arXiv subjects

Srinivasan Arunachalam

Publications and source records attributed to Srinivasan Arunachalam.

At least 37 records · Page 2Linked to original sources

arXiv2024

Matrix hypercontractivity, streaming algorithms and LDCs: the large alphabet case

We prove a hypercontractive inequality for matrix-valued functions defined over large alphabets. In order to do so, we prove a generalization of the powerful $2$-uniform convexity inequality for trace norms of Ball, Carlen, Lieb (Inventiones Mathematicae'94). Using our hypercontractive~inequality, we present upper and lower bounds for the communication complexity of the Hidden Hypermatching problem defined over large alphabets. We then consider streaming algorithms for approximating the value of Unique Games on a hypergraph with $t$-size hyperedges. By using our communication lower bound, we show that every streaming algorithm in the adversarial model achieving an $(r-\varepsilon)$-approximation of this value requires $Ω(n^{1-2/t})$ quantum space, where $r$ is the alphabet size. We next present a lower bound for locally decodable codes (LDC) $\mathbb{Z}_r^n\to \mathbb{Z}_r^N$ over large alphabets with recoverability probability at least $1/r + \varepsilon$. Using hypercontractivity, we give an exponential lower bound $N = 2^{Ω(\varepsilon^4 n/r^4)}$ for $2$-query (possibly non-linear) LDCs over $\mathbb{Z}_r$ and using the non-commutative Khintchine inequality we prove an improved lower bound of $N = 2^{Ω(\varepsilon^2 n/r^2)}$.

cs.CC↗

arXiv2024

A note on polynomial-time tolerant testing stabilizer states

We show an improved inverse theorem for the Gowers-$3$ norm of $n$-qubit quantum states $|ψ\rangle$ which states that: for every $γ\geq 0$, if the $\textsf{Gowers}(|ψ\rangle,3)^8 \geq γ$ then the stabilizer fidelity of $|ψ\rangle$ is at least $γ^C$ for some constant $C>1$. This implies a constant-sample polynomial-time tolerant testing algorithm for stabilizer states which accepts if an unknown state is $\varepsilon_1$-close to a stabilizer state in fidelity and rejects when $|ψ\rangle$ is $\varepsilon_2 \leq \varepsilon_1^C$-far from all stabilizer states, promised one of them is the case.

quant-ph↗

arXiv2024

Distributed inner product estimation with limited quantum communication

We consider the task of distributed inner product estimation when allowed limited quantum communication. Here, Alice and Bob are given $k$ copies of an unknown $n$-qubit quantum states $\vert ψ\rangle,\vert ϕ\rangle$ respectively. They are allowed to communicate $q$ qubits and unlimited classical communication, and their goal is to estimate $|\langle ψ|ϕ\rangle|^2$ up to constant accuracy. We show that $k=Θ(\sqrt{2^{n-q}})$ copies are essentially necessary and sufficient for this task (extending the work of Anshu, Landau and Liu (STOC'22) who considered the case when $q=0$). Additionally, we consider estimating $|\langle ψ|M|ϕ\rangle|^2$, for arbitrary Hermitian $M$. For this task we show that certain norms on $M$ characterize the sample complexity of estimating $|\langle ψ|M|ϕ\rangle|^2$ when using only classical~communication.

quant-ph↗

arXiv2024

Learning low-degree quantum objects

We consider the problem of learning low-degree quantum objects up to $\varepsilon$-error in $\ell_2$-distance. We show the following results: $(i)$ unknown $n$-qubit degree-$d$ (in the Pauli basis) quantum channels and unitaries can be learned using $O(1/\varepsilon^d)$ queries (independent of $n$), $(ii)$ polynomials $p:\{-1,1\}^n\rightarrow [-1,1]$ arising from $d$-query quantum algorithms can be classically learned from $O((1/\varepsilon)^d\cdot \log n)$ many random examples $(x,p(x))$ (which implies learnability even for $d=O(\log n)$), and $(iii)$ degree-$d$ polynomials $p:\{-1,1\}^n\to [-1,1]$ can be learned through $O(1/\varepsilon^d)$ queries to a quantum unitary $U_p$ that block-encodes $p$. Our main technical contributions are new Bohnenblust-Hille inequalities for quantum channels and completely bounded~polynomials.

