arXiv subjects

Tibor Rakovszky

Publications and source records attributed to Tibor Rakovszky.

At least 19 recordsLinked to original sources

arXiv2026

LDPC stabilizer codes as gapped quantum phases: stability under graph-local perturbations

We generalize the proof of stability of topological order, due to Bravyi, Hastings and Michalakis, to stabilizer Hamiltonians corresponding to low-density parity check (LDPC) codes without the restriction of geometric locality in Euclidean space. We consider Hamiltonians $H_0$ defined by $[[N,K,d]]$ LDPC codes which obey certain topological quantum order conditions: (i) code distance $d \geq c \log(N)$, implying local indistinguishability of ground states, and (ii) a mild condition on local and global compatibility of ground states; these include good quantum LDPC codes, and the toric code on a hyperbolic lattice, among others. We consider stability under weak perturbations that are quasi-local on the interaction graph defined by $H_0$, and which can be represented as sums of bounded-norm terms. As long as the local perturbation strength is smaller than a finite constant, we show that the perturbed Hamiltonian has well-defined spectral bands originating from the $O(1)$ smallest eigenvalues of $H_0$. The band originating from the smallest eigenvalue has $2^K$ states, is separated from the rest of the spectrum by a finite energy gap, and has exponentially narrow bandwidth $δ= C N e^{-Θ(d)}$, which is tighter than the best known bounds even in the Euclidean case. We also obtain that the new ground state subspace is related to the initial code subspace by a quasi-local unitary, allowing one to relate their physical properties. Our proof uses an iterative procedure that performs successive rotations to eliminate non-frustration-free terms in the Hamiltonian. Our results extend to quantum Hamiltonians built from classical LDPC codes, which give rise to stable symmetry-breaking phases. These results show that LDPC codes very generally define stable gapped quantum phases, even in the non-Euclidean setting, initiating a systematic study of such phases of matter.

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arXiv2026

Rare Events and Griffiths Phases in Topological Quantum Error Correction

The performance of quantum error correcting (QEC) codes are often studied under the assumption of spatio-temporally uniform error rates. On the other hand, experimental implementations almost always produce heterogeneous error rates, in either space or time, as a result of effects such as imperfect fabrication and/or cosmic rays. It is therefore important to understand if and how their presence can affect the performance of QEC in qualitative ways. In this work, we study effects of non-uniform error rates in the representative examples of the 1D repetition code and the 2D toric code, focusing on when they have extended spatio-temporal correlations; these may arise, for instance, from rare events (such as cosmic rays) that temporarily elevate error rates over the entire code patch. These effects can be described in the corresponding statistical mechanics models for decoding, where long-range correlations in the error rates lead to extended rare regions of weaker coupling. For the 1D repetition code where the rare regions are linear, we find two distinct decodable phases: a conventional ordered phase in which logical failure rates decay exponentially with the code distance, and a rare-region dominated Griffiths phase in which failure rates are parametrically larger and decay as a stretched exponential. In particular, the latter phase is present when the error rates in the rare regions are above the bulk threshold. For the 2D toric code where the rare regions are planar, we find no decodable Griffiths phase: rare events which boost error rates above the bulk threshold lead to an asymptotic loss of threshold and failure to decode. Unpacking the failure mechanism implies that techniques for suppressing extended sequences of repeated rare events (which, without intervention, will be statistically present with high probability) will be crucial for QEC with the toric code.

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arXiv2026

Finite-temperature quantum topological order in three dimensions

We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature--it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic point-like excitations. The fermionic toric code, importantly, possesses an anomalous 2-form symmetry, associated with the space-like Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to an equilibrium phase of matter that only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous 2-form symmetries. Our work therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.

cond-mat.str-el↗

arXiv2025

Expansion creates spin-glass order in finite-connectivity models: a rigorous and intuitive approach from the theory of LDPC codes

