  The advent of quantum computers promises exponential speed ups in the
execution of various computational tasks. While their capabilities are hindered
by quantum decoherence, they can be exactly simulated on classical hardware at
the cost of an exponential scaling in terms of number of qubits. To circumvent
this, quantum states can be represented as matrix product states (MPS), a
product of tensors separated by so-called bond dimensions. Limiting bond
dimensions growth approximates the state, but also limits its ability to
represent entanglement. Methods based on this representation have been the most
popular tool at simulating large quantum systems. But how to trust resulting
approximate quantum states for such intractable systems sizes ? I propose here
a method for inferring the fidelity of an approximate quantum state without
direct comparison to its exact counterpart, and use it to design an
``entanglement-aware'' (EA) algorithm for both pure and mixed states. As
opposed to state of the art methods which limit bond dimensions up to an
arbitrary maximum value, this algorithm receives as input a fidelity, and
adapts dynamically its bond dimensions to both local entanglement and noise
such that the final quantum state fidelity at least reaches the input fidelity.
I show that this algorithm far surpasses standard fixed bond dimension
truncation schemes. In particular, a noiseless random circuit of 300 qubits and
depth 75 simulated using MPS methods takes one week of computation time, while
EA-MPS only needs 2 hours to reach similar quantum state fidelity.
