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arxiv: 2605.25089 · v1 · pith:IO76TFK2new · submitted 2026-05-24 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.str-el

Dissipative preparation of injective tensor network states

Drishti Baruah , Georgios Styliaris , J. Ignacio Cirac , Rahul Trivedi This is my paper

Pith reviewed 2026-06-30 00:43 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.str-el
keywords dissipative state preparationtensor network statesmatrix product statesrapid mixingopen quantum systemsinjectivityquantum simulation
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The pith

Geometrically local dissipative processes prepare any injective tensor network state as their unique steady state in logarithmic time.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper constructs both continuous-time and discrete-time geometrically local dissipative processes whose unique steady state is any prescribed injective tensor network state. For all injective matrix product states the protocol reaches error ε after time scaling as O(log(N/ε)), an exponential improvement over earlier dissipative schemes. In two and higher dimensions the same logarithmic scaling holds when the tensors are highly injective, delivering polynomial speedups over unitary preparation on lattices and exponential speedups on general bounded-degree graphs. Numerical simulations indicate the protocol often succeeds even when the high-injectivity assumption is relaxed.

Core claim

The authors demonstrate that the injectivity property of the tensors permits construction of a geometrically local Lindbladian (or discrete channel) for which the target tensor network state is the unique fixed point; the resulting spectral gap of the generator guarantees that the system converges to within ε of the N-site target in O(log(N/ε)) time for every injective MPS and for every highly injective higher-dimensional tensor network state.

What carries the argument

A geometrically local dissipative generator (continuous Lindbladian or discrete-time map) engineered so that the injective tensor network state is its unique steady state, with the injectivity condition ensuring a sufficiently large spectral gap for rapid mixing.

If this is right

  • Every injective one-dimensional matrix product state reaches ε accuracy after O(log(N/ε)) evolution time.
  • Highly injective tensor network states in two and higher dimensions exhibit rapid mixing under the constructed processes, again in O(log(N/ε)) time.
  • The approach yields a polynomial speedup relative to known unitary protocols on lattices and an exponential speedup on general bounded-degree graphs.
  • Numerical evidence indicates that the dissipative protocol can still prepare the target state rapidly even when the tensors fall outside the strict high-injectivity regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local dissipation might remain effective for states that are only approximately injective or that arise as ground states of gapped Hamiltonians.
  • On noisy intermediate-scale quantum hardware the inherent robustness of dissipative preparation could reduce the overhead of active error correction during state preparation.
  • The construction suggests a systematic route for turning tensor-network injectivity into fast-mixing open-system dynamics on arbitrary graphs.
  • One could test whether relaxing geometric locality while preserving the steady-state condition further improves the scaling.

Load-bearing premise

The local tensors of the network must satisfy the injectivity (or high-injectivity) condition, which is required both for uniqueness of the steady state and for the logarithmic mixing-time bound.

What would settle it

An explicit injective tensor network state on N sites together with a concrete lower bound showing that every geometrically local dissipative process having that state as unique steady state requires mixing time super-logarithmic in N (or in 1/ε) would falsify the claimed scaling.

Figures

Figures reproduced from arXiv: 2605.25089 by Drishti Baruah, Georgios Styliaris, J. Ignacio Cirac, Rahul Trivedi.

Figure 1
Figure 1. Figure 1: FIG. 1: Projected entangled pair state (PEPS) on an [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Numerical study of the dissipative protocol beyond the high-injectivity assumption. (a) A schematic illus [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

The preparation of tensor network states is a fundamental prerequisite for a wide range of quantum simulation tasks. While many unitary protocols for preparing these states have been investigated, dissipative state preparation provides a powerful alternative since it can be robust to noise and initialization errors. In this paper, we construct both continuous-time and discrete-time geometrically local dissipative processes whose unique steady state is a given injective tensor network state. Our method prepares all injective matrix product states on $N$ sites to an error $\varepsilon$ in $O(\log (N/\varepsilon))$ time, yielding an exponential improvement over previously known dissipative preparation schemes. For two and higher-dimensional tensor network states, we prove that when the tensors of the state are \emph{highly injective}, the constructed dissipative processes are rapid-mixing i.e., they prepare a state $\varepsilon$-close to the $N$-site target state in $O( \log (N/\varepsilon))$ time. For these states, our approach provides a polynomial speedup over known unitary methods for states defined on lattices and an exponential speedup for states on general bounded-degree graphs. We corroborate our theoretical results with numerical studies that indicate that the dissipative protocol can rapidly prepares states outside the high-injectivity assumption.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs geometrically local dissipative processes (continuous-time and discrete-time) whose unique steady state is any given injective tensor network state. For injective matrix product states on N sites it proves preparation to error ε in O(log(N/ε)) time, an exponential improvement over prior dissipative schemes. For two- and higher-dimensional tensor networks whose tensors satisfy a high-injectivity condition, the constructed processes are rapid-mixing and achieve the same O(log(N/ε)) scaling; this yields a polynomial speedup relative to known unitary protocols on lattices and an exponential speedup on general bounded-degree graphs. Numerical experiments are presented to indicate that the protocol remains effective even when the high-injectivity assumption is relaxed.

Significance. If the proofs are correct, the result supplies a substantial improvement in the efficiency of dissipative preparation of tensor-network states, which are ubiquitous in quantum simulation and many-body physics. The logarithmic scaling under a well-defined injectivity condition, together with the explicit construction of both continuous- and discrete-time generators, is a clear technical advance. The numerical evidence of robustness beyond the strict assumption adds practical value.

minor comments (3)
  1. [Abstract] The abstract contains an awkward line break inside the sentence describing the higher-dimensional result; this should be corrected for readability.
  2. The precise quantitative comparison to previous dissipative preparation times (the claimed exponential improvement) should be stated explicitly, with citations, in the introduction or a dedicated comparison subsection.
  3. Notation for the high-injectivity condition and the associated spectral-gap lower bound should be introduced once and used consistently throughout the proofs and statements of theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation for minor revision. The report correctly identifies the exponential improvement in dissipative preparation time for 1D injective MPS and the rapid mixing under high injectivity in higher dimensions.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper explicitly constructs new continuous- and discrete-time geometrically local dissipative processes with the target injective TN state as unique steady state (conditioned on injectivity). Logarithmic mixing bounds are derived for the high-injectivity case via rapid-mixing analysis. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or renamed input; the O(log(N/ε)) claims are proven under stated assumptions rather than re-expressing prior results. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all technical details of the constructions and proofs are absent.

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Forward citations

Cited by 1 Pith paper

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  1. Static features from mixing in short- and long-range Lindbladians: Markov property and correlations

    quant-ph 2026-06 unverdicted novelty 6.0

    Rapid mixing and frustration-freeness in short- and long-range Lindbladians imply polynomial decay of MI and CMI in fixed points, and long-range non-commuting Gibbs states satisfy local Markov property at any temperature.

Reference graph

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