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arXiv:2608.08821

State preparation via measurement and feedback: pushing relations, state structures, and non-invertible symmetries

Yabo Li, Aditi Mitra, Tsung-Cheng Lu

quant-phcond-mat.str-elSubmitted Aug 9, 2026• Announced Aug 11, 2026• 0 citations
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Abstract

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 $L_α L_{α'} = L_{α+α'} + L_{α-α'}$, together with their generalizations up to (not necessarily transversal) gates.

Comments
30+12 pages, 3 figures

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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⟩|A\rangle∣A⟩, we associate a state ∣B⟩|B\rangle∣B⟩ from which ∣A⟩|A\rangle∣A⟩ can be prepared using a 1-round MF circuit; in particular, ∣A⟩|A\rangle∣A⟩ is preparable from a product state using a circuit with 1 round of MF whenever ∣B⟩|B\rangle∣B⟩ 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 LαLα′=Lα+α′+Lα−α′L_α L_{α'} = L_{α+α'} + L_{α-α'}Lα​Lα′​=Lα+α′​+Lα−α′​, together with their generalizations up to (not necessarily transversal) gates.
Preparing long-range entangled states poses significant challenges for near-term quantum devices. It is known that measurement and feedback (MF) can aid this task by allowing the preparation of certain paradigmatic long-range entangled states with only constant circuit depth. Here, we systematically explore the structure of states that can be prepared using constant-depth local circuits and a single MF round. Using the framework of tensor networks, the preparability under MF translates to tensor symmetries. We detail the structure of matrix-product states (MPSs) and projected entangled-pair states (PEPSs) that can be prepared using MF, revealing the coexistence of Clifford-like properties and magic. In one dimension, we show that states with Abelian-symmetry-protected topological order are a restricted class of MF-preparable states. In two dimensions, we parametrize a subset of states with Abelian topological order that are MF preparable. Finally, we discuss the analogous implementation of operators via MF, providing a structural theorem that connects to the well-known Clifford teleportation. Published by the American Physical Society 2024
Two prevalent approaches for preparing long-range entangled quantum states are (i) linear-depth sequential unitary (SU) circuits, which apply local unitary gates sequentially, and (ii) constant-depth measurement-feedback (MF) circuits, which employ mid-circuit measurements and conditional feedback based on measurement outcomes. Here, we establish that a broad class of SU and MF circuits are dual to each other under a spacetime rotation. We investigate this spacetime duality in the preparation of various long-range entangled states, including GHZ states, topologically ordered states, and fractal symmetry-breaking states. As an illustration, applying a spacetime rotation to a linear-depth SU circuit that implements a non-invertible Kramers-Wannier duality, originally used to prepare a 1D GHZ state, yields a constant-depth MF circuit that implements a Z2\mathbb{Z}_2Z2​ symmetry gauging map, which equivalently prepares the GHZ state. Leveraging this duality, we further propose experimental protocols that require only a constant number of qubits to measure unconventional properties of 1D many-body states. These include (i) measurement of disorder operators, which diagnose the absence of spontaneous symmetry breaking, and (ii) postselection-free detection of measurement-induced long-range order, which emerges in certain symmetry-protected topological phases. We also show that measurement-induced long-range order provides a lower bound for strange correlators, which may be of independent interest.
Adaptive quantum circuits, which combine local unitary gates, midcircuit measurements, and feedforward operations, have recently emerged as a promising avenue for efficient state preparation, particularly on near-term quantum devices limited to shallow-depth circuits. Matrix product states (MPS) comprise a significant class of many-body entangled states, efficiently describing the ground states of one-dimensional gapped local Hamiltonians and finding applications in a number of recent quantum algorithms. Recently, it has been shown that the Affleck-Kennedy-Lieb-Tasaki state—a paradigmatic example of an MPS—can be exactly prepared with an adaptive quantum circuit of constant depth, an impossible feat with local unitary gates alone due to its nonzero correlation length [Smith , PRX Quantum 4, 020315 (2023)]. In this work, we broaden the scope of this approach and demonstrate that a diverse class of MPS can be exactly prepared using constant-depth adaptive quantum circuits, outperforming theoretically optimal preparation with unitary circuits. We show that this class includes short- and long-ranged entangled MPS, symmetry-protected topological (SPT) and symmetry-broken states, MPS with finite Abelian, non-Abelian, and continuous symmetries, resource states for MBQC, and families of states with tunable correlation length. Moreover, we illustrate the utility of our framework for designing constant-depth sampling protocols, such as for random MPS or for generating MPS in a particular SPT phase. We present sufficient conditions for particular MPS to be preparable in constant time, with global on-site symmetry playing a pivotal role. Altogether, this work demonstrates the immense promise of adaptive quantum circuits for efficiently preparing many-body entangled states and provides explicit algorithms that outperform known protocols to prepare an essential class of states. Published by the American Physical Society 2024
Sequential circuits provide an optimal linear-depth route to unitarily preparing long-range ordered quantum states, but circuit-level noise can be far more damaging than noise applied after preparation: local errors may be propagated by subsequent gates into nonlocal defects that destroy the target order. In this work, we present classes of unitary sequential circuits that prepare certain non-trivial orders in the presence of Pauli noise. Our constructions exploit the spatial structure of the syndromes of the target state, which in certain cases allows us to systematically suppress the propagation of errors using strictly local gates. The strategy can be applied both to symmetry-breaking order, for which we present a 3D example, and to topological order, which we showcase on a 4D version of the toric code. We also highlight how this stability against errors necessitates non-Clifford gates, and how measurement-feedback loops can be used to stabilize lower-dimensional states as well.
Although tensor network states constitute a broad range of exotic quantum states, their realization is challenging and often requires resources whose depth scales with system size. In this work, we explore criteria on the local tensors for enabling deterministic state preparation via a single round of measurements and on-site unitary feedback. We use these criteria to construct families of measurement-preparable states in one and two dimensions, tuning between distinct symmetry-breaking, symmetry-protected, and intrinsic topological phases of matter. For instance, in one dimension we chart out a three-parameter family of preparable states which interpolate between the AKLT, cluster, GHZ and other states of interest. Our protocol even allows one to engineer preparable quantum states with a range of desired correlation lengths and entanglement properties. In addition to such constructive approaches, we present diagnostics for verifying whether a given tensor network state is preparable using measurements. We conclude by charting out generalizations, such as considering multiple rounds of measurements, implementing matrix product operators, and using incomplete basis measurements.
The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback ("adaptive circuits") can evade such restrictions.We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs).The three classes are inspired by distinct physical insights, including tensor-network constructions, the multiscale entanglement renormalization ansatz, and parton constructions.A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and onedimensional CFT states and non-Abelian topological orders with both solvable and nonsolvable groups can be prepared in depth scaling logarithmically with system size.We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary Calderbank-Shor-Steane codes.Our work illustrates the practical and conceptual versatility of measurement for state preparation.