectral representations of the split operators, which we demonstrate by deriving efficient quantum circuits for simulating the damped-wave equation with up to sixth-order accuracy in time. A single sixth-order step in three dimensions on 35 trillion cells requires 1,562 CNOT gates, which can be executed within the coherence time of modern quantum processors.

Figures (4)

• Figure 1: Quantum circuit for solving the wave equation in Fourier space using little-endian ordering with the input state in Eq. \ref{['eq:spectral_input_state']}. The non-dimensional evolution time $\zeta t = 4\pi ct/L$.
• Figure 2: Quantum circuits for implementing one splitting step of the damped-wave equation in Fourier space with little-endian ordering for (a) Lie-Trotter, (b) Strang, and (c) the fourth-order scheme of Castella2009Castella2009. The circuit in (c) generalizes to higher orders with the same construction, including the sixth-order scheme of Bernier2023Bernier2023. The non-dimensional splitting step size is $\zeta t = 4\pi ct/L$ wave propagation time scales scaled by $4\pi$, or $\gamma t$ damping time scales.
• Figure 3: Statevector simulations of the damped-wave equation evolution showing (a) the displacement vector corresponding to selector qubit $\ket{0}_S$, and (b) the function of the velocity vector in Eq. \ref{['eq:physical_space_encoding']} corresponding to $\ket{1}_S$. The simulations use nine qubits and four sixth-order splitting steps. Both plots show the physical-space quantities, corresponding to the QFT$^\dagger$ applied to the $\ket{\text{dat}}_D$ register.
• Figure 4: Error norm $\epsilon = \|\ket{\phi(t)}-\vec{\phi}(t)\|$ from statevector simulations of the damped-wave simulation in Fig. \ref{['fig:damped_wave_evolution']} using various splitting step sizes. The analytical solution $\vec{\phi}(t)$ was evaluated from Eq. \ref{['eq:damped_wave_ode']}, then scaled to have a unit norm. The error norm is shown against the number of operator splitting steps in (a) and the number of CNOT gates in (b).