Complex free-energy landscapes with many local minima separated by large barriers are believed to underlie glassy behavior across diverse physical systems. This is the heuristic picture associated with replica symmetry breaking (RSB) in spin glasses, but RSB has only been rigorously verified for certain mean-field models with all-to-all connectivity. In this work, we give a rigorous proof of finite temperature spin glass order for a family of models with local interactions on finite-connectivity, non-Euclidean expander graphs. To this end, we bypass the RSB formalism entirely, and instead exploit the mathematical equivalence of such models to certain low-density parity check (LDPC) codes. We use code expansion, a property of LDPC codes which guarantees extensive energy barriers around ground states. Together with mild additional assumptions, this allows us to construct an explicit decomposition of the low-temperature Gibbs state into disjoint components, each hosting an asymptotically long-lived state associated with a local minimum of the landscape. Each component carries at most an exponentially small fraction of the total weight, and almost all components do not contain ground states -- which we take together to define spin-glass order. The proof is elementary, and treats various expanding graph topologies on the same footing, including those with short loops where existing approaches such as the cavity method fail. Our results apply rigorously to diluted p-spin glasses for sufficiently large p, and while unproven, we also expect our assumptions to hold in a broader family of codes. Motivated by this, we numerically study two simple models, on random regular graphs and a regular tesselation of hyperbolic space. We show that both models undergo two transitions as a function of temperature, corresponding to the onset of weak ergodicity breaking and spin glass order, respectively.

cond-mat.stat-mech↗

arXiv2025

Spectral Properties Versus Magic Generation in $T$-doped Random Clifford Circuits

We study the emergence of complexity in deep random $N$-qubit $T$-gate doped Clifford circuits, as reflected in their spectral properties and in magic generation, characterized by the stabilizer Rényi entropy distribution and the non-stabilizing power of the circuit. For pure (undoped) Clifford circuits, a unique periodic orbit structure in the space of Pauli strings implies peculiar spectral correlations and level statistics with large degeneracies. $T$-gate doping induces an exponentially fast transition to chaotic behavior, described by random matrix theory. We compare these complexity indicators with magic generation properties of the Clifford+$T$ ensemble, and determine the distribution of magic, as well as the average non-stabilizing power of the quantum circuit ensemble. In the dilute limit, $N_T \ll N$, magic generation is governed by single-qubit behavior. Magic is generated in approximate quanta, increases approximately linearly with the number of $T$-gates, $N_T$, and displays a discrete distribution for small $N_T$. At $N_T\approx N$, the distribution becomes quasi-continuous, and for $N_T\gg N$ it converges to that of Haar-random unitaries, and averages to a finite magic density, $m_2$, $\lim_{N\to\infty} \langle m_2 \rangle_\text{Haar} = 1$. This is in contrast to the spectral transition, where ${\cal O} (1)$ $T$-gates suffice to remove spectral degeneracies and to induce a transition to chaotic behavior in the thermodynamic limit. Magic is therefore a more sensitive indicator of complexity.

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arXiv2024

Topological Quantum Spin Glass Order and its realization in qLDPC codes

Ordered phases of matter have close connections to computation. Two prominent examples are spin glass order, with wide-ranging applications in machine learning and optimization, and topological order, closely related to quantum error correction. Here, we introduce the concept of topological quantum spin glass (TQSG) order which marries these two notions, exhibiting both the complex energy landscapes of spin glasses, and the quantum memory and long-range entanglement characteristic of topologically ordered systems. Using techniques from coding theory and a quantum generalization of Gibbs state decompositions, we show that TQSG order is the low-temperature phase of various quantum LDPC codes on expander graphs, including hypergraph and balanced product codes. Our work introduces a topological analog of spin glasses that preserves quantum information, opening new avenues for both statistical mechanics and quantum computer science.

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arXiv2024

Bottlenecks in quantum channels and finite temperature phases of matter

We prove an analogue of the "bottleneck theorem", well-known for classical Markov chains, for Markovian quantum channels. In particular, we show that if two regions (subspaces) of Hilbert space are separated by a region that has very low weight in the channel's steady state, then states initialized on one side of this barrier will take a long time to relax, putting a lower bound on the mixing time in terms of an appropriately defined "quantum bottleneck ratio". Importantly, this bottleneck ratio involves not only the probabilities of the relevant subspaces, but also the size of off-diagonal matrix elements between them. For low-temperature quantum many-body systems, we use the bottleneck theorem to bound the performance of any quasi-local Gibbs sampler. This leads to a new perspective on thermally stable quantum phases in terms of a decomposition of the Gibbs state into multiple components separated by bottlenecks. As a concrete application, we show rigorously that weakly perturbed commuting projector models with extensive energy barriers (including certain classical and quantum expander codes) have exponentially large mixing times.

