Speaker
Description
In the absence of error correction, noisy intermediate-scale quantum devices are typically operated using PQCs trained to minimize a loss function. Finding the circuits' optimal parameters is a hard optimization problem, where global guarantees are known only for highly structured cases of limited practical relevance, and first-order methods can fail even locally due to barren plateaus. In this work, we study the training of practical classes of PQCs, namely polynomial-depth circuits with a constant number of trainable parameters. This captures widely used PQC families, including fixed-depth QAOA, hardware-efficient ansätze with a controlled number of parameters per layer, and Fixed Parameter Count QAOA. Our main technical result is an additive fully polynomial randomized approximation scheme (FPRAS), which, for every $\epsilon > 0$, returns an $\epsilon$-approximate solution to the problem’s global optimum with high probability, and has runtime and query complexity polynomial in $1 / \epsilon$ and the number of qubits. Unlike standard VQA training, where the quantum device is queried repeatedly throughout the training, our approach has two stages: (1) an initial quantum data-acquisition phase, followed by (2) a classical global-optimization phase based on the trigonometric moment/sum-of-squares hierarchies of semidefinite programs. Under a standard flat-extension condition, which can be checked numerically, the method also supports the extraction of globally optimal circuit parameters. The FPRAS implies that the promise problem associated with the optimization of poly-depth constant-parameter PQCs is tractable. This limits the class's expressive power, namely, it cannot encode combinatorial optimization problems whose objective values are separated by an inverse-polynomial gap.
| Topic | Hybrid quantum-classical approaches for learning and optimization |
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