How measurement error works in PRNG operation? Suppose, I have PRNG with seed key length of $|K_s|$ and output sequence length of $2^{|K_s|}$, so the complexity of the PRNG is of the order of $2^{|K_s|}$. Now, if there is any measurement error in the output of the PRNG, how this measurement error works in the overall complexity of the PRNG?
Unless the measurement error is huge (e.g. you read every bit correctly with probability $p = 0.5$), it doesn't add significantly to the work required to find the key.

To give a concrete example, let us consider a measurement error where we read each bit correctly with probability $p = 0.75$ (and for simplicity, let us assume that the errors are independent of each other, and whether the correct bit was a 0 or a 1).

Then, what the attacker can do is cycle through every possible key, and generate the first 1000 bits of the PRNG with that key, and check if at least 600 of those bits agree with the first 1000 bits of the noisy sample he was given.  With extremely high probability, the correct key will generate a sample with that many bits set, and no incorrect key will.

This procedure still takes $O(2^{|K_s|})$ time, and so all you've done is increase the time required by a constant factor.