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Ryan O'Donnell

Publications and source records attributed to Ryan O'Donnell.

At least 37 records · Page 2Linked to original sources
arXiv2024

Coboundary expansion inside Chevalley coset complex HDXs

Recent major results in property testing~\cite{BLM24,DDL24} and PCPs~\cite{BMV24} were unlocked by moving to high-dimensional expanders (HDXs) constructed from $\widetilde{C}_d$-type buildings, rather than the long-known $\widetilde{A}_d$-type ones. At the same time, these building quotient HDXs are not as easy to understand as the more elementary (and more symmetric/explicit) \emph{coset complex} HDXs constructed by Kaufman--Oppenheim~\cite{KO18} (of $A_d$-type) and O'Donnell--Pratt~\cite{OP22} (of $B_d$-, $C_d$-, $D_d$-type). Motivated by these considerations, we study the $B_3$-type generalization of a recent work of Kaufman--Oppenheim~\cite{KO21}, which showed that the $A_3$-type coset complex HDXs have good $1$-coboundary expansion in their links, and thus yield $2$-dimensional topological expanders. The crux of Kaufman--Oppenheim's proof of $1$-coboundary expansion was: (1)~identifying a group-theoretic result by Biss and Dasgupta~\cite{BD01} on small presentations for the $A_3$-unipotent group over~$\mathbb{F}_q$; (2)~``lifting'' it to an analogous result for an $A_3$-unipotent group over polynomial extensions~$\mathbb{F}_q[x]$. For our $B_3$-type generalization, the analogue of~(1) appears to not hold. We manage to circumvent this with a significantly more involved strategy: (1)~getting a computer-assisted proof of vanishing $1$-cohomology of $B_3$-type unipotent groups over~$\mathbb{F}_5$; (2)~developing significant new ``lifting'' technology to deduce the required quantitative $1$-cohomology results in $B_3$-type unipotent groups over $\mathbb{F}_{5^k}[x]$.

math.GR↗
arXiv2024

Uniformity testing when you have the source code

We study quantum algorithms for verifying properties of the output probability distribution of a classical or quantum circuit, given access to the source code that generates the distribution. We consider the basic task of uniformity testing, which is to decide if the output distribution is uniform on $[d]$ or $ε$-far from uniform in total variation distance. More generally, we consider identity testing, which is the task of deciding if the output distribution equals a known hypothesis distribution, or is $ε$-far from it. For both problems, the previous best known upper bound was $O(\min\{d^{1/3}/ε^{2},d^{1/2}/ε\})$. Here we improve the upper bound to $O(\min\{d^{1/3}/ε^{4/3}, d^{1/2}/ε\})$, which we conjecture is optimal.

quant-ph↗
arXiv2024

Computability Theory of Closed Timelike Curves

We study the question of what is computable by Turing machines equipped with time travel into the past; i.e., with Deutschian closed timelike curves (CTCs) having no bound on their width or length. An alternative viewpoint is that we study the complexity of finding approximate fixed points of computable Markov chains and quantum channels of countably infinite dimension. Our main result is that the complexity of these problems is precisely $Δ_2$, the class of languages Turing-reducible to the Halting problem. Establishing this as an upper bound for qubit-carrying CTCs requires recently developed results in the theory of quantum Markov maps.

quant-ph↗
arXiv2024

Query-optimal estimation of unitary channels in diamond distance

We consider process tomography for unitary quantum channels. Given access to an unknown unitary channel acting on a $\textsf{d}$-dimensional qudit, we aim to output a classical description of a unitary that is $\varepsilon$-close to the unknown unitary in diamond norm. We design an algorithm achieving error $\varepsilon$ using $O(\textsf{d}^2/\varepsilon)$ applications of the unknown channel and only one qudit. This improves over prior results, which use $O(\textsf{d}^3/\varepsilon^2)$ [via standard process tomography] or $O(\textsf{d}^{2.5}/\varepsilon)$ [Yang, Renner, and Chiribella, PRL 2020] applications. To show this result, we introduce a simple technique to "bootstrap" an algorithm that can produce constant-error estimates to one that can produce $\varepsilon$-error estimates with the Heisenberg scaling. Finally, we prove a complementary lower bound showing that estimation requires $Ω(\textsf{d}^2/\varepsilon)$ applications, even with access to the inverse or controlled versions of the unknown unitary. This shows that our algorithm has both optimal query complexity and optimal space complexity.

