Barren Plateau Detection

Barren plateaus are a critical failure mode in variational quantum algorithms: gradients vanish exponentially with circuit width, making training practically impossible.

QuPrep's detect_barren_plateau() estimates this risk analytically at preprocessing time — no circuit simulation, no training required.

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Background

McClean et al. (2018) showed that for a global cost function with an expressive parameterised circuit, the gradient variance scales as:

\[\text{Var}\!\left[\frac{\partial C}{\partial \theta}\right] \leq 2^{1-n}\]

where \(n\) is the number of qubits. This means gradients halve with every additional qubit — for \(n = 12\), the variance is less than \(0.001\).

Cerezo et al. (2021) showed that local cost functions (single-qubit Pauli observables) exhibit only polynomial decay:

\[\text{Var} \approx \frac{1}{n^2}\]

This is the primary mitigation strategy.

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Quick start

import quprep as qd
from quprep.core.dataset import Dataset
import numpy as np

ds = Dataset(data=np.random.default_rng(0).uniform(0, 1, (50, 8)))
report = qd.detect_barren_plateau(qd.IQPEncoder(), ds)
print(report)

BarrenPlateauReport(iqp)
  n_qubits         : 8
  circuit_depth    : 64
  gradient_variance: 7.81e-03  (upper bound)
  risk_level       : mild
  mitigations:
    - Use a local cost function (single-qubit Pauli observables) — polynomial gradient decay instead of exponential

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Risk levels

Risk	Gradient variance	Typical qubit count
none	> 0.05	≤ 5
mild	0.005 – 0.05	6 – 8
high	0.0005 – 0.005	9 – 11
severe	≤ 0.0005	≥ 12

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Global vs local cost

r_global = qd.detect_barren_plateau(qd.IQPEncoder(), ds, cost_type="global")
r_local  = qd.detect_barren_plateau(qd.IQPEncoder(), ds, cost_type="local")

print(r_global.risk_level)   # high
print(r_local.risk_level)    # mild

Pass cost_type="local" when your training objective is a sum of single-qubit expectations (e.g. Pauli-Z on each qubit). This substantially improves the bound and is the recommended default for all circuits with \(n \geq 8\).

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Mitigation strategies

The report's mitigations list grows with risk level:

Risk	Mitigations suggested
mild	Use local cost function
high	+ Layer-wise training, identity-block initialisation
severe	+ Reduce qubit count, prefer shallower encodings

for tip in report.mitigations:
    print("-", tip)

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Sweep across encoders

encoders = {
    "angle"    : qd.AngleEncoder(),
    "iqp"      : qd.IQPEncoder(),
    "reupload" : qd.ReUploadEncoder(),
}

for name, enc in encoders.items():
    r = qd.detect_barren_plateau(enc, ds)
    print(f"{name:12s}  n_qubits={r.n_qubits}  risk={r.risk_level}")

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References

McClean J.R. et al. "Barren plateaus in quantum neural network training landscapes." Nature Communications 9, 4812 (2018). doi:10.1038/s41467-018-07090-4

Cerezo M. et al. "Cost function dependent barren plateaus in shallow parametrized quantum circuits." Nature Communications 12, 1791 (2021). doi:10.1038/s41467-021-21728-w