Learning and Generating Mixed States Prepared by Shallow Channel Circuits

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 (13)

• 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 $\rho'$ in a different phase, any circuit $\mathcal{E}'$ preparing $\rho'$ 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 $\mathcal{E}'$ (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: Learning scheme of a 2D trivial phase mixed state (generalized from Figure 13 of Ref. kim_2024_learning). (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 \ref{['sec:TO-LE']}), 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 \ref{['sec:TO-AM']}). (h) We obtain the overall state $\rho$. All channels act locally and are efficiently learnable from classical shadow snapshots (Section \ref{['sec:learning_locally']}).
• Figure 3: 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 4: (a) Approximate Markovianity under tripartition $\Lambda = A B C$. There exists a channel $\Psi_{B \rightarrow B C}$ such that $\Psi_{B \rightarrow B C} (\rho_{A B}) \approx \rho_{A B C}$. (b) Local extendibility under pentapartition $\Lambda = A B C D E$. For $B' = B E$, there exists a channel $\Phi_{B E \rightarrow B C}$ such that $\Phi_{B E \rightarrow B C} (\rho_{A B E}) \approx \rho_{A B C}$.
• Figure 5: (a) Schematic for lightcone decomposition (Fact \ref{['fac:LCD-1']}). 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 $s = c d$. We recall that $S(s)$ represents the region that extends $S$ by a distance $s$. (b) Schematic for a special case lightcone decomposition (Fact \ref{['fac:LCD-2']}). For any $S \subseteq \Lambda$, the overall circuit has a decomposition $\mathcal{B}_S = \mathcal{Q} \circ \mathcal{B}_{\bar{S}}$ where $\mathrm{Supp}(\mathcal{Q}) \subseteq S$.

Theorems & Definitions (44)

• Theorem 1: Main Theorem, informal
• Definition 2: Backward lightcone
• Definition 5: Approximate Markovianity
• Definition 6: Local extendibility
• Definition 7: Local reversibility
• Definition 8: Mixed state phases via local reversibility
• Lemma 9: Relaxed local reversibility
• proof
• Lemma 10: Local inversion
• proof