arXiv subjects

Timothy H. Hsieh

Publications and source records attributed to Timothy H. Hsieh.

At least 19 recordsLinked to original sources

arXiv2026

A Unified Framework for Locally Stable Phases

We propose a unifying framework for characterizing pure and mixed state phases of matter across equilibrium, non equilibrium, and metastable regimes. We introduce the concept of locally stable states, defined by the operational property that any local operation (including post selection) can be reversed by a local channel. We prove that local stability is equivalent to a state being short range correlated, defined by the decay of both correlations and conditional mutual information. We demonstrate that these properties are invariant under locally reversible channels, thus defining locally stable phases. Furthermore, we prove that local stability implies both the decay of a family of nonlinear correlators, including the fidelity correlator, and the decay of correlations in the canonical purification, thus bridging the gap between mixed and pure states. Along the way, we establish two results which may be of independent interest: we show that post-selection on locally stable (short range correlated) states can be implemented via local channels and that quantum Markov chains can be characterized by the local computability of nonlinear observables.

quant-ph↗

arXiv2026

Unlearnable phases of matter

We identify fundamental limitations in machine learning by demonstrating that non-trivial mixed-state phases of matter are computationally hard to learn. Focusing on unsupervised learning of distributions, we show that autoregressive neural networks fail to learn global properties of distributions characterized by locally indistinguishable (LI) states. We demonstrate that conditional mutual information (CMI) is a useful diagnostic for LI: we show that for classical distributions, long-range CMI of a state implies a spatially LI partner. By introducing a restricted statistical query model, we prove that nontrivial phases with long-range CMI, such as strong-to-weak spontaneous symmetry breaking phases, are hard to learn. We validate our claims by using recurrent, convolutional, and Transformer neural networks to learn the syndrome and physical distributions of toric/surface code under bit flip noise. Our findings suggest hardness of learning as a diagnostic tool for detecting mixed-state phases and transitions and error-correction thresholds, and they suggest CMI and more generally ``non-local Gibbsness'' as metrics for how hard a distribution is to learn.

cond-mat.dis-nn↗

arXiv2026

Mixed-state Quantum Phases: Renormalization and Quantum Error Correction

Open system quantum dynamics can generate a variety of long-range entangled mixed states, yet it has been unclear in what sense they constitute phases of matter. To establish that two mixed states are in the same phase, as defined by their two-way connectivity via local quantum channels, we use the renormalization group (RG) and decoders of quantum error correcting codes. We introduce a real-space RG scheme for mixed states based on local channels which ideally preserve correlations with the complementary system, and we prove this is equivalent to the reversibility of the channel's action. As an application, we demonstrate an exact RG flow of finite temperature toric code in two dimensions to infinite temperature, thus proving it is in the trivial phase. In contrast, for toric code subject to local dephasing, we establish a mixed state toric code phase using local channels obtained by truncating an RG-type decoder and the minimum weight perfect matching decoder. We also discover a precise relation between mixed state phase and decodability, by proving that local noise acting on toric code cannot destroy logical information without bringing the state out of the toric code phase.

quant-ph↗

arXiv2026

Critical non-equilibrium phases from noisy topological memories

We demonstrate the existence of an extended non-equilibrium critical phase, characterized by sub-exponential decay of conditional mutual information (CMI), in the surface code subject to heralded random Pauli measurement channels. By mapping the resulting mixed state to the ensemble of completely packed loops on a square lattice, we relate the extended phase to the Goldstone phase of the loop model. In particular, CMI is controlled by the characteristic length scale of loops, and we use analytic results of the latter to establish polylogarithmic decay of CMI in the critical phase. We find that the critical phase retains partial logical information that can be recovered by a global decoder, but not by any quasi-local decoder. To demonstrate this, we introduce a diagnostic called punctured coherent information which provides a necessary condition for quasi-local decoding.

quant-ph↗

arXiv2025

Autoregressive Typical Thermal States

A variety of generative neural networks recently adopted from machine learning have provided promising strategies for studying quantum matter. In particular, the success of autoregressive models in natural language processing has motivated their use as variational ansätze, with the hope that their demonstrated ability to scale will transfer to simulations of quantum many-body systems. In this paper, we introduce an autoregressive framework to calculate finite-temperature properties of a quantum system based on the imaginary-time evolution of an ensemble of pure states. We find that established approaches based on minimally entangled typical thermal states (METTS) have numerical instabilities when an autoregressive recurrent neural network is used as the variational ansätz. We show that these instabilities can be mitigated by evolving the initial ensemble states with a unitary operation, along with applying a threshold to curb runaway evolution of ensemble members. By comparing our algorithm to exact results for the spin 1/2 quantum XY chain, we demonstrate that autoregressive typical thermal states are capable of accurately calculating thermal observables.

