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

TL;DR

The paper addresses whether 2D chiral topological states can be prepared using strictly local, sequential quantum circuits. It develops a channel-state correspondence that links 1D local quantum channel dynamics, 2D isometric tensor-network states (isoTNS), and sequential circuits, and uses this framework to derive two no-go theorems. For Gaussian fermion systems, translationally invariant local channels cannot sustain the chiral single-particle entanglement spectrum required for 2D chiral states, so Gaussian isoTNS cannot host algebraically decaying correlations or chiral edges. For generic interacting circuits, causality forbids the tripartite entanglement structure characteristic of ungappable chiral edges, proving that such circuits cannot realize chiral topological states on finite cylinders. Together, these results impose fundamental limits on preparing 2D chiral states with strictly local, Markovian sequential dynamics and point to the necessity of nonlocal or non-Markovian mechanisms to realize chiral topology on lattices.

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.

Figures (6)

• Figure 1: Correspondence between (a) 1+1D brick-wall channel circuit, (b) 2D isometric tensor network states and (c) sequential unitary circuits.
• Figure 2: Illustration of Gaussian fermion matrix product state. We interpret each tensor (blue box) as an isometric map from its bottom virtual leg to its physical and top virtual leg. This isometric map corresponds to a quantum channel from the bottom virtual leg to the top virtual leg of each tensor. Red (yellow) lines represent dissipative (preserved) modes of the quantum channel. Only the dissipative modes (red lines) are coupled to the physical legs in the bulk. The boundary condition of the lowest tensor (green triangle) corresponds to the initial state of the quantum channel.
• Figure 3: Single-particle entanglement spectrum of the ground state of a p+ip superconductor and its tensor network approximations. (a) Entanglement spectrum of the ground state of the Hamiltonian in Eq. \ref{['Eq:pipHamiltonian']}. (b) Entanglement spectrum of a chiral tensor network state Dubail:2013pda. The entanglement spectrum is discontinuous at $k_x = 0$. (c) Entanglement spectrum of an isoTNS approximation of the ground state. The isoTNS entanglement spectrum mimics the chiral mode. However, the continuity of isoTNS entanglement spectrum requires extra anti-chiral modes, leading to a trivial entanglement spectrum.
• Figure 4: (a-c) virtual states along the space direction, a space-like direction, and the light direction in isoTNS. For any space-like direction, the evolution of the virtual state from one cut to the cut above is given by a finite-depth local channel, where elementary channels in different unit cells (yellow-shaded area) can be applied in parallel. For the light-like direction, the evolution of a virtual state to the cut above is given by a sequential channel. In panel (c), we use $l,r,t,b$ to label the left, right, top, and bottom virtual legs of each tensor. The yellow shaded channels act sequentially from the left to the right, and they compose into a channel from all '$b$' legs along the cut to all '$t$' legs along the cut above. (d) alt-isoTNS: isometric arrows along the red lines are reversed relative to panel (c). The resulting alternating-isoTNS corresponds to a depth-$L^2$ sequential circuit. The evolution of the virtual state from each cut to the cut above is given by a sequential channel, but the order in which local channels act alternates between even and odd steps.
• Figure 5: Sequential circuit on a ring, $L_y< L_x/6$ with the periodic boundary condition in the $x$ direction. The cylinder is divided into three blocks of size $L_x/3\times L_y$; the left half of $A$ and the right half of $C$ are omitted in the figure. The grey-colored gates above the red cuts form three clusters $U_A$, $U_B$, and $U_C$. The pink-colored gates below the red cuts prepare a triangle state from a product state or a sum of triangle states from a cat state.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof