Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz

A large ongoing research effort focuses on variational quantum algorithms (VQAs), representing leading candidates to achieve computational speed-ups on current quantum devices. The scalability of VQAs to a large number of qubits, beyond the simulation capabilities of classical computers, is still debated. Two major hurdles are the proliferation of low-quality variational local minima, and the exponential vanishing of gradients in the cost-function landscape, a phenomenon referred to as barren plateaus. In this work, we show that by employing iterative search schemes, one can effectively prepare the ground state of paradigmatic quantum many-body models, also circumventing the barren plateau phenomenon. This is accomplished by leveraging the transferability to larger system sizes of a class of iterative solutions, displaying an intrinsic smoothness of the variational parameters, a result that does not extend to other solutions found via random-start local optimization. Our scheme could be directly tested on near-term quantum devices, running a refinement optimization in a favorable local landscape with nonvanishing gradients.

Figure 1

Smooth optimal parameters $({\mathit{\beta }}^{*},{\mathit{\alpha }}^{*}){|}_{P,N}=({\beta }_1^{*}\cdots {\beta }_P^{*},{\alpha }_1^{*}\cdots {\alpha }_P^{*})$ obtained with INTERP (see main text), plotted vs the rescaled index $\tilde{m}\equiv (m-1)/(P-1)$ in the $x$-axis range [0,1]. Results are shown for the Heisenberg model (${\mathrm{\Delta }}_{\mathrm{Y}}=1, {\mathrm{\Delta }}_{\mathrm{Z}}=1$) (left) and the LTFIM ($g_x=1, g_z=1$) (right) for $P=16$, and they are qualitatively similar for different sizes $N$. These solutions are stable by further increasing the number of layers, $P$. Similar smooth solutions can be found for different points of the phase diagram.

Figure 2

Residual energies [Eq. (10)] for different system sizes using parameters from a small-size “guess system” ($N_G=8$) computed for different flavors of the two models. We show the transferability of smooth optimal solutions $({\mathit{\beta }}^{*},{\mathit{\alpha }}^{*}){|}_{P,N_G}$ with $P=10$ for XYZ models (left) and the LTFIM (right).

Figure 3

Barren plateaus in the whole search space (data denoted as “Global”), contrasted with a qualitatively different trend in the $\epsilon$-neighborhood of the transferred smooth solution $({\mathit{\beta }}^{*},{\mathit{\alpha }}^{*}){|}_{P,N_G}$, obtained with INTERP for a small system size $N_G=8$ (data denoted as “Smooth”). Here, we focus on a single partial derivative with respect to $\theta ={\alpha }_1$ [see Eqs. (7) and (9)] of the cost function in Eq. (8), rescaled in terms of the few independent correlators (see SM [54]) and dubbed $C$. We plot the sample variance of the partial derivative as a function of the number $P$ of HVA layers in the circuit. We fix $\epsilon =0.05$ and a batch of 1000 samples for each value of $P$ and $N$.

Figure 4

Same quantity as in Fig. 3, here plotted vs the qubit number for a fixed circuit depth $P=40$ ($P=100$) for the Heisenberg model (LTFIM). The labels “Global” and “Smooth” refer to the same data of Fig. 3. We denote with “Local $\epsilon$” a sampling performed in an $\epsilon$-region centered around a random point. Data are averages over 20 random points (generated independently for each value of $N$) with $\epsilon =0.05$. We use a constant batch size of 1000 samples, for each $N$ and sampling region. A clear trend appears: the neighborhood of a random point (or that of a transferred nonsmooth solution; see SM [54]) exhibits the same exponential decay as the whole space, whereas this is not present in the neighborhood of the transferred smooth solution.