Frustration-Free Control and Absorbing-State Transport in Entangled State Preparation

Abstract

We study frustration-free control, a measurement-feedback protocol for quantum state preparation that extends the concept of frustration-free Hamiltonians to stochastic dynamics. The protocol drives many-body systems into highly entangled target states, common dark states of all measurement projectors, through minimal local unitary corrections that realize an absorbing-state dynamics without post-selection. We show that relaxation to the target state is governed by emergent transport of nonlocal charges, such as singlet excitations in SU$(2)$-symmetric dynamics. While measurement-feedback annihilates compatible charge configurations, both measurement and scrambling unitaries induce charge transport and thus determine the convergence time. Mapping a baseline model of SU$(N)$ SWAP measurements with local corrections to a solvable absorbing random walk yields a runtime scaling $t \sim L^z$ with transport exponent $z=2$. Simulations of Motzkin and Fredkin chains reveal subdiffusive scaling $z \ge \tfrac{8}{3}$, confirming the transport picture and suggesting strategies for controlled entangled-state preparation and charge-transport probing in monitored quantum dynamics.

Figures (5)

• Figure 1: Frustration-free control and absorbing random walks. Schematic circuit that drives a spin chain into the entangled ground state manifold of the Heisenberg ferromagnet. Local measurements (red) compatible with the target leave the state unchanged, while incompatible outcomes trigger minimal unitary corrections (blue) that steer it toward the absorbing target state. Optional scrambling unitaries (green) preserve the target and accelerate convergence. (b) SU$(2)$-symmetric measurement and scrambling unitaries induce Brownian motion of singlet endpoints. Feedback annihilates short singlets of length $r(t)=1$. The diffusion-annihilation dynamics map to an absorbing random walk, governed by the transport exponent $z$.
• Figure 2: Frustration-free control of the Heisenberg chain. (a) Exact simulations of the measurement-feedback protocol show a hydrodynamic regime with $\mathds{E}(\langle \hat{P}_{\ell,\ell+1} \rangle) \sim t^{-1/z}$ and an asymptotic exponential decay $\sim \exp(-t/L^z)$ for system sizes $L \leq 24$, averaged over $10^5$ trajectories ($10^3$ for L=24). The yellow line is a single run of $\mathcal{P}_1$ in Eq. \ref{['eq:rw_master']} for $L=24$, the same initial state and no free parameters. (b) The variance of the order parameter exhibits strong fluctuations at the onset of the hydrodynamic regime ($t \sim O(L^0)$), plateaus during hydrodynamic evolution and is exponential cutoff at $t \sim L^z$ (grey region). (c,d) Scaling exponents $\mu(L)$ of the cutoff time $t\sim L^{\mu(L)}$ as functions of system size and long-range exponent $\Delta$ for absorbing random walks in one (c) and two (d) dimensions, obtained from solving Eq. \ref{['eq:effME']}. Inset: Total decay rate as a function of $\Delta$.
• Figure 3: Anomalous transport in frustration-free control. Measurement-feedback dynamics for (a,b) the spin-$\tfrac{1}{2}$ Fredkin and (c,d) spin-1 Motzkin chains exhibit an intermediate regime $\mathds{E}(\langle \hat{Q}_\alpha \rangle)\!\sim\! t^{-1/z}$ followed by exponential decay $\mathds{E}(\langle \hat{Q}_\alpha \rangle)\!\sim\!\exp(-t/L^{z})$, signaling subdiffusive transport with $z>2$. For the Motzkin chain, data up to $L\!\le\!14$ yield $z\!\approx\!3.6$. The Fredkin chain shows a hydrodynamic regime $\mathds{E}(\langle \hat{P}_\alpha \rangle)\!\sim\! t^{-z}$ with $z\!=\!\tfrac{8}{3}$, crossing over to an asymptotic decay consistent with $z\!\gtrsim\!3.3$Fradkin.
• Figure 4: Stationary-state half-chain entanglement. Numerical half-chain entanglement in the $S_z=0$ sectors versus analytical expressions. (a) The spin-$\tfrac{1}{2}$ SU$(2)$ model (circles) coincides with the entanglement of the Dicke state $S_{L/2}(\ket*{D_L^{L/2}})$ (lower solid line). The Fredkin model (triangles) converges to $\ket{A}-\ket*{D_L^{L/2}}$ (upper solid line). (b) Consistent with analytical results for the open Motzkin chain DellAnna2019, the stationary state of the periodic Motzkin chain exhibits logarithmic entanglement scaling (solid line: linear fit on a semi-log plot).
• Figure 5: Imperfect control. Scaling of the stationary singlet expectation $\mathcal{P}_r$ with noise rate $\eta = L^{-2}$ versus exact simulations. The gray lines show $M^{-1}\vec{\eta}$. (a) Length-1 singlets scale as $\mathcal{P}_1 \sim \eta L^{z-1}$; (b) longer singlets obey $\mathcal{P}_r \sim \eta L^z$ for $r>1$. Dots display the stationary singlet expectations for the corresponding $r$, exhibiting the same scaling.