We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard the computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log(n)) bits. Let E(f) be the Shannon entropy of the random variable f(X), where X is uniformly random in f's domain. Our main result is that it takes Ω(E(f)) queries to compute any \emph{total} function f. It is interesting to contrast this bound with the Ω(E(f)/log(n)) bound, which is tight for \emph{partial} functions. Our approach is the polynomial method.
We prove a general lower bound of quantum decision tree complexity in terms of some entropy notion. We regard the computation as a communication process in which the oracle and the computer exchange several rounds of messages, each round consisting of O(log(n)) bits. Let E(f) be the Shannon entropy of the random variable f(X), where X is uniformly random in f's domain. Our main result is that it takes Ω(E(f)) queries to compute any \emph{total} function f. It is interesting to contrast this bound with the Ω(E(f)/log(n)) bound, which is tight for \emph{partial} functions. Our approach is the polynomial method.
Let P = (X, < P) be a partial order on a set of n elements X = x1, x2,..., xn. Define the quantum sorting problem QSORTP as: given n distinct numbers x1, x2,..., xn consistent with P, sort them by a quantum decision tree using comparisons of the form "xi: xj". Let Qε(P) be the minimum number of queries used by any quantum decision tree for solving QSORTP with error less than ε (where 0 < ε < 1/10 is fixed). It was proved by Hoyer, Neerbek and Shi (Algorithmica 34 (2002), 429--448) that, when P0 is the empty partial order, Qε(P0) ≥ Ω (n log n), i. e., the classical information lower bound holds for quantum decision trees when the input permutations are unrestricted.In this paper we show that the classical information lower bound holds, up to an additive linear term, for quantum decision trees for any partial order P. Precisely, we prove Qε(P) ≥ c log2 e(P)-c'n where c,c' > 0 are constants and e(P) is the number of linear orderings consistent with P. Our proof builds on an interesting connection between sorting and Korner's graph entropy that was first noted and developed by Kahn and Kim (JCSS 51(1995), 390--399).
We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
We reformulate the notion of quantum query complexity in terms of inequalities and equations for a set of positive matrices, which we view as a quantum analogue of a decision tree. Using the new formulation we show that: 1. every quantum query algorithm needs to use at most n quantum bits in addition to the query register. 2. For any function f there is an algorithm that runs in polynomial time in terms the truth table of f and (for {var_epsilon} > 0) computes the {var_epsilon}-error quantum decision tree complexity of f. 3. Using the dual of our system we can treat lower bound methods on a uniform platform, which paves the way to their future comparison. In particular we describe Ambainis's bound in our framework. 4. The output condition on quantum algorithms used by Ambainis and others is not sufficient for an algorithm to compute a function with {var_epsilon}-bounded error: we show the existence of algorithms whose final entanglement matrix satisfies the condition, but for which the value of f cannot be determined from a quantum measurement on the accessible part of the computer.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
Given a classical query algorithm as a decision tree, when does there exist a quantum query algorithm with a speed-up over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up. In particular, we obtain a bounded-error quantum query algorithm of cost O(s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s. Lin and Lin [ToC'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor, and gave upper bounds in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Inf. Comp.'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any complexity-theoretic measure. We also show a polynomial separation between rank and randomized rank for the complete binary AND-OR tree. Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to its edges. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are asymptotically stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes could yield better upper bounds.