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arXiv:2511.19612

No-go theorems for sequential preparation of two-dimensional chiral states via channel-state correspondence

Ruihua Fan, Yantao Wu, Yimu Bao, Zhehao Dai

quant-phcond-mat.quant-gascond-mat.str-elSubmitted Nov 24, 2025• Announced Nov 26, 2025• 0 citations
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Abstract

We investigate whether sequential unitary circuits can prepare two-dimensional chiral states, using a correspondence between sequentially prepared states, isometric tensor network states, and one-dimensional quantum channel circuits. We establish two no-go theorems, one for Gaussian fermion systems and one for generic interacting systems. In Gaussian fermion systems, the correspondence relates the defining features of chiral wave functions in their entanglement spectrum to the algebraic decaying correlations in the steady state of channel dynamics. We establish the no-go theorem by proving that local channel dynamics with translational invariance cannot support such correlations. As a direct implication, two-dimensional Gaussian fermion isometric tensor network states cannot support algebraically decaying correlations in all directions or represent a chiral state. In generic interacting systems, we establish a no-go theorem by showing that the state prepared by sequential circuits cannot host the tripartite entanglement of a chiral state due to the constraints from causality.

Comments
19 pages + 6 pages appendices, 6 figures

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We investigate whether sequential unitary circuits can prepare two-dimensional chiral states, using a correspondence between sequentially prepared states, isometric tensor network states, and one-dimensional quantum channel circuits. We establish two no-go theorems, one for Gaussian fermion systems and one for generic interacting systems. In Gaussian fermion systems, the correspondence relates the defining features of chiral wave functions in their entanglement spectrum to the algebraic decaying correlations in the steady state of channel dynamics. We establish the no-go theorem by proving that local channel dynamics with translational invariance cannot support such correlations. As a direct implication, two-dimensional Gaussian fermion isometric tensor network states cannot support algebraically decaying correlations in all directions or represent a chiral state. In generic interacting systems, we establish a no-go theorem by showing that the state prepared by sequential circuits cannot host the tripartite entanglement of a chiral state due to the constraints from causality.
Trial wave functions that can be represented by summing over locally coupled degrees of freedom are called tensor network states (TNSs); they have seemed difficult to construct for two-dimensional topological phases that possess protected gapless edge excitations. We show it can be done for chiral states of free fermions, using a Gaussian Grassmann integral, yielding ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ and Chern insulator states, in the sense that the fermionic excitations live in a topologically nontrivial bundle of the required type. We prove that any strictly short-range quadratic parent Hamiltonian for these states is gapless; the proof holds for a class of systems in any dimension of space. The proof also shows, quite generally, that sets of compactly supported Wannier-type functions do not exist for band structures in this class. We construct further examples of TNSs that are analogs of fractional (including non-Abelian) quantum Hall phases; it is not known whether parent Hamiltonians for these are also gapless.
Tensor network quantum states are powerful tools for strongly correlated systems, tailored to capture local correlations such as in ground states with entanglement area laws. When applying tensor network states to interacting fermionic systems, a proper choice of the basis or orbitals can reduce the bond dimension of tensors and provide physically relevant orbitals. We introduce such a change of basis with unitary gates obtained from compressing fermionic Gaussian states into quantum circuits corresponding to various tensor networks. These circuits can reduce the ground-state entanglement entropy and improve the performance of algorithms such as the density matrix renormalization group. We study the Anderson impurity model with one and two impurities to show the potential of the method for improving computational efficiency and interpreting impurity physics. Furthermore, fermionic Gaussian circuits can also suppress entanglement during the time evolution out of low-energy state. Last, we consider Gaussian multiscale entanglement renormalization ansatz (GMERA) circuits which compress fermionic Gaussian states hierarchically. The emergent coarse-grained physical models from these GMERA circuits are studied in terms of their entanglement properties and suitability for performing time evolution.
