Quantum Computers Sidestep Major Flaw, Paving Way for Larger, More Accurate Calculations

Scientists are increasingly exploring variational quantum eigensolvers as practical approaches to prepare ground states, but their potential for quantum advantage remains unclear.

Baptiste Anselme Martin from Eviden Quantum Lab and Thomas Ayral from CPHT, CNRS, Ecole Polytechnique, IP Paris, alongside et al., demonstrate a novel method utilising differentiable 2D tensor networks to optimise parameterised circuits for the transverse field Ising model.

This research is significant because it enables the preparation of highly accurate ground states for systems exceeding one dimension and crucially, mitigates the detrimental barren plateau issue by identifying enhanced gradient zones that maintain performance as system size increases. By evaluating the classical simulation cost at these optimised starting points, the team delineate regimes where quantum hardware may ultimately outperform tensor network simulations. Tensor network pre-optimisation overcomes barren plateaus in variational quantum circuits by improving initial parameterisation Researchers are pioneering a new approach to harness the power of quantum computing by integrating classical tensor network algorithms with parameterized quantum circuits. This work details the use of differentiable two-dimensional tensor networks to optimize circuits designed to prepare the ground state of the transverse field Ising model, achieving high energy accuracy even for complex systems exceeding one-dimensional limitations. The study demonstrates that pre-optimization using tensor networks effectively mitigates the barren plateau issue, a significant obstacle in quantum computation, by unlocking enhanced gradient zones that maintain their size even as system complexity increases. Specifically, the research focuses on optimizing quantum circuits using projected entangled pair states, a type of two-dimensional tensor network, combined with automatic differentiation techniques. This method allows for the efficient preparation of ground states and provides a means to initialize deeper quantum circuits in regions with substantial energy gradients, thereby improving accuracy and avoiding the exponential flattening of the energy landscape associated with barren plateaus. Investigations into the classical simulation cost of evaluating energies at these optimized starting points reveal regimes where quantum hardware could potentially outperform tensor network simulations.

The team quantified the cost of tensor network simulations and compared it to the sampling cost on a quantum computer, identifying scenarios where quantum devices may offer a polynomial advantage within the framework of variational quantum algorithms. By leveraging the strengths of both classical and quantum computational methods, this work presents a promising practical approach to ground state preparation and parameter initialization. The integration of tensor networks with quantum computing offers a pathway to enhance quantum computations, whether through encoding tensor network states, compressing quantum circuits, or optimizing parameterized quantum circuits for ground state preparation. Parameter optimisation of projected entangled pair states for transverse field Ising model ground state preparation is a challenging task Differentiable two-dimensional tensor networks (TN) underpin the methodology used to optimize parameterized quantum circuits for preparing the ground state of the transverse field Ising model (TFIM). Projected entangled pair states (PEPS) serve as the specific TN implementation, leveraging automatic differentiation to refine the parameters within the quantum circuits. This approach facilitates the preparation of states exhibiting high energy accuracy, even for systems exceeding one-dimensional limitations. The research involved optimizing circuits of varying depths, denoted by ‘D’, to assess the impact on energy convergence. Energy errors were calculated as a relative value, δE, and monitored against the coupling strength, ‘g’, and the number of iterations during the optimization process. Specifically, simulations were conducted on a 127-qubit heavy-hex lattice and a 5×5 square lattice to evaluate performance across different geometries. The TN optimization was performed with a maximum bond dimension, χ, of 8 for the heavy-hex lattice. A key innovation lies in the use of TN pre-optimization to mitigate the barren plateau issue. This was achieved by identifying and accessing enhanced gradient zones within the parameter space, which do not diminish exponentially with increasing system size. The study then evaluated the classical simulation cost of assessing energies within these identified regions, comparing it to the sampling cost associated with quantum hardware to determine regimes where quantum computation offers a scaling advantage. Energy convergence was assessed by monitoring δE against both ‘g’ and iteration number, allowing for a detailed analysis of optimization performance for different circuit depths and bond dimensions. Optimised brickwall circuits demonstrate low logical error rates and ground state preparation fidelity Logical error rates reached 2.9% per cycle within the study’s optimized circuits. These circuits utilized brickwall configurations composed of SO(4) gates, each parameterized by six parameters and applied to neighboring qubits. Mitigation of truncation errors was achieved through a SU- regauging process applied after each layer of the brickwall circuit. Expectation values were computed using a SU-type approach for both lattice topologies investigated. For the 127-qubit heavyhex topology with a bond dimension of 8, results were obtained referencing energies from converged imaginary time-evolution using PEPS+SU. Optimization procedures across a range of values for g and varying depths allowed for the preparation of ground states with high energy accuracy. Specifically, at the critical point g ≃ 1.5, increasing circuit depth consistently yielded lower energy states, even while maintaining a low bond dimension. The square lattice presented greater challenges, though optimization of short-depth quantum circuits remained possible, particularly away from gc. With a depth of 3, standard tensor network approximations failed to optimize circuits initialized with random parameters, but this was resolved by utilizing parameters from previously optimized circuits of depth 2. Energy variance was then assessed as a function of distance r in parameter space from the initial point, revealing a trainable region around the PEPS-optimized warm-starts. For the square lattice, the size of the region with sizable gradients, termed rmax, scaled linearly with system size and proportionally to 1/ √ D. The heavyhex topology enabled exploration of larger system sizes, demonstrating that rmax hardly depended on system size. Classical simulations were then used to assess the time complexity of evaluating energy expectation values, comparing it to the sampling cost of equivalent quantum computation. The TN simulation error scaled with bond dimension, allowing for error assessment relative to exact solutions or converged TN simulations. Quantum optimisation via tensor networks and variational circuits offers promising avenues for tackling complex computational problems Researchers have demonstrated a method for optimising quantum circuits using differentiable two-dimensional tensor networks, achieving high energy accuracy in preparing the ground state of the transverse field Ising model even for systems exceeding one dimension. This approach mitigates the barren plateau problem, a significant challenge in quantum machine learning, by identifying and accessing gradient zones that do not diminish exponentially with increasing system size. The study establishes that pre-optimisation with tensor networks can provide advantageous starting points for variational quantum algorithms. Evaluation of the classical computational cost revealed regimes where quantum hardware potentially outperforms tensor network simulations. Both quantum and tensor network methods exhibit polynomial scaling costs, but quantum methods prove superior for more interconnected lattice structures. This work specifically addresses the question of whether a tensor network or a quantum computer is more effective for optimising a quantum circuit to prepare a ground state, finding that tensor networks can successfully locate regions with non-vanishing gradients. The authors acknowledge that the notion of quantum advantage is limited to the context of variational ground state preparation and that other classical algorithms warrant comparison. Limitations include the potential memory overhead associated with automatic differentiation when using large bond dimensions in tensor networks. Future research may focus on exploring the performance of this method with different quantum circuit architectures and on larger, more complex systems, potentially extending the scope to other quantum many-body problems and assessing the broader applicability of tensor network pre-optimisation. 👉 More information 🗞 Pre-optimization of quantum circuits, barren plateaus and classical simulability: tensor networks to unlock the variational quantum eigensolver 🧠 ArXiv: https://arxiv.org/abs/2602.04676 Tags: