Protocol for Implementing Quantum Nonparametric Learning with Trapped Ions

Nonparametric learning is able to make reliable predictions by extracting information from similarities between a new set of input data and all samples. Here we point out a quantum paradigm of nonparametric learning that offers an exponential speedup over the sample size. By encoding data into quantum feature space, the similarity between the data is defined as an inner product of quantum states. A quantum training state is introduced to superpose all data of samples, encoding relevant information for learning in its bipartite entanglement spectrum. We demonstrate that a trained state for prediction can be obtained by entanglement spectrum transformation, using the quantum matrix toolbox. We further work out a feasible protocol to implement the quantum nonparametric learning with trapped ions, and demonstrate the power of quantum superposition for machine learning.

Figure 1

Entanglement entropy of the training state vs the number of samples, where 40 randomly sampled datasets are chosen for each specified number of samples. The insert shows that the mean-square-error decrease with the entropy.

Figure 2

Illustration of the quantum algorithm. (a). Matrix inversion algorithm for a matrix $B={\rho }_K+\chi \mathbf{I}$ that transforms $|{\psi }_{\mathbf{A}}⟩$ into $|{\psi }_{{\mathbf{A}}^{+}}⟩=B^{-1}|{\psi }_{\mathbf{A}}⟩$, using two auxiliary qumodes that are postselected into zero momentum. (b). A swap test that evaluates the inner product between $|{\mathbf{A}}^{+}⟩$ and $|{\psi }_R⟩$, which can be used to infer the prediction for input data $\tilde{\mathbf{a}}$.

Figure 3

(a) A quantum procedure that transforms $|{\psi }_{\mathbf{A}}⟩$ into $|{\psi }_{{\mathbf{A}}^{+}}⟩$ on $b$ ions, assisted by an $a$ ion providing a qubit for control and two qumodes for matrix inversion, and many $c$ ions initialized in $|{\psi }_{\mathbf{A}}⟩$ serving as quantum software states for quantum phase estimation. Note the swap only performs on motional states (red dash lines). (b). Density matrix exponentiation where partial training dataset is superposed. Here $R_M=1$, 2, 3, 4 stands for the number of randomly chosen samples for each copy. The region with dashed lines represents accessible zone for a trade-off between error and the number of ions involved, constraint by the maximum available $c$ ion $N_t=20$.