Noise-Induced Barren Plateaus in Variational Quantum Algorithms

TL;DR

Variational Quantum Algorithms (VQAs) on Noisy Intermediate-Scale Quantum (NISQ) devices face fundamental trainability limits due to local Pauli noise. The authors develop a general Pauli-decomposition framework and prove two key results: (1) a concentration bound showing the noisy cost concentrates near the maximally mixed value, and (2) a gradient bound demonstrating exponential decay of the gradient magnitude with system size when depth scales linearly with $n$, defining Noise-Induced Barren Plateaus (NIBPs). They specialize the results to QAOA and UCC, deriving explicit bounds on gradient suppression that scale with depth and problem size. Numerically, QAOA simulations under hardware noise exhibit exponential gradient decay with the number of rounds, while Hamiltonian Variational Ansatz (HVA) demonstrations on superconducting hardware show comparable NIBP behavior, underscoring practical consequences for near-term quantum advantage.

Abstract

Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits $n$ if the depth of the ansatz grows linearly with $n$. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.

Figures (7)

• Figure 1: Schematic diagram of the Noise-Induced Barren Plateau (NIBP) phenomenon. For various applications such as chemistry and optimization, increasing the problem size often requires one to increase the depth $L$ of the variational ansatz. We show that, in the presence of local noise, the gradient vanishes exponentially in $L$ and hence exponentially in the number of qubits $n$ when $L$ scales linearly in $n$. This can be seen in the plots on the right, which show the cost function landscapes for a simple variational problem with local noise.
• Figure 2: Setting for our analysis. An $n$-qubit input state $\rho$ is sent through a variational ansatz $U(\boldsymbol{\theta})$ composed of $L$ unitary layers $U_l(\boldsymbol{\theta}_l)$ sequentially acting according to Eq. \ref{['eq:ansatz']}. Here, $\mathcal{U}_l$ denotes the quantum channel that implements the unitary $U_l(\boldsymbol{\theta}_l)$. The parameters in the ansatz $\boldsymbol{\theta}=\{\boldsymbol{\theta}_l\}_{l=1}^L$ are trained to minimize a cost function that is expressed as the expectation value of an operator $O$ as in Eq. \ref{['eq:cost']}. We consider a noise model where local Pauli noise channels $\mathcal{N}_j$ act on each qubit $j$ before and after each unitary.
• Figure 3: QAOA heuristics in the presence of realistic hardware noise: increasing number of rounds for fixed problem size. (a) The approximation ratio averaged over 100 random graphs of 5 nodes is plotted versus number of rounds $p$. The black, green, and red curves respectively correspond to noise-free training, noisy training with noise-free final cost evaluation, and noisy training with noisy final cost evaluation. The performance of noise-free training increases with $p$, similar to the results in Ref. Crooks_2018. The green curve shows that the training process itself is hindered by noise, with the performance decreasing steadily with $p$ for $p>4$. The dotted blue lines correspond to known lower and upper bounds on classical performance in polynomial time: respectively the performance guarantee of the Goemans-Williamson algorithm goemans1995improved and the boundary of known NP-hardness arora1998proofhaastad2001some. (b) The deviation of the cost from ${\rm Tr}[H_P]/2^n$ (averaged over graphs and parameter values) is plotted versus $p$. As $p$ increases, this deviation decays approximately exponentially with $p$ (linear on the log scale). (c) The absolute value of the largest partial derivative, averaged over graphs and parameter values, is plotted versus $p$. The partial derivatives decay approximately exponentially with $p$, showing evidence of Noise-Induced Barren Plateaus (NIBPs).
• Figure 4: QAOA heuristics in the presence of realistic hardware noise: increasing problem size for a fixed number of rounds. The approximation ratio averaged over 60 random graphs of increasing number of nodes $n$ and fixed number of rounds $p=4$ is plotted. The black, green, and red curves respectively correspond to noise-free training, noisy training with noise-free final cost evaluation, and noisy training with noisy final cost evaluation. (a) For a problem size of 8 nodes or larger, the noisily-trained approximation ratio falls below the performance guarantee of the classical Goemans-Williamson algorithm. (b) The depth of the circuit (red curve) scales linearly with the number of qubits, confirming we are in a regime where we would expect to observe Noise-Induced Barren Plateaus.
• Figure 5: Implementation on the ibmq_montreal superconducting-qubit device. We consider the HVA with the number of layers growing linearly in the number of qubits, $n$. a) The average magnitude of the partial derivative of the noisy and noise-free cost, with respect to the parameter in the final layer, is plotted versus $n$. The average is taken over 100 randomly selected parameter sets. As $n$ increases, the noisy average partial derivative decreases approximately exponentially, until around $n=9$. This shows evidence of Noise Induced Barren Plateaus on real quantum hardware. b) The deviation from exponential scaling can be understood by observing that it coincides with the point that the variance of the noisy partial derivatives reaches the same order of magnitude as the shot noise given by a finite sample budget of 8192 shots. Thus, from this point onward we expect fluctuations in the partial derivative to be dominated by shot noise, and gradients to be unresolvable. c) The difference of the cost value from its corresponding maximally mixed value is plotted versus $n$. d) The variance of this difference is plotted versus $n$. Both these quantities also show exponential decay until the variance of cost difference approaches the shot noise floor, which shows evidence of exponential cost concentration on this device.

Theorems & Definitions (29)

• Lemma 1: Concentration of the cost function
• Theorem 1: Upper bound on the partial derivative
• Corollary 1: Noise-induced barren plateaus
• Remark 1: Degenerate parameters
• Remark 2: Extensions to the noise model
• Proposition 1: Measurement noise
• Corollary 2: Example: QAOA
• Corollary 3: Example: UCC
• Remark 3: Example: HVA
• Remark 4: Quantum Machine Learning