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Impact of conditional modelling for a universal autoregressive quantum state

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Bortone, Massimo, Rath, Yannic, Booth, George H.·2024·Quantum
DOI 10.22331/q-2024-02-08-1245
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Abstract

We present a generalized framework to adapt universal quantum state approximators, enabling them to satisfy rigorous normalization and autoregressive properties. We also introduce filters as analogues to convolutional layers in neural networks to incorporate translationally symmetrized correlations in arbitrary quantum states. By applying this framework to the Gaussian process state, we enforce autoregressive and/or filter properties, analyzing the impact of the resulting inductive biases on variational flexibility, symmetries, and conserved quantities. In doing so we bring together different autoregressive states under a unified framework for machine learning-inspired ansätze. Our results provide insights into how the autoregressive construction influences the ability of a variational model to describe correlations in spin and fermionic lattice models, as well as ab initio electronic structure problems where the choice of representation affects accuracy. We conclude that, while enabling efficient and direct sampling, thus avoiding autocorrelation and loss of ergodicity issues in Metropolis sampling, the autoregressive construction materially constrains the expressivity of the model in many systems.

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Figure 1 Schematic representations of the correlators used in different states on a 4 × 4 lattice, showing the range of the explicit correlations in each: (a) a GPS with support dimension M = 3 is a state with three correlators each of which is parameterized on the occupation of sites over the whole system without restriction; (b) an MPS is obtained by introducing a one-dimensional ordering of the sites with direct entanglement approximated through sites in this order; (c) masking the correlators up to each site along a one-dimensional ordering and normalizing them leads to an autoregressive state, where each correlator is independently parameterized and depends only on the configuration of sites preceding it in the ordering (for clarity, only the correlators for sites i = 3 , 5 , 11 are shown); (d) if the system has translational symmetry, the same correlator can be translated across the system, reducing the number of independent parameters in the model. This can be used in conjunction with the autoregressive model, as long as the site ordering is preserved by masking out 'higher-ordered' sites in the filter as shown, to obtain an autoregressive filterbased state described in Sec. 2.3 (for clarity, only the correlators for sites i = 11 , 16 are shown). The range of these filters can be restricted if desired (not shown) to impose a further locality constraint.
Figure 1 Schematic representations of the correlators used in different states on a 4 × 4 lattice, showing the range of the explicit correlations in each: (a) a GPS with support dimension M = 3 is a state with three correlators each of which is parameterized on the occupation of sites over the whole system without restriction; (b) an MPS is obtained by introducing a one-dimensional ordering of the sites with direct entanglement approximated through sites in this order; (c) masking the correlators up to each site along a one-dimensional ordering and normalizing them leads to an autoregressive state, where each correlator is independently parameterized and depends only on the configuration of sites preceding it in the ordering (for clarity, only the correlators for sites i = 3 , 5 , 11 are shown); (d) if the system has translational symmetry, the same correlator can be translated across the system, reducing the number of independent parameters in the model. This can be used in conjunction with the autoregressive model, as long as the site ordering is preserved by masking out 'higher-ordered' sites in the filter as shown, to obtain an autoregressive filterbased state described in Sec. 2.3 (for clarity, only the correlators for sites i = 11 , 16 are shown). The range of these filters can be restricted if desired (not shown) to impose a further locality constraint.CC-BY · From the PDF
Figure 2 Relative energy error of different GPS models for the 6 × 6 AFH with PBC as a function of the number of parameters. The exact energy is obtained from Ref. [52]. Models with convolutional filters are shown with dashed lines and are more parameter efficient, while models with independent site conditionals are shown with solid lines. We find that explicit normalization of the autoregressive state is more deleterious to its variational freedom than the other AR requirement of masking conditionals based on a 1D ordering of the lattice sites, for both filter and non-filter models. The explicitly translationally-invariant filter-GPS without masking or normalization constraints are found to be the variationally best ansatz for a given complexity in this system. Inset: Relative energy error of the ARfilter-GPS model for the larger 10 × 10 lattice compare to the stochastic series expansion results of Ref. [51].
