arXiv subjects

Ala Shayeghi

Publications and source records attributed to Ala Shayeghi.

8 recordsLinked to original sources

arXiv2026

Provable Quantum--Classical Separation for Continuous Gibbs Sampling

We prove the first quantum--classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-\beta E}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $\alpha=e^{\beta\Delta}$, where $\Delta = \max E-\min E$, every classical algorithm---querying the value, gradient, or any higher-order derivatives of the log-density---requires $\Omega(\alpha)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrt{\alpha}\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{\Omega(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.

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arXiv2026

Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity

We connect different degrees of smoothness of real-valued periodic functions to the number of qubits required for their high-precision Fourier-basis amplitude encodings as quantum states. Our resulting central observation is that the Gevrey hierarchy, which stratifies the space between smooth and analytic functions, provides a natural class for high-precision quantum algorithms. We then specialize to solving general linear partial differential equations (PDEs) with periodic boundary conditions, showing how our Fourier methods do so efficiently at varying target precisions on a quantum computer. This also demonstrates how our framework enables passage from query-complexity results to explicit elementary gate counts. As an application, we introduce a hierarchy of many-body quantum simulation pipelines that harness these high-precision algorithms to probe the linear response of atomistic systems in first quantization. Each level of the hierarchy unlocks a further polynomial-degree quantum speedup, yielding a gradual improvement in simulation efficiency as quantum computers scale.

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arXiv2026

Nonlinear semigroups with unbounded generators under Carleman linearization

We treat the convergence of Carleman linearization of nonlinear evolutionary equations through the approximation theory of strongly continuous semigroups, by Carleman embedding the underlying nonlinear semigroups as linear semigroups. Linear semigroup theory then lets one replace the norm constraint on the convergence of Carleman linearization in the form used by quantum algorithms for a class of semi-discretized evolution equations by a dissipativity constraint, simplifying arguments for convergence. Applying Trotter-Kato approximation theorem to the linearized semigroup realizes the semigroup as a limit finite dimensional operator exponentials, reducing the question of convergence rate of Carleman linearization to that of the Trotter-Kato approximation. We then examine convergence of the Carleman linearization as the operators become unbounded, treating the hyperviscuous Burger's equation as an example. Next we consider the perturbation theory of the Carleman semigroup and obtain conditions when polynomial nonlinearities correspond to the Carleman linearized semigroup being a $1$-integrated semigroup, so convergence is implied by variants of Trotter-Kato approximation for integrated semigroups.

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arXiv2024

A lower bound on the space overhead of fault-tolerant quantum computation

The threshold theorem is a fundamental result in the theory of fault-tolerant quantum computation stating that arbitrarily long quantum computations can be performed with a polylogarithmic overhead provided the noise level is below a constant level. A recent work by Fawzi, Grospellier and Leverrier (FOCS 2018) building on a result by Gottesman (QIC 2013) has shown that the space overhead can be asymptotically reduced to a constant independent of the circuit provided we only consider circuits with a length bounded by a polynomial in the width. In this work, using a minimal model for quantum fault tolerance, we establish a general lower bound on the space overhead required to achieve fault tolerance. For any non-unitary qubit channel $\mathcal{N}$ and any quantum fault tolerance schemes against $\mathrm{i.i.d.}$ noise modeled by $\mathcal{N}$, we prove a lower bound of $\max\left\{\mathrm{Q}(\mathcal{N})^{-1}n,α_\mathcal{N} \log T\right\}$ on the number of physical qubits, for circuits of length $T$ and width $n$. Here, $\mathrm{Q}(\mathcal{N})$ denotes the quantum capacity of $\mathcal{N}$ and $α_\mathcal{N}>0$ is a constant only depending on the channel $\mathcal{N}$. In our model, we allow for qubits to be replaced by fresh ones during the execution of the circuit and we allow classical computation to be free and perfect. This improves upon results that assumed classical computations to be also affected by noise, and that sometimes did not allow for fresh qubits to be added. Along the way, we prove an exponential upper bound on the maximal length of fault-tolerant quantum computation with amplitude damping noise resolving a conjecture by Ben-Or, Gottesman, and Hassidim (2013).

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arXiv2023

A lower bound on the overhead of quantum error correction in low dimensions

We show that a quantum architecture with an error correction procedure limited to geometrically local operations incurs an overhead that grows with the system size, even if arbitrary error-free classical computation is allowed. In particular, we prove that in order to operate a quantum error correcting code in 2D at a logical error rate of $δ$, a space overhead of $Ω(\sqrt{\log(1/δ)})$ is needed for any constant depolarizing noise $p > 0$.

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arXiv2022

A hierarchy of efficient bounds on quantum capacities exploiting symmetry

Optimal rates for achieving an information processing task are often characterized in terms of regularized information measures. In many cases of quantum tasks, we do not know how to compute such quantities. Here, we exploit the symmetries in the recently introduced $D^{\#}$ in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an $ε$ accuracy in time that is polynomial in $1/ε$.

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arXiv2020

Capacity Approaching Coding for Low Noise Interactive Quantum Communication, Part I: Large Alphabets

We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with $n$ messages, designed for a noiseless qudit channel over a $\mathrm{poly}(n)$ size alphabet, our main result is a simulation method that fails with probability less than $2^{-Θ(nε)}$ and uses a qudit channel over the same alphabet $n\left(1+Θ\left(\sqrtε\right)\right)$ times, of which an $ε$ fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the $\sqrtε$ term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Our work improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS'14] for low $ε$.

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arXiv2015

New one shot quantum protocols with application to communication complexity

In this paper we present the following quantum compression protocol: P : Let $ρ,σ$ be quantum states such that $S(ρ|| σ) = \text{Tr} (ρ\log ρ- ρ\log σ)$, the relative entropy between $ρ$ and $σ$, is finite. Alice gets to know the eigen-decomposition of $ρ$. Bob gets to know the eigen-decomposition of $σ$. Both Alice and Bob know $S(ρ|| σ)$ and an error parameter $ε$. Alice and Bob use shared entanglement and after communication of $\mathcal{O}((S(ρ|| σ)+1)/ε^4)$ bits from Alice to Bob, Bob ends up with a quantum state $\tildeρ$ such that $F(ρ, \tildeρ) \geq 1 - 5ε$, where $F(\cdot)$ represents fidelity. This result can be considered as a non-commutative generalization of a result due to Braverman and Rao [2011] where they considered the special case when $ρ$ and $σ$ are classical probability distributions (or commute with each other) and use shared randomness instead of shared entanglement. We use P to obtain an alternate proof of a direct-sum result for entanglement assisted quantum one-way communication complexity for all relations, which was first shown by Jain, Radhakrishnan and Sen [2005,2008]. We also present a variant of protocol P in which Bob has some side information about the state with Alice. We show that in such a case, the amount of communication can be further reduced, based on the side information that Bob has. Our second result provides a quantum analogue of the widely used classical correlated-sampling protocol. For example, Holenstein [2007] used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.

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