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Quantum State Generation with Structure-Preserving Diffusion Model

Yuchen Zhu, Tianrong Chen, Evangelos A. Theodorou, Xie Chen, Molei Tao

TL;DR

This work tackles the problem of generative modeling for quantum mixed states, where density matrices must satisfy Hermitian, positive semidefinite, and trace-one constraints. It introduces Structure-Preserving Diffusion Model (SPDM), which hard-wires these constraints by mapping density matrices to a dual Euclidean space via the mirror map induced by the negative von Neumann entropy, training a diffusion model there, and mapping samples back to the primal space to obtain valid quantum states. SPDM demonstrates both unconditional generation and classifier-free guided conditional generation, including interpolation to unseen entanglement levels, on synthetic 4-qubit data across product, pairwise, and fully entangled classes. The approach yields accurate eigenvalue and entrywise distributions and preserves entanglement structure, offering a physics-informed pathway for quantum-state generation with potential applications in quantum science when experimental data are scarce or expensive.

Abstract

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage mirror diffusion and borrow the physical notion of von Neumann entropy to design a new map, for enabling strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter enabling the design of new quantum states when generated on unseen labels.

Quantum State Generation with Structure-Preserving Diffusion Model

TL;DR

This work tackles the problem of generative modeling for quantum mixed states, where density matrices must satisfy Hermitian, positive semidefinite, and trace-one constraints. It introduces Structure-Preserving Diffusion Model (SPDM), which hard-wires these constraints by mapping density matrices to a dual Euclidean space via the mirror map induced by the negative von Neumann entropy, training a diffusion model there, and mapping samples back to the primal space to obtain valid quantum states. SPDM demonstrates both unconditional generation and classifier-free guided conditional generation, including interpolation to unseen entanglement levels, on synthetic 4-qubit data across product, pairwise, and fully entangled classes. The approach yields accurate eigenvalue and entrywise distributions and preserves entanglement structure, offering a physics-informed pathway for quantum-state generation with potential applications in quantum science when experimental data are scarce or expensive.

Abstract

This article considers the generative modeling of the (mixed) states of quantum systems, and an approach based on denoising diffusion model is proposed. The key contribution is an algorithmic innovation that respects the physical nature of quantum states. More precisely, the commonly used density matrix representation of mixed-state has to be complex-valued Hermitian, positive semi-definite, and trace one. Generic diffusion models, or other generative methods, may not be able to generate data that strictly satisfy these structural constraints, even if all training data do. To develop a machine learning algorithm that has physics hard-wired in, we leverage mirror diffusion and borrow the physical notion of von Neumann entropy to design a new map, for enabling strict structure-preserving generation. Both unconditional generation and conditional generation via classifier-free guidance are experimentally demonstrated efficacious, the latter enabling the design of new quantum states when generated on unseen labels.
Paper Structure (21 sections, 1 theorem, 27 equations, 9 figures, 1 table)

This paper contains 21 sections, 1 theorem, 27 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Unconditional Generation via Diffusion Model
  4. Diffusion Guidance for Conditional Generation
  5. Density Matrix Representation of Quantum State
  6. Method
  7. Mirror Map with Real Numbers
  8. Mirror Map for Complex-Valued Density Matrices
  9. Generative Modeling in Primal Space Based on Diffusion in Dual Space
  10. Summary of Methodology
  11. Experiments
  12. Problem Setup
  13. Results
  14. Conclusion, Limitation, and Future Work
  15. Appendix
  16. ...and 6 more sections

Key Result

Proposition 3.2

$(\nabla \phi, \nabla \phi^*)$ given by eq:ourMirrorMaps is a pair of bijective mappings between the set of positive definite complex Hermitian matrices $\mathbb{C}_{n}^{+}$ and the set of complex Hermitian matrices $\mathbb{C}_n$. Moreover, $\nabla \phi^{*}$ is the inverse of $\nabla \phi$.

Figures (9)

  • Figure 1: Structure-Preserving Diffusion Model (SPDM) is the first diffusion-based method for quantum state generation. It hard codes physical knowledge into the generative models to strictly satisfy the structural conditions of quantum states (lower path; dynamics inside the constraint cone). In contrast, vanilla Diffusion model (DM) falls short of generating reliable samples due to its failure to detect the sophisticated structures of quantum states in high dimensions (upper path; dynamics ignores the constraint cone, leading to unsatisfactory generation quality).
  • Figure 2: Unconditional generation of the mixture of three classes of quantum states. Each figure shows an observable for comparison between unconditionally generated samples and its corresponding ground truth samples. (1) Leading eigenvalue versus second eigenvalue. (2) Real value of matrix entries in the primal space ($\mathbb{C}_{16}^{+}$). (3) Real value of matrix entries in the dual space ($\mathbb{C}_{16}$). (4) Distribution of entanglement negativity measured between qubit $1$ and the rest of the system.
  • Figure 3: Conditional generation with known labels. Each figure shows a comparison between conditionally generated samples of one class and its corresponding ground truth samples in the primal space $\mathbb{C}_{16}^{+}$. (1) Real value of matrix entries for product states. (2) Real value of matrix entries for pairwisely entangled states. (3) Real value of matrix entries for fully entangled states.
  • Figure 4: Conditional generation with unseen labels. Each figure represents the distribution of the entanglement negativity of samples generated with unseen labels that are convex combinations of one-hot encoding of seen labels. Entanglement negativity is measured between qubit $1$ and the rest of the system. (Left) Interpolation between product state and pairwisely entangled state. (Middle) Interpolation between pairwisely entangled states and fully entangled states. (Right) Interpolation between product states and fully entangled states.
  • Figure 5: SGM can easily recover the distribution of 1 qubit quantum state.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark A.1