arXiv subjects

Yaodong Li

Publications and source records attributed to Yaodong Li.

At least 19 recordsLinked to original sources

arXiv2026

Kinetics of sliding-window quantum error correction

Practical implementations of quantum error correction (QEC) require rapid measurement and continuous processing of the syndrome information in order to prevent a backlog of unprocessed data. While ``static'' QEC is theoretically well understood via mappings to equilibrium statistical mechanics models, such an understanding of ``real-time'' QEC is currently lacking. Here, we study the kinetics of sliding window decoding (SWD), an implementation of real-time decoding that acts on temporally local windows of noisy syndrome information and commits to corrections irreversibly at a nonzero rate. We propose an effective description of SWD in terms of a stochastic kinetic process, where $\mathbb{Z}_2$-charged point particles undergo parity-conserving reaction and diffusion. This model describes dynamics at length and time scales large compared to the window size $W$, whereas physics at scales smaller than $W$ leads to nontrivial $W$-dependent scaling of the effective parameters. We identify the rate of decoding $1/W$ as a relevant perturbation to the decodable phase. We also show broad applicability of our results by changing many microscopic details of SWD without affecting the effective description.

quant-ph↗

arXiv2026

Slow mixing and emergent one-form symmetries in three-dimensional $\mathbb{Z}_2$ gauge theory

Symmetry-breaking order at low temperatures is often accompanied by slow relaxation dynamics, due to diverging free-energy barriers arising from interfaces between different ordered states. Here, we extend this correspondence to classical topological order, where the ordered states are locally indistinguishable, so there is no notion of interfaces between them. We study the relaxation dynamics of the three-dimensional (3D) classical $\mathbb{Z}_2$ lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, $t_{\text{mix}} = \exp [\Omega(L)]$, where L is the linear system size. This bound applies even in the presence of perturbations that explicitly break the one-form symmetry between different long-lived states. This perturbation destroys the energy barriers between ordered states, but we show that entropic effects nevertheless lead to diverging free-energy barriers at nonzero temperature. Our proof establishes the LGT as a robust finite-temperature classical memory. We further prove that entropic effects lead to an emergent one-form symmetry, via a notion that we make precise. We argue that the exponential mixing time follows from universal properties of the deconfined phase, and numerically corroborate this expectation by exploring mixing time scales at the Higgs and confinement transitions out of the deconfined phase. These transitions are found to exhibit markedly different dynamic scaling, even though both have the static critical exponents of the 3D Ising model. We expect this novel entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.

cond-mat.stat-mech↗

arXiv2024

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 $\delta = C N e^{-\Theta(d)}$, 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.

quant-ph↗

arXiv2024

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.

quant-ph↗

arXiv2024

Phase diagram of the three-dimensional subsystem toric code

Subsystem quantum error-correcting codes typically involve measuring a sequence of non-commuting parity check operators. They can sometimes exhibit greater fault-tolerance than conventional subspace codes, which use commuting checks. However, unlike subspace codes, it is unclear if subsystem codes -- in particular their advantages -- can be understood in terms of ground state properties of a physical Hamiltonian. In this paper, we address this question for the three-dimensional subsystem toric code (3D STC), as recently constructed by Kubica and Vasmer [Nat. Comm. 13, 6272(2022)], which exhibits single-shot error correction (SSEC). Motivated by a conjectured relation between SSEC and thermal stability, we study the zero and finite temperature phases of an associated non-commuting Hamiltonian. By mapping the Hamiltonian model to a pair of 3D Z_2 gauge theories coupled by a kinetic constraint, we find various phases at zero temperature, all separated by first-order transitions: there are 3D toric code-like phases with deconfined point-like excitations in the bulk, and there are phases with a confined bulk supporting a 2D toric code on the surface when appropriate boundary conditions are chosen. The latter is similar to the surface topological order present in 3D STC. However, the similarities between the SSEC in 3D STC and the confined phases are only partial: they share the same sets of degrees of freedom, but they are governed by different dynamical rules. Instead, we argue that the process of SSEC can more suitably be associated with a path (rather than a point) in the zero-temperature phase diagram, a perspective which inspires alternative measurement sequences enabling SSEC. Moreover, since none of the above-mentioned phases survives at nonzero temperature, SSEC of the code does not imply thermal stability of the associated Hamiltonian phase.

quant-ph↗

arXiv2024

Perturbative stability and error correction thresholds of quantum codes

Topologically-ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors. We connect the two notions of stability by constructing classical statistical mechanics models for decoding general CSS codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized Z2 lattice gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the discretized imaginary time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative X or Z magnetic field. Motivated by error correction considerations, we define general order parameters for all such generalized Z2 lattice gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the LDPC condition and with a sufficiently large code distance, we prove the existence of a low temperature ordered phase of the corresponding lattice gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence to stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and time-like defects in the lattice gauge theory. A high free-energy cost of space-like defects corresponds to a successful "memory experiment" and suppresses the energy splitting among the ground states, while a high free-energy cost of time-like defects corresponds to a successful "stability experiment" and points to a nonzero gap to local excitations.

