arXiv subjects

Shengyu Zhang

Publications and source records attributed to Shengyu Zhang.

232 records · Page 13Linked to original sources

arXiv2011

On characterizing quantum correlated equilibria

Quantum game theory lays a foundation for understanding the interaction of people using quantum computers with conflicting interests. Recently Zhang proposed a simple yet rich model to study quantum strategic games, and addressed some quantitative questions for general games of growing sizes \cite{Zha10}. However, one fundamental question that the paper did not consider is the characterization of quantum correlated equilibria (QCE). In this paper, we answer this question by giving a sufficient and necessary condition for an arbitrary state $ρ$ being a QCE. In addition, when the condition fails to hold for some player $i$, we give an explicit POVM for that player to achieve a strictly positive gain. Finally, we give some upper bounds for the maximum gain by playing quantum strategies over classical ones, and the bounds are tight for some games.

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arXiv2011

Quantum Strategic Game Theory

We propose a simple yet rich model to extend the notions of Nash equilibria and correlated equilibria of strategic games to the quantum setting, in which we then study the relations between classical and quantum equilibria. Unlike the previous work that focus on qualitative questions on specific games of small sizes, we address the following fundamental and quantitative question for general games: How much "advantage" can playing quantum strategies provide, if any? Two measures of the advantage are studied, summarized as follows. 1. A natural measure is the increase of payoff. We consider natural mappings between classical and quantum states, and study how well those mappings preserve the equilibrium properties. Among other results, we exhibit correlated equilibrium $p$ whose quantum superposition counterpart $\sum_s \sqrt{p(s)}\ket{s}$ is far from being a quantum correlated equilibrium; actually a player can increase her payoff from almost 0 to almost 1 in a [0,1]-normalized game. We achieve this by a tensor product construction on carefully designed base cases. 2. For studying the hardness of generating correlated equilibria, we propose to examine \emph{correlation complexity}, a new complexity measure for correlation generation. We show that there are $n$-bit correlated equilibria which can be generated by only one EPR pair followed by local operation (without communication), but need at least $\log(n)$ classical shared random bits plus communication. The randomized lower bound can be improved to $n$, the best possible, assuming (even a much weaker version of) a recent conjecture in linear algebra. We believe that the correlation complexity, as a complexity-theoretical counterpart of the celebrated Bell's inequality, has independent interest in both physics and computational complexity theory and deserves more explorations.

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arXiv2011

A quantum protocol for sampling correlated equilibria unconditionally and without a mediator

A correlated equilibrium is a fundamental solution concept in game theory that enjoys many desirable properties. However, it requires a trusted mediator, which is a major drawback in many practical applications. A computational solution to this problem was proposed by Dodis, Halevi and Rabin. They extended the original game by adding an initial communication stage and showed that any correlated strategy for 2-player games can be achieved, provided that the players are computationally bounded. In this paper, we show that if the players can communicate via a quantum channel before the game, then any correlated equilibrium for 2-player games can be achieved, without a trusted mediator and unconditionally. This provides another example of a major advantage of quantum information processing. More precisely, we prove that for any correlated equilibrium p of a strategic game G, there exists an extended game (with a quantum communication initial stage) Q with an efficiently computable approximate Nash equilibrium q, such that the expected payoff for both players in q is at least as high as in p. The main cryptographic tool used in the construction is the quantum weak coin flipping protocol of Mochon.

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arXiv2011

The influence lower bound via query elimination

We give a simpler proof, via query elimination, of a result due to O'Donnell, Saks, Schramm and Servedio, which shows a lower bound on the zero-error randomized query complexity of a function f in terms of the maximum influence of any variable of f. Our lower bound also applies to the two-sided error distributional query complexity of f, and it allows an immediate extension which can be used to prove stronger lower bounds for some functions.

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arXiv2011

Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models

We study the communication complexity of symmetric XOR functions, namely functions $f: \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ that can be formulated as $f(x,y)=D(|x\oplus y|)$ for some predicate $D: \{0,1,...,n\} \rightarrow \{0,1\}$, where $|x\oplus y|$ is the Hamming weight of the bitwise XOR of $x$ and $y$. We give a public-coin randomized protocol in the Simultaneous Message Passing (SMP) model, with the communication cost matching the known lower bound for the \emph{quantum} and \emph{two-way} model up to a logarithm factor. As a corollary, this closes a quadratic gap between quantum lower bound and randomized upper bound for the one-way model, answering an open question raised in Shi and Zhang \cite{SZ09}.

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arXiv2011

On the power of a unique quantum witness

In a celebrated paper, Valiant and Vazirani raised the question of whether the difficulty of NP-complete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the d-th tensor power of H. Indeed, the intersection of this subspace with the d-th tensor power of W is one-dimensional, and therefore can play the role of the unique quantum witness.

