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On the Relation between Trainability and Dequantization of Variational Quantum Learning Models
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Formalizes trainability and dequantization for quantum machine learning and proposes "very variational" PQC families that are gradient-trainable yet non-dequantizable, resolving an open question on compatibility.
Problem
Whether variational quantum machine learning models can be simultaneously trainable and non-dequantizable, given that existing natural variational models (e.g., hardware efficient ansatz) appear to be either trainable or non-dequantizable but not both.
Approach
Gives task-dependent formal definitions of trainability, barren plateaus in ML, and dequantization; defines a spectrum of "variationalness" via five structural properties of HEA-like circuits; and builds a recipe embedding a classically hard BQP function (via a DLP-based kernel) as a repeated layer of a deep variational circuit, with gradient-trainable rotation parameters.
Key results
- Shows that a deep HEA-style model with a DLP kernel component has barren plateaus under uniform parameters yet is trainable for cryptographically hard tasks, so absence of BPs is neither necessary nor sufficient for trainability in general.
- Corollary: there exist 'very variational' QML models that are gradient-based trainable and non-dequantizable with up to sub-exponentially many layers in the number of qubits.
- Argues that absence of barren plateaus is necessary (but not sufficient) for gradient-based trainability.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the DLP-kernel-based learning model | DLP kernel · DLP-based learning task | trainability and dequantizability | Mixed |
| Use a vari veryational PQC with a hard quantum function and gradient-trained rotation layers | vari veryational QML model · Supervised learning task defined by Q(x) | trainability and dequantizability | Mixed |
| Use a deep HEA with a reduction to kernel methods for training | Deep HEA · DLP learning task | trainability and dequantizability | Mixed |
| Use a shallow HEA with a local observable | HEA-based QML model · Shallow-circuit learning tasks | dequantizability | Helped |
Surprising findings
- Absence of barren plateaus is neither necessary nor sufficient for trainability in general.
- Classical simulability is neither necessary nor sufficient for dequantization in general, though it is sufficient when the training algorithm is classically efficient.
Limitations
- The constructed models are contrived, with PQCs split into two disconnected parts and the hard function planted in the architecture.
- No experiments; results are purely theoretical and rely on cryptographic hardness assumptions.
- The very variational property list is provisional and the definition of variationalness is not fully well-defined.
Experimental setup
Baselines: DLP kernel Hardware Efficient Ansatz
Builds on: Hardware Efficient Ansatz barren plateaus discrete logarithm problem kernel (Liu et al. 2021) vanishing similarity
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