Weighted Context-Free-Language Ordered Binary Decision Diagrams

This paper presents a new data structure, called Weighted Context-Free-Language Ordered BDDs (WCFLOBDDs), which are a hierarchically structured decision diagram, akin to Weighted BDDs (WBDDs) enhanced with a procedure-call mechanism. For some functions, WCFLOBDDs are exponentially more succinct than WBDDs. They are potentially beneficial for representing functions of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi mathvariant="double-struck">B</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>D</mml:mi> </mml:math> , when a function’s image <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>V</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>D</mml:mi> </mml:math> has many different values. We apply WCFLOBDDs in quantum-circuit simulation, and find that they perform better than WBDDs on certain benchmarks. With a 15-minute timeout, the number of qubits that can be handled by WCFLOBDDs is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>64</mml:mn> <mml:mo>×</mml:mo> </mml:math> that of WBDDs (and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>128</mml:mn> <mml:mo>×</mml:mo> </mml:math> that of CFLOBDDs, which are an unweighted version of WCFLOBDDs). These results support the conclusion that for this application—from the standpoint of problem size, measured as the number of qubits—WCFLOBDDs provide the best of both worlds: performance roughly matches whichever of WBDDs and CFLOBDDs is better. (From the standpoint of running time, the results are more nuanced.)