We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth n tree using O(n2+logω) queries, where ω is independent of n and depends only on the type of subformulas within the tree. We also prove a classical lower bound of nΩ(log log n) queries, thus showing a (small) super-polynomial speed-up.
We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth n tree using O(n2+logω) queries, where ω is independent of n and depends only on the type of subformulas within the tree. We also prove a classical lower bound of nΩ(log log n) queries, thus showing a (small) super-polynomial speed-up.
Recently, Farhi, Goldstone, and Gutmann gave a quantum algorithm for evaluating nand trees that runs in time O ( √ N log N) in the Hamiltonian query model. In this note, we point out that their algorithm can be converted into an algorithm using O(N 1/2+ǫ) queries in the conventional quantum query model, for any fixed ǫ> 0. A nand tree of depth n is a balanced binary tree whose internal vertices represent nand gates. Placing bits x1,...,x2 n at the leaves, the root of the nand tree evaluates to the function fn(x1,...,x2 n), where fn: {0, 1} 2n → {0, 1} is defined recursively as follows. For n = 0, f0(x) = x, and for n> 0, fn(x1,..., x2 n) = ¬ ( fn−1(x1,..., x 2 n−1) ∧ fn−1(x 2 n−1 +1,...,x2 n)). (1) The goal of the nand tree problem is to evaluate fn(x1,..., x2 n), making as few queries to the bits x1,..., x2 n as possible. The optimal classical randomized algorithm for this problem makes Θ(N 0.753) queries, where N = 2 n [7, 8, 9]. Until now, no better quantum algorithm was known, whereas the best known quantum lower bound is only Ω ( √ N) [1]. Here we show that for any fixed ǫ> 0, the quantum query complexity of evaluating nand trees is O(N 1/2+ǫ).
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 give an O(sqrt n log n)-query quantum algorithm for evaluating size-n AND-OR formulas. Its running time is poly-logarithmically greater after efficient preprocessing. Unlike previous approaches, the algorithm is based on a quantum walk on a graph that is not a tree. Instead, the algorithm is based on a hybrid of direct-sum span program composition, which generates tree-like graphs, and a novel tensor-product span program composition method, which generates graphs with vertices corresponding to minimal zero-certificates. For comparison, by the general adversary bound, the quantum query complexity for evaluating a size-n read-once AND-OR formula is at least Omega(sqrt n), and at most O(sqrt{n} log n / log log n). However, this algorithm is not necessarily time efficient; the number of elementary quantum gates applied between input queries could be much larger. Ambainis et al. have given a quantum algorithm that uses sqrt{n} 2^{O(sqrt{log n})} queries, with a poly-logarithmically greater running time.
We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f) , we give an algorithm for evaluating F using O~(Q(f)⋅n) quantum queries. This improves on the bound of O(Q(f)⋅n) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O~(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0∘⊕ circuits of depth d+1 require size Ω~(n1/(1−2−d))≥ω(n1+2−d) to compute the Inner Product function even on average. The previous best size lower bound was Ω(n1+4−(d+1)) and only held in the worst case (Cheraghchi et al., JCSS 2018).
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.
Recent breakthroughs in quantum query complexity have shown that any formula of size n can be evaluated with O(sqrt(n)log(n)/log log(n)) many quantum queries in the bounded-error setting [FGG08, ACRSZ07, RS08b, Rei09]. In particular, this gives an upper bound on the approximate polynomial degree of formulas of the same magnitude, as approximate polynomial degree is a lower bound on quantum query complexity [BBCMW01]. These results essentially answer in the affirmative a conjecture of O'Donnell and Servedio [O'DS03] that the sign degree--the minimal degree of a polynomial that agrees in sign with a function on the Boolean cube--of every formula of size n is O(sqrt(n)). In this note, we show that sign degree is super-multiplicative under function composition. Combining this result with the above mentioned upper bounds on the quantum query complexity of formulas allows the removal of logarithmic factors to show that the sign degree of every size n formula is at most sqrt(n).