Noise-induced shallow circuits and absence of barren plateaus (2403.13927v2)

Abstract: Motivated by realistic hardware considerations of the pre-fault-tolerant era, we comprehensively study the impact of uncorrected noise on quantum circuits. We first show that any noise `truncates' most quantum circuits to effectively logarithmic depth, in the task of estimating observable expectation values. We then prove that quantum circuits under any non-unital noise exhibit lack of barren plateaus for cost functions composed of local observables. But, by leveraging the effective shallowness, we also design an efficient classical algorithm to estimate observable expectation values within any constant additive accuracy, with high probability over the choice of the circuit, in any circuit architecture. The runtime of the algorithm is independent of circuit depth, and for any inverse-polynomial target accuracy, it operates in polynomial time in the number of qubits for one-dimensional architectures and quasi-polynomial time for higher-dimensional ones. Taken together, our results showcase that, unless we carefully engineer the circuits to take advantage of the noise, it is unlikely that noisy quantum circuits are preferable over shallow quantum circuits for algorithms that output observable expectation value estimates, like many variational quantum machine learning proposals. Moreover, we anticipate that our work could provide valuable insights into the fundamental open question about the complexity of sampling from (possibly non-unital) noisy random circuits.

• The paper demonstrates that non-unital noise reduces deep quantum circuits to an effective shallow depth for computing local observables.
• The paper finds that such noise eliminates barren plateaus, ensuring stable gradients and making quantum machine learning algorithms tractable.
• The paper introduces a classical algorithm that efficiently estimates local observable expectation values in one-dimensional circuit architectures.

Noise-Induced Shallow Circuits and Absence of Barren Plateaus

Overview

The paper of quantum circuits operates within an ever-evolving framework, considering the practical constraints imposed by noise in the Noisy Intermediate-Scale Quantum (NISQ) era. The paper explores a nuanced understanding of noise, particularly focusing on non-unital noise and its impact on quantum circuits. It challenges preconceived notions that depolarizing noise models can fully encapsulate the realities of quantum computation devices, stressing the diversity within noise types encountered in practice.

Key Contributions

1. Effective Depth in Quantum Circuits: The research primarily investigates how any non-unital noise transforms ostensibly deep quantum circuits into effectively shallow circuits for computing expectation values of local observables. The authors prove that under non-unital noise, circuits comprising layers of gates only logarithmic in depth play a significant role in determining observable outcomes, effectively reducing their complexity.
2. Absence of Barren Plateaus: Unlike prior studies which indicated that noise could lead to barren plateaus — an area of the optimization landscape where gradients vanish making it difficult for quantum algorithms to train — this work reveals the opposite under non-unital noise conditions. This absence suggests non-unital noise could paradoxically maintain the tractability of cost functions associated with local observables, allowing for manageable rate variations and meaningful training in quantum machine learning tasks.
3. Classical Algorithm for Simulation: Capitalizing on the effective shallow nature of circuits under non-unital noise, the researchers design a classical algorithm capable of estimating local observable expectation values with arbitrary precision. This is particularly highlighted as a tool for one-dimensional circuit architectures where this task becomes polynomially complex.
4. Implications of Expectation Values: By comparing global versus local observables, the research illustrates that local expectation values retain variability in the presence of noise, contradicting scenarios where depolarizing noise leads to predictable convergence towards zero.

Implications and Future Directions

The exploration of quantum circuits under non-unital noise provides a profound understanding of their fundamental characteristics and potentials for optimization without necessitating increased computational destructive interference, as seen with depolarizing noise models. In practice, it could inform better architecture design and the implementation of quantum machine learning algorithms where local, trainable parameters are preferentially used for noisy environments.

The findings suggest paths to richer physics-based simulations where noise isn't merely an impediment but can be strategically integrated. Future work could extend these insights to tackle the complexity of sampling from such circuits extensively, a question that remains open, and explore specific cases that exploit noise to enhance computation, reshaping the narrative around fault-tolerant and practical quantum computing.