Exact stabilization of entangled states in finite time by dissipative quantum circuits

Open quantum systems evolving according to discrete-time dynamics are capable, unlike continuous-time counterparts, to converge to a stable equilibrium in finite time with zero error. We consider dissipative quantum circuits consisting of sequences of quantum channels subject to specified quasi-locality constraints, and determine conditions under which stabilization of a pure multipartite entangled state of interest may be exactly achieved in finite time. Special emphasis is devoted to characterizing scenarios where finite-time stabilization may be achieved robustly with respect to the order of the applied quantum maps, as suitable for unsupervised control architectures. We show that if a decomposition of the physical Hilbert space into virtual subsystems is found, which is compatible with the locality constraint and relative to which the target state factorizes, then robust stabilization may be achieved by independently cooling each component. We further show that if the same condition holds for a scalable class of pure states, a continuous-time quasi-local Markov semigroup ensuring rapid mixing can be obtained. Somewhat surprisingly, we find that the commutativity of the canonical parent Hamiltonian one may associate to the target state does not directly relate to its finite-time stabilizability properties, although in all cases where we can guarantee robust stabilization, a (possibly noncanonical) commuting parent Hamiltonian may be found. Aside from graph states, quantum states amenable to finite-time robust stabilization include a class of universal resource states displaying two-dimensional symmetry-protected topological order, along with tensor network states obtained by generalizing a construction due to Bravyi and Vyalyi [Quantum Inf. Comput. 5, 187 (2005)]. Extensions to representative classes of mixed graph-product and thermal states are also discussed.

Figure 1

Neighborhood structure corresponding to two-body nearest-neighbor (NN) couplings in one dimension (1D), ${\mathcal{N}}_j\equiv \{j,j+1\}, j=1,...,N-1$. Note that, except for the two boundary neighborhoods, associated to $j=1$ and $N-1$, a three-body neighborhood structure ${\mathcal{N}}_j\equiv \{j-1,j,j+1\}$ corresponds instead to graph states on the line, as described in Example t18.

Figure 2

Example of a QLS but non-FTS state: the spin-$\frac{3}{2}$ AKLT state on a bipartite cubic graph. The pairs of nodes connected by an edge are virtual spin-$\frac{1}{2}$ particles in a singlet state. The dashed circles contain the systems which are projected into the spin-$\frac{3}{2}$ subspace. As verified numerically, this AKLT state violates the small Schmidt-span property since for each top-bottom pair of systems (i.e., each neighborhood), the Schmidt-span dimension ($=9$) exceeds half the Hilbert space dimension ($=\frac{16}{2}$).

Figure 3

FTS scheme for the $N=3$ spin-1 VBS state on the line [Eq. (15)], under NN constraints. In each numbered panel, each square represents one of the $D=27$ dimensions in the ${\mathbb{C}}^3\otimes {\mathbb{C}}^3\otimes {\mathbb{C}}^3$ state space, while the dots represent the probabilistic weight of each basis vector for the current state. The task is to move all the probabilistic weight from the initial flat distribution (completely mixed state) into the box in the upper left-hand corner, corresponding to the target state. The Schmidt span on the first two systems is two dimensional, leading to the six-dimensional extended Schmidt span ${\mathrm{\Sigma }}^0$, as represented by the first row of boxes. The remaining rows, labeled ${\mathrm{\Sigma }}^i$, are isometric copies of this subspace, leaving the three-dimensional remainder space $\mathcal{R}\otimes {\mathcal{H}}_{\overline{\mathcal{N}}}$. The dissipative map $\mathcal{W}$ acts only on the first two qutrits, cooling each ${\mathrm{\Sigma }}^i$ to ${\mathrm{\Sigma }}^0$. The unitaries ${\mathcal{U}}_1$ and ${\mathcal{U}}_2$ leave the target state invariant while preparing probabilistic weight to be cooled by $\mathcal{W}$. Since $|{\text{VBS}}_1^N\rangle$ satisfies the unitary generation property, each ${\mathcal{U}}_i$ can be decomposed into a finite sequence of neighborhood-acting invariance-satisfying maps.

Figure 4

Example of the virtual-subsystem decomposition used in constructing a state that cannot be obtained as a simple virtual product state yet is RFTS.

Figure 5

Example structure of a generalized BV state. Dashed circles denote physical particles, nodes correspond to virtual subsystems, and solid lines connect virtual subsystems that are entangled. The hatched semicircles indicate the subspaces ${\mathcal{H}}_2^0$ and ${\mathcal{H}}_5^0$ (${\mathcal{H}}_i^0\oplus {\tilde{\mathcal{H}}}_i={\mathcal{H}}_i$), where the respective reduced states do not have support. Solid curves delineate the four neighborhoods. An entangled state $|{\widehat{\psi }}_k\rangle$ is associated to each of the four groups of virtual particles that are contained in the same ${\mathcal{N}}_k$ of all those connected to them by solid lines. The resulting generalized BV state $|{\psi }_{\text{GBV}}\rangle =(V_1\otimes ...\otimes V_7)|\psi \rangle =|{\widehat{\psi }}_1\rangle \otimes |{\widehat{\psi }}_2\rangle \otimes |{\widehat{\psi }}_3\rangle \otimes |{\widehat{\psi }}_4\rangle$. Entanglement among the virtual subsystems is mapped into multipartite entanglement among the physical particles. Notwithstanding, the state is RFTS. In the RFTS scheme, the job of each neighborhood map ${\mathcal{E}}_k$ is to transfer probabilistic weight into ${\tilde{\mathcal{H}}}_i$ and then prepare the corresponding virtual factor $|{\widehat{\psi }}_k\rangle \in {\widehat{\mathcal{H}}}_k$, while acting trivially on the rest.

Figure 6

Illustrative example of a neighborhood structure on $N=5$ systems that does obey the matching-overlap property (left) versus one that does not (right). By including the dashed (gray) neighborhood, $\mathcal{N}$ would admit a nontrivial cycle, and lose its treelike structure.

Figure 7

Example of a RFTS state with a noncommuting canonical parent Hamiltonian. The three neighborhoods are enclosed by ovals and are most easily described by their respective complements. Letting $S\equiv \{1,...,9\}$, we define ${\mathcal{N}}_A\equiv S\setminus \{6,7\}, {\mathcal{N}}_B\equiv S\setminus \{1,9\}$, and ${\mathcal{N}}_C\equiv S\setminus \{3,4\}$. Note that the matching overlap condition is not obeyed.

Figure 8

Given a 1D lattice of nine systems subject to a next-NN QL constraint, a state which is RFTS can be stabilized with a depth-3 QL dissipative circuit by organizing the application of maps into layers as shown.

Figure 9

Kagome lattice with its unit cell, and neighborhood structure for the ccz state. In constructing such state, the system is initialized in ${|+\rangle }^{\otimes N}$ and a ccz gate is applied to each triangle of adjacent systems [cf. Eq. (18)].