---
title: Noise-Induced Barren Plateaus
url: https://www.emergentmind.com/topics/noise-induced-barren-plateaus
type: topic
---

# Noise-Induced Barren Plateaus

Noise-induced barren plateaus (NIBPs) represent a deterministically induced regime of exponentially vanishing cost-function gradients in the training landscapes of variational quantum algorithms (VQAs) due to the presence of noise. This phenomenon, in contrast to the "intrinsic" or "noise-free" barren plateaus associated with random initialization or excessive circuit expressibility, stems directly from the contractive action of noise channels—particularly local Pauli noise—acting repeatedly within layered quantum circuits. NIBPs pose a critical limitation to the scalability and effectiveness of VQAs on noisy intermediate-scale quantum (NISQ) hardware, as they render gradient-based optimization infeasible when the circuit depth scales with the number of qubits.

## 1. Definition and Distinction from Intrinsic Barren Plateaus

A barren plateau is a region in cost-function parameter space where gradients become exponentially small with increasing problem size, such that
\[
|\nabla_\mu C(\theta)|,\, \operatorname{Var}_\theta[\nabla_\mu C(\theta)] \in O(2^{-\alpha n})
\]
for some observable $C$ and system size $n$ [2007.14384, 2310.08405, 2402.08721]. This vanishing gradient precludes efficient optimization, since detecting meaningful parameter updates would require infeasibly many measurements.

Crucially, NIBPs are *deterministic* and *hardware-induced*: even if parameter initialization avoids flat regions in the noise-free cost landscape, any local noise channel $\mathcal{N}$ with contraction parameter $|q|<1$—such as local Pauli or amplitude damping noise—imposes exponential suppression of gradients through repeated application. This is fundamentally distinct from noise-free barren plateaus, which are probabilistic phenomena linked to random circuit initialization or global cost observables [2007.14384, 2205.05786].

## 2. Mechanism: Contractive Action of Layered Noise Channels

The genesis of NIBPs is the repeated contraction of operator norms in the Pauli decomposition under local noise. For a layered architecture with $L$ blocks of unitaries interleaved with noise, the output state is evolved as
\[
\tilde{\rho} = \mathcal{N} \circ U_L \circ \cdots \circ \mathcal{N} \circ U_1 \circ \mathcal{N}(\rho).
\]
Every noise channel $\mathcal{N}$ acts to shrink the off-diagonal (Pauli) components, and successive layers exponentially concentrate any state towards the maximally mixed state (or another fixed point for non-unital noise) [2007.14384, 2310.08405, 2402.08721]. The cost function $C = \operatorname{Tr}[O \tilde{\rho}]$ thereby becomes nearly independent of the parameters, and all gradients vanish exponentially in $L$ and $n$:
\[
|\partial_{l m} C| \leq \mathcal{O}(n^{1/2})\, q^{c L}\ \ \text{with}\ c = \frac{1}{2\ln 2}.
\]
This is a universal contraction, applying to a broad class of CPTP maps beyond local Pauli noise, including amplitude damping and generalized Hilbert-Schmidt contractive maps [2310.08405, 2402.08721].

## 3. Mathematical Analysis and Gradient Bounds

The formal treatment leverages Pauli expansion and sandwiched 2-Rényi relative entropy contraction [2007.14384]. The central bound on the magnitude of the cost function derivatives is formulated as:
\[
|\partial_{l m} \tilde{C}| \leq \sqrt{8\ln 2\, N_O \,\|\omega\|_\infty\, \|\eta_{l m}\|_1\, n^{1/2}}\, q^{c L + 1},
\]
where $q$ is the contraction parameter and $L$ the number of layers. For ansatzes with depth $L \in \Omega(n)$, this leads to an overall exponential suppression in system size, $F(n) \in O(2^{-\alpha n})$.

In the presence of measurement noise modeled as local bit-flip channels, the gradient suppression is even stronger for observables with high Pauli weight $w$, as the landscape is further multiplied by $q_M^w$ [2007.14384].

## 4. Generality Across Ansatz Classes and Observable Types

The NIBP bound applies to hardware-efficient and physically motivated ansatzes (including QAOA and UCC used in VQE), provided the circuit depth scales with $n$ [2007.14384]. For global observables (high Pauli-weight), such as those encoding highly entangled quantities, the exponential gradient suppression in the presence of noise is particularly acute [2007.14384, 2402.08721, 2310.08405].

Correlated parameterizations (where several variational parameters are set equal) do not avoid NIBPs, as the multiplicative contraction effect persists regardless of parameter redundancy [2007.14384]. For non-unital noise, the cost may not concentrate at a single value but within an interval (a noise-induced limit set, NILS), but trainability is still impaired as the range is noise-parameter-dependent and gradients can remain small [2402.08721].

## 5. Implications for Quantum Algorithm Design and Scaling

In the regime where circuit depth $L$ grows nontrivially with $n$—either due to ansatz structure, compilation overhead, or target state complexity—unavoidable local noise (even weak) will force an exponential suppression of training signals [2007.14384, 2310.08405, 2402.08721]. The resource cost to resolve gradients thus scales exponentially: the required number of samples (shots) for a precision $\epsilon$ obeys
\[
N_\text{shots} \sim \mathcal{O}(1/|\nabla_\mu C|^2),
\]
which becomes impractical for meaningful system sizes unless noise is drastically reduced or circuit depth minimized.

A summary of scaling implications is shown below:

| Ansatz Type            | Depth Scaling      | Effect of Noise (q < 1)        |
|------------------------|-------------------|------------------------|
| Hardware efficient     | $L \sim n$        | NIBP: Exponential gradient suppression |
| QAOA (w. compilation)  | $L \sim n$        | NIBP: Exponential suppression   |
| UCC/VQE (chemistry)    | $L \sim n^k,\ k \geq 1$ | NIBP: Exponential suppression   |
| Shallow local (constant $L$) | $L \sim 1$ | Gradients can remain appreciable |

Designing ansatzes with circuit depth that scales weakly (preferably sublogarithmically) with problem size, together with error mitigation strategies, is thus essential to avoid NIBPs [2007.14384, 2310.08405].

## 6. Extensions, Mitigation Proposals, and Outlook

The contraction mechanism underlying NIBPs generalizes to arbitrary CPTP noise, including both unital and certain non-unital noise maps [2310.08405, 2402.08721]. Non-unital noise such as amplitude damping can, in some regimes, avoid true NIBPs, but often still imposes a noise-induced limit set rather than restoring full trainability [2402.08721, 2403.13927]. Notably, dissipative quantum algorithms employing engineered non-unital channels and periodic reset of ancillary qubits can actively remove entropy, maintaining gradient magnitudes and enabling scalable optimization where unitary VQAs fail in noisy environments [2507.02043].

In summary, noise-induced barren plateaus constitute a deterministic, hardware-driven flattening of the cost function landscape in variational quantum algorithms, distinct from, but potentially compounding, expressibility- and initialization-induced barren plateaus. The rigorous analysis demonstrates that mitigating NIBPs requires a combination of minimizing circuit depth, exploiting problem structure, and, in some settings, leveraging dissipation or non-unital dynamics to actively counteract entropy accumulation [2007.14384, 2507.02043]. These insights underscore a foundational challenge for the scalability of VQAs on NISQ devices and have motivated a broad spectrum of research into architectural, algorithmic, and hardware-level solutions.

Source: https://www.emergentmind.com/topics/noise-induced-barren-plateaus