arXiv subjects

Alexander Zlokapa

Publications and source records attributed to Alexander Zlokapa.

At least 19 recordsLinked to original sources

arXiv2026

When quantum thermal states look classical

At high temperature, quantum Gibbs states retain several classical features of the maximally mixed state: the absence of entanglement, the absence of magic, analyticity of the partition function, correlation decay, and algorithmic tractability. We prove new and sharp bounds showing that these features persist down to finite temperatures independent of system size, but fail at distinct inverse-temperature scales, forming a hierarchy of classical-to-quantum transitions. Our results hold for long-range Pauli interactions with bounded strength at every site. Despite such all-to-all interactions, we show that the death of entanglement occurs at constant temperature, resolving an open question of Rouze, Franca and Alhambra (STOC'25). We give a polynomial-time classical algorithm that prepares Gibbs states up to the death of entanglement transition. Notably, this is asymptotically colder than temperatures at which quantum Gibbs samplers are known to mix quickly, as well as the original separability temperature of Bakshi et al. (FOCS'24), which we improve to be tight up to constants. At asymptotically even colder temperatures, we show that the Gibbs state remains in the thermodynamic infinite-temperature phase. This leads to polynomial-time classical algorithms for estimating thermal expectations despite both entanglement and magic, and the resolution of a correlation decay conjecture of Harrow, Mehraban and Soleimanifar (STOC'20).

quant-ph↗

arXiv2026

Optimal Lower Bounds for Hamiltonian Simulation

For Hamiltonian $H = \sum_j h_j$, we prove asymptotically tight lower bounds on the gate and query complexities of simulating time evolution on a quantum computer. Our bounds hold for arbitrary term norms $\|h_j\|$, time $t$, and trace-distance error $ε$. The matching upper bound (known as composite qDRIFT) consists of high-order Trotterization of the large terms and a randomized first-order Trotterization of the small terms. Unlike prior work that chooses worst-case $\|h_j\|$ to encode the computation of parity or other Boolean functions in time evolution, our proof is elementary and based on a local, bounded-degree classical Hamiltonian. Our work suggests that for many physical systems (e.g., power-law interactions), gate count must scale polynomially in $1/ε$, contrary to the complexity suggested by counting coherent oracle queries such as those in the block-encoding model.

quant-ph↗

arXiv2026

The free energy limit of the SYK model at high temperature

The Sachdev-Ye-Kitaev (SYK) model is a disordered quantum mean-field model studied in condensed matter physics and the holographic theory of black holes. Its structural properties can be derived heuristically using a combination of the replica method and path integration techniques. Analyzing it mathematically rigorously, however, turned out to be notoriously difficult, even for basic questions such as computing the annealed free energy. In this paper we rigorously compute the free energy limit (annealed and quenched) for this model at high enough but constant temperature. Our results are in numerical agreement with the results derived by physics methods. Remarkably, though, our method of proof is novel and is different from the physics approach. It is based on (a) the theory of the component structure of sparse random graphs and (b) a variant of the cavity method, used widely in prior rigorous and heuristic treatments of classical spin glasses.

cond-mat.dis-nn↗

arXiv2026

SYK thermal expectations are classically easy at any temperature

Estimating thermal expectations of local observables is a natural target for quantum advantage. We give a simple classical algorithm that approximates thermal expectations for Gibbs states of local Hamiltonians, and we show it has quasi-polynomial cost $n^{O(\log (n/ε))}$ for all temperatures above a phase transition in the free energy. For many natural models, this coincides with the entire fast-mixing, quantumly easy phase. Our results apply to the Sachdev-Ye-Kitaev (SYK) model at any constant temperature due to its absence of a phase transition -- despite its entanglement, sign problem, and polynomial quantum circuit lower bounds. Beyond SYK, we rigorously establish a universal classically easy high-temperature phase for all local, bounded-degree Hamiltonians and show that it extends to temperatures strictly colder than the death of entanglement transition.

