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The Solovay-Kitaev algorithm is a fundamental result in quantum computation. It gives an algorithm for efficiently compiling arbitrary unitaries using universal gate sets: any unitary can be approximated by short gates sequences, whose length scales merely poly-logarithmically with accuracy. As a consequence, the choice of gate set is typically unimportant in quantum computing. However, the Solovay-Kitaev algorithm requires the gate set to be inverse-closed. It has been a longstanding open question if efficient algorithmic compilation is possible without this condition. In this work, we provide the first inversefree Solovay-Kitaev algorithm, which makes no assumption on the structure within a gate set beyond universality, answering this problem in the affirmative, and providing an efficient compilation algorithm in the absence of inverses for both SU(d) and SL(d, C). The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.

We design and analyze two new low depth algorithms for amplitude estimation (AE) achieving an optimal tradeoff between the quantum speedup and circuit depth. For β∈(0,1], our algorithms require N=O~(1ϵ1+β) oracle calls and require the oracle to be called sequentially D=O(1ϵ1−β) times to perform amplitude estimation within additive error ϵ. These algorithms interpolate between the classical algorithm (β=1) and the standard quantum algorithm (β=0) and achieve a tradeoff ND=O(1/ϵ2). These algorithms bring quantum speedups for Monte Carlo methods closer to realization, as they can provide speedups with shallower circuits.The first algorithm (Power law AE) uses power law schedules in the framework introduced by Suzuki et al \cite{S20}. The algorithm works for β∈(0,1] and has provable correctness guarantees when the log-likelihood function satisfies regularity conditions required for the Bernstein Von-Mises theorem. The second algorithm (QoPrime AE) uses the Chinese remainder theorem for combining lower depth estimates to achieve higher accuracy. The algorithm works for discrete β=q/k where k≥2 is the number of distinct coprime moduli used by the algorithm and 1≤q≤k−1, and has a fully rigorous correctness proof. We analyze both algorithms in the presence of depolarizing noise and provide numerical comparisons with the state of the art amplitude estimation algorithms.