Anchoring games for parallel repetition

Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings

Separating Quantum Communication and Approximate Rank

One of the best lower bound methods for the quantum communication complexity of a function H (with or without shared entanglement) is the logarithm of the approximate rank of the communication matrix of H. This measure is essentially equivalent to the approximate gamma-2 norm and generalized discrepancy, and subsumes several other lower bounds. All known lower bounds on quantum communication complexity in the general unbounded-round model can be shown via the logarithm of approximate rank, and it was an open problem to give any separation at all between quantum communication complexity and the logarithm of the approximate rank. In this work we provide the first such separation: We exhibit a total function H with quantum communication complexity almost quadratically larger than the logarithm of its approximate rank. We construct H using the communication lookup function framework of Anshu et al. (FOCS 2016) based on the cheat sheet framework of Aaronson et al. (STOC 2016). From a starting function F, this framework defines a new function H=F_G. Our main technical result is a lower bound on the quantum communication complexity of F_G in terms of the discrepancy of F, which we do via quantum information theoretic arguments. We show the upper bound on the approximate rank of F_G by relating it to the Boolean circuit size of the starting function F

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

We prove a near optimal round-communication trade off for the two-party quantum communication complexity of disjointness. For protocols with r rounds, we prove a lower bound of Ω(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 Ω(n/r2) due to Jain, Radha krishnan 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 2O(QIC(f)), where QIC(f) is the prior-free quantum information complexity of f (with error 1/3)

Practical Quantum Fingerprinting and Appointment Scheduling

Quantum protocols for many communication tasks have been found which significantly improve on their classical counterparts. However, many of these protocols are beyond the reach of current technology. In this work, we find more readily implementable protocols for the tasks of quantum fingerprinting and appointment scheduling. Our protocols maintain a quantum advantage even under realistic experimental imperfections. In the task of quantum fingerprinting, two parties wish to evaluate the equality function on two n-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of O(log n)-qubit highly entangled signals (PRL.87.167902), whereas a recently-proposed optical protocol uses a tensor product of O(n) single-qubit signals, while maintaining the O(log n) information leakage of the original protocol (PRA.89.062305). The low-dimensionality of each signal in the recently proposed optical protocol makes it more amenable to experimental implementation (ncomms9735, PRL.116.240502), but due to limited coherence times the large number of signals remains a significant barrier to observing a quantum advantage in information leakage. In contrast, the original protocol sends few signals, but the dimension of each signal is prohibitively high. We find a family of protocols which interpolate between the original and optical protocols while maintaining the O(log n) information leakage, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal, and opening the door for experimental implementations to find a ``sweet spot'' for which the number of signals sent and the dimension of each signal are both amenable to current technology. In (ncomms9735, PRL.116.240502) the recently proposed optical protocol is implemented using coherent states. We develop a coherent state protocol which reduces the number of signals by a factor 1/2 from the recently proposed optical protocol, while also reducing the information leakage. We consider several natural generalizations of this protocol to other coherent state protocols which further reduce the number of signals, but find numerical evidence that they have greater information leakage in the ideal setting and also under realistic experimental imperfections. Using a similar technique, we improve a recently proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors (PRA.95.032337) by reducing the number of signals by a factor 1/2 while also reducing the information leakage. We also extend this protocol to handle complex unit vectors. Along the way, we find a simple beamsplitter measurement to perform optimal unambiguous state comparison between two coherent states. In the task of appointment scheduling, two parties each have n-bit strings, and they wish to find a common intersection in the interactive communication model. The known quantum appointment scheduling protocol of (arXiv:quant-ph/9802040) performs this task with O(sqrt(n)log(n)) qubits of communication, a nearly quadratic improvement over the classical lower bound of Omega(n) bits. However, this protocol requires quantum states of high dimension and global unitary operations. We find appointment scheduling protocols which are more feasible for implementation and maintain a quantum advantage over the classical lower bound in terms of information cost, even under experimental imperfections. Our main protocols require the generation of coherent states of a fixed set of amplitudes, along with phase shifters and beamsplitters on two modes with relatively low splitting angle. They also require the parties to transfer two modes back-and-forth multiple times with relatively low loss. Although our protocols make progress towards the experimental implementation of quantum appointment scheduling, we expect that they still remain outside the scope of current technology