quant-ph↗

arXiv2024

Decodable hybrid dynamics of open quantum systems with Z_2 symmetry

We explore a class of "open" quantum circuit models with local decoherence ("noise") and local projective measurements, each respecting a global Z_2 symmetry. The model supports a spin glass phase where the Z_2 symmetry is spontaneously broken (not possible in an equilibrium 1d system), a paramagnetic phase characterized by a divergent susceptibility, and an intermediate "trivial" phase. All three phases are also stable to Z_2-symmetric local unitary gates, and the dynamical phase transitions between the phases are in the percolation universality class. The open circuit dynamics can be purified by explicitly introducing a bath with its own "scrambling" dynamics, as in [Bao, Choi, Altman, arXiv:2102.09164], which does not change any of the universal physics. Within the spin glass phase the circuit dynamics can be interpreted as a quantum repetition code, with each stabilizer of the code measured stochastically at a finite rate, and the decoherences as effective bit-flip errors. Motivated by the geometry of the spin glass phase, we devise a novel decoding algorithm for recovering an arbitrary initial qubit state in the code space, assuming knowledge of the history of the measurement outcomes, and the ability of performing local Pauli measurements and gates on the final state. For a circuit with L^d qubits running for time T, the time needed to execute the decoder scales as O(L^d T) (with dimensionality d). With this decoder in hand, we find that the information of the initial encoded qubit state can be retained (and then recovered) for a time logarithmic in L for a 1d circuit, and for a time at least linear in L in 2d below a finite error threshold. For both the repetition and toric codes, we compare and contrast our decoding algorithm with earlier algorithms that map the error model to the random bond Ising model.

quant-ph↗

arXiv2024

Experimental demonstration of scalable cross-entropy benchmarking to detect measurement-induced phase transitions on a superconducting quantum processor

Quantum systems subject to random unitary evolution and measurements at random points in spacetime exhibit entanglement phase transitions which depend on the frequency of these measurements. Past work has experimentally observed entanglement phase transitions on near-term quantum computers, but the characterization approach using entanglement entropy is not scalable due to exponential overhead of quantum state tomography and postselection. Recently, an alternative protocol to detect entanglement phase transitions using linear cross entropy was proposed, attempting to eliminate both bottlenecks. Here, we report demonstrations of this protocol in systems with one-dimensional and all-to-all connectivities on IBM's quantum hardware on up to 22 qubits, a regime which is presently inaccessible if postselection is required. We demonstrate data collapses onto scaling functions with critical exponents in semiquantitative agreement with theory. Our demonstration of the cross entropy benchmark (XEB) protocol paves the way for studies of measurement-induced entanglement phase transitions and associated critical phenomena on larger near-term quantum systems.

quant-ph↗

arXiv2023

Reachable set-based dynamic quantization for the remote state estimation of linear systems

We employ reachability analysis in designing dynamic quantization schemes for the remote state estimation of linear systems over a finite date rate communication channel. The quantization region is dynamically updated at each transmission instant, with an approximated reachable set of the linear system. We propose a set-based method using zonotopes and compare it to a norm-based method in dynamically updating the quantization region. For both methods, we guarantee that the quantization error is bounded and consequently, the remote state reconstruction error is also bounded. To the best of our knowledge, the set-based method using zonotopes has no precedent in the literature and admits a larger class of linear systems and communication channels, where the set-based method allows for a longer inter-transmission time and lower bit rate. Finally, we corroborate our theoretical guarantees with a numerical example.

math.OC↗

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

Triviality of quantum trajectories close to a directed percolation transition

We study quantum circuits consisting of unitary gates, projective measurements, and control operations that steer the system towards a pure absorbing state. Two types of phase transition occur as the rate of these control operations is increased: a measurement-induced entanglement transition, and a directed percolation transition into the absorbing state (taken here to be a product state). In this work we show analytically that these transitions are generically distinct, with the quantum trajectories becoming disentangled before the absorbing state transition is reached, and we analyze their critical properties. We introduce a simple class of models where the measurements in each quantum trajectory define an Effective Tensor Network (ETN) -- a subgraph of the initial spacetime graph where nontrivial time evolution takes place. By analyzing the entanglement properties of the ETN, we show that the entanglement and absorbing-state transitions coincide only in the limit of infinite local Hilbert-space dimension. Focusing on a Clifford model which allows numerical simulations for large system sizes, we verify our predictions and study the finite-size crossover between the two transitions at large local Hilbert space dimension. We give evidence that the entanglement transition is governed by the same fixed point as in hybrid circuits without feedback.

