Quantum Flow Matching

TL;DR

The Quantum Flow Matching (QFM) is introduced, a fully quantum-circuit realization that offers efficient interpolation between two density matrices, and can be realized on quantum computers without the need for costly circuit redesigns.

Abstract

The flow matching has rapidly become a dominant paradigm in classical generative modeling, offering an efficient way to interpolate between two complex distributions. We extend this idea to the quantum realm and introduce the Quantum Flow Matching (QFM), a quantum-circuit realization that offers efficient interpolation between two density matrices. QFM offers systematic preparation of density matrices and generation of samples for accurately estimating observables, and can be realized on quantum computers without the need for costly circuit redesigns. We validate its versatility on a set of applications: (i) generating target states with prescribed magnetization and entanglement entropy, (ii) estimating nonequilibrium free-energy differences to test the quantum Jarzynski equality, and (iii) expediting the study on superdiffusion. These results position QFM as a unifying and promising framework for generative modeling across quantum systems.

Figures (4)

• Figure 1: Schematic of the quantum flow matching. (a)Inspired by classical flow matching Flow_matching_GM, QFM learns the evolution between two density matrices. Unlike QuDDPM QuDDPM, it does not require starting from a Haar-random ensemble, allowing wider applications. (b) QFM alternates partly measured and unitary circuits to ensure convergence and stability. (c) A single circuit emulates density-matrix dynamics, reducing circuit adjustments; here $i$ labels the time steps and $M$ denotes the ensemble size. (d) Applications include free-energy estimation and superdiffusive scaling (main text), as well as topological-state evolution, entanglement growth, and magnetic phase transitions (Supplemental Material supplementary).
• Figure 2: QFM efficiently estimates nonequilibrium free energy changes in a $6$-qubit TFIM. (a) Free energy is estimated via sampling pseudo-thermal states in METTS. Left: The quantum imaginary time evolution (QITE) QITE_1QITE_2Qu_Jar_exp_1 requires repeated circuit adjustments for each state. Right: QFM generates the ensemble with a fixed circuit after training. (b) For $\beta=1$, the coefficient of variation under time steps $\Delta t=1$ (blue) and $0.1$ (red) shows that QFM converges with $\sim400$ circuit updates (orange), compared with $\sim1000$ for QITE (purple) Qu_Jar_exp_1, with a larger gap for shorter $\Delta t$. (c) The averaged work agrees with the free-energy change versus inverse temperature $\beta$. (d) The QFM circuit depth scales nearly linearly with the qubit number.
• Figure 3: QFM facilitates the estimation of superdiffusive scaling. (a) The conventional method kumaran2025quantum requires adjusting the quantum circuits for a Heisenberg model under various 2D-interaction strengths (black box). (b) QFM evolves the state ensemble with a fixed circuit and adjusts the 2D-interaction strength by measurements on ancilla qubits (gray node in (a)). (c) The diffusive behaviors by QFM for 2D-interaction strengths $J_{\bot}/J\in\{0,1,2,3\}$ in three interaction types, matching the conventional method kumaran2025quantum. Increasing $J_{\bot}/J$ accentuates both diffusive ($\vec{\lambda} = (0,0,1)$) and ballistic ($\vec{\lambda} = (1,0,0)$) breakdown. In the resilient case ($\vec{\lambda} = (1,1,1)$), stronger 2D interactions enlarge the variance of the superdiffusive scaling.
• Figure 5: The training procedure of QFM in step $\tau$. QFM first adopts an $n$-qubit unitary operator for each layer; if the loss for $U_{n}$ does not converge below the threshold $\mathbb{T}_{\tau}$ during training, $n_a$ ancilla qubits are introduced and the training is repeated again.