quant-ph↗

arXiv2025

Mixed-state phases from local reversibility

We propose a refined definition of mixed-state phase equivalence based on locally reversible channel circuits. We show that such circuits preserve topological degeneracy and the locality of all operators including both strong and weak symmetries. Under a locally reversible channel, weak unitary symmetries are locally dressed into channel symmetries, a new generalization of symmetry for open quantum systems. For abelian higher-form symmetries, we show the refined definition preserves anomalies and spontaneous breaking of such symmetries within a phase. As a primary example, a two-dimensional classical loop ensemble is trivial under the previously adopted definition of mixed-state phases. However, it has non-trivial topological degeneracy arising from a mutual anomaly between strong and weak 1-form symmetries, and our results show that it is not connected to a trivial state via locally reversible channel circuits.

quant-ph↗

arXiv2025

Probing mixed-state phases on a quantum computer via Renyi correlators and variational decoding

Recent advances have defined nontrivial phases of matter in open quantum systems, such as many-body quantum states subject to environmental noise. In this work, we experimentally probe and characterize mixed-state phases on Quantinuum's H1 quantum computer using two measures: Renyi correlators and the coding performance of a quantum error-correcting code associated with the phase. As a concrete example, we probe the low-energy states of the critical transverse field Ising model under different dephasing noise channels. First, we employ shadow tomography to observe a newly proposed Renyi correlator in two distinct phases: one exhibiting power-law decay and the other long-ranged. Second, we investigate the decoding fidelity of the associated quantum error-correcting code using a variational quantum circuit, and we find that a shallow circuit is sufficient to distinguish the above-mentioned two mixed-state phases through the decoding performance quantified by entanglement fidelity. Our work is a proof of concept for the quantum simulation and characterization of mixed-state phases.

quant-ph↗

arXiv2025

Higher-form anomaly and long-range entanglement of mixed states

In open quantum systems, we directly relate anomalies of higher-form symmetries to the long-range entanglement of any mixed state with such symmetries. First, we define equivalence classes of long-range entanglement in mixed states via stochastic local channels (SLCs), which effectively ``mod out'' any classical correlations and thus distinguish phases by differences in long-range quantum correlations only. It is then shown that strong symmetries of a mixed state and their anomalies (non-trivial braiding and self-statistics) are intrinsic features of the entire phase of matter. For that, a general procedure of symmetry pullback for strong symmetries is introduced, whereby symmetries of the output state of an SLC are dressed into symmetries of the input state, with their anomaly relation preserved. This allows us to prove that states in (2+1)-D with anomalous strong 1-form symmetries exhibit long-range bipartite entanglement, and to establish a lower bound for their topological entanglement of formation, a mixed-state generalization of topological entanglement entropy. For concreteness, we apply this formalism to the toric code under Pauli-X and Z dephasing noise, as well as under ZX decoherence, which gives rise to the recently discovered intrinsically mixed-state topological order. Finally, we conjecture a connection between higher-form anomalies and long-range multipartite entanglement for mixed states in higher dimensions.

quant-ph↗

arXiv2025

Universal Properties of Critical Mixed States from Measurement and Feedback

We explore the universal properties of mixed quantum matter obtained from "single-shot" adaptive evolution, in which a quantum-critical ground-state is manipulated through a single round of local measurements and local unitary operations conditioned on spatially-distant measurement outcomes. The resulting mixed quantum states are characterized by altered long-distance correlations between local observables, mixed-state entropy, and entanglement negativity. By invoking a coarse-grained, continuum description of single-shot adaptation in (1+1) dimensions, we find that the extensive mixed-state entropy exhibits a sub-leading, constant correction ($γ$), while the entanglement negativity can grow logarithmically with sub-region size, with a coefficient ($α$); both constants can attain universal values which are distinct from the expected behavior in any quantum-critical ground-state. We investigate these properties in single-shot adaptation on ($i$) the critical point between a one-dimensional $Z_{2}\times Z_{2}$ symmetry-protected topological (SPT) order and a symmetry-broken state, and ($ii$) a spinful Tomonaga-Luttinger liquid. In the former case, adaptive evolution that decoheres one sublattice of the SPT can yield a critical mixed-state in which $α$ attains a universal value, which is half of that in the original state. In the latter case, we show how adaptation -- involving feedback on the spin degrees of freedom, after measuring the local charge -- modifies long-distance correlations, and determine via an exact replica field-theoretic calculation that $α$ and $γ$ vary continuously with the strength of feedback. Numerical studies confirm these results.

cond-mat.str-el↗

arXiv2024

Symmetry enforced entanglement in maximally mixed states

Entanglement in quantum many-body systems is typically fragile to interactions with the environment. Generic unital quantum channels, for example, have the maximally mixed state with no entanglement as their unique steady state. However, we find that for a unital quantum channel that is `strongly symmetric', i.e. it preserves a global on-site symmetry, the maximally mixed steady state in certain symmetry sectors can be highly entangled. For a given symmetry, we analyze the entanglement and correlations of the maximally mixed state in the invariant sector (MMIS), and show that the entanglement of formation and distillation are exactly computable and equal for any bipartition. For all Abelian symmetries, the MMIS is separable, and for all non-Abelian symmetries, the MMIS is entangled. Remarkably, for non-Abelian continuous symmetries described by compact semisimple Lie groups (e.g. $SU(2)$), the bipartite entanglement of formation for the MMIS scales logarithmically $\sim \log N$ with the number of qudits $N$.

