Translational-invariant local dissipative preparation of injective matrix-product and projected entangled-pair states with provably fast convergence

Verstraete, Wolf and Cirac (Nature Physics 5, 633–636, 2009) showed that purely dissipative dynamics with time-independent, strictly local couplings can perform universal quantum computation and can prepare ground states of frustration-free Hamiltonians; however, their completely-positive map for an arbitrary injective matrix product state (MPS) — a position-dependent hierarchical scheme built for translationally invariant Hamiltonians — converges only in quasi-polynomial time $O(N^{\log_2 N})$, and they posed the design of more efficient translational-invariant maps for MPS and projected entangled pair states (PEPS). The open core of this problem is: construct, for every injective PEPS with a gapped parent Hamiltonian on a bounded-degree lattice in two or more dimensions, a strictly local translational-invariant dissipative process with that state as unique steady state and provably polylogarithmic mixing time. Non-injective and topologically ordered PEPS form an explicitly open further frontier.

Background

A quantum system coupled to a memoryless environment and driven by time-independent local couplings evolves under a Lindblad master equation $\dot{\rho}=\mathcal{L}(\rho)$, where the Lindbladian $\mathcal{L}$ is assembled from local jump operators acting on a few neighboring sites; in discrete time the same physics is a quantum channel — a completely positive, trace-preserving map (cp-map) — applied repeatedly. Dissipative state engineering (DSE) designs such local dynamics so that the unique steady state (the state annihilated by $\mathcal{L}$, or fixed by the channel) is a target many-body state. Because the dynamics forgets the initial condition, the preparation is robust to initialization errors, in contrast to unitary protocols; the price is the mixing time, the time to reach the steady state to precision $\varepsilon$ from an arbitrary input, which is controlled by the spectral gap of the Lindblidian — a system-size-independent gap gives mixing logarithmic in the system size $N$. Dissipative quantum computation (DQC) is the same idea with the output of a quantum computation encoded in the steady state.

The natural target states are tensor network states. A matrix product state (MPS) is a one-dimensional wavefunction whose coefficients are obtained by contracting a chain of $D\times D$ matrices, where $D$ is the bond dimension; projected entangled pair states (PEPS) generalize this to lattices in two and higher dimensions by placing entangled virtual pairs on edges and mapping them to physical particles by local tensors. A tensor network state is called injective if, after blocking finitely many sites, the linear map from virtual to physical indices is invertible; injective MPS and PEPS are then the unique ground states of a local, frustration-free parent Hamiltonian (a sum of local terms whose ground space minimizes every term separately), as established by Perez-Garcia, Verstraete, Cirac and Wolf (2008). For injective MPS the parent Hamiltonian is provably gapped; in two and higher dimensions injectivity alone does not imply a gap.

Verstraete, Wolf and Cirac (2009) proved that purely dissipative local dynamics is computationally universal, and that ground states of frustration-free Hamiltonians can be prepared by engineered dissipation, with convergence times $O(\log N)$ when Hamiltonian excitations can be annihilated locally and $O(N\log N)$ when they must be transported across the lattice — bounds that apply to commuting, frustration-free cases such as stabilizer states and certain PEPS. For an arbitrary injective MPS they constructed a cp-map with the MPS as unique fixed point, but its hierarchical design converges only in quasi-polynomial time $O(N^{\log_2 N})$. Their concluding section therefore asks, among five future directions, for translational-invariant cp-maps that prepare MPS — and, by extension, higher-dimensional PEPS — more efficiently, with provably polynomial or better convergence.

Several structural results frame what is possible. Kastoryano and Eisert (2013) proved that a local primitive Liouvillian with a system-size-independent spectral gap (or logarithmic-Sobolev constant) forces exponential decay of correlations in its steady state, so rapidly mixing local dynamics can only have short-range-correlated steady states — exactly the regime of injective tensor network states. Cubitt, Lucia, Michalakis and Perez-Garcia (2015) proved that local dissipative systems are stable: perturbing the local terms perturbs the steady state only mildly, giving the robustness relevant to fault-tolerant dissipative computation. On the unitary side, MPS can be generated sequentially site by site (Schön, Solano, Verstraete, Cirac, Wolf, 2005), and Malz, Styliaris, Wei and Cirac (2024) showed every injective MPS is preparable by a log-depth circuit, which is asymptotically optimal in the circuit model.

Most recently, Baruah, Styliaris, Cirac and Trivedi (arXiv:2605.25089, May 2026) resolved the one-dimensional case: for every injective MPS there is a geometrically local Lindbladian (and a discrete-time channel) whose unique steady state is that MPS and which mixes in time $O(\log(N/\varepsilon))$, with the local terms built from blocked tensors and uniform across sites, so translational invariance holds whenever the target is translational-invariant. The same rapid-mixing conclusion is proven for PEPS on bounded-degree graphs whose blocked tensors are $\delta$-isometric (highly injective: close enough to an isometry, quantified by $\delta$ below a threshold set by the lattice degree). For PEPS they emphasize that injectivity alone does not guarantee logarithmic-time preparation — some injective PEPS have gapless parent Hamiltonians, and even commuting frustration-free two-dimensional Hamiltonians with unique product ground states can encode hard classical SAT problems, obstructing efficient generic cooling — and they conjecture, supported by numerics, that a blocking argument should still give rapid mixing whenever the parent Hamiltonian is gapped. The general two-and-higher-dimensional case is thus the surviving open core, with non-injective and topologically ordered PEPS an explicitly open frontier beyond it.

Problem Statement

Given a prescribed injective tensor network state on $N$ sites — an injective matrix product state (MPS) on a chain, or an injective projected entangled pair state (PEPS) on a bounded-degree lattice in two or more dimensions — construct a strictly local, translational-invariant (site-uniform up to a fixed finite blocking length) dissipative process, i.e., a time-independent Lindbladian $\mathcal{L}=\sum_i \mathcal{L}_i$ with edge-local terms or a discrete completely positive trace-preserving map applied repeatedly, such that: (a) the target state is the unique steady state; and (b) starting from an arbitrary initial state, the dynamics reaches trace distance $\varepsilon$ from the target in a mixing time growing at most polylogarithmically in $N$ (a polynomial bound is a weaker partial result, not a complete resolution), with all constants depending only on the tensor data (bond dimension and injectivity condition number), not on $N$. The one-dimensional case is settled — Baruah, Styliaris, Cirac and Trivedi (2026) achieve $O(\log(N/\varepsilon))$ for every injective MPS — and $O(\log(N/\varepsilon))$ is likewise proven for PEPS whose blocked tensors are $\delta$-isometric. A complete answer must resolve the surviving higher-dimensional core for every injective PEPS with a gapped parent Hamiltonian: either by a construction as above or by a proof that no such process exists (the blocking conjecture of Baruah et al.). Extension to non-injective and topologically ordered PEPS, where mixing times growing with the system diameter are expected, is an explicitly open further frontier beyond the completion boundary.

References

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