The implementation of quantum state preparation algorithms, considering the limitations of modern quantum computers

Preparation of an arbitrary initial state of a system of qubits is an important and actual problem of quantum computing. Its importance because many quantum algorithms require preloading classical data to quantum devices, such as quantum neural networks [1] or solving systems of linear equations [3]. The computational cost of data loading can limit potential quantum acceleration, while the accuracy of the prepared state directly impacts the correctness of algorithmic outcomes. This paper analyzes three contemporary approaches to initial quantum state preparation, their translation into QASM code under the constraints of modern quantum hardware, considering the topology and basic sets of gates. The first of them is a modified algorithm for preparing the quantum state using controlled gates [5]. This algorithm does not use additional qubits, and the size and depth of its quantum circuit are estimated at 𝑂(2𝑛). The main idea of the algorithm is that iteratively, considering the first 𝑖+1 qubits at the 𝑖 step, we distribute 1 (corresponding to the initial state, where all qubits are in the state |0⟩) over the entire vector in the required proportions and add a phase at the last iteration. The second algorithm called “divide and conquer”, proposed in the article [5], is an optimization of the previous algorithm, which, by using 2𝑛 − 𝑛 − 1 additional qubits, reduces the asymptotic depth of the circuit to 𝑂(𝑛2), but maintaining its size is 𝑂(2𝑛). The idea is to expand the algorithm in width instead of length due to the tree structure of the algorithm and the gates 𝐶𝑆𝑊𝐴𝑃. This arrangement makes it possible to perform quantum gates in parallel. The third algorithm is the preparation of a quantum state by solving a more general problem of approximation of a unitary operator using template schemes. This approach, unlike the previous ones, is not theoretically accurate and requires resource-intensive classical calculations. In this paper, all algorithms were implemented considering the limitations of the modern ibm_sherbrooke quantum computer [8]. To do this, the basic gates were approximated from theoretical calculations using available gates, the CNOT gates were applied considering the topology, and the problem of the global phase was considered separately. The algorithmic implementation of algorithms for preparing the initial quantum state showed in all cases a shift in the estimation of the asymptotic of the circuit size and its depth for the worse. The algorithm showed the best result in the form of an approximation of the unitary operator with 𝑂(𝑛2𝑛) and 𝑂(2𝑛) accordingly, but it requires preliminary classical preparation of 3𝑛2𝑛 circuit parameters. All algorithms showed low accuracy when algorithmically implemented on the architecture of the ibm_sherbrooke quantum computer. And at the moment, due to the high asymptotic of practical implementation, they are explicitly suitable only for preparing the states of systems of no more than three qubits. Perhaps by applying various optimization methods and error compensation algorithms, it will be possible to increase this number, but without new approaches that reduce the asymptotic size of the circuit or its depth in practice, one should not expect significant improvement. At the same time, it is worth remembering that all the described algorithms allow you to prepare an arbitrary state, that is, they are universal.