Quantum Flow Matching

The emergence of powerful deep generative models based on diffusion and flow matching has enabled the learning and modeling of complex distributions. Learning quantum distributions, however, remains challenging due to the inherent difficulty of accurately modeling the meaningful physical properties of quantum states. We propose Quantum Flow Matching (QFM), a novel generative model designed to learn quantum distribution by utilizing spin Wigner function and flow matching. By converting density matrix into the spin Wigner function and leveraging functional flow matching to learn distributions in function space, QFM enables accurate and effective learning of multi-qubit quantum distributions. We demonstrate the effectiveness of our method by evaluating physical quantities such as trace, purity, and entanglement entropy of the generated quantum states, accurately capturing the underlying physics of the given quantum distributions.

Quantum pure-state ensembles live on complex projective space, making flat Euclidean generative modeling geometrically mismatched. We introduce Intrinsic Flow Matching (IFM), a deterministic transport framework on $\mathbb{CP}^{d-1}$ that learns tangent velocity fields using Pancharatnam phase-aligned conditional paths. IFM replaces local score teachers and reverse-time stochastic sampling with manifold probability flow, while horizontal parameterization removes redundant ambient directions. We show that the IFM objective recovers the induced marginal transport field, represents deterministic projective ensemble flows, and yields endpoint and stability guarantees. Empirically, IFM often improves over ambient Euclidean flow matching across higher-qubit, multimodal, spin-coherent, physics-inspired, and amplitude-encoded MNIST image-vector benchmarks, with strongest gains on high-dimensional and coherence-sensitive tasks but not uniformly across every metric.

Extracting quantum information from a quantum state is a fundamental task of quantum computation, often requiring the estimation of many non-commuting observables under a finite measurement budget. For both near-term and early fault-tolerant settings, the measurement protocol must balance statistical efficiency against implementation resources such as circuit depth, connectivity, and entangling-gate count. Many existing strategies focus on two extremes: hardware-friendly product measurements with high sampling cost, and fully commuting measurements with deep circuits. Here we recast resource-constrained measurement design as a generative learning problem. We introduce FlowMeas, which uses a generative flow network to directly sample finite ensembles of shallow Clifford measurement circuits subject to a prescribed shot budget and hardware constraints. At zero entangling depth, FlowMeas learns qubit-wise commuting measurement schedules and already matches or improves leading product-measurement methods on nearly all molecular benchmarks. Allowing one or two entangling gate layers yields further reductions in energy estimation error of up to $27\%$ relative to the strongest state-independent product-measurement baseline. The learned policy can also be reused across related Hamiltonians, substantially accelerating retraining along a molecular potential-energy surface. We further obtain results for molecular Hamiltonians with up to 20 qubits and apply the framework to a compactly encoded 54-qubit interacting fermionic model, extending the demonstrated scale beyond prior molecular benchmarks. These results establish generative learning as a flexible and unified framework for quantum measurement design under practical resource constraints.