arXiv subjects

Patrick Hayden

Publications and source records attributed to Patrick Hayden.

At least 19 recordsLinked to original sources

arXiv2026

Classical and Quantum Speedups for Non-Convex Optimization via Energy Conserving Descent

The Energy Conserving Descent (ECD) algorithm was recently proposed (De Luca & Silverstein, 2022) as a global non-convex optimization method. Unlike gradient descent, appropriately configured ECD dynamics escape strict local minima and converge to a global minimum, making it appealing for machine learning optimization. We present the first analytical study of ECD, focusing on the one-dimensional setting for this first installment. We formalize a stochastic ECD dynamics (sECD) with energy-preserving noise, as well as a quantum analog of the ECD Hamiltonian (qECD), providing the foundation for a quantum algorithm through Hamiltonian simulation. For positive double-well objectives, we compute the expected hitting time from a local to the global minimum. We prove that both sECD and qECD yield exponential speedup over respective gradient descent baselines--stochastic gradient descent and its quantization. For objectives with tall barriers, qECD achieves a further speedup over sECD.

quant-ph↗

arXiv2026

GKP-inspired high-dimensional superdense coding with energy-time entanglement

Superdense coding, the application of entanglement to boost classical communication capacity, is a cornerstone of quantum communication. In this paper, we propose a high-dimensional superdense coding protocol using energy-time entangled states. These states are biphoton frequency combs, an example of entangled time-frequency Gottesman-Kitaev-Preskill (TFGKP) states or time-frequency grid states. Inspired by GKP codes, our protocol involves discretizing the continuous time and frequency degrees of freedom and encoding information by time-frequency displacements. This approach leverages the inherently large Hilbert space found in quantum frequency combs, with resilience against both temporal and spectral errors. In addition to describing the theoretical structure of the protocol, we propose an experimental implementation using standard telecommunication components, time-resolving single-photon detectors and a frequency beamsplitter. We also analyze the effect of experimental noise and errors on the channel capacity of the protocol. We demonstrate that for realistic experimental parameters, contemporary technologies satisfy the prerequisites for superdense coding with biphoton frequency combs, achieving a transmission rate of approximately 8.91 bits per transmitted photon (equivalent to 481 distinguishable messages with asymptotically vanishing errors). This more than doubles the previously highest transmission rate of 4 bits achieved by the Kwiat-Weinfurter scheme, while also having competitive optical loss. Furthermore, our results beat the rate achievable using a single-photon frequency comb with identical parameters by 4.6 times. Our protocol thus represents an experimentally feasible application of time-frequency grid states to entanglement-assisted communication, contributing to the active fields of continuous-variable and high-dimensional quantum information.

quant-ph↗

arXiv2025

Alpha-bit state merging

State merging is a fundamental protocol in quantum information theory that generalizes quantum teleportation. Traditionally, it is achieved by local operations on shared entanglement and classical communication. In this work, we study state merging done with alpha-bits, a versatile quantum communication resource weaker than qubits. We study alpha-bit state merging with and without catalytic entanglement, and we find a potential gap between the rates of alpha-bits consumed. In light of our result, we discuss how to interpret entanglement wedge reconstruction in AdS/CFT in terms of alpha-bit state merging

quant-ph↗

arXiv2025

Efficiently Computable Limits on EPR Pair Generation in Quantum Broadcast Channels

We investigate the generation of EPR pairs between three observers in a general causally structured setting, where communication occurs via a noisy quantum broadcast channel. The most general quantum codes for this setup take the form of tripartite quantum channels. Since the receivers are constrained by causal ordering, additional temporal relationships naturally emerge between the parties. These causal constraints enforce intrinsic no-signalling conditions on any tripartite operation, ensuring that it constitutes a physically realizable quantum code for a quantum broadcast channel. We analyze these constraints and, more broadly, characterize the most general quantum codes for communication over such channels. We examine the capabilities of codes that are fully no-signalling among the three parties, positive partial transpose (PPT)-preserving, or both, and derive simple semidefinite programs to compute the achievable entanglement fidelity. We then establish a hierarchy of semidefinite programming converse bounds -- both weak and strong -- for the capacity of quantum broadcast channels for EPR pair generation, in both one-shot and asymptotic regimes. Notably, in the special case of a point-to-point channel, our strong converse bound recovers and strengthens existing results. Finally, we demonstrate how the PPT-preserving codes we develop can be leveraged to construct PPT-preserving entanglement combing schemes, and vice versa.

quant-ph↗

arXiv2024

Three-Receiver Quantum Broadcast Channels: Classical Communication with Quantum Non-unique Decoding

