Designing ground states of Hopfield networks for quantum state preparation

We present a protocol to store a polynomial number of arbitrary bit strings, encoded as spin configurations, in the approximately degenerate low-energy manifold of an all-to-all connected Ising spin glass. The iterative protocol is inspired by machine learning techniques utilizing $k$-local Hopfield networks trained with $k$-local Hebbian learning and unlearning. The trained Hamiltonian is the basis of a quantum state-preparation scheme to create quantum many-body superpositions with tunable squared amplitudes using resources available in near term experiments. We find that the number of configurations that can be stored in the ground states and thus turned into superposition scales with the $k$-locality of the Ising interaction.

Figure 1

Schematic of the ground-state design method based on Hopfield networks. (Left panel) Initially, the patterns (indicated by diamonds) are stored in a Hopfield network. Thus, patterns are located at local energy minima [dark blue (upper) curve]. Relearning of a single pattern [transition from dark blue (upper) to light blue (lower) curve] decreases the energy bandwidth ${\mathrm{\Delta }}_p$ of the stored patterns. (Right panel) Unlearning of individual low-energy bulk configurations [transition from light blue (lower) to red (upper) curve], which are typically spurious minima (indicated by crossed circles), increases the energy gap ${\mathrm{\Delta }}_b$ between stored patterns and the bulk. Iteratively applying relearning and unlearning steps results in (approximately) degenerate ground states with ${\mathrm{\Delta }}_p/{\mathrm{\Delta }}_b\ll 1$.

Figure 2

Typical spectra for a ground-state design of a system with $N=10,K=\{2,3\}$, and $M=20$ randomly chosen patterns. The minimum Hamming distance $h$ of every configuration with respect to any stored pattern is color (symbol) coded. (a) Spectrum of the initial Hopfield network trained with all $M$ patterns. (b) Spectrum after the iterative ground-state design protocol with ${\mathrm{\Delta }}^{*}=0.05$. The distribution of energies with respect to the Hamming distance $h$ (dark gray: $h=0$; light gray: $h\ge 1)$ is shown for the initial step of the protocol (c) and after the protocol (d). For this example, we use $h=2$ and thus 654 configurations out of 1024 states. To achieve $\mathrm{\Delta }=0.044,t_f=1820$ iteration steps were needed. Further parameters are $p_{\mathrm{relearn}}=2/3,p_{\mathrm{unlearn}}=1/3,0\le {\varphi }_k,{\eta }_k\le 0.02,r=1$, and $T=1$.

Figure 3

Capacity $\left(\mathrm{SP}=99\%\right)$ of our protocol for given parameters ${\mathrm{\Delta }}^{*}=0.1,h=4,r=1,p_{\mathrm{relearn}}=2/3$, and $T=1$. For every point, we have randomly chosen 1000 realizations of $M$ distinct bit strings of length $N$ and tried to design the desired Hamiltonian for $K=\{1,2\}$ (lower curve) and $K=\{1,2,3\}$ (upper curve). The points in the plot represent the mean of the maximum $M$ (for a given $N)$ reached with $\mathrm{SP}=99\%$ for a significant number of subgroups of size 100 out of 1000 samples. The fit corresponding to the blue (lower) and green (upper) data points is given by $M\left(N\right)=0.57N+0.23$ and $M\left(N\right)=0.53N^2-5.84N+23.02$, respectively.

Figure 4

Illustration of (a) 2D and (b) 3D LHZ architectures. Constraints, consisting either of three- or four-body terms, are visualized by shaded triangles or squares. Qubit labels denote the indices of the interaction matrix elements $J_{ij}$ and $J_{ijk}$, respectively, of the two- and three-local spin-glass Hamiltonians.

Figure 5

Examples of the full state-preparation protocol. The upper (lower) row corresponds to the case of $K=\{1,2\}(K=\{2,3\left\}\right)$ logical Hamiltonians. (a) Logical spectrum obtained via the iterative ground-state design protocol with $N=4$ and $M=3$ for $\mathrm{\Delta }=0.15$ (nondegenerate). (b) Instantaneous energies of the physical system during the sweep described by Eq. (13). (c) Overlap $p_n\left(t\right)=|\langle {\varphi }_n\left(t\right)|\psi \left(t\right)\rangle {|}^2$ of $M$ lowest-energy instantaneous eigenstates with the state of the system during full time evolution for the optimized constraint strengths to obtain uniformly distributed amplitudes $p_n\left(T\right)={\left|a_n\right|}^2=1/M$ (dashed lines: constraint optimization within the effective model; solid lines: exact optimization). (d) Logical spectrum obtained via the iterative ground-state design protocol with $N=4$ and $M=4$ for $\mathrm{\Delta }=0.0035$ (almost degenerate). (e) Instantaneous energies of the physical system as in (b). (f) Successfully programed amplitudes of $|a_n{|}^2=1/M$ as in (c).

Figure 6

Adiabatic and diabatic dynamics for different combinations of $\mathrm{\Delta }={\mathrm{\Delta }}_p/{\mathrm{\Delta }}_b$ and $T$. The upper panels [(a)–(c)] show the overlaps $p_n\left(t\right)=|\langle {\varphi }_n\left(t\right)|\psi \left(t\right)\rangle {|}^2$ of the $M=3$ instantaneous eigenstates which form the low-energy manifold with the state of the system $|\psi \left(t\right)\rangle$ during the sweep of the transverse field and the total population of the bulk $p_{\mathrm{bulk}}={\sum }_{n>3}|\langle {\varphi }_n\left(t\right)|\psi \left(t\right)\rangle {|}^2$. The lower panels [(d)–(f)] show the parameters $A_{12}\left(t\right),A_{13}\left(t\right)$, and $B_1\left(t\right)$ as defined in the main text. The optimization is successful if $p_n\left(T\right)=1/M$ and $p_{\mathrm{bulk}}\left(T\right)=0$ at the end of the sweep. (a) and (d) For the example of Figs. 5–5, the parameters $A_{12}\left(t\right),A_{13}\left(t\right)$ defined in Eq. (19) are strongly peaked with maximum values of $\gtrsim 1$ during short periods of population transfer from levels $1\to 2$ and $1\to 3$. In contrast, $B_1\left(t\right)\ll 1$ at all times. (b) and (d) Optimization of the same example as in (a) does not converge for $\mathrm{\Delta }=1.2$ since necessary diabatic transitions within the low-energy manifold (here $1\to 3)$ are absent $\left(A_{13}\left(t\right)\ll 1\right)$. (e) and (f) Trying to compensate the situation of (b) by decreasing $T$ leads to diabatic transitions into the bulk $B_1\left(t\right)\gtrsim 1$. This results in $p_1\left(T\right)+p_2\left(T\right)+p_3\left(T\right)<1$ and $p_{\mathrm{bulk}}\left(T\right)>0$ at the end of the sweep.