Hari Krovi

Developing quantum algorithms to simulate fluid dynamics has become an active area of research, as accelerating fluid simulations could have significant impact in industry and fundamental science. While many approaches have been proposed for simulating fluid dynamics on quantum computers, it is largely unclear whether these algorithms will provide any speedup over existing classical approaches. In this paper we give evidence that quantum computers cannot significantly outperform classical simulations of fluid dynamics in general. We study two models of fluids: the Korteweg-de Vries (KdV) equation, which models shallow water waves, and the incompressible Euler equations, which model ideal, inviscid fluids. We show that any quantum algorithm simulating the KdV equation or the Euler equations for time T requires Ω(T^2) and exp(Ω(T)) copies of the initial state in the worst case, respectively. These lower bounds hold for the task of preparing the final state, and similar bounds hold for history state preparation. We prove the lower bound for the KdV equation by investigating divergence of solitons. For the Euler equations, we show that instabilities can accelerate state discrimination.

Shared entanglement between two remote parties is a key resource for Quantum Cryptography. Quantum communications capacity using direct transmission over length-$L$ optical fiber scales as $R \sim e^{-\alpha L}$, where $\alpha$ is the fiber's loss coefficient. The rate achieved using a linear chain of quantum repeaters equipped with quantum memories, probabilistic Bell state measurements (BSMs) and switches used for spatial multiplexing, but no quantum error correction was shown to surpass the direct-transmission capacity. However, this rate still decays exponentially with the end-to-end distance, viz., $R \sim e^{-s{\alpha L}}$, with $s < 1$. We show that the introduction of temporal multiplexing---i.e., the ability to perform BSMs among qubits at a repeater node that were successfully entangled with qubits at distinct neighboring nodes at {\em different} time steps---leads to a sub-exponential rate-vs.-distance scaling, i.e., $R \sim e^{-t\sqrt{\alpha L}}$, which is not attainable with just spatial or spectral multiplexing. We evaluate analytical upper and lower bounds to this rate and obtain the exact rate by numerically optimizing the time-multiplexing block length and the number of repeater nodes. We further demonstrate that incorporating losses in the optical switches used to implement time-multiplexing degrades the rate-vs.-distance performance, eventually falling back to exponential scaling for very lossy switches. We also examine models for quantum memory decoherence and describe optimal regimes of operation to preserve the desired boost from temporal multiplexing. QM decoherence is seen to be more detrimental to the repeater's performance over switching losses.