In network communication, it is common in broadcasting scenarios for there to exist a hierarchy among receivers based on information they decode due, for example, to different physical conditions or premium subscriptions. This hierarchy may result in varied information quality, such as higher-quality video for certain receivers. This is modeled mathematically as a degraded message set, indicating a hierarchy between messages to be decoded by different receivers, where the default quality corresponds to a common message intended for all receivers, a higher quality is represented by a message for a smaller subset of receivers, and so forth. We extend these considerations to quantum communication, exploring three-receiver quantum broadcast channels with two- and three-degraded message sets. Our technical tool involves employing quantum non-unique decoding, a technique we develop by utilizing the simultaneous pinching method. We construct one-shot codes for various scenarios and find achievable rate regions relying on various quantum R\'enyi mutual information error exponents. Our investigation includes a comprehensive study of pinching across tensor product spaces, presenting our findings as the asymptotic counterpart to our one-shot codes. By employing the non-unique decoding, we also establish a simpler proof to Marton's inner bound for two-receiver quantum broadcast channels without the need for more involved techniques. Additionally, we derive no-go results and demonstrate their tightness in special cases.

quant-ph↗

arXiv2024

Learning interacting fermionic Hamiltonians at the Heisenberg limit

Efficiently learning an unknown Hamiltonian given access to its dynamics is a problem of interest for quantum metrology, many-body physics and machine learning. A fundamental question is whether learning can be performed at the Heisenberg limit, where the Hamiltonian evolution time scales inversely with the error, $\varepsilon$, in the reconstructed parameters. The Heisenberg limit has previously been shown to be achievable for certain classes of qubit and bosonic Hamiltonians. Most recently, a Heisenberg-limited learning algorithm was proposed for a simplified class of fermionic Hubbard Hamiltonians restricted to real hopping amplitudes and zero chemical potential at all sites, along with on-site interactions. In this work, we provide an algorithm to learn a more general class of fermionic Hubbard Hamiltonians at the Heisenberg limit, allowing complex hopping amplitudes and nonzero chemical potentials in addition to the on-site interactions, thereby including several models of physical interest. The required evolution time across all experiments in our protocol is $\mathcal{O}(1/\varepsilon)$ and the number of experiments required to learn all the Hamiltonian parameters is $\mathcal{O}(\text{polylog}(1/\varepsilon))$, independent of system size as long as each fermionic mode interacts with $\mathcal{O}(1)$ other modes. Unlike prior algorithms for bosonic and fermionic Hamiltonians, to obey fermionic parity superselection constraints in our more general setting, our protocol utilizes $\mathcal{O}(N)$ ancillary fermionic modes, where $N$ is the system size. Each experiment involves preparing fermionic Gaussian states, interleaving time evolution with fermionic linear optics unitaries, and performing local occupation number measurements on the fermionic modes. The protocol is robust to a constant amount of state preparation and measurement error.

quant-ph↗

arXiv2024

Efficient Unitary T-designs from Random Sums

Unitary $T$-designs play an important role in quantum information, with diverse applications in quantum algorithms, benchmarking, tomography, and communication. Until now, the most efficient construction of unitary $T$-designs for $n$-qudit systems has been via random local quantum circuits, which have been shown to converge to approximate $T$-designs in the diamond norm using $O(T^{5+o(1)} n^2)$ quantum gates. In this work, we provide a new construction of $T$-designs via random matrix theory using $\tilde{O}(T^2 n^2)$ quantum gates. Our construction leverages two key ideas. First, in the spirit of central limit theorems, we approximate the Gaussian Unitary Ensemble (GUE) by an i.i.d. sum of random Hermitian matrices. Second, we show that the product of just two exponentiated GUE matrices is already approximately Haar random. Thus, multiplying two exponentiated sums over rather simple random matrices yields a unitary $T$-design, via Hamiltonian simulation. A central feature of our proof is a new connection between the polynomial method in quantum query complexity and the large-dimension ($N$) expansion in random matrix theory. In particular, we show that the polynomial method provides exponentially improved bounds on the high moments of certain random matrix ensembles, without requiring intricate Weingarten calculations. In doing so, we define and solve a new type of moment problem on the unit circle, asking whether a finite number of equally weighted points, corresponding to eigenvalues of unitary matrices, can reproduce a given set of moments.

quant-ph↗

arXiv2024

Security of quantum position-verification limits Hamiltonian simulation via holography

We investigate the link between quantum position-verification (QPV) and holography established in [MPS19] using holographic quantum error correcting codes as toy models. By inserting the "temporal" scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.