quant-ph↗
arXiv2024

Quantum chi-squared tomography and mutual information testing

For quantum state tomography on rank-$r$ dimension-$d$ states, we show that $\widetilde{O}(r^{.5}d^{1.5}/ε) \leq \widetilde{O}(d^2/ε)$ copies suffice for accuracy~$ε$ with respect to (Bures) $χ^2$-divergence, and $\widetilde{O}(rd/ε)$ copies suffice for accuracy~$ε$ with respect to quantum relative entropy. The best previous bound was $\widetilde{O}(rd/ε) \leq \widetilde{O}(d^2/ε)$ with respect to infidelity; our results are an improvement since infidelity is bounded above by both the relative entropy and the $χ^2$-divergence. For algorithms that are required to use single-copy measurements, we show that $\widetilde{O}(r^{1.5} d^{1.5}/ε) \leq \widetilde{O}(d^3/ε)$ copies suffice for $χ^2$-divergence, and $\widetilde{O}(r^{2} d/ε)$ suffice for relative entropy. Using this tomography algorithm, we show that $\widetilde{O}(d^{2.5}/ε)$ copies of a $d\times d$-dimensional bipartite state suffice to test if it has quantum mutual information~$0$ or at least~$ε$. As a corollary, we also improve the best known sample complexity for the \emph{classical} version of mutual information testing to $\widetilde{O}(d/ε)$.

quant-ph↗
arXiv2024

Improved quantum data analysis

We provide more sample-efficient versions of some basic routines in quantum data analysis, along with simpler proofs. Particularly, we give a quantum "Threshold Search" algorithm that requires only $O((\log^2 m)/ε^2)$ samples of a $d$-dimensional state $ρ$. That is, given observables $0 \le A_1, A_2, ..., A_m \le 1$ such that $\mathrm{tr}(ρA_i) \ge 1/2$ for at least one $i$, the algorithm finds $j$ with $\mathrm{tr}(ρA_j) \ge 1/2-ε$. As a consequence, we obtain a Shadow Tomography algorithm requiring only $\tilde{O}((\log^2 m)(\log d)/ε^4)$ samples, which simultaneously achieves the best known dependence on each parameter $m$, $d$, $ε$. This yields the same sample complexity for quantum Hypothesis Selection among $m$ states; we also give an alternative Hypothesis Selection method using $\tilde{O}((\log^3 m)/ε^2)$ samples.

quant-ph↗
arXiv2023

Explicit orthogonal and unitary designs

We give a strongly explicit construction of $\varepsilon$-approximate $k$-designs for the orthogonal group $\mathrm{O}(N)$ and the unitary group $\mathrm{U}(N)$, for $N=2^n$. Our designs are of cardinality $\mathrm{poly}(N^k/\varepsilon)$ (equivalently, they have seed length $O(nk + \log(1/\varepsilon)))$; up to the polynomial, this matches the number of design elements used by the construction consisting of completely random matrices.

cs.CC↗
arXiv2023

Optimizing Strongly Interacting Fermionic Hamiltonians

The fundamental problem in much of physics and quantum chemistry is to optimize a low-degree polynomial in certain anticommuting variables. Being a quantum mechanical problem, in many cases we do not know an efficient classical witness to the optimum, or even to an approximation of the optimum. One prominent exception is when the optimum is described by a so-called "Gaussian state", also called a free fermion state. In this work we are interested in the complexity of this optimization problem when no good Gaussian state exists. Our primary testbed is the Sachdev--Ye--Kitaev (SYK) model of random degree-$q$ polynomials, a model of great current interest in condensed matter physics and string theory, and one which has remarkable properties from a computational complexity standpoint. Among other results, we give an efficient classical certification algorithm for upper-bounding the largest eigenvalue in the $q=4$ SYK model, and an efficient quantum certification algorithm for lower-bounding this largest eigenvalue; both algorithms achieve constant-factor approximations with high probability.

