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 it 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 [L. J. Schulman, Communication on noisy channels: A coding theorem for computation, in Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, IEEE, 1992, pp. 724--733], [L. J. Schulman, Deterministic coding for interactive communication, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, ACM, 1993, pp. 747--756]. We simulate a length $N$ quantum communication protocol by a length $O(N)$ protocol with arbitrarily small error. Under adversarial noise, our strategy can withstand, for arbitrarily small $\varepsilon>0$, error rates as high as $1/2-\varepsilon$ when parties preshare perfect entanglement, but the classical channel is noisy. We show that this is optimal. We provide extension of these results in several other models of communication, including when also the entanglement is noisy, and when there is no preshared 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 preshared entanglement over some quantum channels with quantum capacity $C_Q=0$, proving that $C_Q$ 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 these results hold in particular in the quantum communication complexity settings of the Yao and Cleve--Buhrman models.

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.

Can a classical experimentalist command an untrusted quantum system to realize arbitrary quantum dynamics, aborting if it misbehaves? If so, then we could realize the dream of device-independent quantum cryptography: using untrusted quantum devices to establish a shared random key, with security based on the correctness of quantum mechanics. It would also allow for testing whether a claimed quantum computer is truly quantum. We prove a rigidity theorem for the famous Clauser-Horne-Shimony-Holt (CHSH) game, first formulated to provide a means of experimentally testing the violation of the Bell inequalities. The theorem shows that the only way for the two non-communicating quantum players to win many games played in sequence is if their shared quantum state is close to the tensor product of EPR states (Bell states) and their measurements are the optimal CHSH measurements on successive qubits. This theorem may be viewed as analogous to classical multi-linearity testing, in the sense that the outcome of local checks gives a characterization of a global object.

We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments. This inequality allows us to simplify and strengthen several known direct-product theorems and establish new threshold direct-product theorems. Threshold direct-product theorems are statements of the following form: If one instance of a problem can be solved with probability at most p, then solving significantly more than a p-fraction among multiple instances has negligible probability. Results of this kind are crucial when distinguishing whether a process succeeds with probability s or c, for 0 < s < c < 1. Here standard direct-product theorems are of no help since even a process which can solve one instance with probability c will only be able to solve all k instances with exponentially small probability. Using our concentration inequality we show how to obtain threshold (and standard) direct-product theorems from known XOR Lemmas. We give examples of this approach and obtain (threshold) direct-product theorems for quantum XOR games, quantum random access codes, 2-party and multi-party communication complexity and circuits. Similar results can be obtained for other models of computation, e.g. polynomials over GF(2). It is well-known that direct-product theorems and XOR Lemmas are "essentially" equivalent. We show that one direction is often even tight: going from XOR Lemmas to (threshold) direct-product theorems is possible in an information-theoretically optimal way. We believe that our inequality has applications in other contexts as well.

It is known that epsi-noisy gates with two inputs are universal for arbitrary computation (i.e., can compute any function with bounded error), if all gates fail independently with probability epsi and epsi2= (3 - radic7)/4 ap 8.856%. In this paper, it is shown that this bound is tight for formulas, by proving that gates with two inputs, in which each gate fails with probability at least beta2cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with two-input gates and it is beta2. It is conjectured that the same threshold also holds for circuits.

We consider a class of two-prover interactive proof systems where each prover returns a single bit to the verifier and the verifier's verdict is a function of the XOR of the two bits received. We show that, when the provers are allowed to coordinate their behavior using a shared entangled quantum state, a perfect parallel repetition theorem holds in the following sense. The prover's optimal success probability for simultaneously playing a collection of XOR proof systems is exactly the product of the individual optimal success probabilities. This property is remarkable in view of the fact that, in the classical case (where the provers can only utilize classical information), it does not hold. The theorem is proved by analyzing parities of XOR proof systems using semidefinite programming techniques, which we then relate to parallel repetitions of XOR games via Fourier analysis.

Evans and Pippenger showed in 1998 that e-noisy gates with 2 inputs are universal for arbitrary computation (i.e. can compute any function with bounded error), if all gates fail independently with probability isin and isin0= (3 - radic7)/4 ap8.856%. We show that formulas built from gates with 2 inputs, in which each gate fails with probability at least isin0cannot be universal. Hence, there is a threshold on the tolerable noise for formulas with 2-input gates and it is isin0. We conjecture that the same threshold also holds for circuits.

We show that quantum circuits cannot be made fault-tolerant against a depolarizing noise level of thetas = (6 - 2radic2)/7 ap 45%, thereby improving on a previous bound of 50% (due to Razborov, 2004). More precisely, the circuit model for which we prove this bound contains perfect gates from the Clifford group (CNOT, Hadamard, S, X, Y, Z) and arbitrary additional one-qubit gates that are subject to depolarizing noise thetas. We prove that this set of gates cannot be universal for arbitrary (even classical) computation, from which the upper bound on the noise threshold for fault-tolerant quantum computation follows