arXiv subjects

Yijian Zou

Publications and source records attributed to Yijian Zou.

At least 19 recordsLinked to original sources

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

Polynomial-time certification of fidelity for many-body mixed states and mixed-state universality classes

Computation of Uhlmann fidelity between many-body mixed states generally involves full diagonalization of exponentially large matrices. In this work, we introduce a polynomial-time algorithm to compute certified lower and upper bounds for the fidelity between matrix product density operators (MPDOs). Our method maps the fidelity estimation problem to a variational optimization of sequential quantum circuits, allowing for systematic improvement of the lower bounds by increasing the circuit depth. Complementarily, we obtain certified upper bounds on fidelity by variational lower bounds on the trace distance through the same framework. We demonstrate the power of this approach with two examples: fidelity correlators in critical mixed states, and codeword distinguishability in an approximate quantum error-correcting code. Remarkably, the variational lower bound accurately track the universal scaling behavior of the fidelity with a size-consistent relative error, allowing for the extraction of previously unknown critical exponents. Our results offer an exponential improvement in precision over known moment-based bounds and establish a scalable framework for the verification of many-body quantum systems.

quant-ph↗

arXiv2025

Universal quantum phase classification on quantum computers from machine learning

The classification of quantum phases of matter remains a fundamental challenge in condensed matter physics. We present a novel framework that combines shadow tomography with modern time-series machine learning models to enable efficient and practical quantum phase classification. Our approach leverages the definition of quantum phases based on connectivity through finite-depth local unitary circuits, generating abundant training data by applying Haar random evolution to representative quantum states for a given phase. In this way, the training data can be efficiently obtained from a quantum simulator. Additionally, we demonstrate that advanced time-series models can be used to process the training data and achieve universal quantum phase classification that does not rely on local order parameters. To validate the universality and versatility of our method, we test the model against one-dimensional quantum spin chains such as the Ising model and the axial next-nearest-neighbor Ising (ANNNI) model, showing excellent agreement with known phase boundaries.

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

Chirality, magic, and quantum correlations in multipartite quantum states

We introduce a notion of chirality for generic quantum states. A chiral state is defined as a state which cannot be transformed into its complex conjugate in a local product basis using local unitary operations. We introduce a number of quantitative measures of chirality which vanish for non-chiral states. A faithful measure called the "chiral log-distance" is defined in terms of the maximum fidelity between the state and its complex conjugate under local unitary transformations. We further introduce a set of computable measures that involve nested commutators of modular Hamiltonians. We show general relations between chirality and the amount of "magic" or non-stabilizer resource in a state. We first show that stabilizer states of qubits are non-chiral. Furthermore, magic monotones such as the stabilizer nullity and the stabilizer fidelity are lower-bounded by the chiral log-distance. We further relate chirality to discord-like quantum correlations. We define a measure of discord-like correlations called the intrinsic interferometric power, and show that it is lower-bounded by a nested commutator chirality measure. One interesting aspect of chirality is that it is not monotonic under local partial traces, indicating that it is a unique kind of resource that is not captured by traditional quantum resource theories.

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

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

Studying the 3d Ising surface CFTs on the fuzzy sphere

Boundaries not only are fundamental elements in nearly all realistic physical systems, but also greatly enrich the structure of quantum field theories. In this paper, we demonstrate that conformal field theory (CFT) with a boundary, known as surface CFT in three dimensions, can be studied with the setup of fuzzy sphere. We consider the example of surface criticality of the 3D Ising CFT. We propose two schemes by cutting a boundary in the orbital space or the real space to realise the ordinary and the normal surface CFTs on the fuzzy sphere. We obtain the operator spectra through state-operator correspondence. We observe integer spacing of the conformal multiplets, and thus provide direct evidence of conformal symmetry. We identify the ordinary surface primary $o$, the displacement operator $\mathrm{D}$ and their conformal descendants and extract their scaling dimensions. We also study the one-point and two-point correlation functions and extract the bulk-to-surface OPE coefficients, some of which are reported for the first time. In addition, using the overlap of the bulk CFT state and the polarised state, we calculate the boundary central charges of the 3D Ising surface CFTs non-perturbatively. Other conformal data obtained in this way also agrees with prior methods.