quant-ph↗
arXiv2022

Explicit near-Ramanujan graphs of every degree

For every constant $d \geq 3$ and $ε> 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Θ(n)$ vertices that is $ε$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + ε$ (excluding the single trivial eigenvalue of~$d$).

cs.DS↗
arXiv2022

Mean estimation when you have the source code; or, quantum Monte Carlo methods

Suppose $\boldsymbol{y}$ is a real random variable, and one is given access to ``the code'' that generates it (for example, a randomized or quantum circuit whose output is $\boldsymbol{y}$). We give a quantum procedure that runs the code $O(n)$ times and returns an estimate $\widehat{\boldsymbolμ}$ for $μ= \mathrm{E}[\boldsymbol{y}]$ that with high probability satisfies $|\widehat{\boldsymbolμ} - μ| \leq σ/n$, where $σ= \mathrm{stddev}[\boldsymbol{y}]$. This dependence on $n$ is optimal for quantum algorithms. One may compare with classical algorithms, which can only achieve the quadratically worse $|\widehat{\boldsymbolμ} - μ| \leq σ/\sqrt{n}$. Our method improves upon previous works, which either made additional assumptions about $\boldsymbol{y}$, and/or assumed the algorithm knew an a priori bound on $σ$, and/or used additional logarithmic factors beyond $O(n)$. The central subroutine for our result is essentially Grover's algorithm but with complex phases.ally Grover's algorithm but with complex phases.

quant-ph↗
arXiv2022

High-Dimensional Expanders from Chevalley Groups

Let $Φ$ be an irreducible root system (other than $G_2$) of rank at least $2$, let $\mathbb{F}$ be a finite field with $p = \operatorname{char} \mathbb{F} > 3$, and let $\mathrm{G}(Φ,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(Φ)$, where $\mathrm{G}(Φ,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $λ$-spectral HDXs with $λ\to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $Φ= A_d$. Our work gives three new families of spectral HDXs of any dimension $\ge 2$, and four exceptional constructions of dimension $4$, $6$, $7$, and $8$.

cs.DM↗
arXiv2022

Fooling Gaussian PTFs via Local Hyperconcentration

We give a pseudorandom generator that fools degree-$d$ polynomial threshold functions over $n$-dimensional Gaussian space with seed length $\mathrm{poly}(d)\cdot \log n$. All previous generators had a seed length with at least a $2^d$ dependence on $d$. The key new ingredient is a Local Hyperconcentration Theorem, which shows that every degree-$d$ Gaussian polynomial is hyperconcentrated almost everywhere at scale $d^{-O(1)}$.

cs.CC↗
arXiv2021

Explicit Abelian Lifts and Quantum LDPC Codes

For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and each edge by a matching corresponding to the action of an element of $H$. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group $H \leqslant \text{Sym}(\ell)$, constant degree $d \ge 3$ and $ε> 0$, we construct explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a (suitable) base $n$-vertex expander $G_0$ with the following parameters: (i) $λ(G) \le 2\sqrt{d-1} + ε$, for any lift size $\ell \le 2^{n^δ}$ where $δ=δ(d,ε)$, (ii) $λ(G) \le ε\cdot d$, for any lift size $\ell \le 2^{n^{δ_0}}$ for a fixed $δ_0 > 0$, when $d \ge d_0(ε)$, or (iii) $λ(G) \le \widetilde{O}(\sqrt{d})$, for lift size ``exactly'' $\ell = 2^{Θ(n)}$. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items $(i)$ and $(ii)$ above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for $2$-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing'" depth-first search traversals. Result $(iii)$ is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.

