In this work, we study the limitations of the Quantum Approximate Optimization Algorithm (QAOA) through the lens of statistical physics and show that there exists > 0, such that log(n) depth QAOA cannot arbitrarily-well approximate the ground state energy of random diluted k-spin glasses when k ≥ 4 is even. This is equivalent to the weak approximation resistance of logarithmic depth QAOA to the Max-k-XOR problem. We further extend the limitation to other boolean constraint satisfaction problems as long as the problem satisfies a combinatorial property called the coupled overlap-gap property (OGP) [Chen et al., Annals of Probability, 47(3), 2019]. As a consequence of our techniques, we confirm a conjecture of Brandao et al. [arXiv:1812.04170, 2018 asserting that the landscape independence of QAOA extends to logarithmic depth-in other words, for every fixed choice of QAOA angle parameters, the algorithm at logarithmic depth performs almost equally well on almost all instances. Our results provide a new way to study the power and limit of QAOA through statistical physics methods and combinatorial properties.

We study random constraint satisfaction problems (CSPs) in the unsatis able regime. We relate the structure of near-optimal solutions for any Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP's predicate is, up to a constant, the mixture polynomial of the associated spin glass. We prove two main consequences:1. We relate the maximum fraction of constraints that can be satis ed in a random Max-CSP to the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula [Par80, Tal06, AC17], we provide numerical values for some popular CSPs. 2. We prove that a Max-CSP possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from [HS21] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms [CLSS22].

Ensuring security and integrity of elections constitutes an important challenge with wide-ranging societal implications. Classically, security guarantees can be ensured based on computational complexity, which may be challenged by quantum computers. We show that the use of quantum networks can enable information-theoretic security for the desirable aspects of a distributed voting scheme in a resource-efficient manner. In our approach, ballot information is encoded in quantum states that enable an exponential reduction in communication complexity compared to classical communication. In addition, we provide an efficient and secure anonymous queuing protocol. As a result, our scheme only requires modest quantum memories with size scaling logarithmically with the number of voters. This intrinsic efficiency together with certain noise-robustness of our protocol paves the way for its physical implementation in realistic quantum networks.

We present a general framework for analyzing high-probability bounds for stochastic dynamics in learning algorithms. Our framework composes standard techniques such as a stopping time, a martingale concentration and a closed-from solution to give a streamlined three-step recipe with a general and flexible principle to implement it. To demonstrate the power and the flexibility of our framework, we apply the framework on three very different learning problems: stochastic gradient descent for strongly convex functions, streaming principal component analysis and linear bandit with stochastic gradient descent updates. We improve the state of the art bounds on all three dynamics.