Measurement disturbance tradeoffs in three-qubit unsupervised quantum classification

We consider measurement disturbance tradeoffs in quantum machine learning protocols which seek to learn about quantum data. We study the simplest example of a binary classification task in the unsupervised regime. Specifically, we investigate how a classification of two qubits, that can each be in one of two unknown states, affects our ability to perform a subsequent classification on three qubits when a third is added. Surprisingly, we find a range of strategies in which a nontrivial first classification does not affect the success rate of the second classification. There is, however, a nontrivial measurement disturbance tradeoff between the success rate of the first and second classifications, and we fully characterize this tradeoff analytically.

Figure 1

Plots showing the effect that a measurement on the first two qubits of a system, with outcome ${\widehat{\pi }}_{-}$ (for various values of $\alpha ,\beta$), has on the three-qubit states and measurement operators $\{{\widehat{\rho }}_{001}^{-},{\widehat{\rho }}_{010}^{-},{\widehat{\rho }}_{011}^{-}\}$ and $\{{\widehat{\pi }}_{001}^{-},{\widehat{\pi }}_{010}^{-},{\widehat{\pi }}_{011}^{-}\}$, respectively. Note that the states and measurement operators vary with the strength of the two-qubit measurement until (c) $\alpha =\frac{2}{3},\beta =\frac{1}{3}$ after which, while the states continue to change, the measurement operators stay constant. (c) corresponds to the boundary between the constant and nonconstant regions in Fig. 2. Further, (a) corresponds to the case in which no measurement is performed on the first two qubits and (d) to when the optimal measurement is performed on the first two qubits.

Figure 2

Plot of the tradeoff between the success rates of the first binary classification of two qubits and the second binary classification when a third qubit is added. The probability of success of the first (second) measurement is denoted $P_{\text{succ}}^{\text{1st}}\left(P_{\text{succ}}^{\text{2nd}}\right)$.