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arXiv:2307.02292 · Physical review. B./Physical review. B

Measurement-induced phase transitions in the toric code

Amir-Reza Negari, Subhayan Sahu, Timothy H. Hsieh

Published Mar 27, 2024• 18 citations
arXivHTMLPDF

Abstract

We show how distinct phases of matter can be generated by performing random single-qubit measurements on a subsystem of toric code. Using a parton construction, such measurements map to random Gaussian tensor networks, and in particular, random Pauli measurements map to a classical loop model in which watermelon correlators precisely determine measurement-induced entanglement. Measuring all but a 1d boundary of qubits realizes hybrid circuits involving unitary gates and projective measurements in $1+1$ dimensions. We find that varying the probabilities of different Pauli measurements can drive transitions in the unmeasured boundary between phases with different orders and entanglement scaling, corresponding to short- and long-loop phases in the classical model. Furthermore, by utilizing single-site boundary unitaries conditioned on the bulk measurement outcomes, we generate mixed-state ordered phases and transitions that can be experimentally diagnosed via linear observables. This demonstrates how parton constructions provide a natural framework for measurement-based quantum computing setups to produce and manipulate phases of matter.

DOI
10.1103/physrevb.109.125148

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We show how distinct phases of matter can be generated by performing random single-qubit measurements on a subsystem of toric code. Using a parton construction, such measurements map to random Gaussian tensor networks, and in particular, random Pauli measurements map to a classical loop model in which watermelon correlators precisely determine measurement-induced entanglement. Measuring all but a 1d boundary of qubits realizes hybrid circuits involving unitary gates and projective measurements in 1+11+11+1 dimensions. We find that varying the probabilities of different Pauli measurements can drive transitions in the unmeasured boundary between phases with different orders and entanglement scaling, corresponding to short- and long-loop phases in the classical model. Furthermore, by utilizing single-site boundary unitaries conditioned on the bulk measurement outcomes, we generate mixed-state ordered phases and transitions that can be experimentally diagnosed via linear observables. This demonstrates how parton constructions provide a natural framework for measurement-based quantum computing setups to produce and manipulate phases of matter.
We extensively study long-time dynamics and the fate of a topologically ordered state in the toric code model evolving through a projective measurement-only circuit. The circuit is composed of several measurement operators corresponding to each term of the toric code Hamiltonian with magnetic-field perturbations, which is a gauge-fixed version of a (2+12+12+1)-dimensional gauge Higgs model. We employ a cylinder geometry with distinct upper and lower boundaries to classify stationary states after long-time measurement dynamics. The appearing stationary states depend on measurement probabilities for each measurement operator. The Higgs, confined, and deconfined phases emerge in the time evolution by the circuit. We find that both Higgs and confined phases are separated from the deconfined phase by topological entanglement entropy. We numerically clarify that both Higgs and confined phases are characterized by a long-range order on the boundaries accompanying edge modes, and they shift with each other by a crossover reflecting properties in the bulk phase diagram.
Monitored random circuits, consisting of alternating layers of entangling two-qubit gates and projective single-qubit measurements applied to some fraction p of the qubits, have been a topic of recent interest. In particular, the resulting steady state exhibits a phase transition from highly correlated states with "volume-law" entanglement at p<p_{c} to localized states with "area-law" entanglement at p>p_{c}. It is hard to access this transition experimentally, as it cannot be seen at the ensemble level. Naively, to observe it one must repeat the experiment until the set of measurement results repeats itself, with likelihood that is exponentially small in the number of measurements. To overcome this issue, we present a hybrid quantum-classical algorithm which creates a matrix product state (MPS) based "unitary mirror" of the projected circuit. Polynomial-sized tensor networks can represent quantum states with area-law entanglement, and so the unitary mirror can well approximate the experimental state above p_{c} but fails exponentially below it. The breaking of this mirror can thus pinpoint the critical point. We outline the algorithm and how such results would be obtained. We present a bound on the maximum entanglement entropy of any given state that is well represented by an MPS, and from the bound suggest how the volume-law phase can be bounded. We consider whether the entanglement could similarly be bounded from below where the MPS fails. Finally, we present numerical results for small qubit numbers and for monitored circuits with random Clifford gates.
In quantum many-body spin systems, the interplay between the entangling effect of multiqubit Pauli measurements and the disentangling effect of single-qubit Pauli measurements may give rise to two competing effects. By introducing a randomized measurement pattern with such bases, a phase transition can be induced by altering the ratio between them. In this work, we numerically investigate a measurement-based model associated with the (2+1)DZ2(2+1)\mathrm{D}\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}(2+1)DZ2​ Fradkin-Shenker Hamiltonian that encompasses the deconfining, confining, and Higgs phases. We determine the phase diagram in our measurement-only model by employing entanglement measures. For the bulk topological order, we use the topological entanglement entropy. We also use the mutual information between separated boundary regions to diagnose the boundary phase transition associated with the Higgs or the bulk symmetry-protected topological (SPT) phase. We observe the structural similarity between our phase diagram and the one in the standard quantum Hamiltonian formulation of the Fradkin-Shenker model with the open rough boundary. First, a deconfining phase is detected by nonzero and constant topological entanglement entropy. Second, we find a (boundary) phase transition curve separating the Higgs=SPT phase from the rest. In certain limits, the topological phase transitions reside at the critical point of the formation of giant homological cycles in the bulk three-dimensional (3D) space-time lattice, as well as the bond percolation threshold of the boundary 2D space-time lattice when it is effectively decoupled from the bulk. Additionally, there are analogous mixed-phase properties at a certain region of the phase diagram, emerging from how we terminate the measurement-based procedure. Our findings pave an alternative pathway to study the physics of Higgs=SPT phases on quantum devices in the near future.
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 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 (LLL log LLL) violation of the area law, and a topologically ordered phase reminiscent of the toric code phase.
Quantum measurements performed on a subsystem of a quantum many-body state can generate entanglement for its remaining constituents. The whole system including the measurement record is described by a hybrid mixed state, which can exhibit exotic phase transitions and critical phenomena. We demonstrate that generic measurement-induced phase transitions (MIPTs) can be cast as decoherence-induced critical mixed states in one higher dimension, by constructing a projected entangled pair state (PEPS) prior to decoherence or measurement. In this context, a deeper conceptual understanding of such mixed-state criticality is called for, particularly with regard to algebraic symmetry as an advanced organizing principle for such entangled states of matter. Integrating these connections we investigate the role of self-dual symmetry -- a fundamental notion in theoretical physics -- in mixed states, showing that the decoherence of electric (e) and magnetic (m) vortices from the 2D bulk of the toric code, or equivalently, a 2D cluster state with symmetry-protected topological order, can leave a (1+1)D quantum critical mixed state protected by a weak Kramers-Wannier self-dual symmetry. The corresponding self-dual critical bulk is described by the N->1 limit of the 2D Non-linear Sigma Model in symmetry class D with target space SO(2N)/U(N) at ΘΘΘ-angle πππ, and represents a "measurement-version" of the Cho-Fisher network model subjected to Born-rule randomness...