# Query Complexity in Expectation

@article{Kaniewski2015QueryCI, title={Query Complexity in Expectation}, author={Jędrzej Kaniewski and Troy Lee and Ronald de Wolf}, journal={ArXiv}, year={2015}, volume={abs/1411.7280} }

We study the query complexity of computing a function \(f:\{0,1\}^n\rightarrow \mathbb {R}_+\) in expectation. This requires the algorithm on input \(x\) to output a nonnegative random variable whose expectation equals \(f(x)\), using as few queries to the input \(x\) as possible. We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-of-squares degree, respectively. We observe that the quantum… Expand

#### Topics from this paper

#### 11 Citations

Post-selected Classical Query Complexity

- Computer Science, Mathematics
- ArXiv
- 2018

It is shown that the zero-error variants of post-selected query algorithms are equivalent to non-deterministic classical query algorithms, which in turn are characterised by nonnegative polynomials, and for some problems require exponentially more queries to the input than their bounded-error counterparts. Expand

On the sum-of-squares degree of symmetric quadratic functions

- Mathematics, Computer Science
- Computational Complexity Conference
- 2016

Three complexity-theoretic applications are described: a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further, and bounds on the query complexity of quantum algorithms whose expected output approximates such functions. Expand

When Is Amplification Necessary for Composition in Randomized Query Complexity?

- Computer Science, Mathematics
- APPROX-RANDOM
- 2020

It is shown that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Expand

Quantum Query Algorithms are Completely Bounded Forms

- Computer Science, Physics
- ITCS
- 2018

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. Expand

Communication Lower Bounds via Query Complexity

- Computer Science
- 2016

This thesis proves lower bounds in communication complexity by exploiting new connections to query complexity by proving a general theorem stating that for a large class of communication problems F, any protocol for F can be efficiently simulated by a decision tree solving a related problem f . Expand

Nearly optimal separations between communication (or query) complexity and partitions

- Computer Science, Physics
- Computational Complexity Conference
- 2015

We show a nearly quadratic separation between deterministic communication complexity and the logarithm of the partition number, which is essentially optimal. This improves upon a recent power 1.5… Expand

A Composition Theorem for Conical Juntas

- Computer Science, Mathematics
- Electron. Colloquium Comput. Complex.
- 2015

A general method of proving degree lower bounds for conical juntas (nonnegative combinations of conjunctions) that compute recursively defined boolean functions that compute recursive NAND functions is described. Expand

A Majority Lemma for Randomised Query Complexity

- Computer Science
- Electron. Colloquium Comput. Complex.
- 2021

We show that computing the majority of n copies of a boolean function g has randomised query complexity R(Maj ◦ g) = Θ(n · R1/n(g)). In fact, we show that to obtain a similar result for any composed… Expand

Randomized Communication versus Partition Number

- Mathematics, Computer Science
- ACM Trans. Comput. Theory
- 2018

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the… Expand

Randomized Communication vs. Partition Number

- Computer Science, Mathematics
- Electron. Colloquium Comput. Complex.
- 2015

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the… Expand

#### References

SHOWING 1-10 OF 47 REFERENCES

Quantum lower bounds by polynomials

- Mathematics, Computer Science
- JACM
- 2001

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

Nondeterministic Quantum Query and Communication Complexities

- Mathematics, Computer Science
- SIAM J. Comput.
- 2003

The nondeterministic quantum algorithms for Boolean functions f have positive acceptance probability on input x iff f(x)=1, which implies that the quantum communication complexities of the equality and disjointness functions are n+1 if the authors do not allow any error probability. Expand

Nondeterministic Quantum Query and Quantum Communication Complexities

- Mathematics, Computer Science
- ArXiv
- 2000

The nondeterministic quantum algorithms for Boolean functions f have positive acceptance probability on input x iff f(x)=1, which implies that the quantum communication complexities of the equality and disjointness functions are n+1 if the authors do not allow any error probability. Expand

Sums of squares on the hypercube

- Mathematics
- 2014

Let X be a finite set of points in $${\mathbb {R}}^n$$Rn. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of… Expand

The matching polytope does not admit fully-polynomial size relaxation schemes

- Computer Science, Mathematics
- SODA
- 2015

It turns out that the high extension complexity for the matchingpolytope stem from the same source of hardness as for the correlation polytope: a direct sum structure. Expand

Bounds for small-error and zero-error quantum algorithms

- Mathematics, Computer Science
- 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

We present a number of results related to quantum algorithms with small error probability and quantum algorithms that are zero-error. First, we give a tight analysis of the trade-offs between the… Expand

The matching polytope has exponential extension complexity

- Mathematics, Computer Science
- STOC
- 2014

By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2Ω(√n) to 2 Ω(n). Expand

Lower Bounds on the Size of Semidefinite Programming Relaxations

- Computer Science, Mathematics
- STOC
- 2015

It is proved that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations, and this result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Expand

Quantum vs. classical communication and computation

- Mathematics, Computer Science
- STOC '98
- 1998

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. Expand

Communication complexity lower bounds by polynomials

- Mathematics, Physics
- Proceedings 16th Annual IEEE Conference on Computational Complexity
- 2001

The "log rank" lower bound extends to the strongest variant of quantum communication complexity (qubit communication+unlimited prior entanglement) and the polynomial equivalence of quantum and classical communication complexity for various classes of functions is proved. Expand