Provably efficient variational generative modeling of quantum many-body systems via quantum-probabilistic information geometry

Abstract

The dual tasks of quantum Hamiltonian learning and quantum Gibbs sampling are relevant to many important problems in physics and chemistry. In the low temperature regime, algorithms for these tasks often suffer from intractabilities, for example from poor sample-or time-complexity. With the aim of addressing such intractabilities, we introduce a generalization of quantum natural gradient descent to parameterized mixed states, as well as provide a robust first-order approximating algorithm, Quantum-Probabilistic Mirror Descent. We prove data sample efficiency for the dual tasks using tools from information geometry and quantum metrology, thus generalizing the seminal result of classical Fisher efficiency to a variational quantum algorithm for the first time. Our approaches extend previously sample-efficient techniques to allow for flexibility in model choice, including to spectrally-decomposed models like Quantum Hamiltonian-Based Models, which may circumvent intractable time complexities. Our first-order algorithm is derived using a novel quantum generalization of the classical mirror descent duality. Both results require a special choice of metric, namely, the Bogoliubov-Kubo-Mori metric. To test our proposed algorithms numerically, we compare their performance to existing baselines on the task of quantum Gibbs sampling for the transverse field Ising model. Finally, we propose an initialization strategy leveraging geometric locality for the modelling of sequences of states such as those arising from quantum-stochastic processes. We demonstrate its effectiveness empirically for both real and imaginary time evolution while defining a broader class of potential applications.

Variational Quantum Time Evolution without the Quantum Geometric Tensor

“…Note that this dual formulation can alternatively be obtained from the derivation of quantum natural gradients [22,24], which is detailed in Appendix B. For an intuitive understanding of the relationship of the infidelity and QGT, Appendix C demonstrates the effect of approximating ||δθ|| g(θ) ≈ 1 − F (θ, θ + δθ) in an illustrative example.…”

“…Furthermore, the p-VQD algorithm is not directly applicable to imaginary-time evolution. Another line of work directly focuses on the preparation of thermal states by minimizing the free energy of a variational ansatz [22]. This, approach however, does not implement general quantum time evolution.…”

Quantum information criteria for model selection in quantum state estimation

“…QHBMs can turn the quantum log-partition function into the classical one, which actually mitigates the computational hardness of the partition function. We will use QHBM in the numerical demonstration of section 4; it will be shown there that we can obtain the analytic form of terms like ∂ i log σ QHBM (θ) and ∂ i ∂ j log σ QHBM (θ), as detailed in [32,33].…”

“…We use the analytic form of the first and second derivatives of the logarithm of the density matrix, ∂ i log σ QHBM (θ) and ∂ i ∂ j log σ QHBM (θ), derived in [32,33], for these steps. We note that [32,33] proposed a method to compute the derivatives via the parameter shift rule on a quantum computer, but the derivatives are supposed to be computed on classical computers in our setting to ensure that all the estimates are based on the same measurement data of n classical snapshots. This computation of derivative is consistent with the definition of the quantum information criteria introduced in section 3.…”

“…Moreover, in terms of information geometry, only the BKM metric introduces a dually flat structure for a family of density matrices, which is the novel property that the classical Fisher information has for a family of probability distributions. As an application, the BKM metric is chosen to develop the quantum generalization of the classical mirror descent algorithm [33].…”