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Register a new userAdiabatic optimization without local minima
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Several previous works have investigated the circumstances under which quantum adiabatic optimization algorithms can tunnel out of local energy minima that trap simulated annealing or other classical local search algorithms. Here we investigate the even more basic question of whether adiabatic optimization algorithms always succeed in polynomial time for trivial optimization problems in which there are no local energy minima other than the global minimum. Surprisingly, we find a counterexample in which the potential is a single basin on a graph, but the eigenvalue gap is exponentially small as a function of the number of vertices. In this counterexample, the ground state wavefunction consists of two lobes separated by a region of exponentially small amplitude. Conversely, we prove if the ground state wavefunction is single-peaked then the eigenvalue gap scales at worst as one over the square of the number of vertices.
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Many physically interesting models show a quantum phase transition when a single parameter is varied through a critical point, where the ground state and the first excited state become degenerate. When this parameter appears as a coupling constant, these models can be understood as straight-line interpolations between different Hamiltonians $H_{rm I}$ and $H_{rm F}$. For finite-size realizations however, there will usually be a finite energy gap between ground and first excited state. By slowly changing the coupling constant through the point with the minimum energy gap one thereby has an adiabatic algorithm that prepares the ground state of $H_{rm F}$ from the ground state of $H_{rm I}$. The adiabatic theorem implies that in order to obtain a good preparation fidelity the runtime $tau$ should scale with the inverse energy gap and thereby also with the system size. In addition, for open quantum systems not only non-adiabatic but also thermal excitations are likely to occur. It is shown that -- using only local Hamiltonians -- for the 1d quantum Ising model and the cluster model in a transverse field the conventional straight line path can be replaced by a series of straight-line interpolations, along which the fundamental energy gap is always greater than a constant independent on the system size. The results are of interest for adiabatic quantum computation since strong similarities between adiabatic quantum algorithms and quantum phase transitions exist.
It is believed that the presence of anticrossings with exponentially small gaps between the lowest two energy levels of the system Hamiltonian, can render adiabatic quantum optimization inefficient. Here, we present a simple adiabatic quantum algorithm designed to eliminate exponentially small gaps caused by anticrossings between eigenstates that correspond with the local and global minima of the problem Hamiltonian. In each iteration of the algorithm, information is gathered about the local minima that are reached after passing the anticrossing non-adiabatically. This information is then used to penalize pathways to the corresponding local minima, by adjusting the initial Hamiltonian. This is repeated for multiple clusters of local minima as needed. We generate 64-qubit random instances of the maximum independent set problem, skewed to be extremely hard, with between 10^5 and 10^6 highly-degenerate local minima. Using quantum Monte Carlo simulations, it is found that the algorithm can trivially solve all the instances in ~10 iterations.
Most experimental and theoretical studies of adiabatic optimization use stoquastic Hamiltonians, whose ground states are expressible using only real nonnegative amplitudes. This raises a question as to whether classical Monte Carlo methods can simulate stoquastic adiabatic algorithms with polynomial overhead. Here, we analyze diffusion Monte Carlo algorithms. We argue that, based on differences between L1 and L2 normalized states, these algorithms suffer from certain obstructions preventing them from efficiently simulating stoquastic adiabatic evolution in generality. In practice however, we obtain good performance by introducing a method that we call Substochastic Monte Carlo. In fact, our simulations are good classical optimization algorithms in their own right, competitive with the best previously known heuristic solvers for MAX-k-SAT at k=2,3,4.
We propose a protocol for quantum adiabatic optimization, whereby an intermediary Hamiltonian that is diagonal in the computational basis is turned on and off during the interpolation. This `diagonal catalyst serves to bias the energy landscape towards a given spin configuration, and we show how this can remove the first-order phase transition present in the standard protocol for the ferromagnetic $p$-spin and the Weak-Strong Cluster problems. The success of the protocol also makes clear how it can fail: biasing the energy landscape towards a state only helps in finding the ground state if the Hamming distance from the ground state and the energy of the biased state are correlated. We present examples where biasing towards low energy states that are nonetheless very far in Hamming distance from the ground state can severely worsen the efficiency of the algorithm compared to the standard protocol. Our results for the diagonal catalyst protocol are analogous to results exhibited by adiabatic reverse annealing, so our conclusions should apply to that protocol as well.
Variational Quantum Algorithms have emerged as a leading paradigm for near-term quantum computation. In such algorithms, a parameterized quantum circuit is controlled via a classical optimization method that seeks to minimize a problem-dependent cost function. Although such algorithms are powerful in principle, the non-convexity of the associated cost landscapes and the prevalence of local minima means that local optimization methods such as gradient descent typically fail to reach good solutions. In this work we suggest a method to improve gradient-based approaches to variational quantum circuit optimization, which involves coupling the output of the quantum circuit to a classical neural network. The effect of this neural network is to peturb the cost landscape as a function of its parameters, so that local minima can be escaped or avoided via a modification to the cost landscape itself. We present two algorithms within this framework and numerically benchmark them on small instances of the Max-Cut optimization problem. We show that the method is able to reach deeper minima and lower cost values than standard gradient descent based approaches. Moreover, our algorithms require essentially the same number of quantum circuit evaluations per optimization step as the standard approach since, unlike the gradient with respect to the circuit, the neural network updates can be estimated in parallel via the backpropagation method. More generally, our approach suggests that relaxing the cost landscape is a fruitful path to improving near-term quantum computing algorithms.
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