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arXiv2024

Probing hydrodynamic crossovers with dissipation-assisted operator evolution

Using artificial dissipation to tame entanglement growth, we chart the emergence of diffusion in a generic interacting lattice model for varying U(1) charge densities. We follow the crossover from ballistic to diffusive transport above a scale set by the scattering length, finding the intuitive result that the diffusion constant scales as $D \propto 1/ρ$ at low densities $ρ$. Our numerical approach generalizes the Dissipation-Assisted Operator Evolution (DAOE) algorithm: in the spirit of the BBGKY hierarchy, we effectively approximate non-local operators by their ensemble averages, rather than discarding them entirely. This greatly reduces the operator entanglement entropy, while still giving accurate predictions for diffusion constants across all density scales. We further construct a minimal model for the transport crossover, yielding charge correlation functions which agree well with our numerical data. Our results clarify the dominant contributions to hydrodynamic correlation functions of conserved densities, and serve as a guide for generalizations to low temperature transport.

cond-mat.str-el↗

arXiv2024

Entanglement Properties of Gauge Theories from Higher-Form Symmetries

We explore the relationship between higher-form symmetries and entanglement properties in discrete lattice gauge theories, which can exhibit both topologically ordered phases and higher-form symmetry-protected topological (SPT) phases. Our study centers on generalizing the Fradkin-Shenker model, where the Gauss law constraint can be either emergent or exact. The phase diagram includes a topologically ordered phase and a non-trivial SPT phase protected by a 1-form and a 0-form symmetry. We obtain the following key findings: First, the entanglement properties depend on whether the 1-form symmetries and the Gauss law are exact or emergent. For the emergent Gauss law, the entanglement spectrum (ES) of the non-trivial SPT phase exhibits degeneracies, which are robust at low energies against weak perturbations that explicitly break the exact 1-form symmetry. When the Gauss law and the 1-form symmetry are both exact, the ES degeneracy is extensive. This extensive degeneracy is fragile and can be removed completely by infinitesimal perturbations that explicitly break the exact 1-form symmetry while keeping the Gauss law exact. Second, we consider the ES in the topologically ordered phase where 1-form symmetries are spontaneously broken. In contrast to the ES of the non-trivial SPT phase, we find that spontaneous higher-form symmetry breaking removes "half" of the ES levels, leading to a non-degenerate ES in the topologically ordered phase in general. Third, we derive a connection between spontaneous higher-form symmetry breaking and the topological entanglement entropy (TEE). Using this relation, we investigate the entanglement entropy that can be distilled in the deconfined phase of the original Fradkin-Shenker model using gauge-invariant measurements. We show that the TEE is robust against the measurement when the 1-form symmetry is emergent but fragile when the 1-form symmetry is exact.

cond-mat.str-el↗

arXiv2024

The Physics of (good) LDPC Codes II. Product constructions

We continue the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective. We focus on constructive approaches and formulate a general framework for systematically constructing codes with various features on generic Euclidean and non-Euclidean graphs. These codes can serve as fixed-point limits for phases of matter. To build our machinery, we unpack various product constructions from the coding literature in terms of physical principles such as symmetries and redundancies, introduce a new cubic product, and combine these products with the ideas of gauging and Higgsing introduced in Part I. We illustrate the usefulness of this approach in finite Euclidean dimensions by showing that using the one-dimensional Ising model as a starting point, we can systematically produce a very large zoo of classical and quantum phases of matter, including type I and type II fractons and SPT phases with generalized symmetries. We also use the balanced product to construct new Euclidean models, including one with topological order enriched by translation symmetry, and another exotic fracton model whose excitations are formed by combining those of a fractal spin liquid with those of a toric code, resulting in exotic mobility constraints. Moving beyond Euclidean models, we give a review of existing constructions of good qLDPC codes and classical locally testable codes and elaborate on the relationship between quantum code distance and classical energy barriers, discussed in Part I, from the perspective of product constructions.

