Simulation of many-qubit quantum computation with matrix product states

Matrix product states provide a natural entanglement basis to represent a quantum register and operate quantum gates on it. This scheme can be materialized to simulate a quantum adiabatic algorithm solving hard instances of an $NP$-complete problem. Errors inherent to truncations of the exact action of interacting gates are controlled by the size of the matrices in the representation. The property of finding the right solution for an instance and the expected value of the energy (cost function) are found to be remarkably robust against these errors. As a symbolic example, we simulate the algorithm solving a 100-qubit hard instance, that is, finding the correct product state out of $\sim {10}^{30}$ possibilities. Accumulated statistics for up to $60$ qubits seem to point at a subexponential growth of the average minimum time to solve hard instances with highly truncated simulations of adiabatic quantum evolution.