Learning and Generating Mixed States Prepared by Shallow Channel Circuits

Fangjun Hu; Christian Kokail; Milan Kornjača; Pedro L. S. Lopes; Weiyuan Gong; Sheng-Tao Wang; Xun Gao; Stefan Ostermann

quant-phApr 1, 2026

Learning and Generating Mixed States Prepared by Shallow Channel Circuits

Authors: Fangjun Hu, Christian Kokail, Milan Kornjača, Pedro L. S. Lopes, Weiyuan Gong, Sheng-Tao Wang, Xun Gao, Stefan Ostermann

Organizations: QuEra Computing Inc. · School of Engineering and Applied Sciences, Harvard University · JILA and Department of Physics, CU Boulder

Abstract

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 $\rho$ and $\sigma$ in the same phase, one can transform $\rho$ and $\sigma$ mutually using shallow-and-local channel circuits $\mathcal{E}$ and $\tilde{\mathcal{E}}$ along one single path (red). For $\rho^{\prime}$ in a different phase, any circuit $\mathcal{E}^{\prime}$ preparing $\rho^{\prime}$ from $\sigma$ (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 $\mathcal{E}$ (red) that prepares $\rho$ from $\rho_{0}=\ket{0}\!\bra{0}^{\otimes n}$ with local reversibility, our work always produces a new generation channel $\mathcal{W}$ (orange) that generates $\rho$ , without knowing anything about $\mathcal{E}$ . Notice that we do not exclude the possibility that there is some other preparation channel $\mathcal{E}^{\prime}$ (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 $\mathcal{W}$ may not exhibit local reversibility and can cross the phase boundary (not shown in the figure). Also, the preparation dynamics $\mathcal{E}$ can be time-continuous, but the dynamics of $\mathcal{W}$ in our construction is always time-discrete with $k+1$ steps.

Figure 2: Schematic of the general circuit setting. We consider a channel circuit $\mathcal{E}$ 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 $\rho$ 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 $\rho$ 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 $\Phi_{BE\to BC}$ (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 $\Psi_{B\to BC}$ supported on $BC$ (Section 2.3 ). (h) We obtain the overall state $\rho$ . All channels act locally and are efficiently learnable from classical shadow snapshots (Section 6.2 ).

Figure 4: Schematic for local reversibility: any channel gate $\mathcal{E}_{\ell,x}$ can always be effectively canceled by some other channel gate $\tilde{\mathcal{E}}_{\ell,x}$ .

Figure 5: (a) Approximate Markovianity under tripartition $\Lambda=ABC$ . There exists a channel $\Psi_{B\rightarrow BC}$ such that $\Psi_{B\rightarrow BC}(\rho_{AB})\approx\rho_{ABC}$ . (b) Local extendibility under pentapartition $\Lambda=ABCDE$ . For $B^{\prime}=BE$ , there exists a channel $\Phi_{BE\rightarrow BC}$ such that $\Phi_{BE\rightarrow BC}(\rho_{ABE})\approx\rho_{ABC}$ .

Figure 6: (a) Schematic for lightcone decomposition (Fact 3 ). For any $S=S_{1}S_{2}$ , the backward lightcone has a decomposition $\mathcal{B}_{S}=\mathcal{Q}\circ\mathcal{B}_{S_{1}}$ where $\mathrm{Supp}(\mathcal{Q})\subseteq S_{2}(s)\backslash S_{1}$ 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 $S\subseteq\Lambda$ , the overall circuit has a decomposition $\mathcal{B}_{S}=\mathcal{Q}\circ\mathcal{B}_{\bar{S}}$ where $\mathrm{Supp}(\mathcal{Q})\subseteq S$ . (c) A trivial phase mixed state has no long-range correlation $\rho_{S_{1}S_{2}}=\rho_{S_{1}}\otimes\rho_{S_{2}}$ if $\mathrm{dist}(S_{1},S_{2})\geq 2s$ (Fact 5 ), because their backward lightcones $\mathcal{B}_{S_{1}}$ and $\mathcal{B}_{S_{2}}$ have no overlap.

Figure 7: Schematic for Lemma 10 . Local reversibility still holds for any extra channel $\mathcal{C}$ acting on $\mathcal{E}_{\leq\ell-1}(\sigma)$ , as long as with $\mathrm{Supp}(\mathcal{C})\cap\mathrm{Supp}(\mathcal{E}_{\ell,x})=\varnothing$ .

