Reversible logic synthesis is a crucial component in quantum electronic design automation. While rule-based methodologies have gained prominence in reversible circuit optimization, the completeness of the transformation rule systems is a longstanding problem in this domain. In this work, we propose the first complete set of transformation rules for reversible circuits, comprising five fundamental rules: any two equivalent reversible circuits can be transformed into each other using the rules. To prove the completeness, a canonical circuit representation for reversible functions is introduced, and we show that every reversible function is computed by a unique reversible circuit in the canonical form, and any reversible circuit can be transformed into its canonical form by applying the rules.

Blind quantum computation (BQC) enables clients with limited quantum capabilities to protect the privacy of inputs, outputs and algorithms during the computation process. However, if a client’s private information is exposed to a server after the computation, the server can deduce the client’s output or even input. Quantum encryption with certified deletion (QECD) offers a potential solution by enabling the data owner to generate a deletion certificate, making the original plaintext inaccessible, provided the certificate is valid. Nevertheless, current QECD can only handle classical data and cannot be applied to BQC with quantum inputs. This paper first introduces the concept of certified deletion for quantum states and then proposes a single-client BQC protocol with certified deletion, where the client can use the classical certificate generated by the server to confirm whether her quantum inputs have been deleted after computation.We also give a specific example and simulate it using Qiskit to show its feasibility. In addition, the proposed protocol can also be extended to a multi-client environment in which honest clients request certified deletion if any client disconnects or behaves maliciously, thereby terminating the protocol and protecting their privacy.

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 $Θ(\frac{nd}{\log nd} + n)$.

Quantum homomorphic encryption (QHE) can allow clients to directly perform quantum computation on encrypted data with the assistance of a remote quantum server. However, existing QHE schemes often overlook the crucial property of verifiability which enables clients to validate the correctness of computation results provided by the server. In this paper, we propose a verifiable QHE scheme based on the universal quantum gate set {H,P,Toffoli}. At first, a specialized gadget is designed to eliminate the errors that may arise during the homomorphic evaluation of non-Clifford Toffoli gates in a non-interactive manner. Furthermore, the designed gadget is versatile and can be seamlessly integrated into an existing QHE scheme that implements quantum gates in another universal quantum gate set {H,T,CNOT} to homomorphically evaluate more quantum gates. Subsequently, a verifiable method is introduced to the QHE scheme based on {H,P,Toffoli} for the client to detect whether the server is honest during the homomorphic computation by employing three types of indistinguishable quantum circuits.

Variational quantum algorithm (VQA) derives advantages from its error resilience and high flexibility in quantum resource requirements, rendering it broadly applicable in the noisy intermediate-scale quantum era. As the performance of VQA highly relies on the structure of the parameterized quantum circuit, it is worthwhile to propose quantum architecture search (QAS) algorithms to automatically search for high-performance circuits. Nevertheless, existing QAS methods are time-consuming, requiring circuit training to assess circuit performance. This study pioneers training-free QAS by utilizing two training-free proxies to rank quantum circuits, in place of the expensive circuit training employed in conventional QAS. Taking into account the precision and computational overhead of the path-based and expressibility-based proxies, we devise a two-stage progressive training-free QAS (TF-QAS). Initially, directed acyclic graphs (DAGs) are employed for circuit representation, and a zero-cost proxy based on the number of paths in the DAG is designed to filter out a substantial portion of unpromising circuits. Subsequently, an expressibility-based proxy, finely reflecting circuit performance, is employed to identify high-performance circuits from the remaining candidates. These proxies evaluate circuit performance without circuit training, resulting in a remarkable reduction in computational cost compared to current training-based QAS methods. Simulations on three VQE tasks demonstrate that TF-QAS achieves a substantial enhancement of sampling efficiency ranging from 5 to 57 times compared to state-of-the-art QAS, while also being 6 to 17 times faster.

We explore whether quantum advantages can be found for the zeroth-order feedback online exp-concave optimization problem, which is also known as bandit exp-concave optimization with multi-point feedback. We present quantum online quasi-Newton methods to tackle the problem and show that there exists quantum advantages for such problems. Our method approximates the Hessian by quantum estimated inexact gradient and can achieve $O(n\log T)$ regret with $O(1)$ queries at each round, where $n$ is the dimension of the decision set and $T$ is the total decision rounds. Such regret improves the optimal classical algorithm by a factor of $T^{2/3}$.

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.

Blind quantum computation (BQC) provides a solution for clients with limited quantum capabilities to delegate their quantum computational tasks to remote quantum servers while keeping their own data private. In this paper, we first propose three multi-party verifiable blind quantum computation (MPVBQC) protocols, each of which can handle one type of clients with certain simple quantum capabilities such as making single-qubit measurements, preparing single qubits, or performing a few single-qubit gates. Then a flexible and hybrid MPVBQC framework for multi-type clients in quantum networks is given by combining the three proposed MPVBQC protocols. It simultaneously allows at least three types of clients in quantum networks to achieve BQC depending on their own quantum devices. Furthermore, all the proposed protocols can achieve identity authentication, resist both insider and outsider attacks, and be verifiable which means that the clients can verify the correctness of their computational results.

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.

Determining whether nodes can be localized, called localizability detection, is essential for wireless sensor networks (WSNs). This step is required for localizing nodes, achieving low-cost deployments, and identifying prerequisites in location-based applications. Centralized graph algorithms are inapplicable to a resource-limited WSN because of their high computation and communication costs, whereas distributed approaches may miss a large number of theoretically localizable nodes in a resource-limited WSN. In this paper, we propose an efficient and effective distributed approach in order to address this problem. Furthermore, we prove the correctness of our algorithm and analyze the reasons our algorithm can find more localizable nodes while requiring fewer known location nodes than existing algorithms, under the same network configurations. The time complexity of our algorithm is linear with respect to the number of nodes in a network. We conduct both simulations and real-world WSN experiments to evaluate our algorithm under various network settings. The results show that our algorithm significantly outperforms the existing algorithms in terms of both the latency and the accuracy of localizability detection.

This paper investigates the minimization problem of fuzzy automata, aiming to obtain a procedure for finding a minimal state fuzzy automaton equivalent to a given one. The decision version of the minimization problem is as follows: Given a fuzzy automaton A and a natural number k, i.e., a pair (A, k), is there a k-state fuzzy automaton equivalent to A? We prove that the above problem is decidable for fuzzy automata over totally ordered lattices and then obtain a procedure for minimizing a given fuzzy automaton. To this end, we introduce the concept of systems of fuzzy polynomial equations, present a procedure for finding solutions of these systems and, finally, reduce the above decision problem to finding a solution of a system of fuzzy polynomial equations. It is worth pointing out that although some algorithms in the literature were claimed to be minimization algorithms, the term “minimization” there did not mean state minimization in our sense, since these algorithms did not aim at a minimal fuzzy automaton but found “reasonably” small fuzzy automata.