Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.03939

Disentangling Haldane Phase by Generalized Clifford Circuits

Minsoo Kim, Changhun Oh, Donghoon Kim

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Jul 4, 2026• Announced Jul 7, 2026• 0 citations
arXivHTMLPDF

Abstract

Disentangling transformations play a central role in the classical simulation of quantum many-body systems, yet their analytic structure and underlying mechanism remain largely unexplored. Here, we study the structure of the disentangler in the Haldane phase of spin-1 systems using generalized Clifford circuits. To this end, we extend the Clifford-circuit-augmented matrix product states (CAMPS)-based density-matrix renormalization group (DMRG) method to spin-1 systems. Within this framework, we find that the local disentanglers optimized for the Haldane phase implement the generalized Kramers--Wannier (KW) transformation, and we analytically verify its optimality for the Affleck--Kennedy--Lieb--Tasaki (AKLT) state. Beyond reducing entanglement, the KW transformation maps the Haldane phase to a phase with spontaneously broken $\mathbb{Z}_{2}$ symmetry. This mapping is distinct from the Kennedy--Tasaki transformation and provides a new unitary route from symmetry-protected topological order to symmetry breaking.

Related papers

Disentangling critical quantum spin chains with Clifford circuits

Chaohui Fan, Xiangjian Qian, Hua-Chen Zhang +3 more

Clifford circuits can be utilized to disentangle quantum states with polynomial cost, thanks to the Gottesman-Knill theorem. Based on this idea, the Clifford Circuits Augmented Matrix Product States (CAMPS) method, which is a seamless integration of Clifford circuits within the density-matrix renormalization group algorithm, was proposed recently and was shown to be able to reduce entanglement in various quantum systems. In this work, we further explore the power of the CAMPS method in critical spin chains described by conformal field theories (CFTs) in the scaling limit. We find that the optimized disentanglers correspond to duality transformations, which significantly reduce the entanglement entropy in the ground state. For the critical quantum Ising spin chain governed by the Ising CFT with self-duality, the Clifford circuits found by CAMPS coincide with the duality transformation, i.e., the Kramer-Wannier self-duality in the critical Ising chain. It reduces the entanglement entropy by mapping the free conformal boundary condition to the fixed one. In the more general case of the XXZ chain, the CAMPS gives rise to a duality transformation mapping the model to the quantum Ashkin-Teller spin chain. Our results highlight the potential of the framework as a versatile tool for uncovering hidden dualities and simplifying the entanglement structure of critical quantum systems.

arXiv:2411.12683•Physical review. B./Physical review. B•Feb 10, 2025•12 citations

Disentangling Kitaev Quantum Spin Liquid

Xiang Li, Xiangjian Qian, Mingpu Qin

In this work, we investigate the Kitaev honeycomb model employing the recently developed Clifford Circuits Augmented Matrix Product States (CAMPS) method. While the model in the gapped phase is known to reduce to the toric code model - whose ground state is entirely constructible from Clifford circuits - we demonstrate that the very different gapless quantum spin liquid (QSL) phase can also be significantly disentangled with Clifford circuits. Specifically, CAMPS simulations reveal that approximately two-thirds of the entanglement entropy in the isotropic point arises from Clifford-circuit contributions, enabling dramatically more efficient computations compared to conventional matrix product state (MPS) methods. Crucially, this finding implies that the Kitaev QSL state retains significant Clifford-simulatable structure, even in the gapless phase with non-abelian anyon excitations when time reversal symmetry is broken. This property not only enhances classical simulation efficiency significantly but also suggests substantial resource reduction for preparing such states on quantum devices. As an application, we leverage CAMPS to study the Kitaev-Heisenberg model and determine the most accurate phase boundary between the anti-ferromagnetic phase and the Kitaev QSL phase in the model. Our results highlight how Clifford circuits can effectively disentangle the intricate entanglement of Kitaev QSLs, opening avenues for efficiently simulating related and similar strongly correlated models.

cond-mat.str-elCross-listedarXiv:2511.20261•Nov 26, 2025•0 citations

Topological disentangler for the valence-bond-solid chain

Kouichi Okunishi

We discuss the topological disentangler for $S=1$ quantum spin chains in the Haldane phase. We first point out that the Kennedy-Tasaki (KT) nonlocal unitary transformation is the perfect disentangler for the Affleck-Kennedy-Lieb-Tasaki model. We then demonstrate that the KT transformation can be reconstructed as an assembly of pair disentanglers. Finally, we show that the KT transformation can be regarded as a topological disentangler, which selectively disentangles the twofold degeneracy in the entanglement spectrum of the $S=1$ Heisenberg chain.

