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arXiv:2507.12523 · Physical review. B./Physical review. B

Spacetime duality between sequential and measurement-feedback circuits

Tsung-Cheng Lu, Sarang Gopalakrishnan, Yizhi You

Published Jul 7, 2026• 0 citations
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Abstract

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 $\mathbb{Z}_2$ 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.

DOI
10.1103/lh1b-g4w8

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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.
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
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.
The quantum Kramers-Wannier (KW) duality is a fundamental transformation mapping short-range entangled (SRE) states to long-range entangled (LRE) states. While spatially local unitary circuits require linear-in-system-size depth to implement this duality, the ultimate speed limit for purely unitary circuits equipped with nonlocal connectivity remains an open question. Here, we explicitly construct logarithmic depth, spatially nonlocal unitary circuits that realize the exact Z2\mathbb{Z}_2Z2​ KW dualities in both one and two spatial dimensions. We further generalize the construction to arbitrary Zn\mathbb{Z}_nZn​ KW dualities. Unlike algorithms tailored to prepare specific target states, our circuits implement complete duality maps. Within the symmetric (charge-neutral) sector, these dualities exactly transform arbitrary non-fixed-point SRE states into their corresponding LRE duals. Consequently, our results establish an efficient, purely coherent pathway for exploring phase transitions and topological dualities on modern quantum platforms.
Finite-depth quantum circuits preserve the long-range entanglement structure in quantum states and map between states within a gapped phase. To map between states of different gapped phases, we can use sequential quantum circuits, which apply unitary transformations to local patches, strips, or other subregions of a system in a sequential way. The sequential structure of the circuit, on the one hand, preserves entanglement area law and hence the gappedness of the quantum states. On the other hand, the circuit has generically a linear depth, hence, it is capable of changing the long-range correlation and entanglement of quantum states and the phase they belong to. In this paper, we systematically discuss the definition, basic properties, and prototypical examples of sequential quantum circuits that map product states to Greenberger-Horne-Zeilinger states, symmetry-protected topological states, intrinsic topological states, and fracton states. We discuss the physical interpretation of the power of the circuits through connection to condensation, Kramers-Wannier duality, and the notion of foliation for fracton phases.
The dynamics of entanglement in "hybrid" nonunitary circuits (for example, involving both unitary gates and quantum measurements) has recently become an object of intense study. A major hurdle toward experimentally realizing this physics is the need to apply postselection on random measurement outcomes in order to repeatedly prepare a given output state, resulting in an exponential overhead. We propose a method to sidestep this issue in a wide class of nonunitary circuits by taking advantage of spacetime duality. This method maps the purification dynamics of a mixed state under nonunitary evolution onto a particular correlation function in an associated unitary circuit. This translates to an operational protocol which could be straightforwardly implemented on a digital quantum simulator. We discuss the signatures of different entanglement phases, and demonstrate examples via numerical simulations. With minor modifications, the proposed protocol allows measurement of the purity of arbitrary subsystems, which could shed light on the properties of the quantum error correcting code formed by the mixed phase in this class of hybrid dynamics.