arXiv subjects

Shima Bab Hadiashar

Publications and source records attributed to Shima Bab Hadiashar.

5 recordsLinked to original sources

arXiv2024

Optimal lower bounds for Quantum Learning via Information Theory

Although a concept class may be learnt more efficiently using quantum samples as compared with classical samples in certain scenarios, Arunachalam and de Wolf (JMLR, 2018) proved that quantum learners are asymptotically no more efficient than classical ones in the quantum PAC and Agnostic learning models. They established lower bounds on sample complexity via quantum state identification and Fourier analysis. In this paper, we derive optimal lower bounds for quantum sample complexity in both the PAC and agnostic models via an information-theoretic approach. The proofs are arguably simpler, and the same ideas can potentially be used to derive optimal bounds for other problems in quantum learning theory. We then turn to a quantum analogue of the Coupon Collector problem, a classic problem from probability theory also of importance in the study of PAC learning. Arunachalam, Belovs, Childs, Kothari, Rosmanis, and de Wolf (TQC, 2020) characterized the quantum sample complexity of this problem up to constant factors. First, we show that the information-theoretic approach mentioned above provably does not yield the optimal lower bound. As a by-product, we get a natural ensemble of pure states in arbitrarily high dimensions which are not easily (simultaneously) distinguishable, while the ensemble has close to maximal Holevo information. Second, we discover that the information-theoretic approach yields an asymptotically optimal bound for an approximation variant of the problem. Finally, we derive a sharper lower bound for the Quantum Coupon Collector problem, via the generalized Holevo-Curlander bounds on the distinguishability of an ensemble. All the aspects of the Quantum Coupon Collector problem we study rest on properties of the spectrum of the associated Gram matrix, which may be of independent interest.

quant-ph↗

arXiv2023

One-shot quantum state redistribution and quantum Markov chains

We revisit the task of quantum state redistribution in the one-shot setting, and design a protocol for this task with communication cost in terms of a measure of distance from quantum Markov chains. More precisely, the distance is defined in terms of quantum max-relative entropy and quantum hypothesis testing entropy. Our result is the first to operationally connect quantum state redistribution and quantum Markov chains, and can be interpreted as an operational interpretation for a possible one-shot analogue of quantum conditional mutual information. The communication cost of our protocol is lower than all previously known ones and asymptotically achieves the well-known rate of quantum conditional mutual information. Thus, our work takes a step towards the important open question of near-optimal characterization of the one-shot quantum state redistribution.

quant-ph↗

arXiv2023

Quantum Inspired Optimization for Industrial Scale Problems

Model-based optimization, in concert with conventional black-box methods, can quickly solve large-scale combinatorial problems. Recently, quantum-inspired modeling schemes based on tensor networks have been developed which have the potential to better identify and represent correlations in datasets. Here, we use a quantum-inspired model-based optimization method TN-GEO to assess the efficacy of these quantum-inspired methods when applied to realistic problems. In this case, the problem of interest is the optimization of a realistic assembly line based on BMW's currently utilized manufacturing schedule. Through a comparison of optimization techniques, we found that quantum-inspired model-based optimization, when combined with conventional black-box methods, can find lower-cost solutions in certain contexts.

quant-ph↗

arXiv2020

On the Entanglement Cost of One-Shot Compression

We revisit the task of visible compression of an ensemble of quantum states with entanglement assistance in the one-shot setting. The protocols achieving the best compression use many more qubits of shared entanglement than the number of qubits in the states in the ensemble. Other compression protocols, with potentially larger communication cost, have entanglement cost bounded by the number of qubits in the given states. This motivates the question as to whether entanglement is truly necessary for compression, and if so, how much of it is needed. Motivated by questions in communication complexity, we lift certain restrictions that are placed on compression protocols in tasks such as state-splitting and channel simulation. We show that an ensemble of the form designed by Jain, Radhakrishnan, and Sen (ICALP'03) saturates the known bounds on the sum of communication and entanglement costs, even with the relaxed compression protocols we study. The ensemble and the associated one-way communication protocol have several remarkable properties. The ensemble is incompressible by more than a constant number of qubits without shared entanglement, even when constant error is allowed. Moreover, in the presence of shared entanglement, the communication cost of compression can be arbitrarily smaller than the entanglement cost. The quantum information cost of the protocol can thus be arbitrarily smaller than the cost of compression without shared entanglement. The ensemble can also be used to show the impossibility of reducing, via compression, the shared entanglement used in two-party protocols for computing Boolean functions.

quant-ph↗

arXiv2018

Communication Complexity of One-Shot Remote State Preparation

Quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of non-oblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate remote state preparation, in which the goal is to prepare an approximation of the desired quantum state. Jain [Quant. Inf. & Comp., 2006] showed that the worst-case communication complexity of approximate RSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this work, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of non-asymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that n bits cannot be communicated with less than n transmitted bits in LOCC protocols. This strengthens a result due to Nayak and Salzman [J. ACM, 2006] and may be of independent interest.

quant-ph↗