quant-ph↗

arXiv2026

A rigorous quasipolynomial-time classical algorithm for SYK thermal expectations

Estimating local observables in Gibbs states is a central problem in quantum simulation. While this task is BQP-complete at asymptotically low temperatures, the possibility of quantum advantage at constant temperature remains open. The Sachdev-Ye-Kitaev (SYK) model is a natural candidate: at any constant temperature, its Gibbs states have polynomial quantum circuit complexity and are not described by Gaussian states. Rigorous analyses of the SYK model are difficult due to the failure of known techniques using random matrix theory, cluster expansions, and rigorous formulations of the quantum path integral and replica trick. Despite this, we give a rigorous proof of a quasipolynomial-time classical algorithm that estimates SYK local thermal expectations at sufficiently high constant temperature. Our result introduces a new Wick-pair cluster expansion that we expect to be broadly useful for disordered quantum many-body systems.

quant-ph↗

arXiv2026

Exponential quantum advantage in processing massive classical data

Broadly applicable quantum advantage, particularly in classical data processing and machine learning, has been a fundamental open problem. In this work, we prove that a small quantum computer of polylogarithmic size can perform large-scale classification and dimension reduction on massive classical data by processing samples on the fly, whereas any classical machine achieving the same prediction performance requires exponentially larger size. Furthermore, classical machines that are exponentially larger yet below the required size need superpolynomially more samples and time. We validate these quantum advantages in real-world applications, including single-cell RNA sequencing and movie review sentiment analysis, demonstrating four to six orders of magnitude reduction in size with fewer than 60 logical qubits. These quantum advantages are enabled by quantum oracle sketching, an algorithm for accessing the classical world in quantum superposition using only random classical data samples. Combined with classical shadows, our algorithm circumvents the data loading and readout bottleneck to construct succinct classical models from massive classical data, a task provably impossible for any classical machine that is not exponentially larger than the quantum machine. These quantum advantages persist even when classical machines are granted unlimited time or if BPP=BQP, and rely only on the correctness of quantum mechanics. Together, our results establish machine learning on classical data as a broad and natural domain of quantum advantage and a fundamental test of quantum mechanics at the complexity frontier.

quant-ph↗

arXiv2025

Quartic quantum speedups for community detection

Community detection is a foundational problem in data science. Its natural extension to hypergraphs captures higher-order correlations beyond pairwise interactions. In this work, we develop a quantum algorithm for hypergraph community detection that achieves a quartic quantum speedup over the best known classical algorithm, along with superpolynomial savings in space. Our algorithm is based on the Kikuchi method, which we extend beyond previously considered problems such as Tensor PCA and $p$XORSAT to a broad family of generalized stochastic block models. To demonstrate (near) optimality of this method, we prove matching lower bounds (up to logarithmic factors) in the low-degree framework, showing that the algorithm saturates a smooth statistical-computational tradeoff. The quantum speedup arises from a quantized version of the Kikuchi method and is based on the efficient preparation of a guiding state correlated with the underlying community structure. Our work suggests that prior quantum speedups using the Kikuchi method are sufficiently robust to encompass a broader set of problems than previously believed; we conjecture that a quantity known as marginal order characterizes the existence of these quantum speedups.

quant-ph↗

arXiv2025

Average-case quantum complexity from glassiness

Glassiness -- a phenomenon in physics characterized by a rough free-energy landscape -- implies hardness for stable classical algorithms. For example, it can obstruct constant-time Langevin dynamics and message-passing in random $k$-SAT and max-cut instances. We provide an analogous framework for average-case quantum complexity showing that a natural family of quantum algorithms (e.g., Lindbladian evolution) fails for natural Hamiltonian ensembles (e.g., random 3-local Hamiltonians). Specifically, we prove that the standard notion of quantum glassiness based on replica symmetry breaking obstructs stable quantum algorithms for Gibbs sampling, which we define by a Lipschitz temperature dependence in quantum Wasserstein complexity. Our proof relies on showing that such algorithms fail to capture a structural phase transition in the Gibbs state, where glassiness causes the Gibbs state to decompose into clusters extensively separated in quantum Wasserstein distance. This yields average-case lower bounds for constant-time local Lindbladian evolution and shallow variational circuits. Unlike mixing time lower bounds, our results hold even when dynamics are initialized from the maximally mixed state. We apply these lower bounds to non-commuting, non-stoquastic Hamiltonians by showing a glass transition via the replica trick. We find that the ensemble of all 3-local Pauli strings with independent Gaussian coefficients is average-case hard, while providing analytical evidence that the general $p$-local Pauli ensemble is non-glassy for sufficiently large constant $p$, in contrast to its classical (Ising $p$-spin, always glassy) and fermionic (SYK, never glassy) counterparts.

