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Imagine a huge puzzle where each piece is a quantum particle. To complete the picture, you don't just fit the pieces together: you need to entangle them in a special way. Usually, this is done by carefully applying operations one after another, like assembling the puzzle strictly by the instructions, row by row. But there's a faster way: occasionally glance at the already-assembled section, and if something's off, immediately adjust the neighboring pieces. In the quantum world, this is called measurement-based feedback.

The authors of the new work figured out how to systematically build such 'fast' circuits for one-dimensional chains of particles. They introduced the concept of pushed defects — something like virtual threads that connect puzzle pieces. If you can pull such a thread through a node, slightly rotating the neighboring pieces, then the state is assembled in a single round of measurements. It turns out that the whole trick lies in which threads can be pushed and how they interact with each other — much like a game where you need to rearrange strings without tangling them.

Interestingly, these pushing rules are closely related to irreversible symmetries — special transformations that cannot simply be undone. For example, if the threads obey a fusion rule like that of the cosine symmetry, the state can be obtained from a simple product using duality operators. This perspective turns quantum state preparation into a kind of algebraic puzzle.

In the end, we get not just a recipe for a specific state, but a whole classification: each class corresponds to its own circuit and its own defect algebra. This is like a puzzle catalog, where for each picture it tells you how many times you need to peek and adjust to assemble it fastest. Such a connection between quantum computing and abstract symmetries could simplify the creation of complex entangled states, a resource for future quantum computers.

Original abstract

as written by the authors arXiv:2608.08821v1

Quantum circuits with measurements and unitary feedback (MF) can prepare long-range entangled states in constant depth, but a systematic construction of the MF preparation circuit for a given target state remains underexplored. We develop such a scheme for one-dimensional states, based on the notion of pushable defects: virtual-bond operators of a matrix product state that can be pushed through the tensor at the price of a physical feedback unitary. We show that the set of pushable defects, together with their pushing relations classifies finite-depth MF-preparable states and dictates their preparation circuits. To each class of the target state $$|A\rangle$$, we associate a state $$|B\rangle$$ from which $$|A\rangle$$ can be prepared using a 1-round MF circuit; in particular, $$|A\rangle$$ is preparable from a product state using a circuit with 1 round of MF whenever $$|B\rangle$$ is preparable by a finite-depth local unitary (FDLU) circuit. For a general target state, the scheme is obtained by iterating this procedure until the associated state is FDLU-preparable. For open-boundary matrix product states, the scheme is complete: it constructs a preparation circuit whenever finite-depth MF preparation with left-conditioned feedback corrections is possible. Pushable defects and pushing relations thus emerge as a unifying principle for quantum state preparation via measurements and feedback. This characterization further reveals an intrinsic connection between MF circuits and non-invertible symmetries: states with certain classes of pushing relations are related to a product state by Tambara-Yamagami duality operators, or by continuous cosine symmetry operators with fusion rules $$L_{\alpha} L_{\alpha'} = L_{\alpha+\alpha'} + L_{\alpha-\alpha'}$$, together with their generalizations up to (not necessarily transversal) gates.