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arXiv2010

Composition theorems in communication complexity

A well-studied class of functions in communication complexity are composed functions of the form $(f \comp g^n)(x,y)=f(g(x^1, y^1),..., g(x^n,y^n))$. This is a rich family of functions which encompasses many of the important examples in the literature. It is thus of great interest to understand what properties of $f$ and $g$ affect the communication complexity of $(f \comp g^n)$, and in what way. Recently, Sherstov \cite{She09b} and independently Shi-Zhu \cite{SZ09b} developed conditions on the inner function $g$ which imply that the quantum communication complexity of $f \comp g^n$ is at least the approximate polynomial degree of $f$. We generalize both of these frameworks. We show that the pattern matrix framework of Sherstov works whenever the inner function $g$ is {\em strongly balanced}---we say that $g: X \times Y \to \{-1,+1\}$ is strongly balanced if all rows and columns in the matrix $M_g=[g(x,y)]_{x,y}$ sum to zero. This result strictly generalizes the pattern matrix framework of Sherstov \cite{She09b}, which has been a very useful idea in a variety of settings \cite{She08b,RS08,Cha07,LS09,CA08,BHN09}. Shi-Zhu require that the inner function $g$ has small {\em spectral discrepancy}, a somewhat awkward condition to verify. We relax this to the usual notion of discrepancy. We also enhance the framework of composed functions studied so far by considering functions $F(x,y) = f(g(x,y))$, where the range of $g$ is a group $G$. When $G$ is Abelian, the analogue of the strongly balanced condition becomes a simple group invariance property of $g$. We are able to formulate a general lower bound on $F$ whenever $g$ satisfies this property.

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arXiv2009

Depth-Independent Lower bounds on the Communication Complexity of Read-Once Boolean Formulas

We show lower bounds of $Ω(\sqrt{n})$ and $Ω(n^{1/4})$ on the randomized and quantum communication complexity, respectively, of all $n$-variable read-once Boolean formulas. Our results complement the recent lower bound of $Ω(n/8^d)$ by Leonardos and Saks and $Ω(n/2^{Ω(d\log d)})$ by Jayram, Kopparty and Raghavendra for randomized communication complexity of read-once Boolean formulas with depth $d$. We obtain our result by "embedding" either the Disjointness problem or its complement in any given read-once Boolean formula.

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arXiv2008

New bounds on classical and quantum one-way communication complexity

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical model, our bound extends the well known upper bound of Kremer, Nisan and Ron to include non-product distributions. We show that for a boolean function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where D_{epsilon}^{1, mu}(f) represents the one-way distributional communication complexity of f with error at most epsilon under mu; vc(f) represents the Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual information, under mu, between the random inputs of the two parties. For a non-boolean function f:X x Y ->[k], we show a similar upper bound on D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' = f/k. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f, in terms the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f . We show for a non-boolean total function f : X x Y -> Z and a product distribution mu on XxY, Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional communication complexity of f with error at most epsilon^3/8 under mu and rec_ epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x 15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).

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arXiv2007

A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation

In measurement-based quantum computation, quantum algorithms are implemented via sequences of measurements. We describe a translationally invariant finite-range interaction on a one-dimensional qudit chain and prove that a single-shot measurement of the energy of an appropriate computational basis state with respect to this Hamiltonian provides the output of any quantum circuit. The required measurement accuracy scales inverse polynomially with the size of the simulated quantum circuit. This shows that the implementation of energy measurements on generic qudit chains is as hard as the realization of quantum computation. Here a ''measurement'' is any procedure that samples from the spectral measure induced by the observable and the state under consideration. As opposed to measurement-based quantum computation, the post-measurement state is irrelevant.

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arXiv2007

Every NAND formula of size N can be evaluated in time N^{1/2+o(1)} on a quantum computer

For every NAND formula of size N, there is a bounded-error N^{1/2+o(1)}-time quantum algorithm, based on a coined quantum walk, that evaluates this formula on a black-box input. Balanced, or ``approximately balanced,'' NAND formulas can be evaluated in O(sqrt{N}) queries, which is optimal. It follows that the (2-o(1))-th power of the quantum query complexity is a lower bound on the formula size, almost solving in the positive an open problem posed by Laplante, Lee and Szegedy.

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arXiv2006

Several natural BQP-Complete problems

A central problem in quantum computing is to identify computational tasks which can be solved substantially faster on a quantum computer than on any classical computer. By studying the hardest such tasks, known as BQP-complete problems, we deepen our understanding of the power and limitations of quantum computers. We present several BQP-complete problems, including Local Hamiltonian Eigenvalue Sampling and Phase Estimation Sampling. Different than the previous known BQP-complete problems (the Quadratically Signed Weight Enumerator problem [KL01] and the Approximation of Jones Polynomials [FKW02, FLW02, AJL06]), our problems are of a basic linear algebra nature and are closely related to the well-known quantum algorithm and quantum complexity theories.

