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We study the entanglement structure of the Greenberger-Horne-Zeilinger (GHZ) state as it thermalizes under a strongly-symmetric quantum channel describing the Metropolis-Hastings dynamics for the $d$-dimensional classical Ising model at inverse temperature $\beta$. This channel outputs the classical Gibbs state when acting on a product state in the computational basis. When applying this channel to a GHZ state in spatial dimension $d>1$, the resulting mixed state changes character at the Ising phase transition temperature from being long-range entangled to short-range-entangled as temperature increases. Nevertheless, we show that the topological entanglement negativity of a large region is insensitive to this transition and takes the same value as that of the pure GHZ state at any finite temperature $\beta>0$. We establish this result by devising a local operations and classical communication (LOCC) ``decoder" that provides matching lower and upper bounds on the negativity in the thermodynamic limit which may be of independent interest. This perspective connects the negativity to an error-correction problem on the $(d-1)$-dimensional bipartitioning surface and explains the persistent negativity in certain correlated noise models found in previous studies. Numerical results confirm our analysis., Comment: 7+5 pages, 5 figures

Certain pure-state symmetry-protected topological orders (SPT) can be used as a resource for transmitting quantum information. Here, we investigate the ability to transmit quantum information using decohered SPT states, and relate this property to the "strange correlation functions" which diagnose quantum many-body orders in these mixed-states. This perspective leads to the identification of a class of quantum channels -- termed symmetry-decoupling channels -- which do not necessarily preserve any weak or strong symmetries of the SPT state, but nevertheless protect quantum many-body order in the decohered mixed-state. We quantify the ability to transmit quantum information in decohered SPT states through the coherent quantum information, whose behavior is generally related to a decoding problem, whereby local measurements in the system are used to attempt to "learn" the symmetry charge of the SPT state before decoherence., Comment: 27 pages, 8 figures

The codespace of a quantum error-correcting code can often be identified with the degenerate ground-space within a gapped phase of quantum matter. We argue that the stability of such a phase is directly related to a set of coherent error processes against which this quantum error-correcting code (QECC) is robust: such a quantum code can recover from adiabatic noise channels, corresponding to random adiabatic drift of code states through the phase, with asymptotically perfect fidelity in the thermodynamic limit, as long as this adiabatic evolution keeps states sufficiently "close" to the initial ground-space. We further argue that when specific decoders -- such as minimum-weight perfect matching -- are applied to recover this information, an error-correcting threshold is generically encountered within the gapped phase. In cases where the adiabatic evolution is known, we explicitly show examples in which quantum information can be recovered by using stabilizer measurements and Pauli feedback, even up to a phase boundary, though the resulting decoding transitions are in different universality classes from the optimal decoding transitions in the presence of incoherent Pauli noise. This provides examples where non-local, coherent noise effectively decoheres in the presence of syndrome measurements in a stabilizer QECC., Comment: 17+14 pages, 12 figures

Quantum point contacts (QPCs) are an essential component in mesoscopic devices. Here, we study the transmission of quantum Hall edge modes through a gate-defined QPC in monolayer graphene. We observe resonant tunneling peaks and a nonlinear conductance pattern characteristic of Coulomb-blockaded localized states. The in-plane electric polarizability reveals the states are localized at a classically-unstable electrostatic saddle point. We explain this unexpected finding within a self-consistent Thomas-Fermi model, finding that localization of a zero-dimensional state at the saddle point is favored whenever the applied confinement potential is sufficiently soft compared to the Coulomb energy. Our results provide a direct demonstration of Coulomb-driven reconstruction at the boundary of a quantum Hall system., Comment: 15 pages, 10 figures. arXiv admin note: text overlap with arXiv:2204.10296

