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

Technical Summary: Pre-optimization of Quantum Circuits, Barren Plateaus, and Classical Simulability

Problem Statement
Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE), are promising approaches for preparing ground states of interacting many-body systems. However, their practical utility is hindered by the "barren plateau" (BP) problem, where the energy landscape flattens exponentially with system size, rendering gradients vanishingly small and optimization infeasible. While recent work suggests the existence of "fertile valleys"—regions in parameter space with non-vanishing gradients that decay only polynomially—these regions are often restricted to effective subspaces that are classically simulable. The central challenge addressed is twofold: (1) Can classical methods effectively initialize quantum circuits within these trainable regions to avoid BPs? and (2) Once initialized, does a quantum processor offer a computational advantage over classical simulations for further optimization, or is the problem still classically simulable?

Methodology
The authors propose a hybrid classical-quantum strategy utilizing two-dimensional tensor networks (TNs), specifically Projected Entangled Pair States (PEPS), to pre-optimize parameterized quantum circuits (PQCs).

• Classical Pre-optimization: The authors employ PEPS combined with the "Simple Update" (SU) algorithm and automatic differentiation to optimize PQCs designed to prepare the ground state of the Transverse Field Ising Model (TFIM). This allows for the determination of optimal parameters ($\theta_{opt}$) for shallow circuits.
• Warm-Start Initialization: These optimized parameters are used to initialize deeper quantum circuits. The remaining parameters in the deeper layers are set to zero (identity initialization), effectively embedding the TN-optimized state into a deeper ansatz.
• Landscape Diagnostics: To verify the avoidance of barren plateaus, the authors probe the energy landscape around the warm-start point. They sample parameters within a hypercube of radius $r$ around $\theta_{opt}$ and measure the energy variance, $\text{Var}(E)$, as a proxy for gradient magnitude.
• Complexity Comparison: The paper compares the scaling of classical TN simulation costs against quantum sampling costs. The classical error scaling is fitted to $\epsilon_{TN} \propto t^{-\beta}$, where $t$ is simulation time. This is compared to the quantum sampling error scaling, $\epsilon_{QC} \propto t^{-1/2}$. A quantum advantage is claimed if the classical scaling exponent $\beta < 0.5$.

Key Results

• Mitigation of Barren Plateaus: The study demonstrates that PEPS pre-optimization successfully identifies "fertile valleys." For both heavyhex (used in superconducting qubit prototypes) and 2D square lattice topologies, the region of substantial gradients ($r_{max}$) around the warm-start point does not shrink exponentially with system size or circuit depth. Instead, $r_{max}$ scales linearly with system size and proportionally to $1/\sqrt{D}$.
• Performance on Different Topologies:
  • Heavyhex Lattice: PEPS optimization is highly effective. For systems up to 127 qubits, the classical TN simulation exhibits a scaling exponent $\beta \approx 1.24 - 1.56$. Since $\beta > 0.5$, the classical method is more efficient than quantum sampling in this regime, suggesting no immediate quantum advantage for these specific topologies and depths.
  • Square Lattice: Due to higher connectivity and loop correlations, TN contraction is more costly. Using an MPS-boundary contraction scheme, the authors found a scaling exponent $\beta \approx 0.15 - 0.17$. Since $\beta < 0.5$, this indicates a polynomial quantum advantage over classical TN simulations for these specific, highly connected topologies, even while avoiding barren plateaus.
• Depth and Initialization: Increasing the pre-optimized depth ($D^*$) from 2 to 6 layers slightly reduced the classical scaling efficiency (increased $\beta$) but did not eliminate the trainable region. The strategy allows for the preparation of states with high energy accuracy even for large systems beyond 1D.

Significance and Claims
The paper claims to provide a viable pathway to circumvent barren plateaus in VQAs by leveraging classical tensor networks for "warm-start" initialization. The authors emphasize that while this strategy successfully places the optimization in trainable regions, the question of quantum advantage is nuanced and topology-dependent.

• Modest Claims on Advantage: The authors explicitly state that the identified "fertile valleys" are classically simulable. A quantum advantage is not guaranteed for all cases; it is observed only for specific topologies (like the square lattice) where the classical contraction cost becomes prohibitively high compared to quantum sampling.
• Scope: The work is framed within the context of variational ground state preparation. The authors note that other classical algorithms (e.g., DMRG) were not the primary comparison point, but rather the specific trade-off between TN simulations and quantum hardware for the same variational task.
• Future Outlook: The authors suggest that moving toward more challenging problems—such as longer-range interactions, more entangled ground states, and more connected lattices—could yield stronger quantum advantages than those observed in the current study. They also highlight that a fair comparison must eventually account for the cost of gradient evaluation on quantum hardware, which scales with the number of parameters.