Pebbling meets coloring: Reversible pebble game on trees

Komarath, Balagopal; Sarma, Jayalal; Sawlani, Saurabh

doi:10.1016/j.jcss.2017.07.009

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…Computing the minimum number of pebbles needed in Bennett's pebble game on an arbitrary DAG is PSPACE-complete; however, it is possible that the spooky pebble game's additional rules make reasoning about the game easier [18]. Previous work has found that applying Bennett's pebble game to a tree requires exactly one more pebble than the tree's edge rank coloring number, and complete trees with n nodes can be pebbled in n O(log log n) steps [19]. We are curious how much the spooky pebble game can improve on these bounds.…”

Section: Future Workmentioning

confidence: 99%

The Spooky Pebble Game

Kornerup¹,

Jonathan²,

Soloveichik³

2021

Preprint

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Pebble games are commonly used to study space-time trade-offs in computation. We present a pebble game that explores this trade-off in quantum simulation of arbitrary classical circuits on non-classical inputs. Many quantum algorithms, such as Shor's algorithm and Grover's search, include subroutines that require simulation of classical functions on inputs in superposition. The current state of the art uses generic reversible simulation through Bennett's pebble game and universal reversible gate sets such as the Toffoli. Using measurement-based uncomputation, we replace many of the ancilla qubits used by existing constructions with classical control bits, which are cheaper. Our pebble game forms a natural framework for reasoning about measurement-based uncomputation, and we prove tight bounds on the time complexity of all algorithms that use our pebble game. For any ε ∈ (0, 1), we present an algorithm that can simulate irreversible classical computation that uses T time and S space in O(With access to more qubits we present algorithms that run in O( 1 ε T ) time with O(S 1−ε T ε ) qubits. Both of these results show an improvement over Bennett's construction, which requires O( T 1+ε S ε ) time when using O(ε2 1/ε S(1 + log T S )) qubits. Additionally the results in our paper combine with Barrington's theorem to provide a general method to efficiently compute any log-depth circuit on quantum inputs using a constant number of ancilla qubits. We also explore a connection between the optimal structure of our pebbling algorithms and backtracking from dynamic programming.

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Section: Future Workmentioning

confidence: 99%

The Spooky Pebble Game

Kornerup¹,

Jonathan²,

Soloveichik³

2021

Preprint

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“…If all predecessors of a pebbled vertex v contain pebbles, the pebble on v may be removed. Reversible pebblings have been studied in [43,39,38] and have been used to prove time-space trade-offs in reversible simulation of irreversible computation in [42,40,61,16]. In a different context, Potechin [52] implicitly used reversible pebbling to obtain lower bounds in monotone space complexity, with the connection made explicit in later works [24,31].…”

Section: Pebble Gamesmentioning

confidence: 99%

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

et al. 2021

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We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t + 1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.

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“…If the underlying graph G is a complete binary tree on n vertices such a polynomial bound is unfortunately not known. While it is known that the smallest number of pebbles required to pebble a binary tree of height h is given by S = log(h) + Θ(log * (h)), where log * denotes the iterated logarithm, to our knowledge the best upper bound on the number of steps is n O(log log(n)) , given in [21]. It is an open problem if a binary tree on n vertices can be pebbled with a polynomial number of steps provided that only S pebbles are available, where S is as above.…”

Section: Preliminariesmentioning

confidence: 99%

“…The pebble games we study are played on directed acylic graphs that have the structure of ternary trees. In related work [21] pebbling of other classes of trees has been considered, in particular that of complete binary trees.…”

Section: Introductionmentioning

confidence: 99%

Improved reversible and quantum circuits for Karatsuba-based integer multiplication

Parent¹,

Roetteler²,

Mosca³

2017

Preprint

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Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to implement arithmetic as low as possible. This applies in particular to the number of qubits required in the implementation as for the foreseeable future this number is expected to be small. We present a reversible circuit for integer multiplication that is inspired by Karatsuba's recursive method. The main improvement over circuits that have been previously reported in the literature is an asymptotic reduction of the amount of space required from O(n 1.585 ) to O(n 1.427 ). This improvement is obtained in exchange for a small constant increase in the number of operations by a factor less than 2 and a small asymptotic increase in depth for the parallel version. The asymptotic improvement are obtained from analyzing pebble games on complete ternary trees.

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Pebbling meets coloring: Reversible pebble game on trees

Cited by 7 publications

References 10 publications

The Spooky Pebble Game

The Spooky Pebble Game

Nullstellensatz Size-Degree Trade-offs from Reversible Pebbling

Improved reversible and quantum circuits for Karatsuba-based integer multiplication

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