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Electronic Colloquium on Computational Complexity

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REPORTS > AUTHORS > ANDRIS AMBAINIS:
All reports by Author Andris Ambainis:

TR17-150 | 26th September 2017
Andris Ambainis, Martins Kokainis, Krisjanis Prusis, Jevgenijs Vihrovs

All Classical Adversary Methods are Equivalent for Total Functions

Revisions: 2

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions, and are equal to the fractional block sensitivity $\text{fbs}(f)$. That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. For partial functions, we show ... more >>>


TR15-200 | 4th December 2015
Andris Ambainis

Almost quadratic gap between partition complexity and query/communication complexity

Revisions: 1

We show nearly quadratic separations between two pairs of complexity measures:
1. We show that there is a Boolean function $f$ with $D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic query complexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$;
2. As a consequence, we obtain that there is ... more >>>


TR15-196 | 4th December 2015
Andris Ambainis

Polynomials, Quantum Query Complexity, and Grothendieck's Inequality

Revisions: 1

We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials.
Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $\epsilon<1/2$ iff $f$ can be approximated by a degree-2 polynomial with error bounded by $\epsilon'<1/2$.
This result holds for two ... more >>>


TR15-098 | 15th June 2015
Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, Juris Smotrovs

Separations in Query Complexity Based on Pointer Functions

Revisions: 2

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized
query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree
of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. ... more >>>


TR14-155 | 21st November 2014
Scott Aaronson, Andris Ambainis

Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum ... more >>>


TR14-154 | 20th November 2014
Andris Ambainis, Yuval Filmus, Francois Le Gall

Fast Matrix Multiplication: Limitations of the Laser Method

Until a few years ago, the fastest known matrix multiplication algorithm, due to Coppersmith and Winograd (1990), ran in time $O(n^{2.3755})$. Recently, a surge of activity by Stothers, Vassilevska-Williams, and Le Gall has led to an improved algorithm running in time $O(n^{2.3729})$. These algorithms are obtained by analyzing higher ... more >>>


TR14-152 | 13th November 2014
Andris Ambainis, Mohammad Bavarian, Yihan Gao, Jieming Mao, Xiaoming Sun, Song Zuo

Tighter Relations Between Sensitivity and Other Complexity Measures

Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is polynomially related to other major complexity measures. Despite much attention to the problem and major advances ... more >>>


TR14-077 | 2nd June 2014
Andris Ambainis, Jevgenijs Vihrovs

Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

Revisions: 2

We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of ... more >>>


TR14-027 | 21st February 2014
Andris Ambainis, Krisjanis Prusis

A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity

Revisions: 1

Sensitivity, certificate complexity and block sensitivity are widely used Boolean function complexity measures. A longstanding open problem, proposed by Nisan and Szegedy, is whether sensitivity and block sensitivity are polynomially related. Motivated by the constructions of functions which achieve the largest known separations, we study the relation between 1-certificate complexity ... more >>>


TR13-164 | 28th November 2013
Scott Aaronson, Andris Ambainis, Kaspars Balodis, Mohammad Bavarian

Weak Parity

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that ... more >>>


TR11-116 | 17th August 2011
Andris Ambainis, Xiaoming Sun

New separation between $s(f)$ and $bs(f)$

In this note we give a new separation between sensitivity and block sensitivity of Boolean functions: $bs(f)=\frac{2}{3}s(f)^2-\frac{1}{3}s(f)$.

more >>>

TR10-191 | 9th December 2010
Andris Ambainis, Loïck Magnin, Martin Roetteler, Jérémie Roland

Symmetry-assisted adversaries for quantum state generation

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum ... more >>>


TR09-110 | 5th November 2009
Scott Aaronson, Andris Ambainis

The Need for Structure in Quantum Speedups

Revisions: 1

Is there a general theorem that tells us when we can hope for exponential speedups from quantum algorithms, and when we cannot? In this paper, we make two advances toward such a theorem, in the black-box model where most quantum algorithms operate.

First, we show that for any problem that ... more >>>


TR07-013 | 6th February 2007
Andris Ambainis, Joseph Emerson

Quantum t-designs: t-wise independence in the quantum world

A t-design for quantum states is a finite set of quantum states with the property of simulating the Haar-measure on quantum states w.r.t. any test that uses at most t copies of a state. We give efficient constructions for approximate quantum t-designs for arbitrary t.

We then show that an ... more >>>


TR04-120 | 22nd November 2004
Andris Ambainis, William Gasarch, Aravind Srinivasan, Andrey Utis

Lower bounds on the Deterministic and Quantum Communication Complexity of HAM_n^a

Alice and Bob want to know if two strings of length $n$ are
almost equal. That is, do they differ on at most $a$ bits?
Let $0\le a\le n-1$.
We show that any deterministic protocol, as well as any
error-free quantum protocol ($C^*$ version), for this problem
requires at ... more >>>


TR03-082 | 22nd November 2003
Andris Ambainis, Ke Yang

Towards the Classical Communication Complexity of Entanglement Distillation Protocols with Incomplete Information

Entanglement is an essential resource for quantum communication and quantum computation, similar to shared random bits in the classical world. Entanglement distillation extracts nearly-perfect entanglement from imperfect entangled state. The classical communication complexity of these protocols is the minimal amount of classical information that needs to be exchanged for the ... more >>>


TR01-019 | 2nd March 2001
Andris Ambainis, Harry Buhrman, William Gasarch, Bala Kalyansundaram, Leen Torenvliet

The Communication Complexity of Enumeration, Elimination, and Selection

Normally, communication Complexity deals with how many bits
Alice and Bob need to exchange to compute f(x,y)
(Alice has x, Bob has y). We look at what happens if
Alice has x_1,x_2,...,x_n and Bob has y_1,...,y_n
and they want to compute f(x_1,y_1)... f(x_n,y_n).
THis seems hard. We look at various ... more >>>


TR99-012 | 19th April 1999
Eric Allender, Andris Ambainis, David Mix Barrington, Samir Datta, Huong LeThanh

Bounded Depth Arithmetic Circuits: Counting and Closure

Comments: 1

Constant-depth arithmetic circuits have been defined and studied
in [AAD97,ABL98]; these circuits yield the function classes #AC^0
and GapAC^0. These function classes in turn provide new
characterizations of the computational power of threshold circuits,
and provide a link between the circuit classes AC^0 ... more >>>


TR98-020 | 10th April 1998
Andris Ambainis, David Mix Barrington, Huong LeThanh

On Counting $AC^0$ Circuits with Negative Constants

Continuing the study of the relationship between $TC^0$,
$AC^0$ and arithmetic circuits, started by Agrawal et al.
(IEEE Conference on Computational Complexity'97),
we answer a few questions left open in this
paper. Our main result is that the classes Diff$AC^0$ and
Gap$AC^0$ ... more >>>




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