All reports by Author Andris Ambainis:

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TR21-115
| 6th August 2021
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Scott Aaronson, Andris Ambainis, Andrej Bogdanov, Krishnamoorthy Dinesh, Cheung Tsun Ming#### On quantum versus classical query complexity

Revisions: 2

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TR17-150
| 26th September 2017
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Andris Ambainis, Martins Kokainis, Krisjanis Prusis, Jevgenijs Vihrovs#### All Classical Adversary Methods are Equivalent for Total Functions

Revisions: 2

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TR15-200
| 4th December 2015
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Andris Ambainis#### Almost quadratic gap between partition complexity and query/communication complexity

Revisions: 1

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TR15-196
| 4th December 2015
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Andris Ambainis#### Polynomials, Quantum Query Complexity, and Grothendieck's Inequality

Revisions: 1

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TR15-098
| 15th June 2015
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Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, Juris Smotrovs#### Separations in Query Complexity Based on Pointer Functions

Revisions: 2

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TR14-155
| 21st November 2014
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Scott Aaronson, Andris Ambainis#### Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

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TR14-154
| 20th November 2014
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Andris Ambainis, Yuval Filmus, Francois Le Gall#### Fast Matrix Multiplication: Limitations of the Laser Method

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TR14-152
| 13th November 2014
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Andris Ambainis, Mohammad Bavarian, Yihan Gao, Jieming Mao, Xiaoming Sun, Song Zuo#### Tighter Relations Between Sensitivity and Other Complexity Measures

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TR14-077
| 2nd June 2014
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Andris Ambainis, Jevgenijs Vihrovs#### Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma

Revisions: 2

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TR14-027
| 21st February 2014
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Andris Ambainis, Krisjanis Prusis#### A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity

Revisions: 1

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

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TR11-116
| 17th August 2011
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Andris Ambainis, Xiaoming Sun#### New separation between $s(f)$ and $bs(f)$

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TR10-191
| 9th December 2010
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Andris Ambainis, Loïck Magnin, Martin Roetteler, Jérémie Roland#### Symmetry-assisted adversaries for quantum state generation

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TR09-110
| 5th November 2009
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Scott Aaronson, Andris Ambainis#### The Need for Structure in Quantum Speedups

Revisions: 1

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TR07-013
| 6th February 2007
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Andris Ambainis, Joseph Emerson#### Quantum t-designs: t-wise independence in the quantum world

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TR04-120
| 22nd November 2004
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Andris Ambainis, William Gasarch, Aravind Srinivasan, Andrey Utis#### Lower bounds on the Deterministic and Quantum Communication Complexity of HAM_n^a

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TR03-082
| 22nd November 2003
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Andris Ambainis, Ke Yang#### Towards the Classical Communication Complexity of Entanglement Distillation Protocols with Incomplete Information

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TR01-019
| 2nd March 2001
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Andris Ambainis, Harry Buhrman, William Gasarch, Bala Kalyansundaram, Leen Torenvliet#### The Communication Complexity of Enumeration, Elimination, and Selection

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TR99-012
| 19th April 1999
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Eric Allender, Andris Ambainis, David Mix Barrington, Samir Datta, Huong LeThanh#### Bounded Depth Arithmetic Circuits: Counting and Closure

Comments: 1

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TR98-020
| 10th April 1998
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Andris Ambainis, David Mix Barrington, Huong LeThanh#### On Counting $AC^0$ Circuits with Negative Constants

Scott Aaronson, Andris Ambainis, Andrej Bogdanov, Krishnamoorthy Dinesh, Cheung Tsun Ming

Aaronson and Ambainis (STOC 2015, SICOMP 2018) claimed that the acceptance probability of every quantum algorithm that makes $q$ queries to an $N$-bit string can be estimated to within $\epsilon$ by a randomized classical algorithm of query complexity $O_q((N/\epsilon^2)^{1-1/2q})$. We describe a flaw in their argument but prove that the ... more >>>

Andris Ambainis, Martins Kokainis, Krisjanis Prusis, Jevgenijs Vihrovs

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 >>>

Andris Ambainis

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 ...
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Andris Ambainis

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 ...
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Andris Ambainis, Kaspars Balodis, Aleksandrs Belovs, Troy Lee, Miklos Santha, Juris Smotrovs

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 >>>

Scott Aaronson, Andris Ambainis

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 >>>

Andris Ambainis, Yuval Filmus, Francois Le Gall

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 >>>

Andris Ambainis, Mohammad Bavarian, Yihan Gao, Jieming Mao, Xiaoming Sun, Song Zuo

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 >>>

Andris Ambainis, Jevgenijs Vihrovs

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 >>>

Andris Ambainis, Krisjanis Prusis

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 >>>

Scott Aaronson, Andris Ambainis, Kaspars Balodis, Mohammad Bavarian

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 >>>

Andris Ambainis, Xiaoming Sun

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 >>>Andris Ambainis, Loïck Magnin, Martin Roetteler, Jérémie Roland

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 >>>

Scott Aaronson, Andris Ambainis

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 >>>

Andris Ambainis, Joseph Emerson

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 >>>

Andris Ambainis, William Gasarch, Aravind Srinivasan, Andrey Utis

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 >>>

Andris Ambainis, Ke Yang

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 >>>

Andris Ambainis, Harry Buhrman, William Gasarch, Bala Kalyansundaram, Leen Torenvliet

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 ...
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Eric Allender, Andris Ambainis, David Mix Barrington, Samir Datta, Huong LeThanh

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 ...
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Andris Ambainis, David Mix Barrington, Huong LeThanh

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 >>>