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arXiv:2204.02552 · Quantum Information and Computation

Quantum approximate counting for Markov chains and collision counting

François Le Gall, Iu-Iong Ng

Published Oct 27, 2022• 0 citations
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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 $m\ll N$.

DOI
10.26421/qic22.15-16-1

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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 NNN elements, we obtain a quantum algorithm that estimates the number mmm of collisions of the two functions within relative error ϵ\epsilonϵ by making O~(1ϵ25/24(Nm)2/3)\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)O~(ϵ25/241​(m​N​)2/3) queries, which gives an improvement over the Θ(1ϵNm)\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)Θ(ϵ1​m​N​)-query classical algorithm based on random sampling when m≪Nm\ll Nm≪N.
In 1998, Brassard, Høyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of N items, K of them marked, their algorithm estimates K to within relative error ∊ by making only queries. Although this speedup is of “Grover” type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shor's algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.
We propose a new method for designing quantum search algorithms for finding a element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.
Searching for collisions in random functions is a fundamental computational problem, with many applications in symmetric and asymmetric cryptanalysis. When one searches for a single collision, the known quantum algorithms match the query lower bound. This is not the case for the problem of finding multiple collisions, despite its regular appearance as a sub-component in sieving-type algorithms. At EUROCRYPT 2019, Liu and Zhandry gave a query lower bound Ω(2m/3+2k/3)Ω(2^{m/3 + 2k/3})Ω(2m/3+2k/3) for finding 2k2^k2k collisions in a random function with m-bit output. At EUROCRYPT 2023, Bonnetain et al. gave a quantum algorithm matching this bound for a large range of mmm and kkk, but not all admissible values. Like many previous collision-finding algorithms, theirs is based on the MNRS quantum walk framework, but it chains the walks by reusing the state after outputting a collision. In this paper, we give a new algorithm that tackles the remaining non-optimal range, closing the problem. Our algorithm is tight (up to a polynomial factor) in queries, and also in time under a quantum RAM assumption. The idea is to extend the chained walk to a regime in which several collisions are returned at each step, and the ``walks'' themselves only perform a single diffusion layer.
We will give an introduction into the quantum search algorithms on the Markov chains introduced by Szegedy and recent modifications based on partially absorbing Markov chains due to Krovi et al.Algorithmica 74, 851 (2016) It has been shown that a quantum search can find a set of marked vertices quadratically faster (in units of the hitting time) for any reversible Markov chain.The proofs are based on certain properties of the stationary state of the partially absorbing walk and rely on quantum phase estimation techniques.We will offer an alternative view of the underlying mechanism of the quantum search based on spectral properties of the quantum walk operator.By considering the complete graph as an example, we identify the relevant two-level quantum subspace leading to a Grover-like rotation in the underlying vector space.
In this work, we present a quantum walks based approach for Probabilistic Safety Assessment problems in the dynamic Markovian framework. After presenting well known quantum walk algorithms for detecting or finding marked elements in a graph or for finding paths in specific graphs, we propose two algorithms to address the more general problem of finding paths from some point to marked elements. The first algorithm is based on move and store strategy to built failure sequences from an initial state to the marked states. It uses a full register assembling qubits representing components of the system under study and other control and ancilla qubits (between n and 2n) to identify all the sequences. The second algorithm is based on a hybrid approach to consider di˙erent cascading circuits for solving relatively large instances. We present some tests on a Qiskit simulation library that were performed to compare our algorithms against classical random walks. These tests showed that classical approach has advances in the first iteration but the hybrid approach scales better in the sense that it identifies a large number of paths and converges towards all the possible paths to these marked vertices.
Quantum counting is a fundamental quantum algorithm that estimates the fraction of marked elements using a membership oracle, achieving a quadratic speedup over classical sampling. The membership oracle, however, assumes exact labeling of each element, but this assumption fails when the labels are inherently probabilistic. We study quantum counting with \emph{Bernoulli oracles}, where given mmm Bernoulli distributions with unknown biases p1,…,pmp_1,\dots,p_mp1​,…,pm​ and a gap parameter ΔΔΔ, the goal is to estimate the fraction ρρρ of \emph{positive} distributions (pi≥1/2+Δp_i\ge1/2+Δpi​≥1/2+Δ) to within additive error εεε. We prove an upper bound of O~ ⁣(ρΔε+1Δε)\tilde{O}\!\big(\frac{\sqrtρ}{Δε}+\frac{1}{Δ\sqrtε}\big)O~(Δερ​​+Δε​1​) queries, achieving a quadratic speedup over the classical sample complexity. % of Θ(ρ/Δ2ε2)Θ(ρ/Δ^2ε^2)Θ(ρ/Δ2ε2). Our algorithm first uses the Quantum Singular Value Transformation (QSVT) to coherently amplify the bias gap without collapsing the superposition over distributions, and then applies two-stage adaptive amplitude estimation. We complement this upper bound with a near-matching lower bound of Ω(ρ/(Δε))Ω(\sqrtρ/(Δε))Ω(ρ​/(Δε)) via a new composition theorem for the quantum adversary method in the Boolean-over-average-case direction. For the special case of a constant gap Δ=Θ(1)Δ=Θ(1)Δ=Θ(1), which corresponds to the bounded-error oracle where each query returns the correct label with constant probability, our bounds specialize to O~(ρε+1ε)\tilde{O}\big(\frac{\sqrtρ}ε+\frac{1}{\sqrtε}\big)O~(ερ​​+ε​1​) and Ω(ρε)Ω(\frac{\sqrtρ}ε)Ω(ερ​​), thereby characterizing the query complexity of quantum counting with bounded-error oracles.