The Pancharatnam phase is a generalization of the Berry phase that applies to discrete sequences of quantum states. Here, we show that the Pancharatnam phase is a natural invariant for a wide class of quantum many-body dynamics involving measurements. We specifically investigate how a non-trivial Pancharatnam phase arises in the trajectories of Floquet quantum error-correcting codes and show that this phase can be extracted in a "computationally-assisted" interferometry protocol, involving additional post-processing based on the measurement record that defines a given quantum many-body trajectory. This Pancharatnam phase can also be directly related to the Berry phase accrued by continuous unitary evolution within a gapped phase. For the $\mathbb Z_2$ Floquet code of Hastings and Haah, we show that the associated family of unitary evolutions is the radical chiral Floquet phase. We demonstrate this correspondence explicitly by studying an exactly-solvable model of interacting spins., Comment: 5+4 pages, 2+1 figures

The linear cross-entropy (LXE) has been recently proposed as a scalable probe of the measurement-driven phase transition between volume- and area-law-entangled phases of pure-state trajectories in certain monitored quantum circuits. Here, we demonstrate that the LXE can distinguish distinct area-law-entangled phases of monitored circuits with symmetries, and extract universal behavior at the critical points separating these phases. We focus on (1+1)-dimensional monitored circuits with an on-site $\mathbb{Z}_{2}$ symmetry. For an appropriate choice of initial states, the LXE distinguishes the area-law-entangled spin glass and paramagnetic phases of the monitored trajectories. At the critical point, described by two-dimensional percolation, the LXE exhibits universal behavior which depends sensitively on boundary conditions, and the choice of initial states. With open boundary conditions, we show that the LXE relates to crossing probabilities in critical percolation, and is thus given by a known universal function of the aspect ratio of the dynamics, which quantitatively agrees with numerical studies of the LXE at criticality. The LXE probes correlations of other operators in percolation with periodic boundary conditions. We show that the LXE is sensitive to the richer phase diagram of the circuit model in the presence of symmmetric unitary gates. Lastly, we consider the effect of noise during the circuit evolution, and propose potential solutions to counter it., Comment: 12+6 pages, 16 figures. V2: References added

Adaptive quantum circuits, in which unitary operations, measurements, and feedback are used to steer quantum many-body systems, provide an exciting opportunity to generate new dynamical steady states. We introduce an adaptive quantum dynamics with continuous symmetry where unitary operations, measurements, and local unitary feedback are used to drive ordering. In this setting, we find a pure steady state hosting symmetry-breaking order, which is the ground state of a gapless, local Hamiltonian. We explore the dynamical properties of the approach to this steady state. We find that this steady-state order is fragile to perturbations, even those that respect the continuous symmetry., Comment: 17 pages, 10 figures

We investigate phase transitions in the encoding of quantum information in a quantum many-body system due to the competing effects of unitary scrambling and boundary dissipation. Specifically, we study the fate of quantum information in a one-dimensional qudit chain, subject to local unitary quantum circuit evolution in the presence of depolarizating noise at the boundary. If the qudit chain initially contains a finite amount of locally-accessible quantum information, unitary evolution in the presence of boundary dissipation allows this information to remain partially protected when the dissipation is sufficiently weak, and up to time-scales growing linearly in system size $L$. In contrast, for strong enough dissipation, this information is completely lost to the dissipative environment. We analytically investigate this ``quantum coding transition" by considering dynamics involving Haar-random, local unitary gates, and confirm our predictions in numerical simulations of Clifford quantum circuits. We demonstrate that scrambling the quantum information in the qudit chain with a unitary circuit of depth $ \mathcal{O}(\log L)$ before the onset of dissipation can perfectly protect the information until late times. The nature of the coding transition changes when the dynamics extend for times much longer than $L$. We further show that at weak dissipation, it is possible to code at a finite rate, i.e. a fraction of the many-body Hilbert space of the qudit chain can be used to encode quantum information., Comment: 21 pages, 14 figures

