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Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

by   Kamil Khadiev, et al.

In this paper, we study quantum Ordered Binary Decision Diagrams(OBDD) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2^Ω(n / log n), and the quantum OBDD complexity of it is at most O(n^2/log n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. Another function(shifted equality function) allows us to obtain a gap 2^Ω(n) vs O(n^2). Moreover, we prove the bounded error quantum and probabilistic OBDD width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (k-OBDD) of polynomial width, for k = o(n / log^3 n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017


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1 Introduction

Branching programs are a well-known computation model for discrete functions. This model has been shown useful in a variety of domains, such as hardware verification, model checking, and other CAD applications [Weg00].

One of the most important types of branching programs is oblivious read once branching programs, also known as Ordered Binary Decision Diagrams, or - OBDD  [Weg00]. This model is suitable for studying of data streaming algorithms that are actively used in industry.

One of the most useful measures of complexity of - OBDD s is “width”. This measure is an analog of a number of states for finite automaton and - OBDD s can be seen as nonuniform finite automata (see for example [AG05]). As for many other computation models, it is possible to consider quantum - OBDD s, and during the last decade they have been studied vividly [aazksw2019part1, AGK01, nhk00, ss2005, s06, aakk2018, ikpy2017, ikpy2021, gy2017, gy2015, gy2018, kkm2017, kkkrym2017].

In 2005 Ablayev, Gainutdinova, Karpinski, Moore, and Pollett [AGKMP05] have proven that for any total Boolean function the gap between the width of the minimal quantum - OBDD representing and the width of the minimal deterministic - OBDD representing is at most exponential. However, this is not true for partial functions [AGKY14, g15, agky16]. They have also shown that this bound could be reached for function, that takes the value on an input iff number of s modulo in this input is equal to . Authors have presented a quantum - OBDD of width for (another quantum - OBDD of the same width has been presented in [AV08]). Additionally, they have proven that any deterministic - OBDD representing has the width at least . However, a lower bound for a width of a deterministic - OBDD that represents is tight, and it was unknown if it is possible to construct a function with an exponential gap but an exponential lower bound for the size of a deterministic - OBDD representing this function. It was shown that Boolean function did not have a deterministic - OBDD representation of width less than  [kmw91]. In 2005 Sauerhoff and Sieling [ss2005] presented a quantum - OBDD of width representing and three years later Ablayev, Khasianov, and Vasiliev [akv2008] improved this lower bound and presented a quantum - OBDD for this function of width . But as in the previous case, this separation does not give us a truly exponential lower bound for deterministic - OBDD s.

Nevertheless, if we fix an order of variables in the - OBDD , it is possible to prove the desired statement. For example, it is known that equality function, or , does not have an - OBDD representation of the size less than for some order and it has a quantum - OBDD of width for any order [akv2008]. Unfortunately, for some orders, the equality function has a small deterministic - OBDD s.

Proving lower bounds for different orders is one of the main difficulties of proving lower bounds on width of - OBDD s. In the paper, we present a new technique that allows us to prove such lower bounds. Using the technique, we construct a Boolean function from a Boolean function such that if any deterministic - OBDD representing with the natural order over the variables has width at least , then any deterministic - OBDD representing has width at least for any order of input variables and if there is a quantum - OBDD of width for , then there is a quantum - OBDD of width for the function . It means that if we have a function with some gap between quantum - OBDD complexity and deterministic - OBDD complexity for some order, then we can transform this function into a function with almost the same gap but for all the orders. We call this transformation “reordering”. The idea which is used in the construction of the transformation is similar to the idea of a transformation from [k15, DBLP:journals/jsyml/Krajicek08].

We prove five groups of results using the transformation. At first, we consider the result of the transformation applied to the equality function (we call the new function reordered equality or ). We prove that does not have a deterministic - OBDD representation of width less than and there is a bounded error quantum - OBDD of width , where is a length of an input. As a result, we get a more significant gap between a width of quantum - OBDD s and width of deterministic - OBDD s than this gap for the function. We prove such a gap for all the orders in contrast with a gap for , and we prove a better lower bound for deterministic - OBDD s than the lower bound for the function.

Additionally, we considered shifted equality function (). We prove that does not have a deterministic - OBDD representation of the width less than and there is a bounded error quantum - OBDD with width . Note that the lower bound for the width of the minimal - OBDD representing is better than for but the upper bound for the width of the minimal quantum - OBDD representation is much better.

Using properties of , , and mixed weighted sum function () introduced by [s2005], we prove a width hierarchy for classes of Boolean functions computed by bounded error quantum - OBDD s. We prove three hierarchy theorems:

  1. the first of them and the tightest works for width up to ;

  2. the second of them is slightly worse than the previous one, but it works for width up to ;

  3. and finally the third one with the widest gap works for width up to .

Similar hierarchy theorems are already known for deterministic - OBDD s [AGKY14, AK13], nondeterministic - OBDD s [agky16], and -k-OBDD s [k15, ki2017, aakk2018]. Additionally, we present similar hierarchy theorems for bounded error probabilistic - OBDD s in the paper.

Hierarchies for quantum-model complexity classes and gaps for deterministic and quantum complexities were shown by researchers for automata models [sy2014, kiy2018, qy2009] and other streaming (automata-like) models [l2006, l2009, kkm2018, kk2019, kk2019disj, kk2022, kkzmkry2022].

The fourth group of results is an extension of hierarchies by a number of tests for deterministic and bounded error probabilistic -k-OBDD s of polynomial size. There are two known results of this type:

  • The first is a hierarchy theorem for -k-OBDD s that was proven by Bollig, Sauerhoff, Sieling, and Wegener [bssw96]. They have shown that