[Paper Review] Reconstructing Quantum States Using Basis-Enhanced Born Machines

Abigail McClain Gomez, Susanne F. Yelin+1 more|arXiv (Cornell University)|Jun 2, 2022

Quantum many-body systems4 citations

TL;DR

This paper introduces a basis-enhanced Born machine (BEBM) that reconstructs pure quantum states using only two Pauli measurement bases—significantly reducing sample complexity. By leveraging a complex-valued tensor network model and a priori knowledge of Hamiltonian operators, the BEBM achieves quantum state fidelity above 99% across ordered and critical phases of 1D Rydberg chains and XY spin chains, even in challenging oscillatory regions where single-basis learning fails due to barren plateaus.

ABSTRACT

Rapid improvement in quantum hardware has opened the door to complex problems, but the precise characterization of quantum systems itself remains a challenge. To address this obstacle, novel tomography schemes have been developed that employ generative machine learning models, enabling quantum state reconstruction from limited classical data. In particular, quantum-inspired Born machines provide a natural way to encode measured data into a model of a quantum state. Born machines have shown great success in learning from classical data; however, the full potential of a Born machine in learning from quantum measurement has thus far been unrealized. To this end, we devise a complex-valued basis-enhanced Born machine and show that it can reconstruct pure quantum states using projective measurements from only two Pauli measurement bases. We implement the basis-enhanced Born machine to learn the ground states across the phase diagram of a 1D chain of Rydberg atoms, reconstructing quantum states deep in ordered phases and even at critical points with quantum fidelities surpassing 99%. The model accurately predicts quantum correlations and different observables, and system sizes as large as 37 qubits are considered. Quantum states across the phase diagram of a 1D XY spin chain are also successfully reconstructed using this scheme. Our method only requires simple Pauli measurements with a sample complexity that scales quadratically with system size, making it amenable to experimental implementation.

Motivation & Objective

• To overcome the exponential sample complexity of traditional quantum state tomography in large quantum systems.
• To address the failure of single-basis Born machines in reconstructing quantum states with high Shannon entropy, such as in oscillatory phases of the XY chain.
• To reduce experimental overhead by enabling state reconstruction with only two measurement bases instead of informationally complete POVMs.
• To improve optimization stability and avoid barren plateaus in tensor network training via basis-enhancement and complex-valued parameters.
• To enable efficient reconstruction of quantum states with long-range order and critical behavior using minimal, physically accessible measurements.

Proposed method

• The method employs a complex-valued tensor network-based Born machine that learns probability amplitudes from measured data across two distinct Pauli bases.
• It uses a priori knowledge of the Hamiltonian’s Pauli operators to select optimal measurement bases, minimizing redundancy and Shannon entropy.
• The model is trained via stochastic gradient descent on the negative log-likelihood (NLL) loss function, which approximates maximum-likelihood state reconstruction.
• Basis enhancement allows the model to capture complex quantum correlations and avoid barren plateaus that hinder single-basis training.
• Complex-valued parameters increase the number of equivalent global minima, improving optimization convergence regardless of initialization.
• The approach is validated on 1D Rydberg chains and XY spin chains, with system sizes up to 37 qubits.

Experimental results

Research questions

• RQ1 Can a Born machine reconstruct pure quantum states using only two Pauli measurement bases, rather than informationally complete sets?
• RQ2 Why does single-basis Born machine training fail in the oscillatory phase of the 1D XY chain, and can basis enhancement overcome this?
• RQ3 How does the choice of measurement bases—particularly their Shannon entropy—affect the trainability and fidelity of Born machine state reconstruction?
• RQ4 Can basis-enhanced Born machines achieve high-fidelity reconstruction of critical and ordered phases in strongly correlated spin systems?
• RQ5 To what extent can a priori knowledge of the Hamiltonian’s Pauli terms guide efficient basis selection for minimal yet effective measurement schemes?

Key findings

• The basis-enhanced Born machine reconstructs ground states in a 1D Rydberg chain with quantum fidelity exceeding 99%, even in ordered and critical phases.
• In the oscillatory phase of the 1D XY chain, where single-basis learning fails due to high Shannon entropy and barren plateaus, the BEBM successfully reconstructs the state using only z- and x-basis measurements.
• The x-basis data, which has lower Shannon entropy than the z-basis in the oscillatory phase, significantly improves optimization and avoids barren plateaus.
• The use of complex-valued parameters increases the number of equivalent global minima, enhancing training convergence and robustness to initialization.
• The method achieves accurate prediction of quantum correlations and observables with sample complexity scaling quadratically with system size N.
• The approach is scalable to 37-qubit systems and demonstrates potential for reconstructing non-trivial phases such as topological or spin liquid states.