We propose a general framework for using local measurements, local unitaries, and non-local classical communication to construct quantum channels which can efficiently prepare mixed states with long-range quantum order or quantum criticality. As an illustration, symmetry-protected topological (SPT) phases can be universally converted into mixed-states with long-range entanglement, which can undergo phase transitions with quantum critical correlations of local operators and a logarithmic scaling of the entanglement negativity, despite coexisting with volume-law entropy. Within the same framework, we present two applications using fermion occupation number measurement to convert (i) spinful free fermions in one dimension into a quantum-critical mixed state with enhanced algebraic correlations between spins and (ii) Chern insulators into a mixed state with critical quantum correlations in the bulk. The latter is an example where mixed-state quantum criticality can emerge from a gapped state of matter in constant depth using local quantum operations and non-local classical communication., Comment: 25 pages, 11 figures; updated to the published version

Inspired by the coupled-layer construction of the X-Cube model, we introduce the X-Cube Floquet code, a dynamical quantum error-correcting code where the number of encoded logical qubits grows with system size. The X-Cube Floquet code is defined on a three-dimensional lattice, built from intersecting two-dimensional layers in the $xy$, $yz$, and $xz$ directions, and consists of a periodic sequence of two-qubit measurements which couple the layers together. Within a single Floquet cycle, the codespace switches between that of the X-Cube fracton order and layers of entangled, two-dimensional toric codes. The encoded logical qubits' dynamics are analyzed, and we argue that the new code has a non-zero error threshold. We provide a new Hamiltonian realization of the X-Cube model and, more generally, explore the phase diagram related to the sequence of measurements that define the X-Cube Floquet code., Comment: Main Text (6 pages, 5 figures), Appendices (4 pages, 5 figures)

Quantum circuits -- built from local unitary gates and local measurements -- are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far-from-equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as new dynamical phases in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics., Comment: Review article for Annual Review of Condensed Matter Physics; comments welcome

Quantum circuit dynamics with local projective measurements can realize a rich spectrum of entangled states of quantum matter. Motivated by the physics of the Kitaev quantum spin liquid [1], we study quantum circuit dynamics in (2+1)-dimensions involving local projective measurements, in which the monitored trajectories realize (i) a phase with topological quantum order or (ii) a "critical" phase with a logarithmic violation of area-law-scaling of the entanglement entropy along with long range tripartite entanglement. A Majorana parton description of these dynamics, which provides an out-of-equilibrium generalization of the parton description of the Kitaev honeycomb model, permits an analytic understanding of the universal properties of these two phases, including the entanglement properties of the steady-state, the dynamics of the system on the approach to equilibrium, and the phase transition between these states. In the topologically-ordered phase, two logical qubits can be encoded in an initial state and protected for a time which scales exponentially in the linear dimension of the system, while no robust encoding of quantum information persists in the critical phase. Extensive numerical simulations of these monitored dynamics confirm our analytic predictions., Comment: Main Text (18 pages, 20 figures), Appendices (11 pages, 17 figures)

We study real-time local correlators $\langle\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\rangle$ in chaotic quantum many-body systems. These correlators show universal structure at late times, determined by the dominant operator-space Feynman trajectories for the evolving operator $\mathcal{O}(\mathbf{x},t)$. The relevant trajectories involve the operator contracting to a point at both the initial and final time and so are structurally different from those dominating the out-of-time-order correlator. In the absence of conservation laws, correlations decay exponentially: $\langle\mathcal{O}(\mathbf{x},t)\mathcal{O}(0,0)\rangle\sim\exp(-s_\mathrm{eq} r(\mathbf{v}) t)$, where $\mathbf{v}= \mathbf{x}/ t$ defines a spacetime ray, and $r(\mathbf{v})$ is an associated decay rate. We express $r(\mathbf{v})$ in terms of cost functions for various spacetime structures. In 1+1D, operator histories can show a phase transition at a critical ray velocity $v_c$, where $r(\mathbf{v})$ is nonanalytic. At low $v$, the dominant Feynman histories are "fat": the operator grows to a size of order $t^\alpha\gg 1$ before contracting to a point again. At high $v$ the trajectories are "thin": the operator always remains of order-one size. In a Haar-random unitary circuit, this transition maps to a simple binding transition for a pair of random walks (the two spatial boundaries of the operator). In higher dimensions, thin trajectories always dominate. We discuss ways to extract the butterfly velocity $v_B$ from the time-ordered correlator, rather than the OTOC. Correlators in the random circuit may alternatively be computed with an effective Ising-like model: a special feature of the Ising weights for the Haar brickwork circuit gives $v_c=v_B$. This work addresses lattice models, but also suggests the possibility of morphological phase transitions for real-time Feynman diagrams in quantum field theories., Comment: 29 pages, 15 figures