quant-ph↗

arXiv2023

On the Role of Entanglement and Statistics in Learning

In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-$2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$. $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.

quant-ph↗

arXiv2023

One Clean Qubit Suffices for Quantum Communication Advantage

We study the one-clean-qubit model of quantum communication where one qubit is in a pure state and all other qubits are maximally mixed. We demonstrate a partial function that has a quantum protocol of cost $O(\log N)$ in this model, however, every interactive randomized protocol has cost $Ω(\sqrt{N})$, settling a conjecture of Klauck and Lim. In contrast, all prior quantum versus classical communication separations required at least $Ω(\log N)$ clean qubits. The function demonstrating our separation also has an efficient protocol in the quantum-simultaneous-with-entanglement model of cost $O(\log N )$. We thus recover the state-of-the-art separations between quantum and classical communication complexity. Our proof is based on a recent hypercontractivity inequality introduced by Ellis, Kindler, Lifshitz, and Minzer, in conjunction with tools from the representation theory of compact Lie groups.

quant-ph↗

arXiv2023

Quantum Computing for High-Energy Physics: State of the Art and Challenges. Summary of the QC4HEP Working Group

Quantum computers offer an intriguing path for a paradigmatic change of computing in the natural sciences and beyond, with the potential for achieving a so-called quantum advantage, namely a significant (in some cases exponential) speed-up of numerical simulations. The rapid development of hardware devices with various realizations of qubits enables the execution of small scale but representative applications on quantum computers. In particular, the high-energy physics community plays a pivotal role in accessing the power of quantum computing, since the field is a driving source for challenging computational problems. This concerns, on the theoretical side, the exploration of models which are very hard or even impossible to address with classical techniques and, on the experimental side, the enormous data challenge of newly emerging experiments, such as the upgrade of the Large Hadron Collider. In this roadmap paper, led by CERN, DESY and IBM, we provide the status of high-energy physics quantum computations and give examples for theoretical and experimental target benchmark applications, which can be addressed in the near future. Having the IBM 100 x 100 challenge in mind, where possible, we also provide resource estimates for the examples given using error mitigated quantum computing.

quant-ph↗

arXiv2023

A survey on the complexity of learning quantum states

We survey various recent results that rigorously study the complexity of learning quantum states. These include progress on quantum tomography, learning physical quantum states, alternate learning models to tomography and learning classical functions encoded as quantum states. We highlight how these results are paving the way for a highly successful theory with a range of exciting open questions. To this end, we distill 25 open questions from these results.

quant-ph↗

arXiv2023

Uncertainty Quantification of a Wind Tunnel-Informed Stochastic Wind Load Model for Wind Engineering Applications

The simulation of stochastic wind loads is necessary for many applications in wind engineering. The proper orthogonal decomposition (POD)-based spectral representation method is a popular approach used for this purpose due to its computational efficiency. For general wind directions and building configurations, the data-driven POD-based stochastic model is an alternative that uses wind tunnel smoothed auto- and cross-spectral density as input to calibrate the eigenvalues and eigenvectors of the target load process. Even though this method is straightforward and presents advantages compared to using empirical target auto- and cross-spectral density, the limitations and errors associated with this model have not been investigated. To this end, an extensive experimental study on a rectangular building model considering multiple wind directions and configurations was conducted to allow the quantification of uncertainty related to the use of wind tunnel data for calibration and validation of the data-driven POD-based stochastic model. Errors associated with the use of typical wind tunnel records for model calibration, the model itself, and the truncation of modes were quantified. Results demonstrate that the data-driven model can efficiently simulate stochastic wind loads with negligible model errors, while the errors associated with calibration to typical wind tunnel data can be important.

cs.CE↗

arXiv2023

Generalized Stratified Sampling for Efficient Reliability Assessment of Structures Against Natural Hazards

