Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams

TL;DR

The paper addresses the problem of depth- and width-based separations between quantum and classical OBDDs, seeking explicit functions whose width gaps persist under arbitrary variable orders. It introduces the reordering method, which embeds input addresses into blocks to transform order-specific gaps into order-agnostic gaps, enabling the construction of the REQ function with deterministic width $2^{\Omega(n/\log n)}$ while admitting a quantum OBDD of width $O(n^2/\log^2 n)$. Leveraging REQ and related function families, the work establishes width hierarchies for bounded-error probabilistic and quantum OBDDs across multiple width regimes, and extends these insights to deterministic and probabilistic $k$-OBDDs via reordered Pointer Jumping (RPJ), yielding almost-tight inclusions like $P{-}k$OBDD \subsetneq $P{-}2k$OBDD$ for $k = o(n/\log^3 n)$ and analogous size-class separations. These results provide explicit, order-independent gaps and demonstrate a structured hierarchy of Boolean function classes under OBDD width constraints, advancing understanding of width-based quantum vs. classical separations. Key findings include the explicit REQ function with a substantial width gap and the extended hierarchies across quantum, probabilistic, and $k$-OBDD models.

Abstract

We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to "width" complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called "reordering"), which allows to build Boolean function $g$ from Boolean Function $f$, such that if for $f$ we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function $g$, but for any order. Using it we construct the total function $REQ$ which deterministic OBDD complexity is $2^{Ω(n/\log n)}$ and present quantum OBDD of width $O(n^2)$. It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs. Additionally, we prove the width hierarchy for complexity classes of Boolean functions for bounded error probabilistic OBDDs. And using "reordering" method we extend a hierarchy for $k$-OBDD of polynomial size, for $k=o(n/\log^3n)$. Moreover, we proved a similar hierarchy for bounded error probabilistic $k$-OBDD. And for deterministic and probabilistic $k$-OBDDs of superpolynomial and subexponential size.

Figures (1)

• Figure 1: Input. Blocks of address and value bits.

Theorems & Definitions (14)

• theorem thmcountertheorem
• theorem thmcountertheorem
• Corollary 1
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• Corollary 2
• theorem thmcountertheorem
• lemma thmcounterlemma
• theorem thmcountertheorem
• lemma thmcounterlemma
• theorem thmcountertheorem