Reversible Pebble Game on Trees
04 Aug 2015-pp 83-94
TL;DR: The reversible pebble game is studied and its equivalence between Dymond-Tompa, Raz-McKenzie and Raz- McKenzie games on graphs is established.
Abstract: A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), Raz-McKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]. Motivated by this equivalence, we study the reversible pebble game and establish the following results.
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TL;DR: The synthesis framework is based on lookup-table (LUT) networks, which play a key role in conventional logic synthesis, and can advance over the state-of-the-art hierarchical reversible logic synthesis algorithms.
Abstract: We present a synthesis framework to map logic networks into quantum circuits for quantum computing. The synthesis framework is based on lookup-table (LUT) networks, which play a key role in conventional logic synthesis. Establishing a connection between LUTs in an LUT network and reversible single-target gates in a reversible network allows us to bridge conventional logic synthesis with logic synthesis for quantum computing, despite several fundamental differences. We call our synthesis framework LUT-based hierarchical reversible logic synthesis (LHRS). Input to LHRS is a classical logic network representing an arbitrary Boolean combinational operation; output is a quantum network (realized in terms of Clifford+ T gates). The framework allows one to account for qubit count requirements imposed by the overlying quantum algorithm or target quantum computing hardware. In a fast first step, an initial network is derived that only consists of single-target gates and already completely determines the number of qubits in the final quantum network. Different methods are then used to map each single-target gate into Clifford+ T gates, while aiming at optimally using available resources. We demonstrate the versatility of our method by conducting a design space exploration using different parameters on a set of large combinational benchmarks. On the same benchmarks, we show that our approach can advance over the state-of-the-art hierarchical reversible logic synthesis algorithms.
34 citations
Posted Content•
TL;DR: The synthesized benchmarks provide cost estimates that allow quantum algorithm designers to provide the first complete cost estimates for a host of quantum algorithms, and demonstrate the effectiveness of the method in automatically synthesizing IEEE compliant floating point networks up to double precision.
Abstract: We present a synthesis framework to map logic networks into quantum circuits for quantum computing. The synthesis framework is based on LUT networks (lookup-table networks), which play a key role in conventional logic synthesis. Establishing a connection between LUTs in a LUT network and reversible single-target gates in a reversible network allows us to bridge conventional logic synthesis with logic synthesis for quantum computing, despite several fundamental differences. We call our synthesis framework LUT-based Hierarchical Reversible Logic Synthesis (LHRS). Input to LHRS is a classical logic network; output is a quantum network (realized in terms of Clifford+$T$ gates). The framework offers to trade-off the number of qubits for the number of quantum gates. In a first step, an initial network is derived that only consists of single-target gates and already completely determines the number of qubits in the final quantum network. Different methods are then used to map each single-target gate into Clifford+$T$ gates, while aiming at optimally using available resources. We demonstrate the effectiveness of our method in automatically synthesizing IEEE compliant floating point networks up to double precision. As many quantum algorithms target scientific simulation applications, they can make rich use of floating point arithmetic components. But due to the lack of quantum circuit descriptions for those components, it can be difficult to find a realistic cost estimation for the algorithms. Our synthesized benchmarks provide cost estimates that allow quantum algorithm designers to provide the first complete cost estimates for a host of quantum algorithms. Thus, the benchmarks and, more generally, the LHRS framework are an essential step towards the goal of understanding which quantum algorithms will be practical in the first generations of quantum computers.
14 citations
Cites background from "Reversible Pebble Game on Trees"
...More details can be found in [43], [44], [45], [46]....
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10 Jul 2022
TL;DR: LLVM passes are implemented that can automatically generate QIR functions representing classical Q# functions into QIR code implementing such functions quantumly, using state-of-the-art logic optimization and oracle generation techniques based on XOR-AND graphs for this purpose.
Abstract: Automatic oracle generation techniques can find optimized quantum circuits for classical components in quantum algorithms. However, most implementations of oracle generation techniques require that the classical component is expressed in terms of a conventional logic representation such as logic networks, truth tables, or decision diagrams. We implemented LLVM passes that can automatically generate QIR functions representing classical Q# functions into QIR code implementing such functions quantumly. We are using state-of-the-art logic optimization and oracle generation techniques based on XOR-AND graphs for this purpose. This enables not only a more natural description of the quantum algorithm on a higher level of abstraction, but also enables technology-dependent or application-specific generation of the oracles.
TL;DR: This paper surveys the recent advances in quantum technologies and quantum computation from the design automation perspective and proposes a new approach to design automation of quantum computers.
Abstract: Universal and fault-tolerant quantum computation is a promising new paradigm that may efficiently conquer difficult computation tasks beyond the reach of classical computation. It motivates the development of various quantum technologies. The rapid progress of quantum technologies accelerates the realization of quantum computers. In this paper, we survey the recent advances in quantum technologies and quantum computation from the design automation perspective.
TL;DR: In this paper , the authors survey the recent advances in quantum technologies and quantum computation from the design automation perspective and present a survey of quantum computing from a design automation point of view.
