Quantum state preparation is a fundamental and significant subroutine in quantum computing. In this paper, we conduct a systematic investigation on the circuit size (the total count of elementary gates in the circuit) for sparse quantum state preparation. A quantum state is said to be $d$-sparse if it has only $d$ non-zero amplitudes. For the task of preparing an $n$-qubit $d$-sparse quantum state, we obtain the following results: \textbf{Without ancillary qubits:} Any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log n} + n)$ without using ancillary qubits, which improves the previous best results. It is asymptotically optimal when $d = \mathrm{poly}(n)$, and this optimality holds for a broader scope under some reasonable assumptions. \textbf{With limited ancillary qubits:} (i) Based on the first result, we prove for the first time a trade-off between the number of ancillary qubits and the circuit size: any $n$-qubit $d$-sparse quantum state can be prepared by a quantum circuit of size $O(\frac{nd}{\log (n + m)} + n)$ using $m$ ancillary qubits for any $m \in O(\frac{nd}{\log nd} + n)$. (ii) We establish a matching lower bound $Ω(\frac{nd}{\log {(n + m)} }+ n)$ under some reasonable assumptions, and obtain a slightly weaker lower bound $Ω(\frac{nd}{\log {(n + m)} + \log d} + n)$ without any assumptions. \textbf{With unlimited ancillary qubits:} Given arbitrary amount of ancillary qubits available, the circuit size for preparing $n$-qubit $d$-sparse quantum states is $Θ(\frac{nd}{\log nd} + n)$.

This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm theoretically achieves certainty of success, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the welded tree problem.

We explore potential quantum speedups for the fundamental problem of testing the properties of closeness andk-wise uniformity of probability distributions. •Closeness testingis the problem of distinguishing whether twon-dimensional distributions are identical or at least ε-far in ℓ1- or ℓ2-distance. We show that the quantum query complexities for ℓ1- and ℓ2-closeness testing areO(√n/ε) andO(1/ε), respectively, both of which achieve optimal dependence on ε, improving the prior best results of Gilyén and Li (2019). •k-wise uniformity testingis the problem of distinguishing whether a distribution over {0, 1}nis uniform when restricted to anykcoordinates or ε-far from any such distribution. We propose the first quantum algorithm for this problem with query complexityO(√nk/ε), achieving a quadratic speedup over the state-of-the-art classical algorithm with sample complexityO(nk/ε2) by O’Donnell and Zhao (2018). Moreover, whenk= 2 our quantum algorithm outperforms any classical one because of the classical lower bound Ω(n/ε2). All our quantum algorithms are fairly simple and time-efficient, using only basic quantum subroutines such as amplitude estimation.