It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study $(2+1)\mathrm{D}$ random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to $(2+1)\mathrm{D}$ percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent $z\ensuremath{\approx}1.46$, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study $(2+1)\mathrm{D}$ random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to $(2+1)\mathrm{D}$ percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent $z\ensuremath{\approx}1.46$, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.

Physical Review LettersDec 1, 2021107 citations

Topological order and entanglement dynamics in the measurement-only XZZX quantum code

Kai Klocke, Michael Buchhold

We examine the dynamics of a $(1+1)$ -dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

arXiv:2204.08489Physical review. B./Physical review. BSep 23, 202241 citations

Measurement-induced criticality in ( 2 + 1 ) -dimensional hybrid quantum circuits

Xhek Turkeshi, Rosario Fazio, Marcello Dalmonte

We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to $L=48$ for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.We investigate the dynamics of two-dimensional quantum spin systems under the combined effect of random unitary gates and local projective measurements. When considering steady states, a measurement-induced transition occurs between two distinct dynamical phases, one characterized by a volume-law scaling of entanglement entropy, the other by an area law. Employing stabilizer states and Clifford random unitary gates, we numerically investigate square lattices of linear dimension up to $L=48$ for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.

arXiv:2007.02970Physical review. B./Physical review. BJul 30, 2020177 citations

Phase transitions in ( 2 + 1 ) D subsystem-symmetric monitored quantum circuits

Cole Kelson-Packer, Akimasa Miyake

The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.

arXiv:2407.18340Physical review. B./Physical review. BFeb 4, 20251 citation

Tuning entanglement phases and topological memory in the measurement-only Kitaev model with single and multi-qubit checks

Tushya Kalpada, Aayush Vijayvargia, Ezra Day-Roberts +1 more

Quantum circuits provide an emerging controllable platform to realize novel dynamical non-equilibrium phases including topologically ordered states. The Kitaev model has become a cornerstone of quantum magnetism due to its quantum spin liquid ground state and rich phase diagram. The Kitaev model has also been treated in the monitored circuit setting, giving rise to topological area-law and critical-law entanglement entropy phases. In this article, we study the evolution of its phase diagram under the addition of new terms, motivated by their effects in the Kitaev model. We find that a single-qubit term, analogous to a magnetic field, leads to a trivial state in the high field limit, but with an additional intermediate volume-law phase. A three-qubit operator that commutes with the flux operators has the opposite effect: it stabilizes the critical-law phase against the short ranged area-law entanglement. We also employ a four-qubit plaquette commuting operator that simultaneously measures two opposite identical-type bonds on a plaquette. This generates a distinct volume-law phase and preserves the plaquette fluxes and associated topological order, yielding extensive entanglement while coexisting with the topological memory characteristic of the area-law phase. We quantitatively locate phase boundaries using stabilizer (Clifford) simulations together with tripartite mutual information and entanglement entropy measures. Our results highlight the rich phase diagram accessible from the measurement-only Kitaev model as well as suggesting rules relating the newly added operators to the phases they promote.Quantum circuits provide an emerging controllable platform to realize novel dynamical non-equilibrium phases including topologically ordered states. The Kitaev model has become a cornerstone of quantum magnetism due to its quantum spin liquid ground state and rich phase diagram. The Kitaev model has also been treated in the monitored circuit setting, giving rise to topological area-law and critical-law entanglement entropy phases. In this article, we study the evolution of its phase diagram under the addition of new terms, motivated by their effects in the Kitaev model. We find that a single-qubit term, analogous to a magnetic field, leads to a trivial state in the high field limit, but with an additional intermediate volume-law phase. A three-qubit operator that commutes with the flux operators has the opposite effect: it stabilizes the critical-law phase against the short ranged area-law entanglement. We also employ a four-qubit plaquette commuting operator that simultaneously measures two opposite identical-type bonds on a plaquette. This generates a distinct volume-law phase and preserves the plaquette fluxes and associated topological order, yielding extensive entanglement while coexisting with the topological memory characteristic of the area-law phase. We quantitatively locate phase boundaries using stabilizer (Clifford) simulations together with tripartite mutual information and entanglement entropy measures. Our results highlight the rich phase diagram accessible from the measurement-only Kitaev model as well as suggesting rules relating the newly added operators to the phases they promote.

arXiv:2511.20545Materials Today QuantumMar 11, 20260 citations

Monitored quantum dynamics and the Kitaev spin liquid

Ali Lavasani, Zhu-Xi Luo, Sagar Vijay

The authors study the quantum dynamics of a (2+1)D monitored random circuit involving local projective measurements, inspired by the physics of the Kitaev quantum spin liquid. A Majorana parton description of the dynamics reveals a close connection between the entanglement dynamics of quantum trajectories and the 3D classical fully packed loop model. The authors uncover a critical phase where quantum trajectories exhibit long-range entanglement with a logarithmic ( $L$ log $L$ ) violation of the area law, and a topologically ordered phase reminiscent of the toric code phase.The authors study the quantum dynamics of a (2+1)D monitored random circuit involving local projective measurements, inspired by the physics of the Kitaev quantum spin liquid. A Majorana parton description of the dynamics reveals a close connection between the entanglement dynamics of quantum trajectories and the 3D classical fully packed loop model. The authors uncover a critical phase where quantum trajectories exhibit long-range entanglement with a logarithmic ($L$ log $L$) violation of the area law, and a topologically ordered phase reminiscent of the toric code phase.

arXiv:2207.02877Physical review. B./Physical review. BSep 15, 202353 citations

It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study

(2+1)\mathrm{D}

random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to

(2+1)\mathrm{D}

percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent $z\ensuremath{\approx}1.46$, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.

We examine the dynamics of a

(1+1)

-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a ${D}_{2}={\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\ensuremath{\nu}=\frac{4}{3}$ and the dynamical critical exponent

z=1

are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of

L

qubits. It reveals a modified dynamical critical exponent ${z}^{*}\ensuremath{\ne}z$, which is observable up to times $t\ensuremath{\sim}{L}^{{z}^{*}}$ and is reminiscent of the relaxation of critical systems into a prethermal state.

L=48

for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.

L

log

L

) violation of the area law, and a topologically ordered phase reminiscent of the toric code phase.