cs.DS↗
arXiv2021

Toward Instance-Optimal State Certification With Incoherent Measurements

We revisit the basic problem of quantum state certification: given copies of unknown mixed state $ρ\in\mathbb{C}^{d\times d}$ and the description of a mixed state $σ$, decide whether $σ= ρ$ or $\|σ- ρ\|_{\mathsf{tr}} \ge ε$. When $σ$ is maximally mixed, this is mixedness testing, and it is known that $Ω(d^{Θ(1)}/ε^2)$ copies are necessary, where the exact exponent depends on the type of measurements the learner can make [OW15, BCL20], and in many of these settings there is a matching upper bound [OW15, BOW19, BCL20]. Can one avoid this $d^{Θ(1)}$ dependence for certain kinds of mixed states $σ$, e.g. ones which are approximately low rank? More ambitiously, does there exist a simple functional $f:\mathbb{C}^{d\times d}\to\mathbb{R}_{\ge 0}$ for which one can show that $Θ(f(σ)/ε^2)$ copies are necessary and sufficient for state certification with respect to any $σ$? Such instance-optimal bounds are known in the context of classical distribution testing, e.g. [VV17]. Here we give the first bounds of this nature for the quantum setting, showing (up to log factors) that the copy complexity for state certification using nonadaptive incoherent measurements is essentially given by the copy complexity for mixedness testing times the fidelity between $σ$ and the maximally mixed state. Surprisingly, our bound differs substantially from instance optimal bounds for the classical problem, demonstrating a qualitative difference between the two settings.

quant-ph↗
arXiv2021

Pauli error estimation via Population Recovery

Motivated by estimation of quantum noise models, we study the problem of learning a Pauli channel, or more generally the Pauli error rates of an arbitrary channel. By employing a novel reduction to the "Population Recovery" problem, we give an extremely simple algorithm that learns the Pauli error rates of an $n$-qubit channel to precision $ε$ in $\ell_\infty$ using just $O(1/ε^2) \log(n/ε)$ applications of the channel. This is optimal up to the logarithmic factors. Our algorithm uses only unentangled state preparation and measurements, and the post-measurement classical runtime is just an $O(1/ε)$ factor larger than the measurement data size. It is also impervious to a limited model of measurement noise where heralded measurement failures occur independently with probability $\le 1/4$. We then consider the case where the noise channel is close to the identity, meaning that the no-error outcome occurs with probability $1-η$. In the regime of small $η$ we extend our algorithm to achieve multiplicative precision $1 \pm ε$ (i.e., additive precision $εη$) using just $O\bigl(\frac{1}{ε^2 η}\bigr) \log(n/ε)$ applications of the channel.

quant-ph↗
arXiv2021

The SDP value of random 2CSPs

We consider a very wide class of models for sparse random Boolean 2CSPs; equivalently, degree-2 optimization problems over~$\{\pm 1\}^n$. For each model $\mathcal{M}$, we identify the "high-probability value"~$s^*_{\mathcal{M}}$ of the natural SDP relaxation (equivalently, the quantum value). That is, for all $\varepsilon > 0$ we show that the SDP optimum of a random $n$-variable instance is (when normalized by~$n$) in the range $(s^*_{\mathcal{M}}-\varepsilon, s^*_{\mathcal{M}}+\varepsilon)$ with high probability. Our class of models includes non-regular CSPs, and ones where the SDP relaxation value is strictly smaller than the spectral relaxation value.

cs.DS↗
arXiv2021

Analysis of Boolean Functions

The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : \{0,1\}^n \to \{0,1\}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps the most basic object of study in theoretical computer science, and Fourier analysis has become an indispensable tool in the field. The topic has also played a key role in several other areas of mathematics, from combinatorics, random graph theory, and statistical physics, to Gaussian geometry, metric/Banach spaces, and social choice theory. The intent of this book is both to develop the foundations of the field and to give a wide (though far from exhaustive) overview of its applications. Each chapter ends with a "highlight" showing the power of analysis of Boolean functions in different subject areas: property testing, social choice, cryptography, circuit complexity, learning theory, pseudorandomness, hardness of approximation, concrete complexity, and random graph theory. The book can be used as a reference for working researchers or as the basis of a one-semester graduate-level course. The author has twice taught such a course at Carnegie Mellon University, attended mainly by graduate students in computer science and mathematics but also by advanced undergraduates, postdocs, and researchers in adjacent fields. In both years most of Chapters 1-5 and 7 were covered, along with parts of Chapters 6, 8, 9, and 11, and some additional material on additive combinatorics. Nearly 500 exercises are provided at the ends of the book's chapters.

cs.DM↗