We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω ( √ N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via variety of different techniques. 1

We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1

The collision problem is to decide whether a function X : . . . , n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O , but obtaining any lower bound better than# (1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.

We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.

The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M 1.321...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method. 1

Abstract. We present a quantum algorithm for the dihedral hidden subgroup problem (DHSP) with time and query complexity 2O(√log N). In this problem an oracle computes a function f on the dihedral group DN which is invariant under a hidden reflection in DN. By contrast, the classical query complexity of DHSP is O ( √ N). The algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group. The algorithm begins as usual with a quantum character transform, which in the case of DN is essentially the abelian quantum Fourier transform. This yields the name of a group representation of DN, which is not by itself useful, and a state in the representation, which is a valuable but indecipherable qubit. The algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of DN. Once the algorithm obtains certain target representations, direct measurements reveal the hidden subgroup.

We present a quantum algorithm that verifies a product of two n×n matrices over any integral domain with bounded error in worst-case time O(n 5/3) and expected time O(n 5/3 / min(w, √ n) 1/3), where w is the number of wrong entries. This improves the previous best algorithm [ABH + 02] that runs in time O(n 7/4). We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries. 1

Abstract. We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for non-deterministic protocols of de Wolf. We also give an O ( √ n·c log ∗ n)-qubit bounded-error protocol for disjointness, modifying and improving the earlier O ( √ n log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Ω ( √ n) lower bound for a class of protocols that includes the BCW-protocol as well as our new protocol. 1

Abstract not found

The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambainis’s lower bound theorems. Then it is shown that all these four Ambainis’s lower bounds have an upper bound, which is in terms of certificate complexity. This implies that for some problems such as TRIANGLE, k-CLIQUE, and BIPARTITE/GRAPH MATCHING whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis’s techniques. Another consequence is that all the Ambainis’s lower bounds are not tight. Finally, we show that for total functions, this upper bound for Ambainis’s lower bounds can be further improved. This also implies limitation of Ambainis’s method on some specific problems such as AND-OR TREE, whose precise quantum complexity is still unknown. 1