Understanding Quantum Algorithms via Query Complexity

  title={Understanding Quantum Algorithms via Query Complexity},
  author={Andris Ambainis},
  • A. Ambainis
  • Published 18 December 2017
  • Computer Science
  • ArXiv
Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower… 

Figures from this paper

Variational learning algorithms for quantum query complexity

Quantum query complexity plays an important role in studying quantum algorithms, which captures the most known quantum algorithms, such as search and period finding. A query algorithm applies U t O x

From the sum-of-squares representation of a Boolean function to an optimal exact quantum query algorithm

A primary algorithm framework is proposed with three basic steps that can be used to investigate the quantum query model with low complexity, such as Deutsch’s problem, a five-bit symmetric Boolean function and the characterization of Boolean functions with exact quantum 2-query complexity.

Fast Classical and Quantum Algorithms for Online k-server Problem on Trees

A quantum algorithm to find the first marked element in a collection of $m$ objects, that works even in the presence of two-sided bounded errors on the input oracle, that has worst-case complexity $O(\sqrt{m})$.

A Note on the Quantum Query Complexity of Permutation Symmetric Functions

This paper improves the result of [AA14] and shows that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity.

Unitary property testing lower bounds by polynomials

A generalized polynomial method for unitary property testing problems, leveraging connections with invariant theory, is applied to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximates the entanglement entropy of a marking state.

A Quantum Query Complexity Trichotomy for Regular Languages

The algorithm for star-free languages is viewed as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known.

A tight lower bound for non-coherent index erasure

A tight $\Omega(\sqrt{n})$ lower bound is proved on the quantum query complexity of the non-coherent case of the index erasure problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state.

Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator

We study algorithms for solving three problems on strings. These are sorting of n strings of length k, “the Most Frequent String Search Problem”, and “searching intersection of two sequences of

Extended Learning Graphs for Triangle Finding

New quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances are presented and a framework is presented in order to easily combine and analyze them.

A Tight Lower Bound for Index Erasure

A tight $\Omega(\sqrt{n})$ lower bound is proved on the quantum query complexity of the non-coherent case of the Index Erasure problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state.



The Need for Structure in Quantum Speedups

It is shown that for any problem that is invariant under permuting inputs and outputs, the quantum query complexity is at least the 9 root of the classical randomized query complexity, and that one essentially cannot hope to prove P 6 BQP relative to a random oracle.

Sculpting Quantum Speedups

This work gives a full characterization of sculptable functions in the query complexity setting, and investigates sculpting in the Turing machine model, showing that if there is any BPP-bi-immune language in BQP, then every language outside BPP can be restricted to a promise which places it in PromiseBQP but not in promiseBPP.

Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

It is conjectured that a natural generalization of Forrelation achieves the optimal t versus Ω(N1-1/2t) separation for all t, and it is shown that this generalization is BQP-complete.

Quantum lower bounds by polynomials

This work examines 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 and gives asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings.

Quantum Query Algorithms are Completely Bounded Forms

A characterization of $t-query quantum algorithms in terms of the unit ball of a space of degree-$2t$ polynomials is proved, and it is shown that many polynmials of degree four are far from those coming from two- query quantum algorithms.

New Results on Quantum Property Testing

The third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman.

Separations in query complexity using cheat sheets

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could

Separations in query complexity based on pointer functions

This work shows that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D( f)0.7537…).

Quantum lower bounds for the collision and the element distinctness problems

  • Yaoyun Shi
  • Computer Science
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
It is proved that any quantum algorithm for finding a collision in an r-to-one function must evaluate the function /spl Omega/ ((n/r)/sup 1/3/) times, where n is the size of the domain and r|n.

Quantum vs. classical communication and computation

A simple and general simulation technique is presented that transforms any black-box quantum algorithm to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism, to obtain new positive and negative results.