Barren plateaus analyzed for quantum linear combinations of unitaries
Original title: Barren Plateaus in Quantum Linear Combinations: Interference and Lie-Algebraic Control
A single-pass LCU keeps a gradient-variance lower bound, while interleaved Haar sampling can concentrate exponentially.
The paper separates a single linear combination of unitaries from circuits that repeatedly interleave branch operations and ancilla mixing. For a single-pass LCU, independent zero-mean branches cannot erase a differentiated branch’s gradient variance, and the bound does not require Haar-random unitaries. Exact loss and gradient formulas include interference terms, one of which depends on relative coefficient phases. Matchgate examples show polynomial single-pass variances and exponential concentration under joint Haar-random sampling.
- The variance lower bound is the branch variance times the fourth power of its coefficient magnitude.
- The bound holds at any gate position without Haar-random circuit unitaries.
- A four-qubit matchgate example shows a phase-dependent interference term can be nonzero.
- The paper also gives coefficient-selection rules and bounds for postselection and measurement error.
From the original article
1 min read at arxiv.orgA coherent sum of quantum circuits introduces interference that can alter the training signal. We distinguish a single linear combination of unitaries (LCU) from a circuit that repeatedly alternates branch operations and ancilla mixing.
By Amir Koutahi · arxiv.org
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