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The HHL algorithm for matrix inversion is a landmark algorithm in quantum computation. Its ability to produce a state $$|x\rangle$$ that is the solution of $Ax=b$, given the input state $$|b\rangle$$, is envisaged to have diverse applications. In this paper, we substantially simplify the algorithm, originally formed of a complex sequence of phase estimations, amplitude amplifications and Hamiltonian simulations, by replacing the phase estimations with a continuous time quantum walk. The key technique is the use of weak couplings to access the matrix inversion embedded in perturbation theory. 

When studying the perfect transfer of a quantum state from one site to another, it is typically assumed that one can receive the arriving state at a specific instant in time, with perfect accuracy. Here, we study how sensitive perfect state transfer is to that timing. We design engineered spin chains which reduce their sensitivity, proving that this construction is asymptotically optimal. The same construction is applied to the task of creating superpositions, also known as fractional revival. 

Quantum Hamiltonian simulation, fundamental in quantum algorithm design, extends far beyond its foundational roots, powering diverse quantum computing applications. However, optimizing the compilation of quantum Hamiltonian simulation poses significant challenges. Existing approaches fall short in reconciling deterministic and randomized compilation, lack appropriate intermediate representations, and struggle to guarantee correctness. Addressing these challenges, we present MarQSim, a novel compilation framework. MarQSim leverages a Markov chain-based approach, encapsulated in the Hamiltonian Term Transition Graph, adeptly reconciling deterministic and randomized compilation benefits. Furthermore, we formulate a Minimum-Cost Flow model that can tune transition matrices to enforce correctness while accommodating various optimization objectives. Experimental results demonstrate MarQSim's superiority in generating more efficient quantum circuits for simulating various quantum Hamiltonians while maintaining precision.