Abstract: Universal and fault-tolerant quantum computation is a promising new paradigm that may efficiently conquer difficult computation tasks beyond the reach of classical computation. It motivates the development of various quantum technologies. The rapid progress of quantum technologies accelerates the realization of quantum computers. In this paper, we survey the recent advances in quantum technologies and quantum computation from the design automation perspective.
References
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Book•
14 Nov 1995
TL;DR: In this article, the authors introduce the concept of graph coloring and propose a graph coloring algorithm based on the Eulers formula for k-chromatic graphs, which can be seen as a special case of the graph coloring problem.
Abstract: 1. Fundamental Concepts. Definitions and examples. Paths and proofs. Vertex degrees and counting. Degrees and algorithmic proof. 2. Trees and Distance. Basic properties. Spanning trees and enumeration. Optimization and trees. Eulerian graphs and digraphs. 3. Matchings and Factors. Matchings in bipartite graphs. Applications and algorithms. Matchings in general graphs. 4. Connectivity and Paths. Cuts and connectivity. k-connected graphs. Network flow problems. 5. Graph Coloring. Vertex colorings and upper bounds. Structure of k-chromatic graphs. Enumerative aspects. 6. Edges and Cycles. Line graphs and edge-coloring. Hamiltonian cycles. Complexity. 7. Planar Graphs. Embeddings and Eulers formula. Characterization of planar graphs. Parameters of planarity. 8. Additional Topics. Perfect graphs. Matroids. Ramsey theory. More extremal problems. Random graphs. Eigenvalues of graphs. Glossary of Terms. Glossary of Notation. References. Author Index. Subject Index.
7,126 citations
IBM1
TL;DR: Using a pebbling argument, this paper shows that, for any $\varepsilon > 0$, ordinary multitape Turing machines using time T and space S can be simulated by reversible ones using time $O(T^{1 + \varpsilon } )$ and space $O (S\log T)$ or in linear time and space$O(ST^\varePSilon )$.
Abstract: A reversible Turing machine is one whose transition function is $1:1$, so that no instantaneous description (ID) has more than one predecessor. Using a pebbling argument, this paper shows that, for any $\varepsilon > 0$, ordinary multitape Turing machines using time T and space S can be simulated by reversible ones using time $O(T^{1 + \varepsilon } )$ and space $O(S\log T)$ or in linear time and space $O(ST^\varepsilon )$. The former result implies in particular that reversible machines can simulate ordinary ones in quadratic space. These results refer to reversible machines that save their input, thereby insuring a global $1:1$ relation between initial and final IDs, even when the function being computed is many-to-one. Reversible machines that instead erase their input can of course compute only $1:1$ partial recursive functions and indeed provide a Godel numbering of such functions. The time/space cost of computing a $1:1$ function on such a machine is equal within a small polynomial to the cost of co...
392 citations
"Reversible Pebble Game on Trees" refers background or methods in this paper
...(Reversible Pebbling[1]) Let G be a rooted DAG with root r....
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...([1]) R•(Chn) ≤ log2(n) + 1 for all n...
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...Motivated by applications in the context of reversible computation (for example, quantum computation), Bennett[1] introduced the reversible pebbling game....
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...A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), RazMcKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]....
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TL;DR: A new class of communication complexity search problems is defined, referred to below as DART games, and a tight lower bound for the communication complexity of every member of this class is proved, and lower bounds for the monotone depth of many functions are got.
Abstract: , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes.
1. monotone-NC ≠ monotone-P.
2. For every i≥1, monotone-
≠ monotone-
.
3. More generally: For any integer function D(n), up to (for some e>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const). Only a separation of monotone-
from monotone-
was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds:
1. For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigderson's theorem, as an immediate corollary of our general result.
2. For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known.
214 citations
"Reversible Pebble Game on Trees" refers background in this paper
...A definition for the RazMckenzie pebble game can be found in [10]....
[...]
...A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), RazMcKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]....
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TL;DR: The model used in this paper is a single processor with an arbitraril system that automates the very labor-intensive and therefore time-heavy and expensive process of register allocation.
Abstract: The search for efficient register allocation algorithms dates back to the time of the first FORTRAN compiler for the IBM 704. The model we use in this paper is a single processor with an arbitraril...
201 citations
Additional excerpts
...When we restrict the irreversible black pebbling game to be read-once (each vertex is pebbled only once), then the problem becomes NP-complete (see [11])....
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TL;DR: An algorithm is presented which finds a min-cut linear arrangement of a tree in O(nlogn) time and an extension of the algorithm determines the number of pebbles needed to play the black and white pebble game on a tree.
Abstract: An algorithm is presented that finds a min-cut linear arrangement of a tree in O(n log n) time. An extension of the algorithm determines the number of pebbles needed to play the black and white pebble game on a tree.
132 citations
"Reversible Pebble Game on Trees" refers background in this paper
...However, if we restrict the DAG to a tree, the irreversible black pebble game[9] and black-white pebble game[13] are solvable in polynomial time....
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