Figure 8: Schematic for local inversion (Lemma 11 ). For any $S\subseteq\lambda$ , there exists a channel $\mathcal{P}_{S}$ acting on $S$ , such that $\mathcal{P}_{S}(\rho)\approx\mathcal{B}_{\bar{S}}(\sigma)$ . To construct $\mathcal{P}_{S}$ , we first assign an ordering of indices to gates in $\mathcal{B}_{S}\backslash\mathcal{B}_{\bar{S}}$ – denoted as $\{\tilde{\mathcal{G}}_{m}\}_{m\in[N]}$ (where $N=6$ in this schematic) – based on the ascending order of the layer indices $\ell$ , and then we act the local recovery gates $\{\tilde{\mathcal{G}}_{m}\}_{m\in[N]}$ (the orange channel gates) based on the descending order of the layer indices $\ell$ .

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 $\Psi=\mathcal{Q}_{1}\circ\mathcal{R}_{C(s)\backslash C}\circ\mathcal{P}_{B}$ (light gray region). The step from (a) to (b) is from the local inversion $\mathcal{P}_{B}(\rho)\approx\mathcal{B}_{AC}(\rho_{0})$ . The step from (b) to (c) is from the reset channel property $\mathcal{R}_{C(s)}\circ\mathcal{B}_{C}=\mathcal{R}_{C(s)}$ for $\mathrm{Supp}(\mathcal{B}_{C})\subseteq C(s)$ . The step from (c) to (d) is from the backward lightcone decomposition $\mathcal{E}=\mathcal{Q}_{1}\circ\mathcal{B}_{A}$ .

Figure 10: Schematic of the partition used in the proof of the existence of the local extension map (Theorem 14 ). (a) The pentapartition $\Lambda=ABCDE$ with $\mathrm{dist}(A,C)\geq 2s$ and $\mathrm{dist}(B,D)\geq 2s$ . (b) The region $F$ is defined as $F=CD(s)\backslash A$ (surrounded by the red boundary). It can be obtained by extending $CD$ outwards by $CD(s)\backslash AC$ (gray region). (c) The partition $F=F_{1}F_{2}$ where $F_{1}=D(s)\backslash(A\cup C(s))$ and $F_{2}=C(s)$ . The boundary between $F_{1}$ and $F_{2}$ 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 $\mathrm{Tr}_{DE}\circ\Phi$ where $\Phi=\mathcal{Q}_{1}\circ\mathcal{R}_{F\backslash CD}\circ\mathcal{P}_{BE}$ . Yellow regions represent the regions consistent with $\rho$ ; white regions represent the regions being in $\ket{0}\!\bra{0}$ ; pink regions represent the edge of backward lightcone for $ACD$ (see Eq. ( 9 )). The step from (a) to (b) is from the local inversion $\mathcal{P}_{BE}$ . The step from (b) to (c) is from the reset channel $\mathcal{R}_{F\backslash CD}$ acting on $F\backslash CD$ ( i.e. , the gray region in Figure 10 b). The step from (c) to (d) is from the map $\mathrm{Tr}_{DE}\circ\mathcal{Q}_{1}$ where $\mathcal{Q}_{1}=\mathcal{B}_{BC}\backslash\mathcal{B}_{A}$ acting on $BC(s)\backslash A$ . (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 $\rho$ , if there is a local extension map $\Phi_{BE\to BC}$ learned on the local state $\rho_{A_{\mathrm{in}}BCDE}$ with high accuracy, then $\Phi_{BE\to BC}$ is also a local extension map for global state $\rho_{ABCDE}$ 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 $\mathcal{E}$ into a new circuit $\mathcal{F}$ (we only plot one layer for simplicity). All channel gates in each layer of $\mathcal{F}$ is separated by a distance $2r$ , such that we can find a circuit $\tilde{\mathcal{F}}$ to locally reverse $\mathcal{F}$ with the same depth as $\mathcal{F}$ .

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Representational separation between unitary and channel quantum generative models via shared classical randomness at shallow depth

Arunava Majumder, Marius Krumm, Hendrik Poulsen Nautrup, Hans J. Briegel

University of Innsbruck, Department of Theoretical Physics, Technikerstraße 21a, A-6020 Innsbruck, Austria.

Quantum Generative Modeling Quantum Circuit Expressibility Quantum Machine Learning Generative Modeling

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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.

Abstract

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