Simulating the quantum Fourier transform, Grover's algorithm, and the quantum counting algorithm with limited entanglement using tensor networks

Quantum algorithms reformulate computational problems as quantum evolutions in a large Hilbert space. Most quantum algorithms assume that the time evolution is perfectly unitary and that the full Hilbert space is available. However, in practice, the available entanglement may be limited, leading to a reduced fidelity of the quantum algorithms. To simulate the execution of quantum algorithms with limited entanglement, tensor-network methods provide a useful framework, since they allow us to restrict the entanglement in a quantum circuit. Thus, we here use tensor networks to analyze the fidelity of the quantum Fourier transform, Grover's algorithm, and the quantum counting algorithm as the entanglement is reduced, and we map out the entanglement that is generated during the execution of each algorithm. In all three cases, we find that the algorithms can be executed with high fidelity even if the entanglement is somewhat reduced. For example, the quantum counting algorithm can be executed with a high fidelity beyond a certain threshold level of entanglement. Our results provide an understanding of the entanglement that is required to execute the three algorithms, and our simulations based on tensor networks can easily be applied to other algorithms.