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Noise-Induced Barren Plateaus in Variational Quantum Algorithms (2007.14384v6)

Published 28 Jul 2020 in quant-ph and cs.LG

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.

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Citations (592)
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Summary

  • The paper establishes that local noise causes exponential gradient decay, creating barren plateaus that hinder the trainability of VQAs.
  • It derives analytical bounds showing gradient magnitudes diminish with circuit depth and qubit count, with cost functions concentrating around maximally mixed states.
  • The work demonstrates that even correlated parameters and measurement noise do not mitigate barren plateaus, emphasizing the need for low-depth, hardware-efficient ansatz designs.

Noise-Induced Barren Plateaus in Variational Quantum Algorithms

The paper "Noise-induced Barren Plateaus in Variational Quantum Algorithms" by Wang et al. presents a rigorous analysis of how noise affects the trainability of Variational Quantum Algorithms (VQAs) on Noisy Intermediate-Scale Quantum (NISQ) devices. Specifically, the paper establishes that local noise can lead to barren plateaus in the training landscape, characterized by vanishing gradients, a phenomenon that the authors refer to as Noise-Induced Barren Plateaus (NIBPs).

Summary of Results

  1. Analytical Bounds on Gradients:

The authors derive an upper bound for the partial derivatives of the noisy cost function, showing that the gradient magnitude decays exponentially with both the depth of the quantum circuit and the number of qubits. This decay suggests that NIBPs impose significant challenges for scaling VQAs to larger problem sizes.

  1. Concentration of Cost Functions:

It is shown that the value of the noisy cost function concentrates around the value it would have if the state was maximally mixed, as the number of qubits increases. This concentration occurs at exponential rates under conditions where the circuit depth scales linearly with the number of qubits.

  1. Extensions and Robustness:

The paper extends to cases of degenerate (correlated) parameters and shows that such correlations do not mitigate NIBPs. The authors also provide results under measurement noise conditions, indicating that cost functions involving global observables exacerbate the NIBP issue.

  1. Application to Specific Algorithms:
  • Quantum Approximate Optimization Algorithm (QAOA): For the QAOA, the compilation overhead associated with problem Hamiltonians, such as MaxCut, can result in circuit depths which inherently lead to NIBPs.
  • Unitary Coupled Cluster (UCC) Ansatz: For quantum chemistry problems, the implementation of UCC using 1-D connectivity and SWAP networks results in depths that imply NIBPs for generalized UCC forms.
  1. Numerical and Hardware Demonstrations:

The theoretical findings are supported by numerical simulations using realistic noise models from IBM's superconducting qubit devices. These simulations show performance drops due to exponential concentration and gradient decays.

  1. Implications for Quantum Algorithms:

The paper emphasizes that any meaningful quantum speedup using VQAs requires avoiding exponential gradient decay. This underscores the importance of reducing circuit depth to be less than linear in qubit numbers or significantly lowering noise levels.

Theoretical Implications

The results suggest fundamental limitations on the scalability and applicability of VQAs in the NISQ era. The insights into NIBPs call for careful design of quantum circuits that can evade excessive depths and identify techniques that manage noise more effectively. Furthermore, the work highlights the necessity for exploring shor-depth, hardware-efficient ansatzes that can circumvent both noise-free and noise-induced barren plateaus.

Future Directions

Research should focus on adaptive ansatz designs catering to specific task requirements and problem instances, particularly those that inherently demand less circuit depth. Moreover, exploring advanced error mitigation or correction schemes that can address NIBPs will be crucial for realizing practical quantum advantage using VQAs.

This work serves as a critical reference for researchers aiming to optimize VQAs and lays the groundwork for further studies on the impact of noise on quantum algorithm trainability. The proposed framework and analytical techniques have broad implications across the spectrum of quantum computing applications.

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