Abstract On the basis of a labelled deduction system (LND$_{ED-}$TRS), we demonstrate how to formalize the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. This has allowed us to carry out a formal counterpart to equality between paths which is dealt with in homotopy theory, but this time with an approach using the device of term-rewriting paths. Using such formal calculus dealing with paths, we construct the fundamental groupoid of a path-connected $ X $ type and we define the concept of isomorphism between types. Next, we show that the computational paths determine a weak category, which will be called $ \mathcal {C}_{paths} $. Finally, we show that the weak category $ \mathcal {C}_{paths} $ determines a weak groupoid.

The initialization of quantum states or Quantum State Preparation (QSP) is a basic subroutine in quantum algorithms. In the worst case, general QSP algorithms are expensive due to the application of multi-controlled gates required to build the quantum state. Here, we propose an algorithm that detects whether a given quantum state can be factored into substates, increasing the efficiency of compiling the QSP circuit when we initialize states with some level of unentanglement. The simplification is done by eliminating controls of quantum multiplexers, significantly reducing circuit depth and the number of CNOT gates with a better execution and compilation time than the previous QSP algorithms. Considering efficiency in terms of depth and number of CNOT gates, our method is competitive with the methods in the literature. However, when it comes to run-time and compilation efficiency, our result is significantly better, and the experiments show that by increasing the number of qubits, the gap between the temporal efficiency of the methods increases.

Loading data in a quantum device is required in several quantum computing applications. Without an efficient loading procedure, the cost to initialize the algorithms can dominate the overall computational cost. A circuit-based quantum random access memory named FF-QRAM can load$M$$n$-bit patterns with computational cost$O(CMn)$to load continuous data where$C$depends on the data distribution. In this article, we propose a strategy to load continuous data without post-selection with computational cost$O(Mn$). The proposed method is based on the probabilistic quantum memory, a strategy to load binary data in quantum devices, and the FF-QRAM using standard quantum gates, and is suitable for noisy intermediate-scale quantum computers.