Noise-Induced Barren Plateaus

• Noise-induced barren plateaus are regions where repeated local noise in layered circuits deterministically shrinks gradients, making optimization exponentially difficult.
• They arise from the contractive action of noise channels on the Pauli expansion, distinguishing them from random, noise-free barren plateaus.
• This phenomenon poses a critical challenge for VQAs on NISQ devices, necessitating mitigation strategies like depth reduction and error correction.

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 $n$ (Wang et al., 2020, Schumann et al., 2023, Singkanipa et al., 13 Feb 2024). 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 (Wang et al., 2020, Anschuetz et al., 2022).

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) (Wang et al., 2020, Schumann et al., 2023, Singkanipa et al., 13 Feb 2024). 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 (Schumann et al., 2023, Singkanipa et al., 13 Feb 2024).

3. Mathematical Analysis and Gradient Bounds

The formal treatment leverages Pauli expansion and sandwiched 2-Rényi relative entropy contraction (Wang et al., 2020). 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 $q_M^w$ (Wang et al., 2020).

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 $n$ (Wang et al., 2020). 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 (Wang et al., 2020, Singkanipa et al., 13 Feb 2024, Schumann et al., 2023).

Correlated parameterizations (where several variational parameters are set equal) do not avoid NIBPs, as the multiplicative contraction effect persists regardless of parameter redundancy (Wang et al., 2020). 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 (Singkanipa et al., 13 Feb 2024).

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 (Wang et al., 2020, Schumann et al., 2023, Singkanipa et al., 13 Feb 2024). 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 (Wang et al., 2020, Schumann et al., 2023).

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 (Schumann et al., 2023, Singkanipa et al., 13 Feb 2024). 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 (Singkanipa et al., 13 Feb 2024, Mele et al., 20 Mar 2024). 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 (Zapusek et al., 2 Jul 2025).

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 (Wang et al., 2020, Zapusek et al., 2 Jul 2025). 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.