On the Problem of Approximating the Eigenvalues of Undirected Graphs in Probabilistic Logspace

Doron, Dean; Ta-Shma, Amnon

doi:10.1007/978-3-662-47672-7_34

Cited by 3 publications

(2 citation statements)

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“…Concurrently with our work, Doron, Sarid, and Ta-Shma have shown that analogous problems for stochastic matrices (e.g. computing the eigenvalue gap) are complete for classical randomized logspace, or BPL [16,15]. In addition, Le Gall has shown that analogous problems for Laplacian matrices can be solved in BPL [18].…”

Section: Corollary 1 the Problem Of Approximating To Constant Precisi...mentioning

confidence: 54%

A Complete Characterization of Unitary Quantum Space

Fefferman¹,

Lin²

2016

Preprint

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Motivated by understanding the power of quantum computation with restricted number of qubits, we give two complete characterizations of unitary quantum space bounded computation. First we show that approximating an element of the inverse of a well-conditioned efficiently encoded 2 k(n) × 2 k(n) matrix is complete for the class of problems solvable by quantum circuits acting on O(k(n)) qubits with all measurements at the end of the computation. Similarly, estimating the minimum eigenvalue of an efficiently encoded Hermitian 2 k(n) ×2 k(n) matrix is also complete for this class. In the logspace case, our results improve on previous results of Ta-Shma [32] by giving new space-efficient quantum algorithms that avoid intermediate measurements, as well as showing matching hardness results.Additionally, as a consequence we show that PreciseQMA, the version of QMA with exponentially small completeness-soundess gap, is equal to PSPACE. Thus, the problem of estimating the minimum eigenvalue of a local Hamiltonian to inverse exponential precision is PSPACE-complete, which we show holds even in the frustration-free case. Finally, we can use this characterization to give a provable setting in which the ability to prepare the ground state of a local Hamiltonian is more powerful than the ability to prepare PEPS states.Interestingly, by suitably changing the parameterization of either of these problems we can completely characterize the power of quantum computation with simultaneously bounded time and space.

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Section: Corollary 1 the Problem Of Approximating To Constant Precisi...mentioning

confidence: 54%

A Complete Characterization of Unitary Quantum Space

Fefferman¹,

Lin²

2016

Preprint

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“…However, it is not at all clear how to compute the second eigenvalue of the adjacency matrix to desired accuracy in O(log n) space. [DTS15,DSTS17] study the problem of approximating eigenvalues of an undirected graph in logarithmic space and we might hope to use their algorithms to solve our problem. However, these algorithms, which are randomized and run in logarithmic space, can only approximate the normalized eigenvalues to within constant accuracy.…”

Section: Small Space Clique Completionmentioning

confidence: 99%

Is the space complexity of planted clique recovery the same as that of detection?

Mardia¹

2020

Preprint

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We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph G(n, 12 ), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k = Θ( √ n), and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. In this work, we ask-Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use 'space efficiency' as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Ω( √ n). Our main result comes very close to designing such an algorithm. We show that for k = Ω( √ n), the recovery problem can be solved infor any constant integer ℓ > 0, the space usage is O(log n) bits. 2. If k = Θ( √ n), the space usage is O(log * n • log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Ω( √ n), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity.

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