scholarly journals Quantum states cannot be transmitted efficiently classically

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 154 ◽  
Author(s):  
Ashley Montanaro
Keyword(s):  
Lower Bound ◽  
Classical Solution ◽  
Communication Complexity ◽  
Communication Protocol ◽  
Quantum States ◽  
Classical Communication ◽  
Fourier Sampling ◽  
Simple Protocol ◽  
Sampling Problem ◽  
Quantum Query

We show that any classical two-way communication protocol with shared randomness that can approximately simulate the result of applying an arbitrary measurement (held by one party) to a quantum state of n qubits (held by another), up to constant accuracy, must transmit at least Ω(2n) bits. This lower bound is optimal and matches the complexity of a simple protocol based on discretisation using an ϵ-net. The proof is based on a lower bound on the classical communication complexity of a distributed variant of the Fourier sampling problem. We obtain two optimal quantum-classical separations as easy corollaries. First, a sampling problem which can be solved with one quantum query to the input, but which requires Ω(N) classical queries for an input of size N. Second, a nonlocal task which can be solved using n Bell pairs, but for which any approximate classical solution must communicate Ω(2n) bits.

scholarly journals Detecting cliques in CONGEST networks

2019 ◽  
Vol 33 (6) ◽  
pp. 533-543
Author(s):  
Artur Czumaj ◽  
Christian Konrad
Keyword(s):  
Lower Bound ◽  
Distributed Computing ◽  
Communication Complexity ◽  
Communication Protocol ◽  
Distributed Model ◽  
Input Graph ◽  
Communication Framework ◽  
Communication Topology ◽  
Communication Links ◽  
Underlying Network

AbstractThe problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph H, the input network contains a subgraph isomorphic to H or not. We investigate this problem for H being a clique $$K_{\ell }$$ K ℓ in the classical distributed model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links. Our first and main result is a lower bound, showing that detecting $$K_{\ell }$$ K ℓ requires $$\varOmega (\sqrt{n} / {\mathfrak {b}})$$ Ω ( n / b ) communication rounds, for every $$4 \le \ell \le \sqrt{n}$$ 4 ≤ ℓ ≤ n , and $$\varOmega (n / (\ell {\mathfrak {b}}))$$ Ω ( n / ( ℓ b ) ) rounds for every $$\ell \ge \sqrt{n}$$ ℓ ≥ n , where $${\mathfrak {b}}$$ b is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for $$K_4$$ K 4 detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework.

scholarly journals Quantum communication complexity of distribution testing

2021 ◽  
Vol 21 (15&16) ◽  
pp. 1261-1273
Author(s):  
Aleksandrs Belovs ◽  
Arturo Castellanos ◽  
Francois Le Gall ◽  
Guillaume Malod ◽  
Alexander A. Sherstov
Keyword(s):  
Lower Bound ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Discrete Distributions ◽  
Classical Communication ◽  
Testing Problem ◽  
Quantum Protocols ◽  
Quantum Communication Complexity ◽  
Pattern Matrix ◽  
Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.

scholarly journals Robust Bell inequalities from communication complexity

Quantum ◽  
2018 ◽  
Vol 2 ◽  
pp. 72 ◽  
Author(s):  
Sophie Laplante ◽  
Mathieu Laurière ◽  
Alexandre Nolin ◽  
Jérémie Roland ◽  
Gabriel Senno
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Bell Inequality ◽  
Bell Inequalities ◽  
Local Distribution ◽  
Classical Communication ◽  
Quantum Communication Complexity ◽  
Complexity Lower Bounds ◽  
Quantum Distributions

The question of how large Bell inequality violations can be, for quantum distributions, has been the object of much work in the past several years. We say that a Bell inequality is normalized if its absolute value does not exceed 1 for any classical (i.e. local) distribution. Upper and (almost) tight lower bounds have been given for the quantum violation of these Bell inequalities in terms of number of outputs of the distribution, number of inputs, and the dimension of the shared quantum states. In this work, we revisit normalized Bell inequalities together with another family: inefficiency-resistant Bell inequalities. To be inefficiency-resistant, the Bell value must not exceed 1 for any local distribution, including those that can abort. This makes the Bell inequality resistant to the detection loophole, while a normalized Bell inequality is resistant to general local noise. Both these families of Bell inequalities are closely related to communication complexity lower bounds. We show how to derive large violations from any gap between classical and quantum communication complexity, provided the lower bound on classical communication is proven using these lower bound techniques. This leads to inefficiency-resistant violations that can be exponential in the size of the inputs. Finally, we study resistance to noise and inefficiency for these Bell inequalities.

