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arXiv:2306.12103

Quantum and classical query complexities for determining connectedness of matroids

Xiaowei Huang, Shiguang Feng, Lvzhou Li

Published Jun 21, 2023• 0 citations
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Abstract

Connectivity is a fundamental structural property of matroids, and has been studied algorithmically over 50 years. In 1974, Cunningham proposed a deterministic algorithm consuming $O(n^{2})$ queries to the independence oracle to determine whether a matroid is connected. Since then, no algorithm, not even a random one, has worked better. To the best of our knowledge, the classical query complexity lower bound and the quantum complexity for this problem have not been considered. Thus, in this paper we are devoted to addressing these issues, and our contributions are threefold as follows: (i) First, we prove that the randomized query complexity of determining whether a matroid is connected is $Ω(n^2)$ and thus the algorithm proposed by Cunningham is optimal in classical computing. (ii) Second, we present a quantum algorithm with $O(n^{3/2})$ queries, which exhibits provable quantum speedups over classical ones. (iii) Third, we prove that any quantum algorithm requires $Ω(n)$ queries, which indicates that quantum algorithms can achieve at most a quadratic speedup over classical ones. Therefore, we have a relatively comprehensive understanding of the potential of quantum computing in determining the connectedness of matroids.\

DOI
10.48550/arxiv.2306.12103

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Connectivity is a fundamental structural property of matroids, and has been studied algorithmically over 50 years. In 1974, Cunningham proposed a deterministic algorithm consuming O(n2)O(n^{2})O(n2) queries to the independence oracle to determine whether a matroid is connected. Since then, no algorithm, not even a random one, has worked better. To the best of our knowledge, the classical query complexity lower bound and the quantum complexity for this problem have not been considered. Thus, in this paper we are devoted to addressing these issues, and our contributions are threefold as follows: (i) First, we prove that the randomized query complexity of determining whether a matroid is connected is Ω(n2)Ω(n^2)Ω(n2) and thus the algorithm proposed by Cunningham is optimal in classical computing. (ii) Second, we present a quantum algorithm with O(n3/2)O(n^{3/2})O(n3/2) queries, which exhibits provable quantum speedups over classical ones. (iii) Third, we prove that any quantum algorithm requires Ω(n)Ω(n)Ω(n) queries, which indicates that quantum algorithms can achieve at most a quadratic speedup over classical ones. Therefore, we have a relatively comprehensive understanding of the potential of quantum computing in determining the connectedness of matroids.\
In this paper, we study for the first time how well quantum algorithms perform on matroid properties. We show that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats, hyperplanes) of a matroid, and the decision problem of deciding whether a matroid is uniform or Eulerian, by giving a uniform lower bound Ω((n⌊n/2⌋))\Omega(\sqrt{\binom{n}{\lfloor n/2\rfloor}})Ω((⌊n/2⌋n​)​) on the query complexity for all these problems. On the other hand, for the uniform matroid decision problem, we present an asymptotically optimal quantum algorithm which achieves the lower bound, and for the girth problem, we give an almost optimal quantum algorithm with query complexity O(log⁡n(n⌊n/2⌋))O(\log n\sqrt{\binom{n}{\lfloor n/2\rfloor}})O(logn(⌊n/2⌋n​)​). In addition, for the paving matroid decision problem, we prove a lower bound Ω((n⌊n/2⌋)/n)\Omega(\sqrt{\binom{n}{\lfloor n/2\rfloor}/n})Ω((⌊n/2⌋n​)/n​) on the query complexity, and give an O((n⌊n/2⌋))O(\sqrt{\binom{n}{\lfloor n/2\rfloor}})O((⌊n/2⌋n​)​) quantum algorithm.
This paper initiates the study of quantum algorithms for matroid property problems. It is shown that quadratic quantum speedup is possible for the calculation problem of finding the girth or the number of circuits (bases, flats, hyperplanes) of a matroid, and for the decision problem of deciding whether a matroid is uniform or Eulerian, by giving a uniform lower bound Ω((n⌊n/2⌋))Ω(\sqrt{\binom{n}{\lfloor n/2\rfloor}})Ω((⌊n/2⌋n​)​) on the query complexity for all these problems. On the other hand, for the uniform matroid decision problem, an asymptotically optimal quantum algorithm is proposed which achieves the lower bound, and for the girth problem, an almost optimal quantum algorithm is given with query complexity O(log⁡n(n⌊n/2⌋))O(\log n\sqrt{\binom{n}{\lfloor n/2\rfloor}})O(logn(⌊n/2⌋n​)​). In addition, for the paving matroid decision problem, a lower bound Ω((n⌊n/2⌋)/n)Ω(\sqrt{\binom{n}{\lfloor n/2\rfloor}/n})Ω((⌊n/2⌋n​)/n​) on the query complexity is obtained, and an O((n⌊n/2⌋))O(\sqrt{\binom{n}{\lfloor n/2\rfloor}})O((⌊n/2⌋n​)​) quantum algorithm is presented.
A matroid generalizes linear independence, being constructed on (E, I) with ground set E and independent sets I as a subset of 2E. Matroid intersection looks for the largest independent set in two matroids on E. Classical algorithms, like Edmonds’ and Cunningham’s, solve this using independence queries, but the minimum needed is unestablished. To establish this, we selected a target function, constructed query gates for non-adaptive query words, and introduced adaptivity, where the next query depends on previous answers. Using v(M1, M2) as the largest common independent set’s size, we examine fk(M1, M2):= [v >= k], because v is the common refinement of all {fk}. Independence queries have three models: onesided (tests one matroid), pair-valued (tests both matroids), and common-independence (tests independence in both). In the non-adaptive query-only model, one fk needs 2 (n/k) one-sided queries but only (n/k) jointly, while v needs 2(2n - 1) one-sided queries, an exact exponential lower bound. This common-independence oracle computes every fk, yet cannot reconstruct the exchange graph classical algorithms depend on, implying other types of algorithms. We recovered Edmonds’ algorithm as an adaptive query word matching its known nr2 bound, then built optimal decision trees for small ground sets. We found that, at n = 4, the optimal worst case is 5 queries versus 17 and 18 for Edmonds’ and Cunningham’s, demonstrating that classical algorithms are not query-optimal despite running in polynomial time. We are working to generalize this classical workflow to discover quantum query circuits.
Query complexity is a model of computation in which we have to compute a function f(x1,…,xN)f(x_1, \ldots, x_N)f(x1​,…,xN​) of variables xix_ixi​ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring Ω(N)Ω(\sqrt{N})Ω(N​) queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with MMM queries quantumly but requiring Ω~(M2.5)\tildeΩ(M^{2.5})Ω~(M2.5) queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with MMM queries probabilistically but requiring Ω~(M2)\tildeΩ(M^{2})Ω~(M2) queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most qqq oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision εεε requires Θ(1qε2)Θ(\frac{1}{q ε^2})Θ(qε21​) queries; (ii) (Unbiased) Amplitude estimation up to a precision εεε requires Θ(p(1−p)qε2)Θ(\frac{p(1-p)}{q ε^2})Θ(qε2p(1−p)​) queries; (iii) Search among NNN items requires Θ(Nq)Θ(\frac{N}{q})Θ(qN​) queries; (iv) Two level AND-OR tree (AND of mmm ORs, with nnn inputs to each OR) requires Θ(nmq)Θ(\frac{nm}{q})Θ(qnm​) queries. All of these bounds are optimal up to a constant factor, for all qqq from 1, corresponding to the classical complexity, to Q(f)Q(f)Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.