In an all-van der Waals heterostructure, the active layer, gate dielectrics and gate electrodes are assembled from two-dimensional crystals that have a low density of atomic defects. This design allows two-dimensional electron systems with very low disorder to be created, particularly in heterostructures where the active layer also has intrinsically low disorder, such as crystalline graphene layers or metal dichalcogenide heterobilayers. A key missing ingredient has been nanoscale electrostatic control, with existing methods for fabricated local gates typically introducing unwanted contamination. Here we describe a resist-free local anodic oxidation process for patterning sub 100nm features in graphite gates, and their subsequent integration into an all-van der Waals heterostructure. We define a quantum point contact in the fractional quantum Hall regime as a benchmark device and observe signatures of chiral Luttinger liquid behaviour, indicating an absence of extrinsic scattering centres in the vicinity of the point contact. In the integer quantum Hall regime, we demonstrate in situ control of the edge confinement potential, a key requirement for the precision control of chiral edge states. This technique may enable the fabrication of devices capable of single anyon control and coherent edge-state interferometry in the fractional quantum Hall regime., Comment: Main Text: 11 pages, 4 figures | Extended Data: 4 figures | Supplementary Info: 6 pages, 4 figures; updated manuscript post review

Topologically-ordered phases of matter at non-zero temperature are conjectured to exhibit universal patterns of long-range entanglement which may be detected by a mixed-state entanglement measure known as entanglement negativity. We show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order which is localized on the entanglement bipartition. This connection leads to an understanding of ($i$) universal contributions to the entanglement negativity which diagnose finite-temperature topological order, and ($ii$) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries -- which arise, for example, in the description of the entanglement negativity for $\mathbb{Z}_{2}$ topological order in $d=4$ spatial dimensions -- remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at non-zero temperature for certain topological orders., Comment: 18 pages, 3 figures

We introduce a model of interacting bosons exhibiting an infinite collection of fractal symmetries -- termed "Pascal's triangle symmetries" -- which provides a natural $U(1)$ generalization of a spin-(1/2) system with Sierpinski triangle fractal symmetries. The Pascal's triangle symmetry gives rise to exact degeneracies, as well as a manifold of low-energy states which are absent in the Sierpinski triangle model. Breaking the $U(1)$ symmetry of this model to $Z_p$, with prime integer $p$, yields a lattice model with a unique fractal symmetry which is generated by an operator supported on a fractal subsystem with Hausdorff dimension $d_H = \ln (p(p+1)/2)/\ln p$. The Hausdorff dimension of the fractal can be probed through correlation functions at finite temperature. The phase diagram of these models at zero temperature in the presence of quantum fluctuations, as well as the potential physical construction of the $U(1)$ model are discussed., Comment: 10 pages, 5 figure