Performance-based engineering for natural hazards facilitates the design and appraisal of structures with rigorous evaluation of their uncertain structural behavior under potentially extreme stochastic loads expressed in terms of failure probabilities against stated criteria. As a result, efficient stochastic simulation schemes are central to computational frameworks that aim to estimate failure probabilities associated with multiple limit states using limited sample sets. In this work, a generalized stratified sampling scheme is proposed in which two phases of sampling are involved: the first is devoted to the generation of strata-wise samples and the estimation of strata probabilities whereas the second aims at the estimation of strata-wise failure probabilities. Phase-I sampling enables the selection of a generalized stratification variable (i.e., not necessarily belonging to the input set of random variables) for which the probability distribution is not known a priori. To improve the efficiency, Markov Chain Monte Carlo Phase-I sampling is proposed when Monte Carlo simulation is deemed infeasible and optimal Phase-II sampling is implemented based on user-specified target coefficients of variation for the limit states of interest. The expressions for these coefficients are derived with due regard to the sample correlations induced by the Markov chains and the uncertainty in the estimated strata probabilities. The proposed stochastic simulation scheme reaps the benefits of near-optimal stratified sampling for a broader choice of stratification variables in high-dimensional reliability problems with a mechanism to approximately control the accuracy of the failure probability estimators. The practicality of the scheme is demonstrated using two examples involving the estimation of failure probabilities associated with highly nonlinear responses induced by wind and seismic excitations.

cs.CE↗

arXiv2023

Optimal algorithms for learning quantum phase states

We analyze the complexity of learning $n$-qubit quantum phase states. A degree-$d$ phase state is defined as a superposition of all $2^n$ basis vectors $x$ with amplitudes proportional to $(-1)^{f(x)}$, where $f$ is a degree-$d$ Boolean polynomial over $n$ variables. We show that the sample complexity of learning an unknown degree-$d$ phase state is $Θ(n^d)$ if we allow separable measurements and $Θ(n^{d-1})$ if we allow entangled measurements. Our learning algorithm based on separable measurements has runtime $\textsf{poly}(n)$ (for constant $d$) and is well-suited for near-term demonstrations as it requires only single-qubit measurements in the Pauli $X$ and $Z$ bases. We show similar bounds on the sample complexity for learning generalized phase states with complex-valued amplitudes. We further consider learning phase states when $f$ has sparsity-$s$, degree-$d$ in its $\mathbb{F}_2$ representation (with sample complexity $O(2^d sn)$), $f$ has Fourier-degree-$t$ (with sample complexity $O(2^{2t})$), and learning quadratic phase states with $\varepsilon$-global depolarizing noise (with sample complexity $O(n^{1+\varepsilon})$). These learning algorithms give us a procedure to learn the diagonal unitaries of the Clifford hierarchy and IQP~circuits.

quant-ph↗

arXiv2022

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Štefankovič, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Štefankovič, Vempala and Vigoda with the paired-product estimator of Huber (\emph{Ann.\ Appl.\ Probab.}, 25(2),~2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Štefankovič, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

quant-ph↗

arXiv2022

Reliability-Based Collapse Assessment of Wind-Excited Steel Structures within Performance-Based Wind Engineering

As inelastic design for wind is embraced by the engineering community, there is an increasing demand for computational tools that enable the investigation of the nonlinear behavior of wind-excited structures and subsequent development of performance criteria. To address this need, a probabilistic collapse assessment framework for steel structures is proposed in this paper. The framework is based on the integration of a high-fidelity fiber-based nonlinear structural modeling environment with a wind-tunnel-informed stochastic wind load model to perform nonlinear time history analysis. General uncertainty is propagated using a stratified sampling scheme enabling the efficient estimation of reliabilities associated with rare events. The adopted models for simulating high-fidelity nonlinear structural behavior were found, in general, to be adequate for capturing phenomena, including progressive yielding, buckling, and low-cycle fatigue, that are essential for wind induced collapse analysis. In particular, the adopted fatigue model was found to be capable of predicting damage and potential fiber/section fracture associated with non-fully reversing stress-strain cycles that are characteristic of wind loading. Through illustration on a 45-story archetype steel building, critical discussions on the types of observed collapse mechanisms, the difference between alongwind and acrosswind nonlinear behavior, reliabilities associated with first yield, and collapse are presented. A probabilistic description of the residual and peak story drifts is also provided through development of fragility functions.