quant-ph↗

arXiv2024

Entanglement cost for infinite-dimensional physical systems

We prove that the entanglement cost equals the regularized entanglement of formation for any infinite-dimensional quantum state $\rho_{AB}$ with finite quantum entropy on at least one of the subsystems $A$ or $B$. This generalizes a foundational result in quantum information theory that was previously formulated only for operations and states on finite-dimensional systems. The extension to infinite-dimensional systems is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typicality, monotonicity, and asymptotic continuity, are no longer directly applicable. To address this problem, we construct a new entanglement dilution protocol for infinite-dimensional states implementable by local operations and a finite amount of one-way classical communication (one-way LOCC), using weak and strong typicality multiple times. We also prove the optimality of this protocol among all protocols, even under infinite-dimensional separable operations, by developing an argument based on alternative forms of monotonicity and asymptotic continuity of the entanglement of formation for infinite-dimensional states. Along the way, we derive a new integral representation for the quantum entropy of infinite-dimensional states, which we believe to be of independent interest. Our results allow us to fully characterize an important operational entanglement measure -- the entanglement cost -- for all infinite-dimensional physical systems.

quant-ph↗

arXiv2023

Efficient Quantum Algorithm for Port-based Teleportation

In this paper, we provide the first efficient algorithm for port-based teleportation, a unitarily equivariant version of teleportation useful for constructing programmable quantum processors and performing instantaneous nonlocal computation (NLQC). The latter connection is important in AdS/CFT, where bulk computations are realized as boundary NLQC. Our algorithm yields an exponential improvement to the known relationship between the amount of entanglement available and the complexity of the nonlocal part of any unitary that can be implemented using NLQC. Similarly, our algorithm provides the first nontrivial efficient algorithm for an approximate universal programmable quantum processor. The key to our approach is a generalization of Schur-Weyl duality we call twisted Schur-Weyl duality, as well as an efficient algorithm we develop for the twisted Schur transform, which transforms to a subgroup-reduced irrep basis of the partially transposed permutation algebra, whose dual is the $U^{\otimes n-k} \otimes (U^*)^{\otimes k}$ representation of the unitary group.

quant-ph↗

arXiv2023

What exactly does Bekenstein bound?

The Bekenstein bound posits a maximum entropy for matter with finite energy confined to a spatial region. It is often interpreted as a fundamental limit on the information that can be stored by physical objects. In this work, we test this interpretation by asking whether the Bekenstein bound imposes constraints on a channel's communication capacity, a context in which information can be given a mathematically rigorous and operationally meaningful definition. We study specifically the \emph{Unruh channel} that describes a stationary Alice exciting different species of free scalar fields to send information to an accelerating Bob, who is confined to a Rindler wedge and exposed to the noise of Unruh radiation. We show that the classical and quantum capacities of the Unruh channel obey the Bekenstein bound that pertains to the decoder Bob. In contrast, even at high temperatures, the Unruh channel can transmit a significant number of \emph{zero-bits}, which are quantum communication resources that can be used for quantum identification and many other primitive protocols. Therefore, unlike classical bits and qubits, zero-bits and their associated information processing capability are generally not constrained by the Bekenstein bound. However, we further show that when both the encoder and the decoder are restricted, the Bekenstein bound does constrain the channel capacities, including the zero-bit capacity.

hep-th↗

arXiv2023

Reflected entropy is not a correlation measure

By explicit counterexample, we show that the "reflected entropy" defined by Dutta and Faulkner is not monotonically decreasing under partial trace, and so is not a measure of physical correlations. In fact, our counterexamples show that none of the R\'enyi reflected entropies $S_{R}^{(\alpha)}$ for $0 < \alpha < 2$ is a correlation measure; the usual reflected entropy is realized as the $\alpha=1$ member of this family. The counterexamples are given by quantum states that correspond to classical probability distributions, so reflected entropy fails to measure correlations even at the classical level.

hep-th↗

arXiv2022

Concatenation Schemes for Topological Fault-tolerant Quantum Error Correction

We investigate a family of fault-tolerant quantum error correction schemes based on the concatenation of small error detection or error correction codes with the three-dimensional cluster state. We propose fault-tolerant state preparation and decoding schemes that effectively convert every circuit-level error into an erasure error, leveraging the cluster state's high threshold against such errors. We find a set of codes for which such a conversion is possible, and study their performance against the standard circuit-level depolarizing model. Our best performing scheme, which is based on a concatenation with a classical code, improves the threshold by $16.5\%$ and decreases the spacetime overhead by $32\%$ compared to the scheme without concatenation, with each scheme subject to a physical error rate of $10^{-3}$ and achieving a logical error rate of $10^{-6}$.

