arXiv subjects

Alessandro Andrea Zecchi

Publications and source records attributed to Alessandro Andrea Zecchi.

3 recordsLinked to original sources

arXiv2026

Block encoding of sparse matrices with a periodic diagonal structure

Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $ω$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.

quant-ph↗

arXiv2026

Quantum Algorithm for the Fixed-Radius Neighbor Search

Neighbor search is a computationally demanding problem, usually both time- and memory-consuming. The main problem of this kind of algorithms is the long execution time due to cache misses. In this work, we propose a quantum algorithm for the Fixed RAdius Neighbor Search problem (FRANS) based on the fixed-point version of Grover's algorithm. We propose an efficient circuit for solving the FRANS with linear query complexity with the number of particles $N$. The quantum circuit returns the list of all the neighbors' pairs within the fixed radius, together with their distance, avoiding the slow down given by cache miss. We analyzed the gate and the query complexity of the circuit. Our FRANS algorithm presents a query complexity of $\mathcal{O}(N/\sqrt{M})$, where $M$ is the number of solutions, reaching the optimal lower bound of the Grover's algorithm. We propose different implementations of the oracle, which must be chosen depending on the precise structure of the database. Among these, we present an implementation using the Chebyshev distance with depth $\mathcal{O}(q_1)$, where $2^{q_1}$ is the number of grid points used to discretize a spatial dimension. State-of-the-art algorithms for state preparation allow for a trade-off between depth and width of the circuit, with a volume (depth$\times$ width) of $\mathcal{O}(N\log(N))$. This unfavorable scaling can be brought down to $\mathcal{O}(\text{poly}(\log N))$ in case of structured datasets. We proposed a stopping criterion based on Bayes interference and tested its validity on $1D$ simulations. Finally, we accounted for the readout complexity and assessed the resilience of the model to the readout error, suggesting an error correction-free strategy to check the accuracy of the results.

quant-ph↗

arXiv2025

Improved amplitude amplification strategies for the quantum simulation of classical transport problems

The quantum simulation of classical fluids often involves the use of probabilistic algorithms that encode the result of the dynamics in the form of the amplitude of the selected quantum state. In most cases, however, the amplitude probability is too low to allow an efficient use of these algorithms, thereby hindering the practical viability of the quantum simulation. The oblivious amplitude amplification algorithm is often presented as a solution to this problem, but to no avail for most classical problems, since its applicability is limited to unitary dynamics. In this paper, we show analytically that oblivious amplitude amplification when applied to non-unitary dynamics leads to a distortion of the quantum state and to an accompanying error in the quantum update. We provide an analytical upper bound of such error as a function of the degree of non-unitarity of the dynamics and we test it against a quantum simulation of an advection-diffusion-reaction equation, a transport problem of major relevance in science and engineering. Finally, we also propose an amplification strategy that helps mitigate the distortion error, while still securing an enhanced success probability.

quant-ph↗