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

Abstract

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 3 n). Moreover, we proved a similar hierarchy for bounded error probabilistic k-OBDD. And for deterministic and probabilistic k-OBDDs of superpolynomial and subexponential size.

Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test

“…Note that due to the definition, k-OBDD is polynomial width iff it is polynomial size. Similar hierarchies are known for classical cases [11,6,17,19]. But for k-QOBDD it is a new result.The paper has the following structure.…”

“…Hierarchy for Polynomial Size. Let us consider a Boolean function XRP J k,n , it is a modification of boolean version of P J k,n function using reordering method from [19]. We add address for each bit of input and compute with respect to the address in original input.…”

“…If we meet bits with the same address, then we consider their XOR. XRP J k,n is a total version of xor-reordered P J k,n , details in [19]. Let us define this formally.…”

“…The proof of the first claim is based on lower bound for quantum communication complexity from [24,25]. We apply the bound in the similar way as in [19]. Assume that XRP J 2k−1,n is computed by k-QOBDD P of width w = 2 o(r) .…”

“…The main idea is to store a pointer for current steps and use new qubits for a new step. And we apply reordering method from [19]. See Appendix F for the full proof.…”