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On the Trainability and Classical Simulability of Learning Matrix Product States Variationally
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Proves that MPS ansatz training with global observables suffers cost concentration and barren plateaus while local observables avoid them, and provides numerical evidence of effective subspaces suggesting classical simulability.
Problem
Trainability of variational learning of weakly entangled quantum state approximations with the matrix product state (MPS) ansatz, and whether such trainable objectives are classically simulable with few quantum resources.
Approach
Rigorous Haar-integration proofs for the MPS ansatz built from unitary 2-design subcircuits showing global-observable fidelity costs concentrate exponentially while local-observable costs are lower bounded linearly in n; plus a C-K norm to numerically detect effective subspaces in the Pauli basis.
Key results
- Global-observable MPS fidelity objective has variance bounded above by h1(sigma)/4^(n-k-1), so it concentrates exponentially (barren plateau).
- Local-observable objective variance is lower bounded by [1/(n(2^(2k+1)+4))] minus h2(sigma)/(2^n), scaling linearly in n.
- Simulations show C-K norms of the MPS ansatz with local observables grow exponentially with n, suggesting effective subspaces in the Pauli basis.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a global observable instead of local observables (ablation) | MPS ansatz for state approximation · State approximation (global observable) | objective-function variance | Hurt |
| Use local observables instead of a global observable (ablation) | MPS ansatz for state approximation · State approximation (local observables) | objective-function variance | Helped |
| Use a global observable instead of local observables (ablation) | MPS ansatz for state approximation · State approximation (global observable) | partial-derivative variance | Hurt |
| Use local observables instead of a global observable (ablation) | MPS ansatz for state approximation · State approximation (local observables) | partial-derivative variance | Helped |
| Increase subcircuit width from k = 2 to k = 3 or k = 4 (ablation) | MPS ansatz optimized with SPSA · State approximation, n = 12 | infidelity | Hurt |
| Use local observables instead of global observables (ablation) | MPS ansatz · Random HEA-generated input states | objective-function second moment | Helped |
| Use local observables instead of global observables (ablation) | MPS ansatz · Pauli-basis effective-subspace analysis | C-Pn norm | Helped |
| Move the 1-local Pauli Z component closer to the last qubit | MPS ansatz with local observable |0⟩⟨0|16 · 16-qubit simulation | Pauli probability | Helped |
Surprising findings
- Numerical C-K norm results suggest effective subspaces exist for MPS with local observables but are absent for global observables, contrary to no prior MPS-specific evidence.
- Optimization with local observables still degrades as subcircuit width k increases in the learning-curve experiments.
Limitations
- Classical simulation algorithm is not explicitly constructed; only evidence for effective subspaces is provided.
- Proofs assume subcircuits form unitary 2-designs and only treat a single MPS layer (T = n-k+1 layers, not multiple layers of C_T).
- Theorem 2 bound requires h2(sigma) small to be meaningful; general input states not covered.
Experimental setup
Metrics: variance of objective function variance of partial derivatives infidelity C-K norm second moment
Baselines: Global observable MPS training Hardware Efficient Ansatz with local observables QCNN with local observables
Builds on: Cerezo et al. 2021 barren plateaus with local vs global cost Cerezo et al. 2024 effective subspaces conjecture Ben-Dov et al. 2022 MPS state approximation Liu et al. 2022 tensor network barren plateaus
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