Ryan O’Donnell
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View article: Non-iid hypothesis testing: from classical to quantum
Non-iid hypothesis testing: from classical to quantum Open
We study hypothesis testing (aka state certification) in the non-identically distributed setting. A recent work (Garg et al. 2023) considered the classical case, in which one is given (independent) samples from $T$ unknown probability dist…
View article: SPAM Tolerance for Pauli Error Estimation
SPAM Tolerance for Pauli Error Estimation Open
The Pauli channel is a fundamental model of noise in quantum systems, motivating the task of Pauli error estimation. We present an algorithm that builds on the reduction to Population Recovery introduced in [FO21]. Addressing an open quest…
View article: Instance-Optimal Quantum State Certification with Entangled Measurements
Instance-Optimal Quantum State Certification with Entangled Measurements Open
We consider the task of quantum state certification: given a description of a hypothesis state $σ$ and multiple copies of an unknown state $ρ$, a tester aims to determine whether the two states are equal or $ε$-far in trace distance. It is…
View article: Learning the Closest Product State
Learning the Closest Product State Open
We study the problem of finding a (pure) product state with optimal fidelity to an unknown $n$-qubit quantum state $ρ$, given copies of $ρ$. This is a basic instance of a fundamental question in quantum learning: is it possible to efficien…
View article: Quartic Quantum Speedups for Planted Inference
Quartic Quantum Speedups for Planted Inference Open
We describe a quantum algorithm for the Planted Noisy kXOR Problem (also known as Sparse Learning Parity with Noise) that achieves a nearly (fourth-power) speedup over the best known classical algorithm while using exponentially less space…
View article: Pseudorandomness Properties of Random Reversible Circuits
Pseudorandomness Properties of Random Reversible Circuits Open
Motivated by practical concerns in cryptography, we study pseudorandomness properties of permutations on $\{0,1\}^n$ computed by random circuits made from reversible $3$-bit gates (permutations on $\{0,1\}^3$). Our main result is that a ra…
View article: Uniformity Testing When You Have the Source Code
Uniformity Testing When You Have the Source Code Open
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 testin…
View article: Sparsifying Suprema of Gaussian Processes
Sparsifying Suprema of Gaussian Processes Open
We give a dimension-independent sparsification result for suprema of centered Gaussian processes: Let $T$ be any (possibly infinite) bounded set of vectors in $\mathbb{R}^n$, and let $\{\boldsymbol{X}_t := t \cdot \boldsymbol{g} \}_{t\in T…
View article: Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier
Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier Open
We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every s…
View article: Coboundary expansion inside Chevalley coset complex HDXs
Coboundary expansion inside Chevalley coset complex HDXs Open
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 $\widetilde{A}_d$-…
View article: Pseudorandom Permutations from Random Reversible Circuits
Pseudorandom Permutations from Random Reversible Circuits Open
We study pseudorandomness properties of permutations on $\{0,1\}^n$ computed by random circuits made from reversible $3$-bit gates (permutations on $\{0,1\}^3$). Our main result is that a random circuit of depth $n \cdot \tilde{O}(k^2)$, w…
View article: Improved quantum data analysis
Improved quantum data analysis Open
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 $O((\log^2 m)/\epsilon^2)$ samples of a $…
View article: Explicit orthogonal and unitary designs
Explicit orthogonal and unitary designs Open
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 $\mathrm{poly}(N^k/\varepsilon)$ …
View article: Quantum chi-squared tomography and mutual information testing
Quantum chi-squared tomography and mutual information testing Open
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 $\widetilde{O}(rd/ε)$ copi…
View article: Query-optimal estimation of unitary channels in diamond distance
Query-optimal estimation of unitary channels in diamond distance Open
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 un…
View article: Locally Covert Learning
Locally Covert Learning Open
The goal of a covert learning algorithm is to learn a function f by querying it, while ensuring that an adversary, who sees all queries and their responses, is unable to (efficiently) learn any more about f than they could learn from rando…
View article: Mean estimation when you have the source code; or, quantum Monte Carlo methods
Mean estimation when you have the source code; or, quantum Monte Carlo methods Open
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 $…
View article: Optimizing strongly interacting fermionic Hamiltonians
Optimizing strongly interacting fermionic Hamiltonians Open
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 …
View article: High-Dimensional Expanders from Chevalley Groups
High-Dimensional Expanders from Chevalley Groups Open
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 de…
View article: The Quantum Union Bound made easy
The Quantum Union Bound made easy Open
We give a short proof of Gao's Quantum Union Bound and Gentle Sequential Measurement theorems.
View article: Explicit Abelian Lifts and Quantum LDPC Codes
Explicit Abelian Lifts and Quantum LDPC Codes Open
For an abelian group H acting on the set [𝓁], an (H,𝓁)-lift of a graph G₀ is a graph obtained by replacing each vertex by 𝓁 copies, and each edge by a matching corresponding to the action of an element of H. Expanding graphs obtained via a…
View article: Log-Sobolev inequality for the multislice, with applications
Log-Sobolev inequality for the multislice, with applications Open
Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a …
View article: Optimizing Strongly Interacting Fermionic Hamiltonians
Optimizing Strongly Interacting Fermionic Hamiltonians Open
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 …
View article: The SDP value of random 2CSPs
The SDP value of random 2CSPs Open
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 nat…
View article: Improved Quantum data analysis
Improved Quantum data analysis Open
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((log2 m)/є2) samples of a d-dimensional…
View article: Fooling Gaussian PTFs via Local Hyperconcentration
Fooling Gaussian PTFs via Local Hyperconcentration Open
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 $2^d$ depe…
View article: Toward Instance-Optimal State Certification With Incoherent Measurements
Toward Instance-Optimal State Certification With Incoherent Measurements Open
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 ma…
View article: Explicit Near-Ramanujan Graphs of Every Degree
Explicit Near-Ramanujan Graphs of Every Degree Open
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…
View article: Quantum Approximate Counting with Nonadaptive Grover Iterations
Quantum Approximate Counting with Nonadaptive Grover Iterations Open
Approximate Counting refers to the problem where we are given query access to a function f : [N] → {0,1}, and we wish to estimate K = #{x : f(x) = 1} to within a factor of 1+ε (with high probability), while minimizing the number of queries…