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.

quant-ph↗

arXiv2023

Continuous symmetry breaking in adaptive quantum dynamics

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.

quant-ph↗

arXiv2022

Entanglement phase transitions in random stabilizer tensor networks

We explore a class of random tensor network models with "stabilizer" local tensors which we name Random Stabilizer Tensor Networks (RSTNs). For RSTNs defined on a two-dimensional square lattice, we perform extensive numerical studies of entanglement phase transitions between volume-law and area-law entangled phases of the one-dimensional boundary states. These transitions occur when either (a) the bond dimension $D$ of the constituent tensors is varied, or (b) the tensor network is subject to random breaking of bulk bonds, implemented by forced measurements. In the absence of broken bonds, we find that the RSTN supports a volume-law entangled boundary state with bond dimension $D\geq3$ where $D$ is a prime number, and an area-law entangled boundary state for $D=2$. Upon breaking bonds at random in the bulk with probability $p$, there exists a critical measurement rate $p_c$ for each $D\geq 3$ above which the boundary state becomes area-law entangled. To explore the conformal invariance at these entanglement transitions for different prime $D$, we consider tensor networks on a finite rectangular geometry with a variety of boundary conditions, and extract universal operator scaling dimensions via extensive numerical calculations of the entanglement entropy, mutual information and mutual negativity at their respective critical points. Our results at large $D$ approach known universal data of percolation conformal field theory, while showing clear discrepancies at smaller $D$, suggesting a distinct entanglement transition universality class for each prime $D$. We further study universal entanglement properties in the volume-law phase and demonstrate quantitative agreement with the recently proposed description in terms of a directed polymer in a random environment.

cond-mat.stat-mech↗

arXiv2021

Statistical mechanics model for Clifford random tensor networks and monitored quantum circuits

We introduce an exact mapping of Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, onto a statistical mechanics model. With Haar unitaries, the fundamental degrees of freedom ('spins') are permutations because all operators commuting with the action of the unitaries on a tensor product arise from permutations of the tensor factors ('Schur-Weyl duality'). For unitaries restricted to the smaller Clifford group, the set of commuting operators, the 'commutant', forming the new 'spin' degrees of freedom, will be larger. We use the recent full characterization of this commutant by Gross et al., Comm. Math. Phys. 385, 1325 (2021), to construct the Clifford statistical mechanics models for on-site Hilbert space dimensions which are powers of a prime number $p$. We show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field ${\bf F}_p$. This implies that the symmetry group, and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will in general depend on, and only on the prime $p$. We show that Clifford monitored circuits with on-site Hilbert space dimension $d=p^M$ are described by percolation in the limits $d \to \infty$ at (a) $p=$ fixed but $M\to \infty$, and at (b) $M= 1$ but $p \to \infty$. In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_A =(\sqrt{3}/\pi)\ln p \ln L_A$ for $d=p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert space dimension $d=p$ for a variety of different values of $p$, and find that they approach percolation values at large $p$.

cond-mat.stat-mech↗

arXiv2021

Conformal invariance and quantum non-locality in critical hybrid circuits

We establish the emergence of a conformal field theory (CFT) in a (1+1)-dimensional hybrid quantum circuit right at the measurement-driven entanglement transition by revealing space-time conformal covariance of entanglement entropies and mutual information for various subregions at different circuit depths. While the evolution takes place in real time, the spacetime manifold of the circuit appears to host a Euclidean field theory with imaginary time. Throughout the paper we investigate Clifford circuits with several different boundary conditions by injecting physical qubits at the spatial and/or temporal boundaries, all giving consistent characterizations of the underlying "Clifford CFT." We emphasize (super)universal results that are consequences solely of the conformal invariance and do not depend crucially on the precise nature of the CFT. Among these are the infinite entangling speed as a consequence of measurement-induced quantum nonlocality and the critical purification dynamics of a mixed initial state.

quant-ph↗

arXiv2021

Entanglement Domain Walls in Monitored Quantum Circuits and the Directed Polymer in a Random Environment

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.

cond-mat.stat-mech↗

arXiv2021

Statistical mechanics of quantum error correcting codes

We study stabilizer quantum error correcting codes (QECC) generated under hybrid dynamics of local Clifford unitaries and local Pauli measurements in one dimension. Building upon 1) a general formula relating the error-susceptibility of a subregion to its entanglement properties, and 2) a previously established mapping between entanglement entropies and domain wall free energies of an underlying spin model, we propose a statistical mechanical description of the QECC in terms of "entanglement domain walls". Free energies of such domain walls generically feature a leading volume law term coming from its "surface energy", and a sub-volume law correction coming from thermodynamic entropies of its transverse fluctuations. These are most easily accounted for by capillary-wave theory of liquid-gas interfaces, which we use as an illustrative tool. We show that the information-theoretic decoupling criterion corresponds to a geometric decoupling of domain walls, which further leads to the identification of the "contiguous code distance" of the QECC as the crossover length scale at which the energy and entropy of the domain wall are comparable. The contiguous code distance thus diverges with the system size as the subleading entropic term of the free energy, protecting a finite code rate against local undetectable errors. We support these correspondences with numerical evidence, where we find capillary-wave theory describes many qualitative features of the QECC; we also discuss when and why it fails to do so.

quant-ph↗