We develop a framework for Matrix Product Quantum Channels (MPQCs), a one-dimensional tensor-network description of completely positive, trace-preserving maps. We focus on translation-invariant channels, generated by a single repeated tensor, that admit a local purification. We show that their purifying isometry can always be implemented by a constant-depth brickwork quantum circuit, implying that such channels generate only short-range correlations when applied on product states. In contrast to the unitary setting, where one-dimensional quantum cellular automata (in one-to-one correspondence with matrix product unitaries) carry a nontrivial index, we prove that all locally purified channels belong to a single phase, that is, they can be continuously deformed into one another. We then extend the framework to a broader class of translation-invariant channels capable of generating long-range entanglement and show that these remain deterministically implementable in constant depth using two rounds of measurements and feedforward.
In previous work, it was shown that non-trivial gapped states can be generated from a product state using a sequential quantum circuit. Explicit circuit constructions were given for a variety of gapped states at exactly solvable fixed points. In this paper, we show that a similar generation procedure can be established for chiral topological states as well, despite the fact that they lack a zero-correlation-length exactly solvable form. Instead of sequentially applying local unitary gates, we sequentially evolve the Hamiltonian by changing local terms in one subregion and then the next. The Hamiltonian remains gapped throughout the process, giving rise to an adiabatic evolution mapping the ground state from a product state to a chiral topological state. We demonstrate such a sequential adiabatic generation process for free fermion chiral states like the Chern Insulator and the p+ip p + i p superconductor. Moreover, we show that coupling a quantum state to a discrete gauge group can be achieved through a sequential quantum circuit, thereby generating interacting chiral topological states from the free fermion ones.
Efficient preparation of many-body ground states is key to harnessing the power of quantum computers in studying quantum many-body systems. In this work, we propose a simple method to design exact linear-depth parameterized quantum circuits which prepare a family of ground states across topological quantum phase transitions in 2D. We achieve this by constructing ground states represented by isometric tensor networks (isoTNS), which form a subclass of tensor network states that are efficiently preparable. By continuously tuning a parameter in the wavefunction, the many-body ground state undergoes quantum phase transitions, exhibiting distinct 2D quantum phases. We illustrate this by constructing an isoTNS path with bond dimension D = 2 interpolating between distinct symmetry-enriched topological (SET) phases. At the transition point, the wavefunction is related to a gapless point in the classical six-vertex model. Furthermore, the critical wavefunction supports a power-law correlation along one spatial direction while remaining long-range ordered in the other spatial direction. We provide an explicit parametrized local quantum circuit for the path and show that the 2D isoTNS can also be efficiently simulated by a holographic quantum algorithm requiring only an 1D array of qubits. Published by the American Physical Society 2024
Isometric tensor networks in two dimensions enable efficient and accurate study of quantum many-body states, yet the effect of the isometric restriction on the represented quantum states is not fully understood. We address this question in two main contributions. First, we introduce an improved variant of isometric tensor network states (isoTNS) in two dimensions, where the isometric arrows on the columns of the network alternate between pointing upward and downward; hence the name . Second, we introduce a numerical tool—the (isoGfTNS)—that incorporates isometric constraints into the framework of Gaussian fermionic tensor network states. We demonstrate in numerous ways that alternating isoTNSs represent many-body ground states of two-dimensional quantum systems significantly better than the original isoTNSs. First, we show that the entanglement in an isoTNS is mediated along the isometric arrows and that alternating isoTNSs mediate entanglement more efficiently than conventional isoTNSs. Second, alternating isoTNSs correspond to a deeper, and thus more representative, sequential-circuit construction of depth O ( L x ⋅ L y ) compared to the original isoTNSs of depth O ( L x + L y ) . Third, using the Gaussian framework and gradient-based energy minimization, we provide numerical evidence of better bond-dimension scaling and variational energy of alternating isoGfTNSs for ground states of various free-fermionic models, including the Fermi surface, the band insulator, and the p x + i p y mean-field superconductor. Finally, benchmarking on the transverse-field Ising model, we demonstrate that an alternating isoTNS provides substantially improved performance and stability relative to the original isoTNS for the ground-state search algorithm in interacting systems.