We introduce lattice gauge theories which describe three-dimensional, gapped quantum phases exhibiting the phenomenology of both conventional three-dimensional topological orders and fracton orders, starting from a finite group $G$, a choice of an Abelian normal subgroup $N$, and a choice of foliation structure. These hybrid fracton orders -- examples of which were introduced in arXiv:2102.09555 -- can also host immobile, point-like excitations that are non-Abelian, and therefore give rise to a protected degeneracy. We construct solvable lattice models for these orders which interpolate between a conventional, three-dimensional $G$ gauge theory and a pure fracton order, by varying the choice of normal subgroup $N$. We demonstrate that certain universal data of the topological excitations and their mobilities are directly related to the choice of $G$ and $N$, and also present complementary perspectives on these orders: certain orders may be obtained by gauging a global symmetry which enriches a particular fracton order, by either fractionalizing on or permuting the excitations with restricted mobility, while certain hybrid orders can be obtained by condensing excitations in a stack of initially decoupled, two-dimensional topological orders., Comment: 17+11 pages, 4 figures, 7 tables. Added discussion on planons supported in multiple layers. Published version

Monitored quantum dynamics reveal quantum state trajectories which exhibit a rich phenomenology of entanglement structures, including a transition from a weakly-monitored volume law entangled phase to a strongly-monitored area law phase. For one-dimensional hybrid circuits with both random unitary dynamics and interspersed measurements, we combine analytic mappings to an effective statistical mechanics model with extensive numerical simulations on hybrid Clifford circuits to demonstrate that the universal entanglement properties of the volume law phase can be quantitatively described by a fluctuating entanglement domain wall that is equivalent to a "directed polymer in a random environment" (DPRE). This relationship improves upon a qualitative "mean-field" statistical mechanics of the volume-law-entangled phase [1, 2]. For the Clifford circuit in various geometries, we obtain agreement between the subleading entanglement entropies and error correcting properties of the volume-law phase (which quantify its stability to projective measurements) with predictions of the DPRE. We further demonstrate that depolarizing noise in the hybrid dynamics near the final circuit time can drive a continuous phase transition to a non-error correcting volume law phase that is not immune to the disentangling action of projective measurements. We observe this transition in hybrid Clifford dynamics, and obtain quantitative agreement with critical exponents for a "pinning" phase transition of the DPRE in the presence of an attractive interface., Comment: 22 pages, 18 figures

We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) point-like topological excitations with restricted mobility, with non-trivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore, they serve as parent orders for both long-range-entangled quantum liquid and non-liquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional $\mathbb Z_2$ topological order with (i) the X-Cube fracton order, or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry $G$ and subsystem symmetries of a normal subgroup $N$ along lower-dimensional sub-regions. A further generalization of this construction to non-Abelian gauge groups is presented in arXiv:2106.03842., Comment: 19+10 pages, 17 figures, 5 tables. Published version

Classical reversible cellular automata (CAs), which describe the discrete-time dynamics of classical degrees of freedom in a finite state-space, can exhibit exact, nonthermal quantum eigenstates despite being classically chaotic. We show that every classical CA defines a family of generically non-integrable, periodically-driven (Floquet) quantum dynamics with exact, nonthermal eigenstates. These Floquet dynamics are nonergodic in the sense that certain product states on a periodic classical orbit fail to thermalize, while generic initial states thermalize as expected in a quantum chaotic system. We demonstrate that some signatures of these effects can be probed in quantum simulators based on Rydberg atoms in the blockade regime. These results establish classical CAs as parent models for a class of quantum chaotic systems with rare nonthermal eigenstates., Comment: 5+7 pages, 3+5 figures

We identify a phase transition between two kinds of volume-law entangled phases in non-local but few-body unitary dynamics with local projective measurements. In one phase, a finite fraction of the system belongs to a fully-entangled state, one for which no subsystem is in a pure state, while in the second phase, the steady-state is a product state over extensively many, finite subsystems. We study this "separability" transition in a family of solvable models in which we analytically determine the transition point, the evolution of certain entanglement properties of interest, and relate this to a mean-field percolation transition. Since the entanglement entropy density does not distinguish these phases, we introduce the entangling power - which measures whether local measurements outside of two finite subsystems can boost their mutual information - as an order parameter, after considering its behavior in tensor network states, and numerically studying its behavior in a model of Clifford dynamics with measurements. We argue that in our models, the separability transition coincides with a transition in the computational "hardness" of classically determining the output probability distribution for the steady-state in a certain basis of product states. A prediction for this distribution, which is accurate in the separable phase, and should deviate from the true distribution in the fully-entangled phase, provides a possible benchmarking task for quantum computers., Comment: 14 pages, 8 figures

Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=\kappa L_A+\frac{3}{2}\log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I(\{x\}:\bar{A})\propto x^{-3/2}$ with distance $x$ from the region's boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$: $p_{c}\log[(d^{2}-1)({p_{c}^{-1}-1})]\le \log[(1-p_{c})d]$. The bound is saturated at $p_c(d\rightarrow\infty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) \le 0.1893$., Comment: 26 pages, 10 figures

We propose a set of constraints on the ground-state wavefunctions of fracton phases, which provide a possible generalization of the string-net equations used to characterize topological orders in two spatial dimensions. Our constraint equations arise by exploiting a duality between certain fracton orders and quantum phases with "subsystem" symmetries, which are defined as global symmetries on lower-dimensional manifolds, and then studying the distinct ways in which the defects of a subsystem symmetry group can be consistently condensed to produce a gapped, symmetric state. We numerically solve these constraint equations in certain tractable cases to obtain the following results: in $d=3$ spatial dimensions, the solutions to these equations yield gapped fracton phases that are distinct as conventional quantum phases, along with their dual subsystem symmetry-protected topological (SSPT) states. For an appropriate choice of subsystem symmetry group, we recover known fracton phases such as Haah's code, along with new, symmetry-enriched versions of these phases, such as non-stabilizer fracton models which are distinct from both the X-cube model and the checkerboard model in the presence of global time-reversal symmetry, as well as a variety of fracton phases enriched by spatial symmetries. In $d=2$ dimensions, we find solutions that describe new weak and strong SSPT states, such as ones with both line-like subsystem symmetries and global time-reversal symmetry. In $d=1$ dimension, we show that any group cohomology solution for a symmetry-protected topological state protected by a global symmetry, along with lattice translational symmetry necessarily satisfies our consistency conditions., Comment: 26 pages, 5 figures, 15 tables. v2: published version

We introduce quantum circuits in two and three spatial dimensions which are classically simulable, despite producing a high degree of operator entanglement. We provide a partial characterization of these "automaton" quantum circuits, and use them to study operator growth, information spreading, and local charge relaxation in quantum dynamics with subsystem symmetries, which we define as overlapping symmetries that act on lower-dimensional submanifolds. With these symmetries, we discover the anomalous subdiffusion of conserved charges; that is, the charges spread slower than diffusion in the dimension of the subsystem symmetry. By studying an effective operator hydrodynamics in the presence of these symmetries, we predict the charge autocorrelator to decay ($i$) as $\log(t)/\sqrt{t}$ in two dimensions with a conserved $U(1)$ charge along intersecting \emph{lines}, and ($ii$) as $1/t^{3/4}$ in three spatial dimensions with intersecting \emph{planar} $U(1)$ symmetries. Through large-scale studies of automaton dynamics with these symmetries, we numerically observe charge relaxation that is consistent with these predictions. In both cases, the spatial charge distribution is distinctly non-Gaussian, and reminiscent of the diffusion of charges along a fractal surface. We numerically study the onset of quantum chaos in the spreading of local operators under these automaton dynamics, and observe power-law broadening of the ballistically-propagating fronts of evolving operators in two and three dimensions, and the saturation of out-of-time-ordered correlations to values consistent with quantum chaotic behavior., Comment: 17 pages, 10 figures

