Quantum pattern recognition with liquid-state nuclear magnetic resonance

A quantum pattern recognition scheme is presented, which combines the idea of a classic Hopfield neural network with adiabatic quantum computation. Both the input and the memorized patterns are represented by means of the problem Hamiltonian. In contrast to classic neural networks, the algorithm can return a quantum superposition of multiple recognized patterns. A proof of principle for the algorithm for two qubits is provided using a liquid-state NMR quantum computer.

Figure 1

Illustrative representation of the energy landscapes for the memory Hamiltonian ${\mathcal{H}}_{\text{mem}}$ (a), the weighted input Hamiltonian $\Gamma {\mathcal{H}}_{\text{inp}}$ (b) and the total problem Hamiltonian ${\mathcal{H}}_p$ (c). Dashed circles represent memory patterns stored as energy minima of ${\mathcal{H}}_{\text{mem}}$; crossed dashed circle represents the input pattern as energy minimum of ${\mathcal{H}}_{\text{inp}}$.

Figure 2

Illustrative representation of classic and quantum dynamics within the energy landscape. Classic dynamics drives the initial state (crossed circle) which represents the input pattern toward the final state (solid circle) which represents a nearby memory pattern. Quantum dynamics transforms the energy landscape from the initial Hamiltonian close to ${\mathcal{H}}_i$ (a) via intermediate Hamiltonian $\mathcal{H}\left(t\right)$ (b) toward the final Hamiltonian equivalent to the problem Hamiltonian ${\mathcal{H}}_p$ (c). The current state of the system remains the ground state of the instantaneous Hamiltonian with the “empty memory” state $|{\psi }_{\text{un}}⟩$ [Eq. (7)] for $t=0$ (represented by the star in A) and the final state of the system corresponding to the outcome memory pattern for $t=T$ (represented by the solid circle in C).

Figure 3

Experimental ${}^1\text{H}$ and ${}^{13}\text{C}$ spectra for five combinations of ${\xi }_1^{\text{inp}}$ and ${\xi }_2^{\text{inp}}$ and two possible values of $w$, where $w=-1$ and $w=+1$ correspond to stored patterns with opposite and equal bit values, respectively. The spectral range in each spectrum is $\pm 240\text{Hz}$.

Figure 4

Final measurement probabilities according to memory set (A7) and input pattern (A8). The peaks correspond to memory patterns ${\xi }^1=|-1,-1,-1,-1,-1⟩\equiv |1⟩$ (peak A), ${\xi }^2=|-1,-1,-1,+1,-1⟩\equiv |3⟩$ (peak B), and ${\xi }^3=|-1,-1,+1,-1,+1⟩\equiv |6⟩$ (peak C) in binary and decimal representation, respectively.