qml.resource.estimate_shots¶
-
estimate_shots(coeffs, variances=None, error=0.0016)[source]¶ Estimate the number of measurements required to compute an expectation value with a target error.
See also
estimate_error().- Parameters
coeffs (list[tensor_like]) – list of coefficient groups
variances (list[float]) – variances of the Pauli word groups
error (float) – target error in computing the expectation value
- Returns
the number of measurements
- Return type
int
Example
>>> coeffs = [np.array([-0.32707061, 0.7896887]), np.array([0.18121046])] >>> qml.resource.estimate_shots(coeffs) 419218
Theory
An estimation for the number of measurements \(M\) required to predict the expectation value of an observable \(H = \sum_i A_i\), with \(A = \sum_j c_j O_j\) representing a linear combination of Pauli words, can be obtained following Eq. (34) of [PRX Quantum 2, 040320 (2021)] as
\[M = \frac{\left ( \sum_i \sqrt{\text{Var}(A_i)} \right )^2}{\epsilon^2},\]with \(\epsilon\) and \(\text{Var}(A)\) denoting the target error in computing \(\left \langle H \right \rangle\) and the variance in computing \(\left \langle A \right \rangle\), respectively. It has been shown in Eq. (10) of [arXiv:2201.01471v3] that the variances can be computed from the covariances between the Pauli words as
\[\text{Var}(A_i) = \sum_{jk} c_j c_k \text{Cov}(O_j, O_k),\]where
\[\text{Cov}(O_j, O_k) = \left \langle O_j O_k \right \rangle - \left \langle O_j \right \rangle \left \langle O_k \right \rangle.\]The values of \(\text{Cov}(O_j, O_k)\) are not known a priori and should be either computed from affordable classical methods, such as the configuration interaction with singles and doubles (CISD), or approximated with other methods. If the variances are not provided to the function as input, they will be approximated following Eqs. (6-7) of [Phys. Rev. Research 4, 033154, 2022] by assuming \(\text{Cov}(O_j, O_k) =0\) for \(j \neq k\) and using \(\text{Var}(O_i) \leq 1\) from
\[\text{Var}(O_i) = \left \langle O_i^2 \right \rangle - \left \langle O_i \right \rangle^2 = 1 - \left \langle O_i \right \rangle^2.\]This approximation gives
\[M \approx \frac{\left ( \sum_i \sqrt{\sum_j c_{ij}^2} \right )^2}{\epsilon^2},\]where \(i\) and \(j\) run over the observable groups and the Pauli words inside the group, respectively.