Polynomial scaling of QAOA for ground-state preparation of the fully-connected p-spin ferromagnet

Abstract

We show that the quantum approximate optimization algorithm (QAOA) can construct with polynomially scaling resources the ground state of the fully-connected p-spin Ising ferromagnet, a problem that notoriously poses severe difficulties to a Quantum Annealing (QA) approach, due to the exponentially small gaps encountered at first-order phase transition for p ≥ 3. For a generic target state, we find that an appropriate QAOA parameter initialization is necessary to achieve a good performance of the algorithm when the number of variational parameters 2P is much smaller that the system size N, because of the large number of sub-optimal local minima. We find that when P > P * N ∝ N, the structure of the parameter space simplifies, as all minima become degenerate. This allows to achieve the ground state with perfect fidelity with a number of parameters scaling extensively with N, and with resources scaling polynomially with N.

Solving correlation clustering with QAOA and a Rydberg qudit system: a full-stack approach

“…We will be interested in performance at low-depth and hence we only consider r = 1 in Eq. (11) for the simulations in sections IV and V. Through numerical evaluation of the vanilla QAOA, we found that the following strategies considerably improved the QAOA's performance for our problem:…”

“…The current body of literature points into mixed directions as far as the utility of QAOA is concerned: whilst it has provable advantages such as recovering near optimal query complexity in Grover's search [2], exhibiting universality [3,4] and the possibility for quantum supremacy [5], there are also known limitations in the low depth regime [6][7][8][9]. However, current analytical tool for analysing the performance of QAOA have only been able to investigate very specific problem instances, predominantly at low depth [10][11][12][13][14]. A general analytical approach remains to be found, which is why a large portion of the literature resorts to numerical simulation.…”

Quantifying the efficiency of state preparation via quantum variational eigensolvers

“…IV, which was obtained as a Trotter splitting of the Hamiltonian (10) into its translation invariant components: the protocol is unable to prepare ground states that are far from being Gaussian (as measured by interaction distance). This limitation is fundamentally related to the probability 11)- (13).…”

“…4,5 QAOA was originally proposed to tackle classical optimization problems, such as the MaxCut problem, 6 and several others. [7][8][9][10] More recently, it has been pointed out that QAOA could also serve as a tool for exactly preparing quantum many-body states, such as the GHZ state, the ground state of the Ising model at the critical point, for both short-range 11 and long-range interactions, 12,13 the ground state of the toric code, 11 the ground state of the two-dimensional Hubbard model, 14 and the thermofield double states. 15,16 In this paper we focus on the latter type of applications of QAOA in the context of preparing ground states of non-integrable quantum Hamiltonians.…”

Digitized-counterdiabatic quantum approximate optimization algorithm

“…One compelling outcome of VQA is the development of the quantum approximate optimization algorithm (QAOA) [26], which provides an alternative for solving combinatorial optimization problems using shallow quantum circuits with classically optimized parameters. In the past few years, there has been a rapid development in QAOA-based techniques that have been applied not only for solving conventional optimization problems like MaxCut but also for solving ground state problems in different physical systems [27][28][29]. Improved versions of QAOA, like ADAPT-QAOA [30] and Digital-Analog QAOA [31] have also been reported recently.…”