From Bell Products to GHZ: Quantum Memories via Emergent Hamiltonians

TL;DR

The paper introduces the Emergent Hamiltonian framework to freeze unitary quantum dynamics and store time-evolved states by quenching from an entangling Hamiltonian $\hat{H}_f$ to $\hat{\mathcal{M}}(t)$. In 1D, an exact, local Emergent Hamiltonian is derived, enabling efficient storage of highly entangled states; in 2D, exact compact forms exist only for single-particle dynamics, with accurate approximations for many-body systems. The authors demonstrate applications to storing tensor products of Bell states and GHZ states, provide spectral analyses showing regular but dense operator structure, and illustrate a GHZ-state preparation and indefinite freezing via the Emergent Hamiltonian. The framework offers a flexible design tool for quantum memories, with practical considerations on decoherence and potential connections to ETH and ergodicity in emergent Hamiltonians.

Abstract

With the advent of exquisite quantum emulators, storing highly entangled many-body states becomes essential. While entanglement typically builds over time when evolving a quantum system initialized in a product state, freezing that information at any given instant requires quenching to a Hamiltonian with the time-evolved state as an eigenstate, a concept we realize via an Emergent Hamiltonian framework. While the Emergent Hamiltonian is generically non-local and may lack a closed form, we show examples where it is exact and local, thereby enabling, in principle, indefinite state storage limited only by experimental imperfections. Unlike other phenomena, such as many-body localization, our method preserves both local and global properties of the quantum state. In some of our examples, we demonstrate that this protocol can be used to store maximally entangled multi-qubit states, such as tensor products of Bell states, or fragile, globally distributed entangled states, in the form of GHZ states, which are often challenging to initialize in actual devices.

Figures (7)

• Figure 1: Schematics of the protocol. We freeze the dynamics at time $t=t_1$ via carrying out a quench from the entangling Hamiltonian $\hat{H}_f$ to $\hat{\mathcal{M}}(t_1)$ --- the Emergent or freezing Hamiltonian defined at $t_1$.
• Figure 2: Dynamics of the half-chain entanglement entropy, scaled by the maximal entropy, considering different chain sizes, for a density-wave (a) [domain wall (b)] initial product state $|\psi(0)\rangle$; a quench $\hat{H}_{f}\to\hat{\cal M}$ is performed at $t_{\rm freeze}=3\pi/2$. (c) and (d) show the distribution of Hamming distances and the corresponding weights $|\langle n|\psi(t)\rangle|^2$ in the time-dependent state $|\psi(t)\rangle$ of the Fock states $|n\rangle$, at representative times in the dynamics. Schmidt coefficients of the half-chain decomposition at similar times in (e) and (f). In (c)--(d), the system size is $L=16$; as indicated, (a), (c) and (e) [(b), (d), and (f)] refer to the density-wave [domain wall] initial product state.
• Figure 3: Dynamics of the half-system entanglement entropy considering different lattice sizes $L_x\times L_y$, for the [next-] nearest-neighbor entangling Hamiltonian (a) [(b)] taking as initial product state $|\psi(0)\rangle$ a single excitation at the lower left corner of the lattice. (c) and (d) provide snapshots of the site resolved probabilities $|\psi_j|^2$ at the times where the entanglement entropy is maximal. (e) and (f) display the expectation values of $\{\langle \hat{S}_{1\alpha}\rangle\}$ and $\{\langle \hat{S}_{2\alpha}\rangle\}$ ($\alpha=x,y,z$) over time, defining a trajectory in the Bloch sphere --- here, the colorbar maps the time $t$. As indicated, (a), (c), and (e) [(b), (d), and (f)] refer to the entangling Hamiltonian $\hat{H}_f$ in Eq. \ref{['eq:Hf_2dNN']} [\ref{['eq:Hf_2dNNN']}].
• Figure 4: (a) Time evolution of the overlap between the time-evolved state $|\psi(t)\rangle$ and the normalized $\hat{\mathcal{M}}(t)|\psi(t)\rangle$ under the first-order emergent Hamiltonian $\hat{\mathcal{M}}^{(1)}$ for different system sizes $L_x\times L_y$. The initial state consists of two particles positioned at opposite corners of the square lattice. (b) [(c)] The same, but considering the second-order emergent Hamiltonian $(\hat{\mathcal{M}}^{(2)}$ [the spin emergent Hamiltonian $\hat{\mathcal{M}}_{\text{spin}}$]. The inset in (a) contrasts the overlap at short times among the different approximate emergent Hamiltonian for an $10\times 10$ lattice. The shadings in all panels highlight the different regimes where the dynamics are effectively single-particle and when such a distinction can no longer be made --- see text.
• Figure 5: (a) Comparison of the overlap between $|\psi(t)\rangle$ and the normalized $\hat{\mathcal{M}}(t)|\psi(t)\rangle$ under the second order Emergent Hamiltonian $\hat{\mathcal{M}}^{(2)}$, for different diagonal hoppings $J_\times$ in $\hat{H}_f$, on a $6 \times 6$ lattice. The initial state consists of three particles at the lower corner of the square lattice, specifically at sites $(0,0)$, $(1,0)$, and $(0,1)$. (b) The same as in (a), but contrasting different system sizes for $J_\times = 0.2$.