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arXiv2024

Defining stable phases of open quantum systems

The steady states of dynamical processes can exhibit stable nontrivial phases, which can also serve as fault-tolerant classical or quantum memories. For Markovian quantum (classical) dynamics, these steady states are extremal eigenvectors of the non-Hermitian operators that generate the dynamics, i.e., quantum channels (Markov chains). However, since these operators are non-Hermitian, their spectra are an unreliable guide to dynamical relaxation timescales or to stability against perturbations. We propose an alternative dynamical criterion for a steady state to be in a stable phase, which we name uniformity: informally, our criterion amounts to requiring that, under sufficiently small local perturbations of the dynamics, the unperturbed and perturbed steady states are related to one another by a finite-time dissipative evolution. We show that this criterion implies many of the properties one would want from any reasonable definition of a phase. We prove that uniformity is satisfied in a canonical classical cellular automaton, and provide numerical evidence that the gap determines the relaxation rate between nearby steady states in the same phase, a situation we conjecture holds generically whenever uniformity is satisfied. We further conjecture some sufficient conditions for a channel to exhibit uniformity and therefore stability.

quant-ph↗

arXiv2023

The Physics of (good) LDPC Codes I. Gauging and dualities

Low-depth parity check (LDPC) codes are a paradigm of error correction that allow for spatially non-local interactions between (qu)bits, while still enforcing that each (qu)bit interacts only with finitely many others. On expander graphs, they can give rise to ``good codes'' that combine a finite encoding rate with an optimal scaling of the code distance, which governs the code's robustness against noise. Such codes have garnered much recent attention due to two breakthrough developments: the construction of good quantum LDPC codes and good locally testable classical LDPC codes, using similar methods. Here we explore these developments from a physics lens, establishing connections between LDPC codes and ordered phases of matter defined for systems with non-local interactions and on non-Euclidean geometries. We generalize the physical notions of Kramers-Wannier (KW) dualities and gauge theories to this context, using the notion of chain complexes as an organizing principle. We discuss gauge theories based on generic classical LDPC codes and make a distinction between two classes, based on whether their excitations are point-like or extended. For the former, we describe KW dualities, analogous to the 1D Ising model and describe the role played by ``boundary conditions''. For the latter we generalize Wegner's duality to obtain generic quantum LDPC codes within the deconfined phase of a Z_2 gauge theory. We show that all known examples of good quantum LDPC codes are obtained by gauging locally testable classical codes. We also construct cluster Hamiltonians from arbitrary classical codes, related to the Higgs phase of the gauge theory, and formulate generalizations of the Kennedy-Tasaki duality transformation. We use the chain complex language to discuss edge modes and non-local order parameters for these models, initiating the study of SPT phases in non-Euclidean geometries.

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arXiv2023

The ballistic to diffusive crossover in a weakly-interacting Fermi gas

Charge and energy are expected to diffuse in interacting systems of fermions at finite temperatures, even in the absence of disorder, with the interactions inducing a crossover from the coherent and ballistic streaming of quasi-particles at early times, to incoherent diffusive behavior at late times. The relevant crossover timescales and the transport coefficients are both controlled by the strength of interactions. In this work we develop a numerical method to simulate such systems at high temperatures, applicable in a wide range of interaction strengths, by adapting Dissipation-assisted Operator Evolution (DAOE) to fermions. Our fermion DAOE, which approximates the exact dynamics by systematically discarding information from high $n$-point functions, is tailored to capture non-interacting dynamics exactly, thus providing a good starting point for the weakly interacting problem. Applying our method to a microscopic model of weakly interacting fermions, we numerically demonstrate that the crossover from ballistic to diffusive transport happens at a time $t_D\sim1/Δ^{2}$ and that the diffusion constant similarly scales as $D \sim 1/Δ^2$, where $Δ$ is the interaction strength. We substantiate this scaling with a Fermi's golden rule calculation in the operator spreading picture, interpreting $t_D$ as the fermion-fermion scattering time and lifetime of the single-particle Green's function.