hep-th↗

arXiv2024

Petz map recovery for long-range entangled quantum many-body states

Given a tripartite quantum state on $A,B,C$ and the erasure channel on $C$, the rotated Petz map is a recovery channel that acts on $B$ to recover the erased quantum information. The infidelity of the best recovery is upper-bounded by the conditional mutual information (CMI). In this work, we study the infidelity of the rotated Petz map on several physically-relevant long-range entangled quantum states. Specifically, we study three classes of quantum phases: (i) steady states of measurement-induced phase transitions, (ii) critical ground state under local measurements, and (iii) chiral states under local measurements. We find that the averaged infidelity of the Petz map recovery sharply distinguishes the three classes: (i) and (ii) are distinguished by the scaling of the infidelity with CMI and (iii) is characterized by an asymmetry of the infidelity with the rotation parameter. We also study Petz map recovery for topological order and find an operational interpretation of the topological entanglement entropy. Our result indicates that recovery fidelity of the Petz map is a useful diagnostic of quantum phases of matter.

quant-ph↗

arXiv2024

The $g$-function and Defect Changing Operators from Wavefunction Overlap on a Fuzzy Sphere

Defects are common in physical systems with boundaries, impurities or extensive measurements. The interaction between bulk and defect can lead to rich physical phenomena. Defects in gapless phases of matter with conformal symmetry usually flow to a defect conformal field theory (dCFT). Understanding the universal properties of dCFTs is a challenging task. In this paper, we propose a computational strategy applicable to a line defect in arbitrary dimensions. Our main assumption is that the defect has a UV description in terms of a local modification of the Hamiltonian so that we can compute the overlap between low-energy eigenstates of a system with or without the defect insertion. We argue that these overlaps contain a wealth of conformal data, including the $g$-function, which is an RG monotonic quantity that distinguishes different dCFTs, the scaling dimensions of defect creation operators $Δ^{+0}_α$ and changing operators $Δ^{+-}_α$ that live on the intersection of different types of line defects, and various OPE coefficients. We apply this method to the fuzzy sphere regularization of 3D CFTs and study the magnetic line defect of the 3D Ising CFT. Using exact diagonalization and DMRG, we report the non-perturbative results $g=0.602(2),Δ^{+0}_0=0.108(5)$ and $Δ^{+-}_0=0.84(5)$ for the first time. We also obtain other OPE coefficients and scaling dimensions. Our results have significant physical implications. For example, they constrain the possible occurrence of spontaneous symmetry breaking at line defects of the 3D Ising CFT. Our method can be potentially applied to various other dCFTs, such as plane defects and Wilson lines in gauge theories.

hep-th↗

arXiv2023

Petz recovery from subsystems in conformal field theory

We probe the multipartite entanglement structure of the vacuum state of a CFT in 1+1 dimensions, using recovery operations that attempt to reconstruct the density matrix in some region from its reduced density matrices on smaller subregions. We use an explicit recovery channel known as the twirled Petz map, and study distance measures such as the fidelity, relative entropy, and trace distance between the original state and the recovered state. One setup we study in detail involves three contiguous intervals $A$, $B$ and $C$ on a spatial slice, where we can view these quantities as measuring correlations between $A$ and $C$ that are not mediated by the region $B$ that lies between them. We show that each of the distance measures is both UV finite and independent of the operator content of the CFT, and hence depends only on the central charge and the cross-ratio of the intervals. We evaluate these universal quantities numerically using lattice simulations in critical spin chain models, and derive their analytic forms in the limit where $A$ and $C$ are close using the OPE expansion. In the case where $A$ and $C$ are far apart, we find a surprising non-commutativity of the replica trick with the OPE limit. For all values of the cross-ratio, the fidelity is strictly better than a general information-theoretic lower bound in terms of the conditional mutual information. We also compare the mutual information between various subsystems in the original and recovered states, which leads to a more qualitative understanding of the differences between them. Further, we introduce generalizations of the recovery operation to more than three adjacent intervals, for which the fidelity is again universal with respect to the operator content.

