Quantum Coupon Collector

• URL: http://arxiv.org/abs/2002.07688v1
• Date: Tue, 18 Feb 2020 16:14:55 GMT
• Title: Quantum Coupon Collector
• Authors: Srinivasan Arunachalam and Aleksandrs Belovs and Andrew M. Childs and Robin Kothari and Ansis Rosmanis and Ronald de Wolf
• Abstract summary: We study how efficiently a $k$-element set $Ssubseteq[n]$ can be learned from a uniform superposition $|Srangle of its elements. We give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms.
• Score: 62.58209964224025
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study how efficiently a $k$-element set $S\subseteq[n]$ can be learned from a uniform superposition $|S\rangle$ of its elements. One can think of $|S\rangle=\sum_{i\in S}|i\rangle/\sqrt{|S|}$ as the quantum version of a uniformly random sample over $S$, as in the classical analysis of the ``coupon collector problem.'' We show that if $k$ is close to $n$, then we can learn $S$ using asymptotically fewer quantum samples than random samples. In particular, if there are $n-k=O(1)$ missing elements then $O(k)$ copies of $|S\rangle$ suffice, in contrast to the $\Theta(k\log k)$ random samples needed by a classical coupon collector. On the other hand, if $n-k=\Omega(k)$, then $\Omega(k\log k)$ quantum samples are~necessary. More generally, we give tight bounds on the number of quantum samples needed for every $k$ and $n$, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through $|S\rangle$. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.

Related papers

• New Bounds on Quantum Sample Complexity of Measurement Classes [18.531114735719274]
  This paper studies quantum supervised learning for classical inference from quantum states. The hardness of learning is measured via sample complexity under a quantum counterpart of the well-known probably approximately correct (PAC)
  arXiv  Detail & Related papers  (2024-08-22T18:43:13Z)
• Simple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via Gauss-Jordan [49.1574468325115]
  We revisit the sample and computational complexity of completing a rank-1 tensor in $otimes_i=1N mathbbRd$. We present a characterization of the problem which admits an algorithm amounting to Gauss-Jordan on a pair of random linear systems.
  arXiv  Detail & Related papers  (2024-08-10T04:26:19Z)
• Proper vs Improper Quantum PAC learning [3.292420364429101]
  We present an algorithm for the Quantum Coupon Collector problem with sample complexity. We prove that its sample complexity matches that of the classical Coupon Collector problem for both modes of learning. We hope that padding can more generally lift other forms of classical learning behaviour to quantum setting.
  arXiv  Detail & Related papers  (2024-03-05T19:49:44Z)
• Time-Efficient Quantum Entropy Estimator via Samplizer [7.319050391449301]
  Estimating the entropy of a quantum state is a basic problem in quantum information. We introduce a time-efficient quantum approach to estimating the von Neumann entropy $S(rho)$ and R'enyi entropy $S_alpha(rho)$ of an $N$-dimensional quantum state $rho.
  arXiv  Detail & Related papers  (2024-01-18T12:50:20Z)
• Quantum Resources Required to Block-Encode a Matrix of Classical Data [56.508135743727934]
  We provide circuit-level implementations and resource estimates for several methods of block-encoding a dense $Ntimes N$ matrix of classical data to precision $epsilon$. We examine resource tradeoffs between the different approaches and explore implementations of two separate models of quantum random access memory (QRAM) Our results go beyond simple query complexity and provide a clear picture into the resource costs when large amounts of classical data are assumed to be accessible to quantum algorithms.
  arXiv  Detail & Related papers  (2022-06-07T18:00:01Z)
• Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies [68.93512627479197]
  We study quantum Ordered Binary Decision Diagrams($OBDD$) model. We prove lower bounds and upper bounds for OBDD with arbitrary order of input variables. We extend hierarchy for read$k$-times Ordered Binary Decision Diagrams ($k$-OBDD$) of width.
  arXiv  Detail & Related papers  (2022-04-22T12:37:56Z)
• Improved quantum data analysis [1.8416014644193066]
  We give a quantum "Threshold Search" algorithm that requires only $O(log2 m)/epsilon2)$ samples of a $d$-dimensional state. We also give an alternative Hypothesis Selection method using $tildeO((log3 m)/epsilon2)$ samples.
  arXiv  Detail & Related papers  (2020-11-22T01:22:37Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• The Quantum Supremacy Tsirelson Inequality [0.22843885788439797]
  For a quantum circuit $C$ on $n$ qubits and a sample $z in 0,1n$, the benchmark involves computing $|langle z|C|0n rangle|2$. We show that for any $varepsilon ge frac1mathrmpoly(n)$, outputting a sample $z$ is the optimal 1-query for $|langle z|C|0nrangle|2$ on average.
  arXiv  Detail & Related papers  (2020-08-20T01:04:32Z)
• Locally Private Hypothesis Selection [96.06118559817057]
  We output a distribution from $mathcalQ$ whose total variation distance to $p$ is comparable to the best such distribution. We show that the constraint of local differential privacy incurs an exponential increase in cost. Our algorithms result in exponential improvements on the round complexity of previous methods.
  arXiv  Detail & Related papers  (2020-02-21T18:30:48Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.