quant-ph↗

arXiv2021

A canonical Hamiltonian for open quantum systems

If an open quantum system is initially uncorrelated from its environment, then its dynamics can be written in terms of a Lindblad-form master equation. The master equation is divided into a unitary piece, represented by an effective Hamiltonian, and a dissipative piece, represented by a hermiticity-preserving superoperator; however, the division of open system dynamics into unitary and dissipative pieces is non-unique. For finite-dimensional quantum systems, we resolve this non-uniqueness by specifying a norm on the space of dissipative superoperators and defining the canonical Hamiltonian to be the one whose dissipator is minimal. We show that the canonical Hamiltonian thus defined is equivalent to the Hamiltonian initially defined by Lindblad, and that it is uniquely specified by requiring the dissipator's jump operators to be traceless, extending a uniqueness result known previously in the special case of Markovian master equations. For a system weakly coupled to its environment, we give a recursive formula for computing the canonical effective Hamiltonian to arbitrary orders in perturbation theory, which we can think of as a perturbative scheme for renormalizing the system's bare Hamiltonian.

quant-ph↗

arXiv2021

The Markov gap for geometric reflected entropy

The reflected entropy $S_R(A:B)$ of a density matrix $\rho_{AB}$ is a bipartite correlation measure lower-bounded by the quantum mutual information $I(A:B)$. In holographic states satisfying the quantum extremal surface formula, where the reflected entropy is related to the area of the entanglement wedge cross-section, there is often an order-$N^2$ gap between $S_R$ and $I$. We provide an information-theoretic interpretation of this gap by observing that $S_R - I$ is related to the fidelity of a particular Markov recovery problem that is impossible in any state whose entanglement wedge cross-section has a nonempty boundary; for this reason, we call the quantity $S_R - I$ the Markov gap. We then prove that for time-symmetric states in pure AdS$_3$ gravity, the Markov gap is universally lower bounded by $\log(2) \ell_{\text{AdS}}/2 G_N$ times the number of endpoints of the cross-section. We provide evidence that this lower bound continues to hold in the presence of bulk matter, and comment on how it might generalize above three bulk dimensions. Finally, we explore the Markov recovery problem controlling $S_R - I$ using fixed area states. This analysis involves deriving a formula for the quantum fidelity -- in fact, for all the sandwiched R\'enyi relative entropies -- between fixed area states with one versus two fixed areas, which may be of independent interest. We discuss, throughout the paper, connections to the general theory of multipartite entanglement in holography.

hep-th↗

arXiv2021

Perturbative quantum simulation

Approximation based on perturbation theory is the foundation for most of the quantitative predictions of quantum mechanics, whether in quantum many-body physics, chemistry, quantum field theory or other domains. Quantum computing provides an alternative to the perturbation paradigm, yet state-of-the-art quantum processors with tens of noisy qubits are of limited practical utility. Here, we introduce perturbative quantum simulation, which combines the complementary strengths of the two approaches, enabling the solution of large practical quantum problems using limited noisy intermediate-scale quantum hardware. The use of a quantum processor eliminates the need to identify a solvable unperturbed Hamiltonian, while the introduction of perturbative coupling permits the quantum processor to simulate systems larger than the available number of physical qubits. We present an explicit perturbative expansion that mimics the Dyson series expansion and involves only local unitary operations, and show its optimality over other expansions under certain conditions. We numerically benchmark the method for interacting bosons, fermions, and quantum spins in different topologies, and study different physical phenomena, such as information propagation, charge-spin separation, and magnetism, on systems of up to $48$ qubits only using an $8+1$ qubit quantum hardware. We experimentally demonstrate our scheme on the IBM quantum cloud, verifying its noise robustness and illustrating its potential for benchmarking large quantum processors with smaller ones.

quant-ph↗

arXiv2021

Automatic Knowledge Extraction with Human Interface

OrbWeaver, an automatic knowledge extraction system paired with a human interface, streamlines the use of unintuitive natural language processing software for modeling systems from their documentation. OrbWeaver enables the indirect transfer of knowledge about legacy systems by leveraging open source tools in document understanding and processing as well as using web based user interface constructs. By design, OrbWeaver is scalable, extensible, and usable; we demonstrate its utility by evaluating its performance in processing a corpus of documents related to advanced persistent threats in the cyber domain. The results indicate better knowledge extraction by revealing hidden relationships, linking co-related entities, and gathering evidence.

cs.HC↗

arXiv2020

Black hole microstates vs. the additivity conjectures

We argue that one of the following statements must be true: (a) extensive violations of quantum information theory's additivity conjectures exist or (b) there exists a set of `disentangled' black hole microstates that can account for the entire Bekenstein-Hawking entropy, up to at most a subleading $O(1)$ correction. Possibility (a) would be a significant result in quantum communication theory, demonstrating that entanglement can enhance the ability to transmit information much more than has currently been established. Option (b) would provide new insight into the microphysics of black holes. In particular, the disentangled microstates would have to have nontrivial structure at or outside the black hole horizon, assuming the validity of the quantum extremal surface prescription for calculating entanglement entropy in AdS/CFT.

hep-th↗