Figure 2 Relative energy error of different GPS models for the 6 × 6 AFH with PBC as a function of the number of parameters. The exact energy is obtained from Ref. [52]. Models with convolutional filters are shown with dashed lines and are more parameter efficient, while models with independent site conditionals are shown with solid lines. We find that explicit normalization of the autoregressive state is more deleterious to its variational freedom than the other AR requirement of masking conditionals based on a 1D ordering of the lattice sites, for both filter and non-filter models. The explicitly translationally-invariant filter-GPS without masking or normalization constraints are found to be the variationally best ansatz for a given complexity in this system. Inset: Relative energy error of the ARfilter-GPS model for the larger 10 × 10 lattice compare to the stochastic series expansion results of Ref. [51].CC-BY · From the PDF
Figure 3 Relative energy error of the autoregressive (red) and masked (violet) filter-based GPS ans¨ atze without symmetrization (dashed) and with normalizationpreserving C 4 v and Z 2 symmetrization (solid) as a function of the support dimension ( M ) on the 6 × 6 AFH with PBC (conservation of total zero magnetization is included in all models). Included is also the relative energy error of a projectively-symmetrized (non-autoregressive) CNN-based NQS [13], which provides similar accuracy and compactness to the projectively-symmetrized GPS [44] model due to the improved symmetrization and lack of masking.
Figure 3 Relative energy error of the autoregressive (red) and masked (violet) filter-based GPS ans¨ atze without symmetrization (dashed) and with normalizationpreserving C 4 v and Z 2 symmetrization (solid) as a function of the support dimension ( M ) on the 6 × 6 AFH with PBC (conservation of total zero magnetization is included in all models). Included is also the relative energy error of a projectively-symmetrized (non-autoregressive) CNN-based NQS [13], which provides similar accuracy and compactness to the projectively-symmetrized GPS [44] model due to the improved symmetrization and lack of masking.CC-BY · From the PDF
Figure 4 Relative energy error of the autoregressive filter-based GPS model with support dimension M = 64 as a function of the interaction strength U/t in a 1D Hubbard system with N = 32 sites and different boundary conditions. Solid lines represent models optimized with Open Boundary Conditions (OBC), while dashed lines are for those obtained with Anti-Periodic Boundary Conditions (APBC). Exact reference energies were obtained from an MPS with bond dimension M = 2500 optimized by DMRG [73].
Figure 4 Relative energy error of the autoregressive filter-based GPS model with support dimension M = 64 as a function of the interaction strength U/t in a 1D Hubbard system with N = 32 sites and different boundary conditions. Solid lines represent models optimized with Open Boundary Conditions (OBC), while dashed lines are for those obtained with Anti-Periodic Boundary Conditions (APBC). Exact reference energies were obtained from an MPS with bond dimension M = 2500 optimized by DMRG [73].CC-BY · From the PDF
Figure 5 Relative energy error of an M = 16 autoregressive GPS on a chain of 16 (top) and a 4 × 4 lattice (bottom) of hydrogen atoms at different interatomic distances, using a canonical (blue circles) and a local (green diamonds) basis representation, in an underlying STO6G basis. Solid and dashed lines represent states optimized with real and complex parameters respectively. Exact energies are obtained via full configuration interaction algorithm from PySCF [59, 60].
Figure 5 Relative energy error of an M = 16 autoregressive GPS on a chain of 16 (top) and a 4 × 4 lattice (bottom) of hydrogen atoms at different interatomic distances, using a canonical (blue circles) and a local (green diamonds) basis representation, in an underlying STO6G basis. Solid and dashed lines represent states optimized with real and complex parameters respectively. Exact energies are obtained via full configuration interaction algorithm from PySCF [59, 60].CC-BY · From the PDF

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Metadata

Authors
Bortone, M., Rath, Y., & Booth, G. H.
Affiliations
King's College London (GB)
ORCID
Venue
Quantum
Publisher
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Published
2024-02-08
Volume
8
Pages
1245
ISSN
2521-327X (ISSN-L)
DOI
10.22331/q-2024-02-08-1245
OpenAlex
W4391653418
Cited by
12
References
7
Citations by year
2026: 4 · 2025: 8
FWCI
1.30
Type
journal article
Language
en
Open access
Licence
Locations
  • Quantum (publishedVersion, cc-by)
  • arXiv (Cornell University) (submittedVersion, cc-by)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
  • Research Portal (King's College London) (submittedVersion, cc-by)
  • arXiv (Cornell University) (cc-by)
Funders
  • Air Force Office of Scientific Research (FA8655-22-1-7011)
  • European Research Council (759063)
  • Engineering and Physical Science Research Council (EP/P020194/1)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
normalization constraint, inductive bias, first integrals, lattice spin models, ab initio electronic structure, Metropolis sampling, ergodicity

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Bortone, M., Rath, Y., & Booth, G. H. (2024). Impact of conditional modelling for a universal autoregressive quantum state. Quantum. https://doi.org/10.22331/q-2024-02-08-1245