Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity

We develop a density-matrix renormalization group (DMRG) algorithm for the simulation of quantum circuits. This algorithm can be seen as the extension of the time-dependent DMRG from the usual situation of Hermitian Hamiltonian matrices to quantum circuits defined by unitary matrices. For small circuit depths, the technique is exact and equivalent to other matrix product state-based techniques. For larger depths, it becomes approximate in exchange for an exponential speed up in computational time. Like an actual quantum computer, the quality of the DMRG results is characterized by a finite fidelity. However, unlike a quantum computer, the fidelity depends strongly on the quantum circuit considered. For the most difficult possible circuit for this technique, the so-called "quantum supremacy" benchmark of Google LLC [Arute et al., Nature 574, 505 (2019)], we find that the DMRG algorithm can generate bit strings of the same quality as the seminal Google experiment on a single computing core. For a more structured circuit used for combinatorial optimization (quantum approximate optimization algorithm), we find a drastic improvement of the DMRG results with error rates dropping by a factor of 100 compared with random quantum circuits. Our results suggest that the current bottleneck of quantum computers is their fidelities rather than the number of qubits.