Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x> to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with $O(\sqrt{N})$ XOR queries but was conjectured to require $Ω(N)$ in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using $O(\sqrt{N})$ in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and $Θ(\sqrt{N})$ in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.Quantum query complexity is typically characterized in terms of XOR queries |x,y> to |x,y+f(x)> or phase queries, which ensure that even queries to non-invertible functions are unitary. When querying a permutation, another natural model is unitary: in-place queries |x> to |f(x)>. Some problems are known to require exponentially fewer in-place queries than XOR queries, but no separation has been shown in the opposite direction. A candidate for such a separation was the problem of inverting a permutation over N elements. This task, equivalent to unstructured search in the context of permutations, is solvable with $O(\sqrt{N})$ XOR queries but was conjectured to require $Ω(N)$ in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using $O(\sqrt{N})$ in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and $Θ(\sqrt{N})$ in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.

arXiv:2504.03620Leibniz international proceedings in informaticsJan 1, 20250 citations

QUANTUM QUERY COMPLEXITIES

Kazuo Iwama

International Symposium on Mesoscopic Superconductivity and Spintronics (MS+S)Aug 1, 20050 citations

Quantum-Quick Queries

Ivars Peterson

Science NewsAug 31, 19964 citations

Complex Quantum Queries

Lov K. Grover

A quantum mechanical computer can search a database consisting of N items in O(sqrt(N)) elementary quantum queries (a query is defined as any question to the database to which a binary (YES/NO) answer is required; an elementary query is a query pertaining to only one of the N possible states). A classical computer would take at least O(N) elementary queries. This paper shows that a quantum mechanical algorithm that can query information relating to multiple states, can search the database in a single query. A classical algorithm will be limited to the information theoretic bound of at least O(log N)queries (which it would achieve by using a binary search). A similar result was recently independently obtained by Terhal & Smolin by a different method.A quantum mechanical computer can search a database consisting of N items in O(sqrt(N)) elementary quantum queries (a query is defined as any question to the database to which a binary (YES/NO) answer is required; an elementary query is a query pertaining to only one of the N possible states). A classical computer would take at least O(N) elementary queries. This paper shows that a quantum mechanical algorithm that can query information relating to multiple states, can search the database in a single query. A classical algorithm will be limited to the information theoretic bound of at least O(log N)queries (which it would achieve by using a binary search). A similar result was recently independently obtained by Terhal & Smolin by a different method.

Jun 3, 19970 citations

UNDERSTANDING QUANTUM ALGORITHMS VIA QUERY COMPLEXITY

Andris Ambainis

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 bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring $Ω(\sqrt{N})$ queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with $M$ queries quantumly but requiring $\tildeΩ(M^{2.5})$ queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with $M$ queries probabilistically but requiring $\tildeΩ(M^{2})$ queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.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 bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring $Ω(\sqrt{N})$ queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with $M$ queries quantumly but requiring $\tildeΩ(M^{2.5})$ queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with $M$ queries probabilistically but requiring $\tildeΩ(M^{2})$ queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.

arXiv:1712.06349International Congress of Mathematicians (ICM)May 1, 201965 citations

Quantum Search

Sergei Kurgalin, Sergei Borzunov

Texts in computer scienceJan 1, 20210 citations

O(\sqrt{N})

XOR queries but was conjectured to require

Ω(N)

in-place queries. We refute this conjecture by designing a quantum algorithm for Permutation Inversion using

O(\sqrt{N})

in-place queries. Our algorithm achieves the same speedup as Grover's algorithm despite the inability to efficiently uncompute queries or perform straightforward oracle-controlled reflections. Nonetheless, we show that there are indeed problems which require fewer XOR queries than in-place queries. We introduce a subspace-conversion problem called Function Erasure that requires 1 XOR query and

Θ(\sqrt{N})

in-place queries. Then, we build on a recent extension of the quantum adversary method to characterize exact conditions for a decision problem to exhibit such a separation, and we propose a candidate problem.

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 bounds on the query complexity, bounding the possible quantum advantage. In the last few years, there have been major advances on several longstanding problems in the query complexity. In this talk, we survey these results and related work, including: - the biggest quantum-vs-classical gap for partial functions (a problem solvable with 1 query quantumly but requiring

Ω(\sqrt{N})

queries classically); - the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for total functions (for example, a problem solvable with

M

queries quantumly but requiring

\tildeΩ(M^{2.5})

queries probabilistically); - the biggest probabilistic-vs-deterministic gap for total functions (a problem solvable with

M

queries probabilistically but requiring

\tildeΩ(M^{2})

queries deterministically); - the bounds on the gap that can be achieved for subclasses of functions (for example, symmetric functions); - the connections between query algorithms and approximations by low-degree polynomials.