Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation.
Figures & tables
Figure 1: Phase diagrams in the space of mixed states. (a) For states ρ and σ in the same phase, one can transform ρ and σ mutually using shallow-and-local channel circuits E and E~ along one single path (red). For ρ′ in a different phase, any circuit E′ preparing ρ′ from σ (blue) must contain some step in the middle where local reversibility fails, denoted as the phase boundary (dashed line). (b) Trivial phase mixed state learnability. As long as a shallow-and-local channel circuits E (red) that prepares ρ from ρ0=∣0⟩⟨0∣⊗n with local reversibility, our work always produces a new generation channel W (orange) that generates ρ , without knowing anything about E . Notice that we do not exclude the possibility that there is some other preparation channel E′ (blue) containing a phase transition, but our result only relies on the existence of a transition-free path (red), hence it still works in this scenario. We remark that W may not exhibit local reversibility and can cross the phase boundary (not shown in the figure). Also, the preparation dynamics E can be time-continuous, but the dynamics of W in our construction is always time-discrete with k+1 steps.
Figure 2: Schematic of the general circuit setting. We consider a channel circuit E of depth d , where each gate acts on at most c qubits. The resulting lightcone extends over a region of radius at most s=cd .
Figure 3: Learning scheme of a 2D trivial phase mixed state (generalized from Figure 13 of Ref. [ KKR24 ] ). (a) Classical shadow by measurement for copies of ρ on a quantum computer (QC). The snapshots of classical shadow on different local regions will be repeatedly used for learning and generation. (b-h) Learning and generating ρ by using a 3 -layer circuit. (b) In layer 1, we learn and generate all local states with spatial support in purple. (c-f) Schematic of layer 2. All channels in layer 2 are local extension maps ΦBE→BC (Section 2.4 ), exemplified in (c), where we learn new regions C and discard E , leading to the state in (d). All local extension maps in layer 2 act in parallel, shown in (e), and the learned state has spatial support on (f). (g) In layer 3, we extend to the remaining hole-shaped region C by using a local recovery map ΨB→BC supported on BC (Section 2.3 ). (h) We obtain the overall state ρ . All channels act locally and are efficiently learnable from classical shadow snapshots (Section 6.2 ).
Figure 4: Schematic for local reversibility: any channel gate Eℓ,x can always be effectively canceled by some other channel gate E~ℓ,x .
Figure 5: (a) Approximate Markovianity under tripartition Λ=ABC . There exists a channel ΨB→BC such that ΨB→BC(ρAB)≈ρABC . (b) Local extendibility under pentapartition Λ=ABCDE . For B′=BE , there exists a channel ΦBE→BC such that ΦBE→BC(ρABE)≈ρABC .
Figure 6: (a) Schematic for lightcone decomposition (Fact 3 ). For any S=S1S2 , the backward lightcone has a decomposition BS=Q∘BS1 where Supp(Q)⊆S2(s)\S1 with s=cd . We recall that S(s) represents the region that extends S by a distance s . (b) Schematic for a special case lightcone decomposition (Fact 4 ). For any S⊆Λ , the overall circuit has a decomposition BS=Q∘BSˉ where Supp(Q)⊆S . (c) A trivial phase mixed state has no long-range correlation ρS1S2=ρS1⊗ρS2 if dist(S1,S2)≥2s (Fact 5 ), because their backward lightcones BS1 and BS2 have no overlap.
Figure 7: Schematic for Lemma 10 . Local reversibility still holds for any extra channel C acting on E≤ℓ−1(σ) , as long as with Supp(C)∩Supp(Eℓ,x)=∅ .
Figure 8: Schematic for local inversion (Lemma 11 ). For any S⊆λ , there exists a channel PS acting on S , such that PS(ρ)≈BSˉ(σ) . To construct PS , we first assign an ordering of indices to gates in BS\BSˉ – denoted as {G~m}m∈[N] (where N=6 in this schematic) – based on the ascending order of the layer indices ℓ , and then we act the local recovery gates {G~m}m∈[N] (the orange channel gates) based on the descending order of the layer indices ℓ .
Figure 9: Schematic for the proof of the existence of local recovery maps (Theorem 12 ). We show the approximate Markovianity by the explicit construction of the local recovery Ψ=Q1∘RC(s)\C∘PB (light gray region). The step from (a) to (b) is from the local inversion PB(ρ)≈BAC(ρ0) . The step from (b) to (c) is from the reset channel property RC(s)∘BC=RC(s) for Supp(BC)⊆C(s) . The step from (c) to (d) is from the backward lightcone decomposition E=Q1∘BA .