arXiv:1011.3277•Physical Review B•Mar 21, 2011•16 citations

Kennedy-Tasaki transformation and noninvertible symmetry in lattice models beyond one dimension

Aswin Parayil Mana, Yabo Li, Hiroki Sukeno +1 more

We give an explicit operator representation (via a sequential circuit and projection to symmetry subspaces) of Kramers-Wannier duality transformation in higher-dimensional subsystem symmetric models, generalizing the construction in the 1D transverse-field Ising model. Using the Kramers-Wannier duality operator, we also construct the Kennedy-Tasaki transformation that maps subsystem symmetry-protected topological phases to spontaneous subsystem symmetry-breaking phases, where the symmetry group for the former is either ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ or ${\mathbb{Z}}_{2}$. This also generalizes the recently proposed picture of the one-dimensional Kennedy-Tasaki transformation as a composition of manipulations involving gauging and stacking symmetry-protected topological phases to higher dimensions.

arXiv:2402.09520•Physical review. B./Physical review. B•Jun 21, 2024•28 citations

Hidden symmetry-breaking picture of symmetry-protected topological order

Dominic V. Else, Stephen D. Bartlett, Andrew C. Doherty

We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including $\mathrm{SO}(2k+1)$ and $\mathrm{SU}(k)$.

arXiv:1304.0783•Physical Review B•Aug 15, 2013•63 citations

Quantum quenches of scar states in the Affleck-Kennedy-Lieb-Tasaki model via Clifford augmented tensor network simulation

Azar C. Nakhl, Ben Harper, Muhammad Usman +1 more

Clifford augmented methods have emerged as a powerful technique to simulate quantum circuits and many-body systems by exploiting the stabiliser structure found in said systems, whilst additionally enabling one to readily quantify the non-stabilizerness (i.e.\ magic) present. In this work, we utilise a qudit Clifford augmented simulation method to probe the presence of magic in the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, finding that its ground state and scar states are highly magical as quantified by their Stabiliser Rényi Entropy (SRE). Nevertheless, owing to its low entanglement structure, these states are still readily expressible as a Clifford augmented Matrix Product State with low bond-dimension and moreover may be time-evolved using the Clifford augmented time-dependent variational principle (TDVP) algorithm. We utilise Clifford augmented TDVP to study the string order dynamics in the scar states of the AKLT model after a quench within the Haldane phase to the anti-ferromagnetic Heisenberg Hamiltonian. We observe that there is a decay of string order following the quench, which is more pronounced with increasing bimagnon number. This finding highlights the sensitivity of string order in the AKLT scar states to symmetry-preserving quenches that break the underlying restricted spectrum generating algebra, even when the initial and final Hamiltonians belong to the same topological phase.