quant-ph↗

arXiv2024

Spacetime Markov length: a diagnostic for fault tolerance via mixed-state phases

We establish a correspondence between the fault-tolerance of local stabilizer codes experiencing measurement and physical errors and the mixed-state phases of decohered resource states in one higher dimension. Drawing from recent developments in mixed-state phases of matter, this motivates a diagnostic of fault-tolerance, which we refer to as the spacetime Markov length. This is a length scale determined by the decay of the (classical) conditional mutual information of repeated syndrome measurement outcomes in spacetime. The diagnostic is independent of the decoder, and its divergence signals the intrinsic breakdown of fault tolerance. As a byproduct, we find that decoherence may be useful for exposing transitions from higher-form symmetry-protected topological phases driven by both incoherent and coherent perturbations.

quant-ph↗

arXiv2024

Approximate quantum error correcting codes from conformal field theory

The low-energy subspace of a conformal field theory (CFT) can serve as a quantum error correcting code, with important consequences in holography and quantum gravity. We consider generic 1+1D CFT codes under extensive local dephasing channels and analyze their error correctability in the thermodynamic limit. We show that (i) there is a finite decoding threshold if and only if the minimal nonzero scaling dimension in the fusion algebra generated by the jump operator of the channel is larger than $1/2$ and (ii) the number of protected logical qubits $k \geq Ω( \log \log n)$, where $n$ is the number of physical qubits. As an application, we show that the one-dimensional quantum critical Ising model has a finite threshold for certain types of dephasing noise. Our general results also imply that a CFT code with continuous symmetry saturates a bound on the recovery fidelity for covariant codes.

quant-ph↗

arXiv2024

Stability of mixed-state quantum phases via finite Markov length

For quantum phases of Hamiltonian ground states, the energy gap plays a central role in ensuring the stability of the phase as long as the gap remains finite. We propose Markov length, the length scale at which the quantum conditional mutual information (CMI) decays exponentially, as an equally essential quantity characterizing mixed-state phases and transitions. For a state evolving under a local Lindbladian, we argue that if its Markov length remains finite along the evolution, then it remains in the same phase, meaning there exists another quasi-local Lindbladian evolution that can reverse the former one. We apply this diagnostic to toric code subject to decoherence and show that the Markov length is finite everywhere except at its decodability transition, at which it diverges. CMI in this case can be mapped to the free energy cost of point defects in the random bond Ising model. This implies that the mixed state phase transition coincides with the decodability transition and also suggests a quasi-local decoding channel.

quant-ph↗

arXiv2024

Measurement-induced phase transitions in the toric code

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 un-measured 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.

quant-ph↗

arXiv2023

Mixed-state long-range order and criticality from measurement and feedback

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.

cond-mat.str-el↗

arXiv2023

Random insights into the complexity of two-dimensional tensor network calculations

Projected entangled pair states (PEPS) offer memory-efficient representations of some quantum many-body states that obey an entanglement area law, and are the basis for classical simulations of ground states in two-dimensional (2d) condensed matter systems. However, rigorous results show that exactly computing observables from a 2d PEPS state is generically a computationally hard problem. Yet approximation schemes for computing properties of 2d PEPS are regularly used, and empirically seen to succeed, for a large subclass of (not too entangled) condensed matter ground states. Adopting the philosophy of random matrix theory, in this work we analyze the complexity of approximately contracting a 2d random PEPS by exploiting an analytic mapping to an effective replicated statistical mechanics model that permits a controlled analysis at large bond dimension. Through this statistical-mechanics lens, we argue that: i) although approximately sampling wave-function amplitudes of random PEPS faces a computational-complexity phase transition above a critical bond dimension, ii) one can generically efficiently estimate the norm and correlation functions for any finite bond dimension. These results are supported numerically for various bond-dimension regimes. It is an important open question whether the above results for random PEPS apply more generally also to PEPS representing physically relevant ground states

quant-ph↗

arXiv2023

Channeling quantum criticality

We analyze the effect of decoherence, modelled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "$g$-function" in conformal field theory (CFT), allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the CFT. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.

quant-ph↗

arXiv2023

Measurement as a shortcut to long-range entangled quantum matter

The preparation of long-range entangled states using unitary circuits is limited by Lieb-Robinson bounds, but circuits with projective measurements and feedback (``adaptive circuits'') can evade such restrictions. We introduce three classes of local adaptive circuits that enable low-depth preparation of long-range entangled quantum matter characterized by gapped topological orders and conformal field theories (CFTs). The three classes are inspired by distinct physical insights, including tensor-network constructions, multiscale entanglement renormalization ansatz (MERA), and parton constructions. A large class of topological orders, including chiral topological order, can be prepared in constant depth or time, and one-dimensional CFT states and non-abelian topological orders with both solvable and non-solvable groups can be prepared in depth scaling logarithmically with system size. We also build on a recently discovered correspondence between symmetry-protected topological phases and long-range entanglement to derive efficient protocols for preparing symmetry-enriched topological order and arbitrary CSS (Calderbank-Shor-Steane) codes. Our work illustrates the practical and conceptual versatility of measurement for state preparation.

cond-mat.str-el↗