Measurement Error

So far the discussion has been focused purely on gathering information of a quantum system. However, a large source of error in current devices (1-5% error rate) are from measurement errors. Naive methods in quantum state (process) tomography are susceptible to such measurement errors, and more generally to state preparation and measurement (SPAM) errors.

A straightforward but effective classical model for measurement error is a Stochastic Assignment matrix

\[ A_{i,j} = P(i|j) \]

which maps the probability of obtaining an incorrect output \(i\) given a correct input \(j\), where \(i\) and \(j\) are bitstrings associated with a qubit measurement.

A complete characterization of the \(A\)-matrix requires preparing \(2^n\) states. For example, to characterize two qubits, four states are needed: \(\{\ket{00}, \ket{01}, \ket{10}, \ket{11}\}\). If the errors are uncorrelated, then the \(A\)-matrix can be constructed as a tensor product. In this case, only 2 states are needed as all probabilities can be estimated simultaneously by preparing all-zero and all-one state. Although the tensor product model is simple, cross-talk errors that are present in real-world setups are ignored. A recent work has introduced a model that takes into account a limited measure of correlated noise based on Continuous Time Markov Processes (CTMP) arXiv:2006.14044. The matrix then has a form \(A = e^G\) where \(G\) is a sum of local operators the generate one and two-qubit readout errors. The total cost of this approach is \(O(e^{5\gamma} poly(n))\) where \(\gamma\) is related to noise strength and assumed small. The authors compare the tensor product noise model and the CTMP noise model with a full characterization model for 4,5,6,7 qubits.