hep-th↗

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

Cross Entropy Benchmark for Measurement-Induced Phase Transitions

We investigate the prospects of employing the linear cross-entropy to experimentally access measurement-induced phase transitions (MIPT) without requiring any postselection of quantum trajectories. For two random circuits that are identical in the bulk but with different initial states, the linear cross-entropy $χ$ between the bulk measurement outcome distributions in the two circuits acts as a boundary order parameter, and can be used to distinguish the volume law from area law phases. In the volume law phase (and in the thermodynamic limit) the bulk measurements cannot distinguish between the two different initial states, and $χ= 1$. In the area law phase $χ< 1$. For circuits with Clifford gates, we provide numerical evidence that $χ$ can be sampled to accuracy $ε$ from $O(1/ε^2)$ trajectories, by running the first circuit on a quantum simulator without postselection, aided by a classical simulation of the second. We also find that for weak depolarizing noise the signature of the MIPT is still present for intermediate system sizes. In our protocol we have the freedom of choosing initial states such that the "classical" side can be simulated efficiently, while simulating the "quantum" side is still classically hard.

quant-ph↗

arXiv2023

Probing sign structure using measurement-induced entanglement

The sign structure of quantum states is closely connected to quantum phases of matter, yet detecting such fine-grained properties of amplitudes is subtle. Here we employ as a diagnostic measurement-induced entanglement (MIE): the average entanglement generated between two parties after measuring the rest of the system. We propose that for a sign-free state, the MIE upon measuring in the sign-free basis decays no slower than correlations in the state before measurement. Concretely, we prove that MIE is upper bounded by mutual information for sign-free stabilizer states (essentially CSS codes), which establishes a bound between scaling dimensions of conformal field theories describing measurement-induced critical points in stabilizer systems. We also show that for sign-free qubit wavefunctions, MIE between two qubits is upper bounded by a simple two-point correlation function, and we verify our proposal in several critical ground states of one-dimensional systems, including the transverse field and tri-critical Ising models. In contrast, for states with sign structure, such bounds can be violated, as we illustrate in critical hybrid circuits involving both Haar or Clifford random unitaries and measurements, and gapless symmetry-protected topological states.

quant-ph↗

arXiv2022

Modular Commutators in Conformal Field Theory

The modular commutator is a recently discovered multipartite entanglement measure that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in $1+1$ dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped $(2+1)$-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.

cond-mat.str-el↗

arXiv2022

Emergence of Kac-Moody Symmetry in Critical Quantum Spin Chains

Given a critical quantum spin chain with a microscopic Lie-group symmetry, corresponding e.g. to $U(1)$ or $SU(2)$ spin isotropy, we numerically investigate the emergence of Kac-Moody symmetry at low energies and long distances. In that regime, one such critical quantum spin chain is described by a conformal field theory where the usual Virasoro algebra associated to conformal invariance is augmented with a Kac-Moody algebra associated to conserved currents. Specifically, we first propose a method to construct lattice operators corresponding to the Kac-Moody generators. We then numerically show that, when projected onto low energy states of the quantum spin chain, these operators indeed approximately fulfill the Kac-Moody algebra. The lattice version of the Kac-Moody generators allow us to compute the so-called level constant and to organize the low-energy eigenstates of the lattice Hamiltonian into Kac-Moody towers. We illustrate the proposal with the XXZ model and the Heisenberg model with a next-to-nearest-neighbor coupling.

cond-mat.str-el↗

arXiv2022

Density Matrix Renormalization Group with Tensor Processing Units

Google's Tensor Processing Units (TPUs) are integrated circuits specifically built to accelerate and scale up machine learning workloads. They can perform fast distributed matrix multiplications and therefore be repurposed for other computationally intensive tasks. In this work we demonstrate the use of TPUs for accelerating and scaling up the density matrix renormalization group (DMRG), a powerful numerical approach to compute the ground state of a local quantum many-body Hamiltonian. The cost of DMRG scales with system size $N$ as $O(ND^3)$, where the so-called bond dimension $D$ regulates how expressive the underlying matrix product state (MPS) variational ansatz is. We consider lattice models in two spatial dimensions, with square lattices of size $10\times 10$ (free fermions) and $20\times 20$ (transverse field Ising model), for which the required MPS bond dimension is known to scale at least as $\exp(\sqrt{N})$. Using half of a TPU v3 pod (namely $1,\!024$ TPU v3 cores) we reached an unprecedentedly large bond dimension $D = 2^{16} = 65,\!536$, for which optimizing a single MPS tensor took about 2 minutes.

cond-mat.str-el↗