Pre-optimization of quantum circuits, barren plateaus and classical simulability: tensor networks to unlock the variational quantum eigensolver

TL;DR

The paper addresses variational quantum algorithms and the barren plateau problem in search of ground states of the 2D transverse-field Ising model (TFIM) by pre-optimizing quantum circuits with two-dimensional tensor networks (PEPS). They show that using PEPS with simple updates to train PQCs yields energy-accurate states for 2D lattices larger than 1D, and that pre-optimization creates fertile valleys with non-vanishing gradients around a warm-start, enabling deeper circuits to be trained. They compare the classical TN simulations with quantum sampling, finding topology-dependent outcomes: in heavyhex topologies TN-based optimization can outperform QC sampling in scaling, while in square lattices there can be polynomial quantum advantage. The work provides practical guidelines for combining classical TN pre-processing with near-term quantum devices and clarifies when quantum hardware may offer a scaling edge for variational GS preparation.

Abstract

Variational quantum algorithms are practical approaches to prepare ground states, but their potential for quantum advantage remains unclear. Here, we use differentiable 2D tensor networks (TN) to optimize parameterized quantum circuits that prepare the ground state of the transverse field Ising model (TFIM). Our method enables the preparation of states with high energy accuracy, even for large systems beyond 1D. We show that TN pre-optimization can mitigate the barren plateau issue by giving access to enhanced gradient zones that do not shrink exponentially with system size. We evaluate the classical simulation cost evaluating energies at these warm-starts, and identify regimes where quantum hardware offers better scaling than TN simulations.

Figures (4)

• Figure 1: (a) A deep parameterized quantum circuit cannot be randomly initialized because of barren plateaus [1'], while a short-depth circuit can be optimized classically [1] and used as a warm-start for a deeper circuit [2], by initializing the remaining parameters around identity. (b) Our scheme allows to run VQE in regions with substantial energy gradients and improve accuracy thanks to deeper circuits [3]. (c) We investigate the cost of evaluating the energy in the green region using TNs, and compare it to the sampling cost of a QC.
• Figure 2: Energy convergence of PEPS optimization for the TFIM. (a) 127 qubit heavyhex lattice: i. Relative energy error $\delta E$ vs $g$ for different circuit depths $D$. ii. $\delta E$ vs. the number of iterations for different circuit depths $D$ at $g_c \simeq 1.5$, and $\chi=8$. (b) 5$\times$5 square lattice: i. $\delta E$ vs $g$ for different circuit depths $D$. Here the energy is computed by statevector simulations obtained from the TN optimizations. ii. $\delta E$ vs the number of iterations and circuit depth $D$ at $g_c \simeq 2.6$, and $\chi$ up to 6 with SU-like expectation values.
• Figure 3: Energy landscape diagnostic. (a) Heavyhex lattice including (i) $\text{Var}(E)$ vs $r$ for $N = 28, 53, 75$ and 127, $r_{\text{max}}$ vs (ii) total depth $D$ with $D^*=2$ and $6$ and (iii) vs system size for $D^*=2$ and $D=20$ on the 53-qubit system. (b) Square lattice with (i) $\text{Var}(E)$ vs $r$ for $N=$$3\times 3$, $4\times 3$, $4\times 4$ and $5\times 4$, $r_{\text{max}}$ vs (ii) total depth $D$ with $D^*=2$ and (iii) $r_{\text{max}}$ vs the system size, relative to $r_{\text{max}}$ for $N= 3\times 3$.
• Figure 4: Average error $\epsilon$ versus time in the trainable region of size $r_{\text{max}}$. PQCs of depth $D$ pre-optimized with a $D^*$ quantum circuit. (a) Heavyhex lattice with (i) 53 qubits with $D^* = 2$ and (ii) $D^* = 6$ and (iii) 127 qubits with $D^* = 2$ and (iv) $D^* = 6$. (b) Square lattice with system sizes (a) $4\times 4$ and (b) $5 \times 5$ with $D^* = 2$ and (i, iv) $D = 10$, (ii,v) 20 and (iii, vi) 30. Dashed lines: quantum sampling error.