Figure 10: Schematic of the partition used in the proof of the existence of the local extension map (Theorem 14 ). (a) The pentapartition Λ=ABCDE with dist(A,C)≥2s and dist(B,D)≥2s . (b) The region F is defined as F=CD(s)\A (surrounded by the red boundary). It can be obtained by extending CD outwards by CD(s)\AC (gray region). (c) The partition F=F1F2 where F1=D(s)\(A∪C(s)) and F2=C(s) . The boundary between F1 and F2 is depicted by dashed lines.
Figure 11: Schematic for the proof of the existence of local extension maps (Theorem 14 ). We show the local extendibility by the explicit construction of the local extension TrDE∘Φ where Φ=Q1∘RF\CD∘PBE . Yellow regions represent the regions consistent with ρ ; white regions represent the regions being in ∣0⟩⟨0∣ ; pink regions represent the edge of backward lightcone for ACD (see Eq. ( 9 )). The step from (a) to (b) is from the local inversion PBE . The step from (b) to (c) is from the reset channel RF\CD acting on F\CD ( i.e. , the gray region in Figure 10 b). The step from (c) to (d) is from the map TrDE∘Q1 where Q1=BBC\BA acting on BC(s)\A . (e) The evolution for the cross-section of ABC (gray dashed lines in a-d), where the circuits are similar to Figure 9 .
Figure 12: Details of the covering scheme for the case k=2 , simplified from Figure 3 . (a) Layer 1: learn and generate all local states in purple, and all patches are separated by a distance at least 2s . (b) Layer 2: local extension maps, which extend to new regions C (pink) from BE (purple and red) and discard E (red). (c) Layer 3: local recovery maps, which extend to regions C (yellow) from B (purple).
Figure 13: For a trivial phase mixed state ρ , if there is a local extension map ΦBE→BC learned on the local state ρAinBCDE with high accuracy, then ΦBE→BC is also a local extension map for global state ρABCDE with high accuracy.
Appendix figures & tables1 asset
Supplementary material from the paper’s appendix.
Appendix
Figure 14: Reorganization trick that transfers from the original circuit E into a new circuit F (we only plot one layer for simplicity). All channel gates in each layer of F is separated by a distance 2r , such that we can find a circuit F~ to locally reverse F with the same depth as F .
Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth Ω(N) in the worst case. We further show that measurement-based quantum computation (MBQC) provides a natural implementation of the required shared classical randomness through suitable adaptation of the random measurement outcomes. Numerical experiments on MBQC-based generative models support the analytical results.
Arunava Majumder, Marius Krumm, Hendrik Poulsen Nautrup +1
University of Innsbruck, Department of Theoretical Physics, Technikerstraße 21a, A-6020 Innsbruck, Austria.
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.
Jun Dai, Olivier Nahman-Lévesque, Guillaume Rabusseau +2
Mila – Québec AI Institute, Montréal, QC, Canada · Département d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, QC, Canada · Institut quantique, Université de Sherbrooke, Sherbrooke, QC, Canada +3
Quantum diffusion models provide a physics-consistent route to generative learning by formulating noising and denoising directly on quantum states. However, applying such models to classical high-dimensional data is constrained by the qubit cost of state encoding and the computational burden of simulating large density operators. We propose a scalable hybrid generative pipeline that combines a classical autoencoder for dimensionality reduction with a mixed-state quantum denoising diffusion probabilistic model (MSQuDDPM) operating in the learned latent space. The autoencoder compresses data into compact latent codes that can be embedded into a small-qubit Hilbert space, after which the quantum diffusion model learns a generative distribution over latent density operators and decodes samples back to the original domain. Algorithmically, we simplify the reverse dynamics by predicting an estimate of the clean state ρ0 at timestep t and computing the one-step reverse update via an analytic backward propagation rule, rather than learning an explicit predictor for ρt−1. We demonstrate the proposed approach on MNIST image generation and discuss how mixed-state quantum diffusion can serve as a practical backbone for hybrid quantum--classical generative modeling under realistic qubit budgets.
Qipeng Qian, Keli Deng, Yuntao Qian
Program of Applied Mathematics, Department of Mathematics, University of Arizona, Tucson, USA · College of Computer Science and Technology, Zhejiang University, Hangzhou, China