  We develop a density-matrix renormalization group (DMRG) algorithm for the
simulation of quantum circuits. This algorithm can be seen as the extension of
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 (MPS) 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 Inc. , 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 or
QAOA), 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.
