Dave Touchette

FOCS Conference 2019 Conference Paper

Quantum Log-Approximate-Rank Conjecture is Also False

• Anurag Anshu
• Naresh Goud Boddu
• Dave Touchette

In a recent breakthrough result, Chattopadhyay, Mande and Sherif [ECCC TR18-17] showed an exponential separation between the log approximate rank and randomized communication complexity of a total function f, hence refuting the log approximate rank conjecture of Lee and Shraibman [2009]. We provide an alternate proof of their randomized communication complexity lower bound using the information complexity approach. Using the intuition developed there, we derive a polynomially-related quantum communication complexity lower bound using the quantum information complexity approach, thus providing an exponential separation between the log approximate rank and quantum communication complexity of f. Previously, the best known separation between these two measures was (almost) quadratic, due to Anshu, Ben-David, Garg, Jain, Kothari and Lee [CCC, 2017]. This settles one of the main question left open by Chattopadhyay, Mande and Sherif, and refutes the quantum log approximate rank conjecture of Lee and Shraibman [2009]. Along the way, we develop a Shearer-type protocol embedding for product input distributions that might be of independent interest.

STOC Conference 2018 Conference Paper

Capacity approaching coding for low noise interactive quantum communication

• Debbie W. Leung
• Ashwin Nayak 0001
• Ala Shayeghi
• Dave Touchette
• Penghui Yao
• Nengkun Yu

STOC Conference 2017 Conference Paper

Exponential separation of quantum communication and classical information

• Anurag Anshu
• Dave Touchette
• Penghui Yao
• Nengkun Yu

FOCS Conference 2015 Conference Paper

Near-Optimal Bounds on Bounded-Round Quantum Communication Complexity of Disjointness

• Mark Braverman
• Ankit Garg 0001
• Young Kun-Ko
• Jieming Mao
• Dave Touchette

We prove a near optimal round-communication tradeoff for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Omega(n/r) on the communication required for computing disjointness of input size n, which is optimal up to logarithmic factors. The previous best lower bound was Omega(n/r̂ 2 ) due to Jain, Radhakrishnan and Sen. Along the way, we develop several tools for quantum information complexity, one of which is a lower bound for quantum information complexity in terms of the generalized discrepancy method. As a corollary, we get that the quantum communication complexity of any boolean function f is at most 2 ̂O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3).

STOC Conference 2015 Conference Paper

Quantum Information Complexity

• Dave Touchette

We define a new notion of information cost for quantum protocols, and a corresponding notion of quantum information complexity for bipartite quantum tasks. These are the fully quantum generalizations of the analogous quantities for bipartite classical tasks that have found many applications recently, in particular for proving communication complexity lower bounds and direct sum theorems. Finding such a quantum generalization of information complexity was one of the open problems recently raised by Braverman (STOC'12). Previous attempts have been made to define such a quantity for quantum protocols, with particular applications in mind; our notion differs from these in many respects. First, it directly provides a lower bound on the quantum communication cost, independent of the number of rounds of the underlying protocol. Secondly, we provide an operational interpretation for quantum information complexity: we show that it is exactly equal to the amortized quantum communication complexity of a bipartite task on a given input. This generalizes a result of Braverman and Rao (FOCS'11) to quantum protocols. Along the way to prove this result, we even strengthens the classical result in a bounded round scenario, and also prove important structural properties of quantum information cost and complexity. We prove that using this definition leads to the first general direct sum theorem for bounded round quantum communication complexity. Previous direct sum results in quantum communication complexity either held for some particular classes of functions, or were general but only held for single-round protocols. We also discuss potential applications of the new quantities to obtain lower bounds on quantum communication complexity.

FOCS Conference 2014 Conference Paper

Noisy Interactive Quantum Communication

• Gilles Brassard
• Ashwin Nayak 0001
• Alain Tapp
• Dave Touchette
• Falk Unger

We study the problem of simulating protocols in a quantum communication setting over noisy channels. This problem falls at the intersection of quantum information theory and quantum communication complexity, and will be of importance for eventual real-world applications of interactive quantum protocols, which can be proved to have exponentially lower communication costs than their classical counterparts for some problems. These are the first results concerning the quantum version of this problem, originally studied by Schulman in a classical setting (FOCS '92, STOC '93). We simulate a length N quantum communication protocol by a length O(N) protocol with arbitrarily small error. Our simulation strategy has a far higher communication rate than a naive one that encodes separately each particular round of communication to achieve comparable success. Such a strategy would have a communication rate going to 0 in the worst interaction case as the length of the protocols increases, in contrast to our strategy, which has a communication rate proportional to the capacity of the channel used. Under adversarial noise, our strategy can withstand, for arbitrarily small ε > 0, error rates as high as 1/2 -- ε when parties preshare perfect entanglement, but the classical channel is noisy. We show that this is optimal. Note that in this model, the naive strategy would not work for any constant fraction of errors. We provide extension of these results in several other models of communication, including when also the entanglement is noisy, and when there is no pre-shared entanglement but communication is quantum and noisy. We also study the case of random noise, for which we provide simulation protocols with positive communication rates and no pre-shared entanglement over some quantum channels with quantum capacity Q = 0, proving that Q is in general not the right characterization of a channel's capacity for interactive quantum communication. Our results are stated for a general quantum communication protocol in which Alice and Bob collaborate, and hold in particular in the quantum communication complexity settings of the Yao and Cleve-Buhrman models.