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arXiv2006

The Communication Complexity of the Hamming Distance Problem

We investigate the randomized and quantum communication complexity of the Hamming Distance problem, which is to determine if the Hamming distance between two n-bit strings is no less than a threshold d. We prove a quantum lower bound of Ω(d) qubits in the general interactive model with shared prior entanglement. We also construct a classical protocol of O(d \log d) bits in the restricted Simultaneous Message Passing model, improving previous protocols of O(d^2) bits (A. C.-C. Yao, Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pp. 77-81, 2003), and O(d\log n) bits (D. Gavinsky, J. Kempe, and R. de Wolf, quant-ph/0411051, 2004).

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arXiv2006

New upper and lower bounds for randomized and quantum Local Search

Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to combinatorial optimization, complexity theory and many other areas in theoretical computer science. In this paper, we study the problem in the randomized and quantum query models and give new lower and upper bound techniques in both models. The lower bound technique works for any graph that contains a product graph as a subgraph. Applying it to the Boolean hypercube {0,1}^n and the constant dimensional grids [n]^d, two particular product graphs that recently drew much attention, we get the following tight results: RLS({0,1}^n) = Θ(2^{n/2}n^{1/2}), QLS({0,1}^n) = Θ(2^{n/3}n^{1/6}), RLS([n]^d) = Θ(n^{d/2}) for d \geq 4, QLS([n]^d) = Θ(n^{d/3}) for d \geq 6. Here RLS(G) and QLS(G) are the randomized and quantum query complexities of Local Search on G, respectively. These improve the previous results by Aaronson [STOC'04], Ambainis (unpublished) and Santha and Szegedy [STOC'04]. Our new algorithms work well when the underlying graph expands slowly. As an application to [n]^2, a new quantum algorithm using O(\sqrt{n}(\log\log n)^{1.5}) queries is given. This improves the previous best known upper bound of O(n^{2/3}) (Aaronson, [STOC'04]), and implies that Local Search on grids exhibits different properties in low dimensions.

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arXiv2005

(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids

The Local Search problem, which finds a local minimum of a black-box function on a given graph, is of both practical and theoretical importance to many areas in computer science and natural sciences. In this paper, we show that for the Boolean hypercube $\B^n$, the randomized query complexity of Local Search is $Θ(2^{n/2}n^{1/2})$ and the quantum query complexity is $Θ(2^{n/3}n^{1/6})$. We also show that for the constant dimensional grid $[N^{1/d}]^d$, the randomized query complexity is $Θ(N^{1/2})$ for $d \geq 4$ and the quantum query complexity is $Θ(N^{1/3})$ for $d \geq 6$. New lower bounds for lower dimensional grids are also given. These improve the previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04]. Finally we show for $[N^{1/2}]^2$ a new upper bound of $O(N^{1/4}(\log\log N)^{3/2})$ on the quantum query complexity, which implies that Local Search on grids exhibits different properties at low dimensions.

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arXiv2004

On the power of Ambainis's lower bounds

The polynomial method and the Ambainis's lower bound (or \emph{Alb}, for short) method are two main quantum lower bound techniques. While recently Ambainis showed that the polynomial method is not tight, the present paper aims at studying the power and limitation of \emph{Alb}'s. We first use known \emph{Alb}'s to derive $Ω(n^{1.5})$ lower bounds for \textsc{Bipartiteness}, \textsc{Bipartiteness Matching} and \textsc{Graph Matching}, in which the lower bound for \textsc{Bipartiteness} improves the previous $Ω(n)$ one. We then show that all the three known Ambainis's lower bounds have a limitation $\sqrt{N\cdot \min\{C_0(f), C_1(f)\}}$, where $C_0(f)$ and $C_1(f)$ are the 0- and 1-certificate complexity, respectively. This implies that for some problems such as \textsc{Triangle}, $k$-\textsc{Clique}, and \textsc{Bipartite/Graph Matching} which draw wide interest and whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis's techniques. Another consequence is that all the Ambainis's lower bounds are not tight. For total functions, this upper bound for \emph{Alb}'s can be further improved to $\min \{\sqrt{C_0(f)C_1(f)}, \sqrt{N\cdot CI(f)}\}$, where $CI(f)$ is the size of max intersection of a 0-and a 1-certificate set. Again this implies that $Alb$'s cannot improve the best known lower bound for some specific problems such as \textsc{And-Or Tree}, whose precise quantum query complexity is still open. Finally, we generalize the three known \emph{Alb}'s and give a new \emph{Alb} style lower bound method, which may be easier to use for some problems.

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