cs.CE↗

arXiv2022

Quantum Query Algorithms are Completely Bounded Forms

We prove a characterization of $t$-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC'16). Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. Using our characterization, we show that many polynomials of degree four are far from those coming from two-query quantum algorithms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials. Revision note: A mistake was found in the proof of the second result on degree-4 polynomials far from 2-query quantum algorithms. An explanation of the issue, a corrected proof and stronger examples are presented in work of Escudero Gutiérrez and the second author.

quant-ph↗

arXiv2022

The Parameterized Complexity of Quantum Verification

We initiate the study of parameterized complexity of $\textsf{QMA}$ problems in terms of the number of non-Clifford gates in the problem description. We show that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size. This result follows from our main result, that for any Clifford + $t$ $T$-gate quantum circuit satisfiability problem, the search space of optimal witnesses can be reduced to a stabilizer subspace isomorphic to at most $t$ qubits (independent of the system size). Furthermore, we derive new lower bounds on the $T$-count of circuit satisfiability instances and the $T$-count of the $W$-state assuming the classical exponential time hypothesis ($\textsf{ETH}$). Lastly, we explore the parameterized complexity of the quantum non-identity check problem.

quant-ph↗

arXiv2022

On the Gaussian surface area of spectrahedra

We show that for sufficiently large $n\geq 1$ and $d=C n^{3/4}$ for some universal constant $C>0$, a random spectrahedron with matrices drawn from Gaussian orthogonal ensemble has Gaussian surface area $Θ(n^{1/8})$ with high probability.

math.PR↗

arXiv2021

Quantum learning algorithms imply circuit lower bounds

We establish the first general connection between the design of quantum algorithms and circuit lower bounds. Specifically, let $\mathfrak{C}$ be a class of polynomial-size concepts, and suppose that $\mathfrak{C}$ can be PAC-learned with membership queries under the uniform distribution with error $1/2 - γ$ by a time $T$ quantum algorithm. We prove that if $γ^2 \cdot T \ll 2^n/n$, then $\mathsf{BQE} \nsubseteq \mathfrak{C}$, where $\mathsf{BQE} = \mathsf{BQTIME}[2^{O(n)}]$ is an exponential-time analogue of $\mathsf{BQP}$. This result is optimal in both $γ$ and $T$, since it is not hard to learn any class $\mathfrak{C}$ of functions in (classical) time $T = 2^n$ (with no error), or in quantum time $T = \mathsf{poly}(n)$ with error at most $1/2 - Ω(2^{-n/2})$ via Fourier sampling. In other words, even a marginal improvement on these generic learning algorithms would lead to major consequences in complexity theory. Our proof builds on several works in learning theory, pseudorandomness, and computational complexity, and crucially, on a connection between non-trivial classical learning algorithms and circuit lower bounds established by Oliveira and Santhanam (CCC 2017). Extending their approach to quantum learning algorithms turns out to create significant challenges. To achieve that, we show among other results how pseudorandom generators imply learning-to-lower-bound connections in a generic fashion, construct the first conditional pseudorandom generator secure against uniform quantum computations, and extend the local list-decoding algorithm of Impagliazzo, Jaiswal, Kabanets and Wigderson (SICOMP 2010) to quantum circuits via a delicate analysis. We believe that these contributions are of independent interest and might find other applications.

quant-ph↗

arXiv2021

Two new results about quantum exact learning

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a $k$-Fourier-sparse $n$-bit Boolean function from $O(k^{1.5}(\log k)^2)$ uniform quantum examples for that function. This improves over the bound of $\widetildeΘ(kn)$ uniformly random \emph{classical} examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our $\widetilde{O}(k^{1.5})$ upper bound by proving an improvement of Chang's lemma for $k$-Fourier-sparse Boolean functions. Second, we show that if a concept class $\mathcal{C}$ can be exactly learned using $Q$ quantum membership queries, then it can also be learned using $O\left(\frac{Q^2}{\log Q}\log|\mathcal{C}|\right)$ \emph{classical} membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a $\log Q$-factor.

quant-ph↗