Quantum Approximate Counting for Markov Chains and Application to Collision Counting

• URL: http://arxiv.org/abs/2204.02552v2
• Date: Thu, 7 Apr 2022 06:25:50 GMT
• Title: Quantum Approximate Counting for Markov Chains and Application to Collision Counting
• Authors: Fran\c{c}ois Le Gall and Iu-Iong Ng
• Abstract summary: We show how to generalize the quantum approximate counting technique developed by Brassard, Hoyer and Tapp [ICALP 1998] to estimating the number of marked states of a Markov chain. This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, H{\o}yer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful "quantum walk based search" framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $\epsilon$ by making $\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.

Related papers

• Low-depth quantum symmetrization [1.5566524830295307]
  We present the first quantum algorithm for the general symmetrization problem. Our algorithm, based on (quantum) sorting networks, uses $O(log n)$ depth and $O(nlog m)$ ancilla qubits. We also propose an $O(log2 n)$-depth quantum algorithm to transform second-quantized states to first-quantized states.
  arXiv  Detail & Related papers  (2024-11-06T16:00:46Z)
• Rank Is All You Need: Estimating the Trace of Powers of Density Matrices [1.5133368155322298]
  Estimating the trace of powers of identical $k$ density matrices is a crucial subroutine for many applications. Inspired by the Newton-Girard method, we developed an algorithm that uses only $mathcalO(r)$ qubits and $mathcalO(r)$ multi-qubit gates.
  arXiv  Detail & Related papers  (2024-08-01T06:23:52Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Approximation Algorithms for Quantum Max-$d$-Cut [42.248442410060946]
  The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy associated with the projector onto the antisymmetric subspace of two, $d$-dimensional qudits over all local interactions. We develop an algorithm that finds product-state solutions of mixed states with bounded purity that achieve non-trivial performance guarantees.
  arXiv  Detail & Related papers  (2023-09-19T22:53:17Z)
• Do you know what q-means? [50.045011844765185]
  Clustering is one of the most important tools for analysis of large datasets. We present an improved version of the "$q$-means" algorithm for clustering. We also present a "dequantized" algorithm for $varepsilon which runs in $Obig(frack2varepsilon2(sqrtkd + log(Nd))big.
  arXiv  Detail & Related papers  (2023-08-18T17:52:12Z)
• On the average-case complexity of learning output distributions of quantum circuits [55.37943886895049]
  We show that learning the output distributions of brickwork random quantum circuits is average-case hard in the statistical query model. This learning model is widely used as an abstract computational model for most generic learning algorithms.
  arXiv  Detail & Related papers  (2023-05-09T20:53:27Z)
• Fast Quantum Algorithms for Trace Distance Estimation [8.646488471216262]
  We propose efficient quantum algorithms for estimating the trace distance within additive error $varepsilon$ between mixed quantum states of rank $r$. We show that the decision version of low-rank trace distance estimation is $mathsfBQP$-complete.
  arXiv  Detail & Related papers  (2023-01-17T10:16:14Z)
• 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)
• Robust Quantum Walk Search Without Knowing the Number of Marked Vertices [0.2320417845168326]
  Existing quantum walk-based search algorithms suffer from the souffl'e problem. We present a new quantum walk-based search framework that achieves robustness without sacrificing the quantum speedup.
  arXiv  Detail & Related papers  (2021-11-17T10:04:44Z)
• Quantum Algorithm for Fidelity Estimation [8.270684567157987]
  For two unknown mixed quantum states $rho$ and $sigma$ in an $N$-dimensional space, computing their fidelity $F(rho,sigma)$ is a basic problem. We propose a quantum algorithm that solves this problem in $namepoly(log (N), r, 1/varepsilon)$ time.
  arXiv  Detail & Related papers  (2021-03-16T13:57:01Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

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.