scholarly journals Better protocol for XOR game using communication protocol and nonlocal boxes

2017 ◽  
Vol 17 (15&16) ◽  
pp. 1261-1276
Author(s):  
Ryuhei Mori
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Quantum Communication ◽  
Communication Protocol ◽  
Constant Factor ◽  
Efficient Communication ◽  
Information Causality ◽  
Special Case

Buhrman showed that an efficient communication protocol implies a reliable XOR game protocol. This idea rederives Linial and Shraibman’s lower bound of randomized and quantum communication complexities, which was derived by using factorization norms, with worse constant factor in much more intuitive way. In this work, we improve and generalize Buhrman’s idea, and obtain a class of lower bounds for randomized communication complexity including an exact Linial and Shraibman’s lower bound as a special case. In the proof, we explicitly construct a protocol for XOR game from a randomized communication protocol by using a concept of nonlocal boxes and Paw lowski et al.’s elegant protocol, which was used for showing the violation of information causality in superquantum theories.

scholarly journals QUANTUM KOLMOGOROV COMPLEXITY AND ITS APPLICATIONS

2007 ◽  
Vol 05 (05) ◽  
pp. 729-750 ◽  
Author(s):  
CATERINA E. MORA ◽  
HANS J. BRIEGEL ◽  
BARBARA KRAUS
Keyword(s):  
Quantum Computation ◽  
Quantum State ◽  
Kolmogorov Complexity ◽  
Communication Complexity ◽  
Quantum States ◽  
Qubit State ◽  
Binary String ◽  
Classical Communication ◽  
Pure Quantum State ◽  
Definition Of

Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.

Quantum lower bound for recursive Fourier sampling

2003 ◽  
Vol 3 (2) ◽  
pp. 165-174
Author(s):  
S. Aaronson
Keyword(s):  
Lower Bound ◽  
Quantum Algorithm ◽  
Independent Interest ◽  
Classical Algorithm ◽  
Fourier Sampling ◽  
Adversary Method

We revisit the oft-neglected `recursive Fourier sampling' (RFS) problem, introduced by Bernstein and Vazirani to prove an oracle separation between BPP and BQP. We show that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place \mathsf{BQP} outside of PH[\log] relative to an oracle, one would need to go outside the RFS framework. Our proof argues that, given any variant of RFS, either the adversary method of Ambainis yields a good quantum lower bound, or else there is an efficient classical algorithm. This technique may be of independent interest.

Lower bounds for cutting planes proofs with small coefficients

1997 ◽  
Vol 62 (3) ◽  
pp. 708-728 ◽  
Author(s):  
Maria Bonet ◽  
Toniann Pitassi ◽  
Ran Raz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cutting Planes ◽  
Communication Complexity ◽  
Proof System ◽  
Small Weight ◽  
Deduction Rule ◽  
Special Case ◽  
Exponential Lower Bound

AbstractWe consider small-weight Cutting Planes (CP*) proofs; that is, Cutting Planes (CP) proofs with coefficients up to Poly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP* proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution.We also prove the following two theorems: (1) Tree-like CP* proofs cannot polynomially simulate non-tree-like CP* proofs. (2) Tree-like CP* proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other.Our proofs also work for some generalizations of the CP* proof system. In particular, they work for CP* with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

Quantumness, generalized spherical 2-design, and symmetric informationally complete POVM

2007 ◽  
Vol 7 (8) ◽  
pp. 730-737
Author(s):  
I.H. Kim
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Equivalence Relation ◽  
Open Problem ◽  
Natural Generalization ◽  
Quantum States ◽  
Minimal Set ◽  
Dimensional Hilbert Space

Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.

Lower Bounds on OBDD Proofs with Several Orders

2021 ◽  
Vol 22 (4) ◽  
pp. 1-30
Author(s):  
Sam Buss ◽  
Dmitry Itsykson ◽  
Alexander Knop ◽  
Artur Riazanov ◽  
Dmitry Sokolov
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Communication Complexity ◽  
Multiparty Communication ◽  
Constant Degree ◽  
Polynomial Size ◽  
Explicit Constant ◽  
Open Question ◽  
System 1 ◽  
Exponential Size

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

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