How many samples are required to estimate the probabilities of a state?



Suppose that we have a quantum state of the form:
$$|\psi\rangle = \sqrt{p}|0\rangle + \sqrt{1-p}|1\rangle$$
In order to get an estimate of the probability of reading $|0\rangle$ or $|1\rangle$, we need to sample $|\psi\rangle$. How many times do we need to sample to have an $\epsilon$-estimate ? Keep in mind that this is different from quantum tomography because i we don't want to reconstruct the state from measurements, i just want to find an approximation of the probability of reading some state. In Supervised Learning with Quantum Computers, Schuld,M,, the authors say that sampling from a qubit is equivalent to sampling from a bernoulli distribution,thus we can use the Wald interval:
$$\epsilon = z\sqrt{{\bar{p}(1-\bar{p})}\over S}$$ where $z$ is the confidence level, $\bar{p}$ is the average and $S$ the number of samples.
In the case of $p$ being close to either 0 or 1, we can use Wilson Score interval:
$$ \epsilon = {z \over {1 + {z^2\over S}}}\sqrt{{{\bar{p}(1-\bar{p})}\over S} + {z^2 \over 4S^2}} $$ Now, the i ask the following question: What if i have a state with multiple qubits? How do i get an $\epsilon$-estimate of the probability of reading some state? If you can suggest some references, i would appreciateit. Thank you very much.


Posted 2020-06-14T15:24:25.540

Reputation: 109



Well, I would say you can consider each possible outcome as a Bernoulli in the very same way!
You can consider a state of $n$ qubits $$|\varphi\rangle = \sum_{i}^k\alpha_i|x_i\rangle$$ Where $|x_i\rangle$ is a vector of the computational basis (i.e. $|x_i\rangle \in \{|0\rangle, |1\rangle\}^{\otimes n}$) and each $|x_i\rangle$ has probability $|\alpha_i|^2$ of being measured.
After creating and sampling such state $S$ times, you could consider $k$ Bernoulli trials - one for each $|x_i\rangle$ - where for each $i^{th}$ trial you consider a success if you measure the state $|x_i\rangle$ and failure if you measure any other state.

In that way you could just reuse the Wald/Wilson intervals for each $|x_i\rangle$.
For instance, using the Wald interval, you could compute for each $|x_i\rangle$ $$\hat{p}_i = \frac{\text{ number of times } |x_i\rangle\text{ appears}}{S}$$ and say with confidence $z$ that $$|\alpha_i|^2 \in [\hat{p}_i - \epsilon, \hat{p}_i + \epsilon]$$ where $$\epsilon = z\sqrt{\frac{\hat{p}_i(1 - \hat{p}_i)}{S}}$$


Posted 2020-06-14T15:24:25.540

Reputation: 96