The quantum approximate optimization algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization that has been a promising avenue for near-term quantum advantage. In this paper, we analyze the performance of the QAOA on the spiked tensor model, a statistical estimation problem that exhibits a large computational-statistical gap classically. We prove that the weak recovery threshold of $1$-step QAOA matches that of $1$-step tensor power iteration. Additional heuristic calculations suggest that the weak recovery threshold of $p$-step QAOA matches that of $p$-step tensor power iteration when $p$ is a fixed constant. This further implies that multi-step QAOA with tensor unfolding could achieve, but not surpass, the asymptotic classical computation threshold $\Theta(n^{(q-2)/4})$ for spiked $q$-tensors. Meanwhile, we characterize the asymptotic overlap distribution for $p$-step QAOA, discovering an intriguing sine-Gaussian law verified through simulations. For some $p$ and $q$, the QAOA has an effective recovery threshold that is a constant factor better than tensor power iteration. Of independent interest, our proof techniques employ the Fourier transform to handle difficult combinatorial sums, a novel approach differing from prior QAOA analyses on spin-glass models without planted structure.

Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding local minima of these Hamiltonians. Therefore, cooling systems to local minima is universal for quantum computation, and, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.

The Quantum Approximate Optimization Algorithm (QAOA) is a general purpose quantum algorithm designed for combinatorial optimization. We analyze its expected performance and prove concentration properties at any constant level (number of layers) on ensembles of random combinatorial optimization problems in the infinite size limit. These ensembles include mixed spin models and Max-q-XORSAT on sparse random hypergraphs. Our analysis can be understood via a saddlepoint approximation of a sum-over-paths integral. This is made rigorous by proving a generalization of the multinomial theorem, which is a technical result of independent interest. We then show that the performance of the QAOA at constant levels for the pure q-spin model matches asymptotically the ones for Max-q XORSAT on random sparse Erdôs-Rényi hypergraphs and every large-girth regular hypergraph. Through this correspondence, we establish that the average-case value produced by the QAOA at constant levels is bounded away from optimality for pure q-spin models when $q\geq 4$ and is even. This limitation gives a hardness of approximation result for quantum algorithms in a new regime where the whole graph is seen.

Analog quantum simulations---simulations of one Hamiltonian by another---is one of the major goals in the noisy intermediate-scale quantum computation (NISQ) era, and has many applications in quantum complexity. We initiate the rigorous study of the physical resources required for such simulations, where we focus on the task of Hamiltonian sparsification. The goal is to find a simulating Hamiltonian $\tilde{H}$ whose underlying interaction graph has bounded degree (this is called degree-reduction) or much fewer edges than that of the original Hamiltonian $H$ (this is called dilution). We set this study in a relaxed framework for analog simulations that we call gap-simulation, where $\tilde{H}$ is only required to simulate the groundstate(s) and spectral gap of $H$ instead of its full spectrum, and we believe it is of independent interest. Our main result is a proof that in stark contrast to the classical setting, general degree-reduction is impossible in the quantum world, even under our relaxed notion of gap-simulation. The impossibility proof relies on devising counterexample Hamiltonians and applying a strengthened variant of Hastings-Koma decay of correlations theorem. We also show a complementary result where degree-reduction is possible when the strength of interactions is allowed to grow polynomially. Furthermore, we prove the impossibility of the related sparsification task of generic Hamiltonian dilution, under a computational hardness assumption. We also clarify the (currently weak) implications of our results to the question of quantum PCP. Our work provides basic answers to many of the "first questions" one would ask about Hamiltonian sparsification and gap-simulation; we hope this serves a good starting point for future research of these topics.