cond-mat.str-elCross-listedarXiv:2609.38848•Oct 1, 2026•0 citations
Disentangling transformations play a central role in the classical simulation of quantum many-body systems, yet their analytic structure and underlying mechanism remain largely unexplored. Here, we study the structure of the disentangler in the Haldane phase of spin-1 systems using generalized Clifford circuits. To this end, we extend the Clifford-circuit-augmented matrix product states (CAMPS)-based density-matrix renormalization group (DMRG) method to spin-1 systems. Within this framework, we find that the local disentanglers optimized for the Haldane phase implement the generalized Kramers--Wannier (KW) transformation, and we analytically verify its optimality for the Affleck--Kennedy--Lieb--Tasaki (AKLT) state. Beyond reducing entanglement, the KW transformation maps the Haldane phase to a phase with spontaneously broken Z2\mathbb{Z}_{2}Z2​ symmetry. This mapping is distinct from the Kennedy--Tasaki transformation and provides a new unitary route from symmetry-protected topological order to symmetry breaking.
Clifford circuits can be utilized to disentangle quantum states with polynomial cost, thanks to the Gottesman-Knill theorem. Based on this idea, the Clifford Circuits Augmented Matrix Product States (CAMPS) method, which is a seamless integration of Clifford circuits within the density-matrix renormalization group algorithm, was proposed recently and was shown to be able to reduce entanglement in various quantum systems. In this work, we further explore the power of the CAMPS method in critical spin chains described by conformal field theories (CFTs) in the scaling limit. We find that the optimized disentanglers correspond to duality transformations, which significantly reduce the entanglement entropy in the ground state. For the critical quantum Ising spin chain governed by the Ising CFT with self-duality, the Clifford circuits found by CAMPS coincide with the duality transformation, i.e., the Kramer-Wannier self-duality in the critical Ising chain. It reduces the entanglement entropy by mapping the free conformal boundary condition to the fixed one. In the more general case of the XXZ chain, the CAMPS gives rise to a duality transformation mapping the model to the quantum Ashkin-Teller spin chain. Our results highlight the potential of the framework as a versatile tool for uncovering hidden dualities and simplifying the entanglement structure of critical quantum systems.
In this work, we investigate the Kitaev honeycomb model employing the recently developed Clifford Circuits Augmented Matrix Product States (CAMPS) method. While the model in the gapped phase is known to reduce to the toric code model - whose ground state is entirely constructible from Clifford circuits - we demonstrate that the very different gapless quantum spin liquid (QSL) phase can also be significantly disentangled with Clifford circuits. Specifically, CAMPS simulations reveal that approximately two-thirds of the entanglement entropy in the isotropic point arises from Clifford-circuit contributions, enabling dramatically more efficient computations compared to conventional matrix product state (MPS) methods. Crucially, this finding implies that the Kitaev QSL state retains significant Clifford-simulatable structure, even in the gapless phase with non-abelian anyon excitations when time reversal symmetry is broken. This property not only enhances classical simulation efficiency significantly but also suggests substantial resource reduction for preparing such states on quantum devices. As an application, we leverage CAMPS to study the Kitaev-Heisenberg model and determine the most accurate phase boundary between the anti-ferromagnetic phase and the Kitaev QSL phase in the model. Our results highlight how Clifford circuits can effectively disentangle the intricate entanglement of Kitaev QSLs, opening avenues for efficiently simulating related and similar strongly correlated models.
We discuss the topological disentangler for S=1S=1S=1 quantum spin chains in the Haldane phase. We first point out that the Kennedy-Tasaki (KT) nonlocal unitary transformation is the perfect disentangler for the Affleck-Kennedy-Lieb-Tasaki model. We then demonstrate that the KT transformation can be reconstructed as an assembly of pair disentanglers. Finally, we show that the KT transformation can be regarded as a topological disentangler, which selectively disentangles the twofold degeneracy in the entanglement spectrum of the S=1S=1S=1 Heisenberg chain.
We give an explicit operator representation (via a sequential circuit and projection to symmetry subspaces) of Kramers-Wannier duality transformation in higher-dimensional subsystem symmetric models, generalizing the construction in the 1D transverse-field Ising model. Using the Kramers-Wannier duality operator, we also construct the Kennedy-Tasaki transformation that maps subsystem symmetry-protected topological phases to spontaneous subsystem symmetry-breaking phases, where the symmetry group for the former is either ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ or Z2{\mathbb{Z}}_{2}Z2​. This also generalizes the recently proposed picture of the one-dimensional Kennedy-Tasaki transformation as a composition of manipulations involving gauging and stacking symmetry-protected topological phases to higher dimensions.
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).
Clifford augmented methods have emerged as a powerful technique to simulate quantum circuits and many-body systems by exploiting the stabiliser structure found in said systems, whilst additionally enabling one to readily quantify the non-stabilizerness (i.e.\ magic) present. In this work, we utilise a qudit Clifford augmented simulation method to probe the presence of magic in the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, finding that its ground state and scar states are highly magical as quantified by their Stabiliser Rényi Entropy (SRE). Nevertheless, owing to its low entanglement structure, these states are still readily expressible as a Clifford augmented Matrix Product State with low bond-dimension and moreover may be time-evolved using the Clifford augmented time-dependent variational principle (TDVP) algorithm. We utilise Clifford augmented TDVP to study the string order dynamics in the scar states of the AKLT model after a quench within the Haldane phase to the anti-ferromagnetic Heisenberg Hamiltonian. We observe that there is a decay of string order following the quench, which is more pronounced with increasing bimagnon number. This finding highlights the sensitivity of string order in the AKLT scar states to symmetry-preserving quenches that break the underlying restricted spectrum generating algebra, even when the initial and final Hamiltonians belong to the same topological phase.