A perturbative gadget for delaying the onset of barren plateaus in variational quantum algorithms

Variational quantum algorithms are being explored as a promising approach to finding useful applications for noisy intermediate-scale quantum computers. However, cost functions corresponding to many problems of interest are inherently global, defined by Hamiltonians with many-body interactions. Consequently, the optimization landscape can exhibit exponentially vanishing gradients, so-called barren plateaus, rendering optimal solutions difficult to find. Strategies for mitigating barren plateaus are therefore needed to make variational quantum algorithms trainable and capable of running on larger-scale quantum computers. In this work, we contribute the toolbox of perturbative gadgets to the portfolio of methods being explored in the quest for making noisy intermediate-scale quantum devices useful. We introduce a novel perturbative gadget, tailored to variational quantum algorithms, that can be used to delay the onset of barren plateaus. Our perturbative gadget encodes an arbitrary many-body Hamiltonian corresponding to a global cost function into the low-energy subspace of a three-body Hamiltonian. Our construction requires $rk$ additional qubits for a $k$-body Hamiltonian comprising $r$ terms. We provide guarantees on the closeness of global minima and prove that the local cost function defined by our three-body Hamiltonian exhibits non-vanishing gradients. We then provide numerical demonstrations to show the functioning of our approach and discuss heuristics that might aid its practical implementation.Comment: Added further discussion of the number of required measurement

Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction

In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of order $\Theta(1/N^{2+\delta})$, for a small parameter $\delta>0$, and for $N$ terms in the target Hamiltonian. In its low-energy subspace, the construction can approximate any normalized target Hamiltonian $H_\mathrm{t}=\sum_{i=1}^N h_i$ with norm ratios $r=\max_{i,j\in\{1,\ldots,N\}}\|h_i\| / \| h_j \|=O(\exp(\exp(\mathrm{poly} n)))$ to within relative precision $O(N^{-\delta})$. This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancilliary system for each coupling; interactions on the ancilliary system are geometrically local, and can be translationally-invariant. As an application, we discuss implications for QMA-hardness of the local Hamiltonian problem, and argue that "almost" translational invariance-defined as arbitrarily small relative variations of the strength of the local terms-is as good as non-translational-invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally-invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAEXP-, or even BQEXPSPACE-complete