We study the error correcting properties of Majorana Surface Codes (MSC), topological quantum codes constructed out of interacting Majorana fermions, which can be used to store quantum information and perform quantum computation. These quantum memories suffer from purely "fermionic" errors, such as quasiparticle poisoning (QP), that have no analog in conventional platforms with bosonic qubits. In physical realizations where QP dominates, we show that errors can be corrected provided that the poisoning rate is below a threshold of $\sim11\%$. When QP is highly suppressed and fermionic bilinear ("bosonic") errors become dominant, we find an error threshold of $\sim16\%$, which is much higher than the threshold for spin-based topological memories like the Surface code or the Color code. In addition, we derive new lattice gauge theories to account for measurement errors. These results, together with the inherent error suppression provided by the superconducting gap in physical realizations of the MSC, makes this a strong candidate for a robust topological quantum memory., Comment: 8 pages, 5 figures

Recently, a new class of three-dimensional spin liquid models have been theoretically discovered, which feature generalized Coulomb phases of emergent symmetric tensor $U(1)$ gauge theories. These "higher rank" tensor models are particularly intriguing due to the presence of quasi-particles with restricted mobility, such as fractons. We investigate universal experimental signatures of tensor Coulomb phases. Most notably, we show that tensor Coulomb spin liquids (both quantum and classical) feature characteristic pinch-point singularities in their spin-spin correlation functions, accessible via neutron scattering, which can be readily distinguished from pinch points in conventional $U(1)$ spin liquids. These pinch points can thus serve as a crisp experimental diagnostic for such phases. We also tabulate the low-temperature heat capacity of various tensor Coulomb phases, which serves as a useful additional diagnostic in certain cases., Comment: 5.5 pages, 1 figure

We study the finite-temperature scrambling behavior of a quantum system described by a Hamiltonian chosen from a random matrix ensemble. This effectively (0+1)-dimensional model admits an exact calculation of various ensemble-averaged out-of-time-ordered correlation functions in the large-$N$ limit, where $N$ is the Hilbert space dimension. For a Hamiltonian drawn from the Gaussian unitary ensemble, we calculate the ensemble averaged OTOC at all temperatures. In addition to an early time quadratic growth of the averaged out-of-time-ordered commutator (OTOC), we determine that the OTOC saturates to its asymptotic value as a power-law in time, with an exponent that depends both on temperature, and on one of four classes of operators appearing in the correlation function, that naturally emerge from this calculation. Out-of-time-ordered correlation functions of operators that are distributed around the thermal circle take a time $t_{s}\sim \beta$ to decay at low temperatures. We generalize these exact results, by demonstrating that out-of-time-ordered correlation functions averaged over any ensemble of Hamiltonians that are invariant under unitary conjugation $H \rightarrow {U} H {U}^{\dagger}$, exhibit power-law decay to an asymptotic value. We argue that this late-time behavior is a generic feature of unitary dynamics with energy conservation. Finally, by comparing the OTOC with a commutator-anticommutator correlation function, we examine whether there is a time window over which a typical Hamiltonian behaves as a "coherent scrambler" in the language of Ref. \cite{Kitaev_IAS, Kitaev_Suh}., Comment: 13 pages, 1 figure

We introduce both an exactly solvable model and a coupled-layer construction for an exotic, three-dimensional phase of matter with immobile topological excitations that carry a protected internal degeneracy. Unitary transformations on this degenerate Hilbert space may be implemented by braiding certain point-like excitations. This provides a new way of extending non-Abelian statistics to three-dimensions., Comment: 9 pages, 7 figures

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, able to capture for example universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1+1D and higher dimensions, and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC, and the butterfly speed $v_{B}$. We find that in 1+1D the `front' of the OTOC broadens diffusively, with a width scaling in time as $t^{1/2}$. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front. Turning to higher D, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as $t^{1/3}$ in 2+1D and $t^{0.24}$ in 3+1D (KPZ exponents). We support our analytic argument with simulations in 2+1D. We point out that, in a lattice model, the late time shape of the spreading operator is in general not spherical. However when full spatial rotational symmetry is present in 2+1D, our mapping implies an exact asymptotic form for the OTOC in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1+1D, we map it to the partition function of an Ising-like model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1+1D circuits., Comment: 29 pages, 16 figures. v2: new appendix on 'mean field'

