qml.resource.estimate_error¶
-
estimate_error(coeffs, variances=None, shots=1000)[source]¶ Estimate the error in computing an expectation value with a given number of measurements.
See also
estimate_shots().- Parameters
coeffs (list[tensor_like]) – list of coefficient groups
variances (list[float]) – variances of the Pauli word groups
shots (int) – the number of measurements
- Returns
target error in computing the expectation value
- Return type
float
Example
>>> coeffs = [np.array([-0.32707061, 0.7896887]), np.array([0.18121046])] >>> qml.resource.estimate_error(coeffs, shots=100000) 0.00327597
Theory
An estimation for the error \(\epsilon\) in predicting 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
\[\epsilon = \frac{\sum_i \sqrt{\text{Var}(A_i)}}{\sqrt{M}},\]with \(M\) and \(\text{Var}(A)\) denoting the number of measurements 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
\[\epsilon \approx \frac{\sum_i \sqrt{\sum_j c_{ij}^2}}{\sqrt{M}},\]where \(i\) and \(j\) run over the observable groups and the Pauli words inside the group, respectively.