cond-mat.str-el↗

arXiv2023

Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model

We consider a paradigmatic solvable model of topological order in two dimensions, Kitaev's honeycomb Hamiltonian, and turn it into a measurement-only dynamics consisting of stochastic measurements of two-qubit bond operators. We find an entanglement phase diagram that resembles that of the Hamiltonian problem in some ways, while being qualitatively different in others. When one type of bond is dominantly measured, we find area-law entangled phases that protect two topological qubits (on a torus) for a time exponential in system size. This generalizes the recently-proposed idea of Floquet codes, where logical qubits are dynamically generated by a time-periodic measurement schedule, to a stochastic setting. When all types of bonds are measured with comparable frequency, we find a critical phase with a logarithmic violation of the area-law, which sharply distinguishes it from its Hamiltonian counterpart. The critical phase has the same set of topological qubits, as diagnosed by the tripartite mutual information, but protects them only for a time polynomial in system size. Furthermore, we observe an unusual behavior for the dynamical purification of mixed states, characterized at late times by the dynamical exponent $z = 1/2$ -- a super-ballistic dynamics made possible by measurements.

quant-ph↗

arXiv2023

Operator relaxation and the optimal depth of classical shadows

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via ``shallow shadows'', a recently-proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth $t$. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (``twirling'') circuit -- namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially-contiguous Pauli operators of weight $k$, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth $t\sim \log(k)$, guarantees an exponential gain in sample complexity over the $t=0$ protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our work connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly-optimized protocols for learning different properties of quantum states.

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arXiv2023

Higgs Condensates are Symmetry-Protected Topological Phases: II. $U(1)$ Gauge Theory and Superconductors

Classifying Higgs phases within the landscape of gapped and symmetry preserving states of matter presents a conceptual challenge. We argue that $U(1)$ Higgs phases are symmetry-protected topological (SPT) phases and we derive their topological response theory and boundary anomaly -- applicable to superconductors treated with dynamical electromagnetic field. This generalizes the discussion of discrete gauge theories by Verresen et al., arXiv:2211.01376. We show that a Higgs phase in $d$ spatial dimensions is in a non-trivial SPT class protected by a global $U(1)$ symmetry associated with the Higgs field, and a $d-2$ form $U(1)$ magnetic symmetry, associated with the absence of magnetic monopoles. In $d=2$, this gives an SPT with a mixed Hall response between conventional symmetries, whereas in $d=3$ we obtain a novel SPT protected by a 0-form and 1-form symmetry whose 2+1d boundary anomaly is satisfied by a superfluid. The signature properties of superconductors -- Higgs phases for electromagnetism -- can be reproduced from this SPT response. For instance, the Josephson effect directly arises from the aforementioned boundary superfluid. In addition to this minimalist approach being complementary to Landau-Ginzburg theory, its non-perturbative nature is useful in situations where fluctuations are significant. We substantiate this by predicting the stability of the Josephson effect upon introducing monopoles in $U(1)$ lattice gauge theory, where tuning from the charge-1 Higgs phase to the confined phase leads to a quantum critical point in the junction. Furthermore, this perspective reveals unexpected connections, such as how persistent currents at the surface of a superconductor arise from generalized Thouless pumps. We also treat generalizations to partial-Higgs phases, including "2e" condensates in electronic superconductors, corresponding to symmetry-enriched topological orders.

cond-mat.str-el↗

arXiv2023

Fragmentation-induced localization and boundary charges in dimensions two and above

We study higher dimensional models with symmetric correlated hoppings, which generalize a one-dimensional model introduced in the context of dipole-conserving dynamics. We prove rigorously that whenever the local configuration space takes its smallest non-trivial value, these models exhibit localized behavior due to fragmentation, in any dimension. For the same class of models, we then construct a hierarchy of conserved quantities that are power-law localized at the boundary of the system with increasing powers. Combining these with Mazur's bound, we prove that boundary correlations are infinitely long lived, even when the bulk is not localized. We use our results to construct quantum Hamiltonians that exhibit the analogues of strong zero modes in two and higher dimensions.

cond-mat.stat-mech↗

arXiv2022

Fractal, logarithmic and volume-law entangled non-thermal steady states via spacetime duality

The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size -- from logarithmic to extensive to \emph{fractal}. We show how these outcomes in non-unitary circuits (that are "spacetime-dual" to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets "radiated away" from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with \emph{fractal} entanglement scaling, $S(\ell) \sim \ell^α$ with tunable $0 < α< 1$ for subsystems of size $\ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of "teleportation" between spacelike and timelike slices of quantum circuits.

quant-ph↗