quant-ph↗

arXiv2024

Slow Mixing of Quantum Gibbs Samplers

Preparing thermal (Gibbs) states is a common task in physics and computer science. Recent algorithms mimic cooling via system-bath coupling, where the cost is determined by mixing time, akin to classical Metropolis-like algorithms. However, few methods exist to demonstrate slow mixing in quantum systems, unlike the well-established classical tools for systems like the Ising model and constraint satisfaction problems. We present a quantum generalization of these tools through a generic bottleneck lemma that implies slow mixing in quantum systems. This lemma focuses on quantum measures of distance, analogous to the classical Hamming distance but rooted in uniquely quantum principles and quantified either through Bohr spectrum jumps or operator locality. Using our bottleneck lemma, we establish unconditional lower bounds on the mixing times of Gibbs samplers for several families of Hamiltonians at low temperatures. For classical Hamiltonians with mixing time lower bounds $T_\mathrm{mix} = \exp[Ω(n^α)]$, we prove that quantum Gibbs samplers also have $T_\mathrm{mix} = \exp[Ω(n^α)]$. This applies to models like random $K$-SAT instances and spin glasses. For stabilizer Hamiltonians, we provide a concise proof of exponential lower bounds $T_\mathrm{mix} = \exp[Ω(n)]$ on mixing times of good $n$-qubit stabilizer codes at low constant temperature. Finally, we consider constant-degree classical Hamiltonians and show how to lift classical slow mixing results in the presence of a transverse field using Poisson Feynman-Kac techniques. We show generic results for models with linear free energy barriers, and we demonstrate that our techniques extend to models with sublinear free energy barriers by proving $T_\mathrm{mix} = \exp[n^{1/2-o(1)}]$ for the ferromagnetic 2D transverse field Ising model.

quant-ph↗

arXiv2024

Hamiltonian simulation for low-energy states with optimal time dependence

We consider the task of simulating time evolution under a Hamiltonian $H$ within its low-energy subspace. Assuming access to a block-encoding of $H'=(H-E)/λ$ for some $E \in \mathbb R$, the goal is to implement an $ε$-approximation to $e^{-itH}$ when the initial state is confined to the subspace corresponding to eigenvalues $[-1, -1+Δ/λ]$ of $H'$. We present a quantum algorithm that uses $O(t\sqrt{λΓ} + \sqrt{λ/Γ}\log(1/ε))$ queries to the block-encoding for any $Γ$ such that $Δ\leq Γ\leq λ$. When $\log(1/ε) = o(tλ)$ and $Δ/λ= o(1)$, this result improves over generic methods with query complexity $Ω(tλ)$. Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform. Using standard access models for $H$, we show that the ability to efficiently block-encode $H'$ is equivalent to $H$ being what we refer to as a "gap-amplifiable" Hamiltonian. This includes physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low-energy simulation algorithms. We also provide lower bounds for low-energy simulation. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to $t$, $Δ$, and $λ$. In the practically relevant regime where $\log (1/ε) = o(tΔ)$ and $Δ/λ= o(1)$, we also prove a matching lower bound in gate complexity (up to log factors). To establish the query lower bounds, we consider $\mathrm{PARITY}\circ\mathrm{OR}$ and degree bounds on trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction acting on a low-energy state.

quant-ph↗

arXiv2024

Bayesian Inference with Deep Weakly Nonlinear Networks

We show at a physics level of rigor that Bayesian inference with a fully connected neural network and a shaped nonlinearity of the form $ϕ(t) = t + ψt^3/L$ is (perturbatively) solvable in the regime where the number of training datapoints $P$ , the input dimension $N_0$, the network layer widths $N$, and the network depth $L$ are simultaneously large. Our results hold with weak assumptions on the data; the main constraint is that $P < N_0$. We provide techniques to compute the model evidence and posterior to arbitrary order in $1/N$ and at arbitrary temperature. We report the following results from the first-order computation: 1. When the width $N$ is much larger than the depth $L$ and training set size $P$, neural network Bayesian inference coincides with Bayesian inference using a kernel. The value of $ψ$ determines the curvature of a sphere, hyperbola, or plane into which the training data is implicitly embedded under the feature map. 2. When $LP/N$ is a small constant, neural network Bayesian inference departs from the kernel regime. At zero temperature, neural network Bayesian inference is equivalent to Bayesian inference using a data-dependent kernel, and $LP/N$ serves as an effective depth that controls the extent of feature learning. 3. In the restricted case of deep linear networks ($ψ=0$) and noisy data, we show a simple data model for which evidence and generalization error are optimal at zero temperature. As $LP/N$ increases, both evidence and generalization further improve, demonstrating the benefit of depth in benign overfitting.

