Trading locality for time: certifiable randomness from low-depth circuits

The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation or computational assumptions. To demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.Comment: 36 pages, 2 figure

Trading isolation for certifiable randomness expansion

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (page 41).A source of random bits is an important resource in modern cryptography, algorithms and statistics. Can one ever be sure that a "random" source is truly random, or in the case of cryptography, secure against potential adversaries or eavesdroppers? Recently the study of non-local properties of entanglement has produced an interesting new perspective on this question, which we will refer to broadly as Certifiable Randomness Expansion (CRE). CRE refers generally to a process by which a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two or more black-box devices do not communicate (i.e. are non-signaling). In this work we make progress on a conjecture of [Col09] which proposes a method for indefinite certifiable randomness expansion using a growing number of devices (we actually prove a slight modification of the original conjecture in which we use the CHSH game as a subroutine rather than the GHZ game as originally proposed). The proof requires a technique not used before in the study of randomness expansion, and inspired by the tools developed in [RUV12]. The result also establishes the existence of a protocol for constant factor CRE using a finite number of devices (here the constant factor can be much greater than 1). While much better expansion rates (polynomial, and even exponential) have been achieved with only two devices, our analysis requires techniques not used before in the study of randomness expansion, and represents progress towards a protocol which is provably secure against a quantum eavesdropper who knows the input to the protocol.by Matthew Ryan Coudron.S.M

Complexity Lower Bounds for Computing the Approximately-Commuting Operator Value of Non-Local Games to High Precision

We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is NP-complete to decide whether the classical value of a non-local game is 1 or 1- epsilon, promised that one of the two is the case. Furthermore, as long as epsilon is small enough, this result does not depend on the gap epsilon. In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap epsilon decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of MIP^{co}(2,1,1,s) (proofs with two provers, one round, completeness probability 1, soundness probability s, and commuting-operator strategies) can increase without bound as the gap 1-s gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class PZK-MIP^{co}_{delta}(2,1,1,s) of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap 1-s gets arbitrarily small. While we do not know any computable time upper bound on the class MIP^{co}, a result of the first author and Vidick shows that for s = 1-1/poly(f(n)) and delta = 1/poly(f(n)), the class MIP^{co}_delta(2,1,1,s), with constant communication from the provers, is contained in TIME(exp(poly(f(n)))). We give a lower bound of coNTIME(f(n)) (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis

Universality of EPR Pairs in Entanglement-Assisted Communication Complexity, and the Communication Cost of State Conversion

In this work we consider the role of entanglement assistance in quantum communication protocols, focusing, in particular, on whether the type of shared entangled state can affect the quantum communication complexity of a function. This question is interesting because in some other settings in quantum information, such as non-local games, or tasks that involve quantum communication between players and referee, or simulating bipartite unitaries or communication channels, maximally entangled states are known to be less useful as a resource than some partially entangled states. By contrast, we prove that the bounded-error entanglement-assisted quantum communication complexity of a partial or total function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be epsilon-approximated by a protocol using O(Q/epsilon+log(1/epsilon)/epsilon) qubits of communication and only EPR pairs as shared entanglement. This conclusion is opposite of the common wisdom in the study of non-local games, where it has been shown, for example, that the I3322 inequality has a non-local strategy using a non-maximally entangled state, which surpasses the winning probability achievable by any strategy using a maximally entangled state of any dimension [Vidick and Wehner, 2011]. We leave open the question of how much the use of a shared maximally entangled state can reduce the quantum communication complexity of a function. Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We give simple and efficiently computable upper and lower bounds. Given two bipartite states |chi> and |upsilon>, we define a natural quantity, d_{infty}(|chi>, |upsilon>), which we call the l_{infty} Earth Mover\u27s distance, and we show that the communication cost of converting between |chi> and |upsilon> is upper bounded by a constant multiple of d_{infty}(|chi>, |upsilon>). Here d_{infty}(|chi>, |upsilon>) may be informally described as the minimum over all transports between the log of the Schmidt coefficients of |chi> and those of |upsilon>, of the maximum distance that any amount of mass must be moved in that transport. A precise definition is given in the introduction. Furthermore, we prove a complementary lower bound on the cost of state conversion by the epsilon-Smoothed l_{infty}-Earth Mover\u27s Distance, which is a natural smoothing of the l_{infty}-Earth Mover\u27s Distance that we will define via a connection with optimal transport theory

Quantum Algorithms and the Power of Forgetting

The so-called welded tree problem provides an example of a black-box problem that can be solved exponentially faster by a quantum walk than by any classical algorithm [Andrew M. Childs et al., 2003]. Given the name of a special entrance vertex, a quantum walk can find another distinguished exit vertex using polynomially many queries, though without finding any particular path from entrance to exit. It has been an open problem for twenty years whether there is an efficient quantum algorithm for finding such a path, or if the path-finding problem is hard even for quantum computers. We show that a natural class of efficient quantum algorithms provably cannot find a path from entrance to exit. Specifically, we consider algorithms that, within each branch of their superposition, always store a set of vertex labels that form a connected subgraph including the entrance, and that only provide these vertex labels as inputs to the oracle. While this does not rule out the possibility of a quantum algorithm that efficiently finds a path, it is unclear how an algorithm could benefit by deviating from this behavior. Our no-go result suggests that, for some problems, quantum algorithms must necessarily forget the path they take to reach a solution in order to outperform classical computation

Quantum algorithms and the power of forgetting

The so-called welded tree problem provides an example of a black-box problem that can be solved exponentially faster by a quantum walk than by any classical algorithm. Given the name of a special ENTRANCE vertex, a quantum walk can find another distinguished EXIT vertex using polynomially many queries, though without finding any particular path from ENTRANCE to EXIT. It has been an open problem for twenty years whether there is an efficient quantum algorithm for finding such a path, or if the path-finding problem is hard even for quantum computers. We show that a natural class of efficient quantum algorithms provably cannot find a path from ENTRANCE to EXIT. Specifically, we consider algorithms that, within each branch of their superposition, always store a set of vertex labels that form a connected subgraph including the ENTRANCE, and that only provide these vertex labels as inputs to the oracle. While this does not rule out the possibility of a quantum algorithm that efficiently finds a path, it is unclear how an algorithm could benefit by deviating from this behavior. Our no-go result suggests that, for some problems, quantum algorithms must necessarily forget the path they take to reach a solution in order to outperform classical computation.Comment: 49 pages, 9 figure