Alexandru Gheorghiu

Nonlocality and its connections to entanglement are fundamental features of quantum mechanics that have found numerous applications in quantum information science. A set of correlations is said to be nonlocal if it cannot be reproduced by spacelike-separated parties sharing randomness and performing local operations. An important practical consideration is that the runtime of the parties has to be shorter than the time it takes light to travel between them. One way to model this restriction is to assume that the parties are computationally bounded. We therefore initiate the study of nonlocality under computational assumptions and derive the following results: (a) We define the set NEL (not-efficiently-local) as consisting of all bipartite states whose correlations arising from local measurements cannot be reproduced with shared randomness and polynomial-time local operations. (b) Under the assumption that the Learning With Errors problem cannot be solved in quantum polynomial-time, we show that NEL=ENT, where ENT is the set of all bipartite entangled states (pure and mixed). This is in contrast to the standard notion of nonlocality where it is known that some entangled states, e.g. Werner states, are local. In essence, we show that there exist (efficient) local measurements producing correlations that cannot be reproduced through shared randomness and quantum polynomial-time computation. (c) We prove that if NEL=ENT unconditionally, then BQP≠PP. In other words, the ability to certify all bipartite entangled states against computationally bounded adversaries gives a non-trivial separation of complexity classes. (d) Using (c), we show that a certain natural class of 1-round delegated quantum computation protocols that are sound against PP provers cannot exist.

Achieving quantum computational advantage requires solving a classically intractable problem on a quantum device. Natural proposals rely upon the intrinsic hardness of classically simulating quantum mechanics; however, verifying the output is itself classically intractable. On the other hand, certain quantum algorithms (e.g. prime factorization via Shor's algorithm) are efficiently verifiable, but require more resources than what is available on near-term devices. One way to bridge the gap between verifiability and implementation is to use "interactions" between a prover and a verifier. By leveraging cryptographic functions, such protocols enable the classical verifier to enforce consistency in a quantum prover's responses across multiple rounds of interaction. In this work, we demonstrate the first implementation of an interactive quantum advantage protocol, using an ion trap quantum computer. We execute two complementary protocols -- one based upon the learning with errors problem and another where the cryptographic construction implements a computational Bell test. To perform multiple rounds of interaction, we implement mid-circuit measurements on a subset of trapped ion qubits, with subsequent coherent evolution. For both protocols, the performance exceeds the asymptotic bound for classical behavior; maintaining this fidelity at scale would conclusively demonstrate verifiable quantum advantage.

We interpret ontological models for finite-dimensional quantum theory as functors from the category of finite-dimensional Hilbert spaces and bounded linear maps to the category of measurable spaces and Markov kernels. This uniformises several earlier results, that we analyse more closely: Pusey, Barrett, and Rudolph's result rules out monoidal functors; Leifer and Maroney's result rules out functors that preserve a duality between states and measurement; Aaronson et al's result rules out functors that adhere to the Schrödinger equation. We also prove that it is possible to have epistemic functors that take values in signed Markov kernels.

The decision problem version of estimating the Shannon entropy is the Entropy Difference problem (ED): given descriptions of two circuits, determine which circuit produces more entropy in its output when acting on a uniformly random input. The analogous problem with quantum circuits (QED) is to determine which circuit produces the state with greater von Neumann entropy, when acting on a fixed input state and after tracing out part of the output. Based on plausible complexity-theoretic assumptions, both of these problems are believed to be intractable for polynomial-time quantum computation. In this paper, we investigate the hardness of these problems in the case where the input circuits have logarithmic and constant depth, respectively. We show that, relative to an oracle, these problems cannot be as hard as their counterparts with polynomial-size circuits. Furthermore, we show that if a certain type of reduction from QED to the log-depth version exists, it implies that any polynomial-time quantum computation can be performed in log depth. While this suggests that having shallow circuits makes entropy estimation easier, we give indication that the problem remains intractable for polynomial-time quantum computation by proving a reduction from Learning-With-Errors (LWE) to constant-depth ED. We then consider a potential application of our results to quantum gravity research. First, we introduce a Hamiltonian version of QED where one is given two local Hamiltonians and asked to estimate the entanglement entropy difference in their ground states. We show that this problem is at least as hard as the circuit version and then discuss a potential experiment that would make use of the AdS/CFT correspondence to solve LWE efficiently. We conjecture that unless the AdS/CFT bulk to boundary map is exponentially complex, this experiment would violate the intractability assumption of LWE.

With experimental quantum computing technologies now in their infancy, the search for efficient means of testing the correctness of these quantum computations is becoming more pressing. An approach to the verification of quantum computation within the framework of interactive proofs has been fruitful for addressing this problem. Specifically, an untrusted agent (prover) alleging to perform quantum computations can have his claims verified by another agent (verifier) who only has access to classical computation and a small quantum device for preparing or measuring single qubits. However, when this quantum device is prone to errors, verification becomes challenging and often existing protocols address this by adding extra assumptions, such as requiring the noise in the device to be uncorrelated with the noise on the prover's devices. In this paper, we present a simple protocol for verifying quantum computations, in the presence of noisy devices, with no extra assumptions. This protocol is based on post hoc techniques for verification, which allow for the prover to know the desired quantum computation and its input. We also perform a simulation of the protocol, for a onequbit computation, and find the error thresholds when using the qubit repetition code as well as the Steane code.

The relationship between correlations and entanglement has played a major role in understanding quantum theory since the work of Einstein, Podolsky and Rosen (1935). Tsirelson (1980) proved that Bell states, shared among two parties, when measured suitably, achieve the maximum non-local correlations allowed by quantum mechanics. Conversely, Reichardt, Unger and Vazirani (2013) showed that observing the maximal correlation value over a sequence of repeated measurements, implies that the underlying quantum state is close to a tensor product of maximally entangled states and, moreover, that it is measured according to an ideal strategy. However, this strong rigidity result comes at a high price, requiring a large number of entangled pairs to be tested. In this paper, we present a significant improvement in terms of the overhead by instead considering quantum steering where the device of the one side is trusted. We first demonstrate a robust one-sided device-independent version of self-testing, which characterises the shared state and measurement operators of two parties up to a certain bound. We show that this bound is optimal up to constant factors and we generalise the results for the most general attacks. This leads us to a rigidity theorem for maximal steering correlations. As a key application we give a one-sided device-independent protocol for verifiable delegated quantum computation, and compare it to other existing protocols, to highlight the cost of trust assumptions. Finally, we show that under reasonable assumptions, the states shared in order to run a certain type of verification protocol must be unitarily equivalent to perfect Bell states.