stat.ML↗

arXiv2024

Long-range wormhole teleportation

We extend the protocol of Gao and Jafferis arXiv:1911.07416 to allow wormhole teleportation between two entangled copies of the Sachdev-Ye-Kitaev (SYK) model communicating only through a classical channel. We demonstrate in finite $N$ simulations that the protocol exhibits the characteristic holographic features of wormhole teleportation discussed and summarized in Jafferis et al. https://www.nature.com/articles/s41586-022-05424-3 . We review and exhibit in detail how these holographic features relate to size winding which, as first shown by Brown et al. arXiv:1911.06314 and Nezami et al. arXiv:2102.01064, encodes a dual description of wormhole teleportation.

quant-ph↗

arXiv2024

Fault-Tolerant Neural Networks from Biological Error Correction Codes

It has been an open question in deep learning if fault-tolerant computation is possible: can arbitrarily reliable computation be achieved using only unreliable neurons? In the grid cells of the mammalian cortex, analog error correction codes have been observed to protect states against neural spiking noise, but their role in information processing is unclear. Here, we use these biological error correction codes to develop a universal fault-tolerant neural network that achieves reliable computation if the faultiness of each neuron lies below a sharp threshold; remarkably, we find that noisy biological neurons fall below this threshold. The discovery of a phase transition from faulty to fault-tolerant neural computation suggests a mechanism for reliable computation in the cortex and opens a path towards understanding noisy analog systems relevant to artificial intelligence and neuromorphic computing.

cs.LG↗

arXiv2023

Bayesian Interpolation with Deep Linear Networks

Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit.

stat.ML↗

arXiv2023

Comment on "Comment on "Traversable wormhole dynamics on a quantum processor" "

We observe that the comment of [1, arXiv:2302.07897] is consistent with [2] on key points: i) the microscopic mechanism of the experimentally observed teleportation is size winding and ii) the system thermalizes and scrambles at the time of teleportation. These properties are consistent with a gravitational interpretation of the teleportation dynamics, as opposed to the late-time dynamics. The objections of [1] concern counterfactual scenarios outside of the experimentally implemented protocol.

quant-ph↗

arXiv2021

TensorFlow Quantum: A Software Framework for Quantum Machine Learning

We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.

quant-ph↗

arXiv2021

A quantum algorithm for training wide and deep classical neural networks

Given the success of deep learning in classical machine learning, quantum algorithms for traditional neural network architectures may provide one of the most promising settings for quantum machine learning. Considering a fully-connected feedforward neural network, we show that conditions amenable to classical trainability via gradient descent coincide with those necessary for efficiently solving quantum linear systems. We propose a quantum algorithm to approximately train a wide and deep neural network up to $O(1/n)$ error for a training set of size $n$ by performing sparse matrix inversion in $O(\log n)$ time. To achieve an end-to-end exponential speedup over gradient descent, the data distribution must permit efficient state preparation and readout. We numerically demonstrate that the MNIST image dataset satisfies such conditions; moreover, the quantum algorithm matches the accuracy of the fully-connected network. Beyond the proven architecture, we provide empirical evidence for $O(\log n)$ training of a convolutional neural network with pooling.

quant-ph↗

arXiv2021

Entangling Quantum Generative Adversarial Networks

Generative adversarial networks (GANs) are one of the most widely adopted semisupervised and unsupervised machine learning methods for high-definition image, video, and audio generation. In this work, we propose a new type of architecture for quantum generative adversarial networks (entangling quantum GAN, EQ-GAN) that overcomes some limitations of previously proposed quantum GANs. Leveraging the entangling power of quantum circuits, EQ-GAN guarantees the convergence to a Nash equilibrium under minimax optimization of the discriminator and generator circuits by performing entangling operations between both the generator output and true quantum data. We show that EQ-GAN has additional robustness against coherent errors and demonstrate the effectiveness of EQ-GAN experimentally in a Google Sycamore superconducting quantum processor. By adversarially learning efficient representations of quantum states, we prepare an approximate quantum random access memory (QRAM) and demonstrate its use in applications including the training of quantum neural networks.

quant-ph↗