[Paper Review] Neural-network states for the classical simulation of quantum computing

Bjàrni Jónsson, Bela Bauer+1 more|arXiv (Cornell University)|Aug 15, 2018

Quantum Computing Algorithms and Architecture53 references45 citations

TL;DR

Introduces neural-network quantum states (NQS) based on complex RBMs to classically simulate general quantum circuits, with exact handling of Z-rotations and CZ gates and a variational scheme to approximate Hadamard gates, enabling simulations beyond brute-force limits.

ABSTRACT

Simulating quantum algorithms with classical resources generally requires exponential resources. However, heuristic classical approaches are often very efficient in approximately simulating special circuit structures, for example with limited entanglement, or based on one-dimensional geometries. Here we introduce a classical approach to the simulation of general quantum circuits based on neural-network quantum states (NQS) representations. Considering a set of universal quantum gates, we derive rules for exactly applying single-qubit and two-qubit Z rotations to NQS, whereas we provide a learning scheme to approximate the action of Hadamard gates. Results are shown for the Hadamard and Fourier transform of entangled initial states for systems sizes and total circuit depths exceeding what can be currently simulated with state-of-the-art brute-force techniques. The overall accuracy obtained by the neural-network states based on Restricted Boltzmann machines is satisfactory, and offers a classical route to simulating highly-entangled circuits. In the test cases considered, we find that our classical simulations are comparable to quantum simulations affected by an incoherent noise level in the hardware of about $10^{-3}$ per gate.

Motivation & Objective

• Motivate and assess classical simulation of general quantum circuits using neural-network representations.
• Develop exact or approximate gate application rules within an RBM-based NQS framework.
• Demonstrate the method on Hadamard and Fourier transforms of entangled TFIM-derived initial states.
• Benchmark scalability to larger system sizes and depths beyond brute-force capabilities.
• Discuss implications for hardware noise tolerance and relevance to quantum supremacy discussions.

Proposed method

• Represent quantum states of N qubits with a complex-valued Restricted Boltzmann Machine (RBM).
• Derive exact RBM weight updates to implement diagonal gates: single-qubit Z rotations and two-qubit CZ gates via visible biases and auxiliary hidden units.
• Approximate Hadamard gates using a variational, stochastic loss minimizing the negative log-overlap between the RBM state and the exact post-Hadamard state.
• Use stochastic gradient descent (Adam/AdaMax) to optimize RBM parameters after Hadamard applications, guided by Monte Carlo sampling of amplitudes.
• Benchmark on Hadamard and truncated Fourier transforms applied to TFIM ground states in 1D and 2D lattices at/near critical points.
• Compare variational RBM results to hardware noise models to relate classical approximation error to depolarizing noise levels.

Experimental results

Research questions

• RQ1 Can neural-network quantum states efficiently approximate the action of universal gate sets on entangled multi-qubit states?
• RQ2 To what extent can diagonal gates be applied exactly within the RBM framework, and how can non-diagonal gates (Hadamard) be approximated variationally?
• RQ3 How large and depth-rich can quantum circuits be simulated classically using NQS before sampling or optimization become intractable?
• RQ4 How does the classical variational error compare to realistic hardware noise in quantum devices?
• RQ5 Do NQS-based simulations of Hadamard and Fourier transforms on TFIM-derived states provide fidelity comparable to noisy quantum hardware?

Key findings

• RBM-based neural-network states allow exact application of Z-rotations and CZ gates by local or minimal network adjustments.
• Hadamard gates are approximated via a stochastic variational scheme minimizing the negative log-overlap.
• Hadamard and truncated Fourier transforms applied to TFIM ground-state inputs in 1D and 2D reach intermediate fidelities above 0.96 with fixed RBM size, exceeding brute-force capabilities.
• Final state overlaps for small 1D systems remain competitive with the cumulative intermediate fidelities, showing error cancellation can occur.
• Comparative analysis with depolarizing noise suggests the variational RBM approach can achieve effective fidelity comparable to hardware noise levels around 10^-3 per gate in Hadamard/FT circuits.
• Results indicate RBM-based simulations can meaningfully approximate highly entangled circuits beyond conventional classical limits, offering insights into quantum supremacy benchmarks.