Variational decision diagrams for quantum-inspired machine learning applications

TL;DR

The paper introduces Variational Decision Diagrams (VDDs), a quantum-inspired framework that embeds variational parameters into decision diagrams to represent quantum states on classical hardware. By analyzing the Accordion ansatz, the authors show that VDDs can avoid barren plateaus, with gradient variance scaling sub-exponentially rather than vanishing exponentially as the system grows. They demonstrate the approach on ground-state estimation for prototypical Hamiltonians, including $H_1 = Z_1Z_2$, the transverse-field Ising model, and the XYZ Heisenberg model, using exact state-vector gradients and, where scalable, variational Monte Carlo. The results suggest VDDs can offer a compact, normalised alternative to tensor networks and neural-network quantum states for certain problem classes, with potential extensions to broader quantum machine learning tasks. Overall, VDDs provide a flexible, quantum-inspired classical tool for simulating quantum systems and exploring QML applications, while highlighting the importance of ansatz design in capturing system-specific correlations.

Abstract

Decision diagrams (DDs) have emerged as an efficient tool for simulating quantum circuits due to their capacity to exploit data redundancies in quantum states and quantum operations, enabling the efficient computation of probability amplitudes. However, their application in quantum machine learning (QML) has remained unexplored. This paper introduces variational decision diagrams (VDDs), a novel graph structure that combines the structural benefits of DDs with the adaptability of variational methods for efficiently representing quantum states. We investigate the trainability of VDDs by applying them to the ground state estimation problem for transverse-field Ising and Heisenberg Hamiltonians. Analysis of gradient variance suggests that training VDDs is possible, as no signs of vanishing gradients--also known as barren plateaus--are observed. This work provides new insights into the use of decision diagrams in QML as an alternative to design and train variational ansätze.

Figures (3)

• Figure 1: Schematic layout of the VDD accordion ansatz for 2-5 qubits. At the top of each diagram is the root node, and at the bottom is the terminal node. The VDD has variational parameters at each edge of the diagram. To obtain the amplitude probability of a given element of the canonical basis of the Hilbert space of a system of $n$ qubits (a bit string of length $n$), all that is needed is to take the path from the root node to the terminal node, where if a zero(one) is encountered in the bit string, the left(right) edge is taken; the probability amplitude is the product of all probability amplitudes lying on the edges of the specified path (see \ref{['def:qdd', 'def:vdd']} for details on how the probability amplitudes are defined and computed). The red path is highlighted in the 3 qubit case is related to an example that is explained in the main text (cf. \ref{['eq:prob_amplitude_example']}).
• Figure 2: Gradient variances, computed with \ref{['eq:variance_of_gradients']}, averaged over random values of some parameters of the accordion ansatz for the expected value of the following Hamiltonians: a) $Z_1Z_2$; b) Heisenberg with $J_x=J_y=J_z=1.0$; TFIM with $g=0.0$ (ordered phase), d) $g=1.0$ (gapless phase) and e) $g=10.0$ (disordered phase). Linear fits are also shown. $\phi_{-1}$ refers to the $\phi$ parameter of the last edge of the VDD.
• Figure 3: Percentual energy error curves as a function of the epoch number for a system with 10 qubits for the following Hamiltonians: a) $Z_1Z_2$, b) Heisenberg with $J_x=J_y=J_z=1.0$, c) TFIM with $g=0.0$ (ordered phase), d) TFIM with $g=1.0$ (gapless phase) and e) TFIM with $g=10.0$ (disordered phase). The optimised loss function is the difference between the expected value of the corresponding Hamiltonian and its ground state energy. $E_0$ is the true ground energy of each Hamiltonian.

Theorems & Definitions (2)

• Definition 1
• Definition 2