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An explicit near-optimal quantum algorithm is proposed for modeling dissipative initial-value problems. This method, based on the Linear Combination of Hamiltonian Simulations (LCHS), approximates a target nonunitary operator as a weighted sum of Hamiltonian evolutions, thereby emulating a dissipative problem by mixing various time scales. We propose an efficient encoding of this algorithm into a quantum circuit based on a simple coordinate transformation that turns the dependence on the summation index into a trigonometric function and significantly simplifies block-encoding. The resulting circuit has high success probability and scales logarithmically with the number of terms in the LCHS sum and linearly with time. We verify the quantum circuit and its scaling by simulating it on a digital emulator of fault-tolerant quantum computers and, as a test problem, solve the advection-diffusion equation. The proposed algorithm can be used for modeling a wide class of nonunitary initial-value problems including the Liouville equation and linear embeddings of nonlinear systems.

We propose an explicit algorithm based on the Linear Combination of Hamiltonian Simulations technique to simulate both the advection-diffusion equation and a nonunitary discretized version of the Koopman-von Neumann formulation of nonlinear dynamics. By including dissipation into the model, through an upwind discretization of the advection operator, we avoid spurious parasitic oscillations which usually accompany standard finite difference discretizations of the advection equation. In contrast to prior works on quantum simulation of nonlinear problems, we explain in detail how different components of the algorithm can be implemented by using the Quantum Signal Processing (QSP) and Quantum Singular Value Transformation (QSVT) methods. In addition, we discuss the general method for implementing the block-encoding (BE) required for QSP and QSVT circuits and provide explicit implementations of the BE oracles tailored to our specific test cases. We simulate the resulting circuit on a digital emulator of quantum fault-tolerant computers and investigate its complexity and success probability. The proposed algorithm is universal and can be used for modeling a broad class of linear and nonlinear differential equations including the KvN and Carleman embeddings of nonlinear systems, the semiclassical Koopman-van Hove (KvH) equation, as well as the advection and Liouville equations.

Maintaining conservation laws in the fully discrete setting is critical for accurate long-time behavior of numerical simulations and requires accounting for discrete conservation properties in both space and time. This paper derives arbitrary order finite element exterior calculus spatial discretizations for the two-dimensional (2D) Navier-Stokes and drift-reduced magnetohydrodynamic equations that conserve both energy and enstrophy to machine precision when coupled with generally symplectic time-integration methods. Both continuous and discontinuous-Galerkin (DG) weak formulations can ensure conservation, but only generally symplectic time integration methods, such as the implicit midpoint method, permit exact conservation in time. Moreover, the symplectic implicit midpoint method yields an order of magnitude speedup over explicit schemes. The methods are implemented using the MFEM library and the solutions are verified for an extensive suite of 2D neutral fluid turbulence test problems. Numerical solutions are verified via comparison to a semi-analytic linear eigensolver as well as to the finite difference Global Drift Ballooning (GDB) code. However, it is found that turbulent simulations that conserve both energy and enstrophy tend to have too much power at high wavenumber and that this part of the spectrum should be controlled by reintroducing artificial dissipation. The DG formulation allows upwinding of the advection operator which dissipates enstrophy while still maintaining conservation of energy. Coupling upwinded DG with implicit symplectic integration appears to offer the best compromise of allowing mid-range wavenumbers to reach the appropriate amplitude while still controlling the high-wavenumber part of the spectrum.

The primary challenge in solving kinetic equations, such as the Vlasov equation, is the high-dimensional phase space. In this context, dynamical low-rank approximations have emerged as a promising way to reduce the high computational cost imposed by such problems. However, a major disadvantage of this approach is that the physical structure of the underlying problem is not preserved. In this paper, we propose a dynamical low-rank algorithm that conserves mass, momentum, and energy as well as the corresponding continuity equations. We also show how this approach can be combined with a conservative time and space discretization.

As a milestone for general-purpose computing machines, we demonstrate that quantum processors can be programmed to efficiently simulate dynamics that are not native to the hardware. Moreover, on noisy devices without error correction, we show that simulation results are significantly improved when the quantum program is compiled using modular gates instead of a restricted set of standard gates. We demonstrate the general methodology by solving a cubic interaction problem, which appears in nonlinear optics, gauge theories, as well as plasma and fluid dynamics. To encode the nonnative Hamiltonian evolution, we decompose the Hilbert space into a direct sum of invariant subspaces in which the nonlinear problem is mapped to a finite-dimensional Hamiltonian simulation problem. In a three-states example, the resultant unitary evolution is realized by a product of ~20 standard gates, using which ~10 simulation steps can be carried out on state-of-the-art quantum hardware before results are corrupted by decoherence. In comparison, the simulation depth is improved by more than an order of magnitude when the unitary evolution is realized as a single cubic gate, which is compiled directly using optimal control. Alternatively, parametric gates may also be compiled by interpolating control pulses. Modular gates thus obtained provide high-fidelity building blocks for quantum Hamiltonian simulations., Comment: 4 figures. pages 1-6 are main text and bibliography, and pages 7-8 are supplemental material

Quantum computers can be used to simulate nonlinear non-Hamiltonian classical dynamics on phase space by using the generalized Koopman-von Neumann formulation of classical mechanics. The Koopman-von Neumann formulation implies that the conservation of the probability distribution function on phase space, as expressed by the Liouville equation, can be recast as an equivalent Schr\"odinger equation on Hilbert space with a Hermitian Hamiltonian operator and a unitary propagator. This Schr\"odinger equation is linear in the momenta because it derives from a constrained Hamiltonian system with twice the classical phase space dimension. A quantum computer with finite resources can be used to simulate a finite-dimensional approximation of this unitary evolution operator. Quantum simulation of classical dynamics is exponentially more efficient than a deterministic Eulerian discretization of the Liouville equation if the Koopman-von Neumann Hamiltonian is sparse. Utilizing quantum walk techniques for state preparation and amplitude estimation for the calculation of observables leads to a quadratic improvement over classical probabilistic Monte Carlo algorithms., Comment: 19 pages, 3 figures