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Why we should use density matrices to simulate quantum systems with noise?

I found that the any QEC circuit is included by some quantum gates just like normal cases, which means the state vector can also be used to simulate it, but all of the simulator only can realize noise model with density matrix, can someone help me with that?

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Noise effects introduce classical uncertainty in what the underlying state is. A mixed state is a statistical ensemble of several quantum states $|\psi_i\rangle$ (not necessarily orthogonal), with respective probabilities $p_i$.

With the state vector you can represent pure states, not mixed ones. Instead, with the density operator you can represent both pure and mixed states.

Noise models are always defined in terms of density operators.

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    $\begingroup$ Good answer, just one remark: density matrices can also describe subsystems of entangled systems. Which results in noise, but it's not really "classical"... $\endgroup$ – M. Stern Jul 19 at 20:20
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You do not need to use the density matrix approach. However, as the most general representation of a quantum state, doing so has several advantages. You can simulate noise using just statevectors using probabilistic approaches, eg wavefunction monte-carlo, that converge to the density matrix results in the limit of many repetitions. Along this same thread of thought, it is important to note that the density matrix approach is not what the system is actually doing. Rather, it is the average of what the system does if the experiment is repeated many (formally infinitely many) times. In contrast, the statevector based methods should well approximate a single realization of the experiment.

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