Quantum Machine Learning in Multi-Qubit Phase-Space Part I: Foundations

TL;DR

This work introduces a phase-space formulation for finite-dimensional quantum systems, mapping qubit operators to smooth functions on the product space $(S^2)^N$ via the Stratonovich–Weyl correspondence. Dynamics are encoded with sine and cosine brackets, yielding unified, differentiable bracket flows for unitary, dissipative, and imaginary-time evolution directly on phase-space functions, while Stinespring dilation guarantees kernel factorisation for open-system dynamics. By developing a product-SW framework, the authors define a physically meaningful, scalable phase-space subspace with a well-defined star-product, phase-space rank, and MGFs to compute observables and correlations, enabling gradient-based learning and Monte Carlo sampling in place of exponential Hilbert-space scaling. The proposed approach reframes the curse of dimensionality as harmonic content on $(S^2)^N$, offering a natural bridge to neural approximation and variational methods for quantum state and dynamics modelling, with applications to tomography, Hamiltonian learning, and open-system control. This foundational work lays the groundwork for neural-phase-space architectures and differentiable learning in quantum dynamics, representing a promising route for scalable QML on finite-dimensional quantum systems.

Abstract

Quantum machine learning (QML) seeks to exploit the intrinsic properties of quantum mechanical systems, including superposition, coherence, and quantum entanglement for classical data processing. However, due to the exponential growth of the Hilbert space, QML faces practical limits in classical simulations with the state-vector representation of quantum system. On the other hand, phase-space methods offer an alternative by encoding quantum states as quasi-probability functions. Building on prior work in qubit phase-space and the Stratonovich-Weyl (SW) correspondence, we construct a closed, composable dynamical formalism for one- and many-qubit systems in phase-space. This formalism replaces the operator algebra of the Pauli group with function dynamics on symplectic manifolds, and recasts the curse of dimensionality in terms of harmonic support on a domain that scales linearly with the number of qubits. It opens a new route for QML based on variational modelling over phase-space.

Figures (8)

• Figure 1: $Q$ representations of the eigenstates of the Pauli operator $\hat{\sigma}_x$ (panels (a), (b)), $\hat{\sigma}_y$ (panels (c), (d)), and $\hat{\sigma}_z$ (panels (e), (f)).
• Figure 2: Diffusion maps between quasi-probability representations for qubit systems.
• Figure 3: $Q$ representations of thermal states of the Hamiltonian $\hat{H} = \hat{\sigma}_x$, for different values of the inverse temperature $\beta$.
• Figure 4: $Q$ representation of the time evolution of the ${| + \rangle}$ state under the Hamiltonian $\hat{H}=\hat{\sigma}_z$, for total time $T=2\pi$. Panels (a) -- (l) represent the $Q$ function of the state at times $t \in \{\frac{k \pi}{11}\}_{k = 0}^{11}$ respectively. Observe that the $Q$ function appears to translate in the $\varphi$ direction, corresponding to a rotation around the $z$-axis.
• Figure 5: $Q$ representation of the time evolution of the ${| {+}{+} \rangle}$ state under the Hamiltonian $\hat{H}=\hat{\sigma}_z\otimes\hat{\sigma}_z$, for total time $T=\pi$, for (a) the $\theta_1=\theta_2=\frac{\pi}{2}$ slice and (b) marginalizing over the second qubit. Panels (a) -- (l) in both of the plots represent the $Q$ function of the state at times $t \in \{\frac{k \pi}{12}\}_{k = 0}^{12}$ respectively. Non-separability implies entanglement, which means that the partial trace is mixed. This is visible in (b) (lower), where we have visualised only the single-qubit marginal. Since the Hamiltonian is entangling, it will change the degree to which this qubit is mixed. We see this in the snap-shots since the panels of (b) show how the second qubit oscillates between being pure and maximally mixed. Specifically, at $t=\pi/2$ (the first plot in the second row of (b)), we see a snapshot of the maximally mixed state (i.e. uniform density), and at the end-points we recognise the heat map of a Pauli eigenstate, which is pure (see Fig. \ref{['fig:q_pauli_eigens']}).

Theorems & Definitions (44)

• Proposition 1: Pauli expectation Q function expansion for single qubit
• Proposition 2: Pauli expectation P function expansion for single spin
• Example 1: Pauli Eigenstates
• Remark : $(S^2)^N \ncong S^{2N}$
• Proposition 3: Many-body Pauli expectation expansion for Q and P functions
• Example 2: Sampling outcomes via expectation values
• Proposition 4: Partial tracing as marginalization
• Example 3: Bell States
• Remark : Q function for product state
• Proposition 5: Separability and $Q$-function factorization