Zusammenfassung
It is well established that simulating a perfect quantum computer with a
classical computer requires computing resources that scale exponentially with
the number of qubits $N$ or the depth $D$ of the circuit. Conversely, a perfect
quantum computer could potentially provide an exponential speed up with respect
to classical hardware. Real quantum computers however are not perfect: they are
characterized by a small error rate $\epsilon$ per operation, so that the
fidelity of the many-qubit quantum state decays exponentially as $ F
(1-\epsilon)^ND$. Here, we discuss a set of classical algorithms based
on matrix product states (MPS) which closely mimic the behavior of actual
quantum computers. These algorithms require resources that scale linearly in
$N$ and $D$ at the cost of making a small error $\epsilon$ per two-qubit gate.
We illustrate our algorithms with simulations of random circuits for qubits
connected in both one and two dimensional lattices. We find that $\epsilon$ can
be decreased at a polynomial cost in computing power down to a minimum error
$\epsilon_ınfty$. Getting below $\epsilon_ınfty$ requires computing resources
that increase exponentially with $\epsilon_ınfty/\epsilon$. For a two
dimensional array of $N=54$ qubits and a circuit with Control-Z gates of depth
$D=20$, a fidelity $F0.002$ can be reached on a single core computer
in a few hours. It is remarkable that such a high fidelity can be obtained
using a variational ansatz that only spans a tiny fraction $(10^-8)$ of
the full Hilbert space. Our results show how the actual computing power
harvested by noisy quantum computers is controlled by the error rate
$\epsilon$.
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