We introduce error-correcting codes that can correct for fermion parity-violating (quasiparticle poisoning) and parity-conserving errors in systems of complex fermions and of Majorana fermions. After establishing properties of fermion codes, we introduce a generic construction of fermion codes from weakly self-dual classical, binary error-correcting codes. We use this method to construct (i) the shortest fermion code to correct for quasiparticle poisoning errors, (ii) translationally-invariant fermion codes and (iii) other codes that correct higher-weight errors. We conclude by discussing a possible physical realizations of one such code in mesoscopic superconductors hosting Majorana zero modes., Comment: 11 pages, 2 figures

Starting from an isotropic configuration of intersecting, two-dimensional toric codes, we construct a fracton topological phase introduced in Ref. [26], which is characterized by immobile, point- like topological excitations ("fractons"), and degenerate ground-states on the torus that are locally indistinguishable. Our proposal leads to a simple description of the fracton excitations and of the ground-state as a "loop" condensate, and provides a basis for building new 3D topological orders such as a natural, $Z_{N}$ generalization of this fracton phase, which we introduce. We describe the rich phase structure of our layered $Z_{N}$ system. By invoking a lattice duality, we demonstrate that when $N \ge 5$, there is an intermediate phase that appears between the decoupled, layered system and the fracton topologically-ordered state, which opens the possibility of a continuous transition into the fracton topological phase. We conclude by presenting a solvable model, that interpolates between the fracton phase and a confined phase in which the phase transition is first-order., Comment: 12 pages, 9 figures

We present a new measurement-based scheme for performing braiding operations on Majorana zero modes and for detecting their non-Abelian statistics without moving or hybridizing them. In our scheme, the topological qubit encoded in any pair of well-separated Majorana zero modes in a mesoscopic superconductor island is read out from the transmission phase shift in electron teleportation through the island in the Coulomb blockade regime. We propose experimental setups to measure the teleportation phase shift via conductance in an electron interferometer or persistent current in a closed loop., Comment: 11 pages, 6 figures

Characterizing how entanglement grows with time in a many-body system, for example after a quantum quench, is a key problem in non-equilibrium quantum physics. We study this problem for the case of random unitary dynamics, representing either Hamiltonian evolution with time--dependent noise or evolution by a random quantum circuit. Our results reveal a universal structure behind noisy entanglement growth, and also provide simple new heuristics for the `entanglement tsunami' in Hamiltonian systems without noise. In 1D, we show that noise causes the entanglement entropy across a cut to grow according to the celebrated Kardar--Parisi--Zhang (KPZ) equation. The mean entanglement grows linearly in time, while fluctuations grow like $(\text{time})^{1/3}$ and are spatially correlated over a distance $\propto (\text{time})^{2/3}$. We derive KPZ universal behaviour in three complementary ways, by mapping random entanglement growth to: (i) a stochastic model of a growing surface; (ii) a `minimal cut' picture, reminiscent of the Ryu--Takayanagi formula in holography; and (iii) a hydrodynamic problem involving the dynamical spreading of operators. We demonstrate KPZ universality in 1D numerically using simulations of random unitary circuits. Importantly, the leading order time dependence of the entropy is deterministic even in the presence of noise, allowing us to propose a simple `minimal cut' picture for the entanglement growth of generic Hamiltonians, even without noise, in arbitrary dimensionality. We clarify the meaning of the `velocity' of entanglement growth in the 1D `entanglement tsunami'. We show that in higher dimensions, noisy entanglement evolution maps to the well-studied problem of pinning of a membrane or domain wall by disorder.