Singkanipa and Lidar (2025) paper DOI: 10.22331/q-2025-01-30-1617
Summary overview
TL;DR:
This paper explores how noise impacts the trainability of variational quantum algorithms (VQAs), revealing that certain noise types (e.g., amplitude damping) can avoid "noise-induced barren plateaus" (NIBPs) but trap cost functions in noise-dependent intervals called noise-induced limit sets (NILS). The findings highlight practical strategies for mitigating noise in near-term quantum applications.
Notable Achievements: First proof that Hilbert-Schmidt contractive non-unital noise can avoid NIBPs, as well as introduction of NILS as a new phenomenon in noisy VQAs.
TL;DRs for Different Audiences:
- Quantum Computing Experts: The study generalizes NIBP theory to non-unital noise, proving that HS-contractive noise avoids exponential gradient decay but confines cost functions to NILS, a noise-dependent interval.
- Algorithm Developers: Use amplitude-damping-like noise (HS-contractive) instead of depolarizing noise to preserve gradients in deep VQA circuits, but expect cost functions to plateau near noise-specific values.
- Practitioners: Simulations show amplitude-damping noise outperforms depolarizing noise in preserving gradients, but neither fully solves the noise problem—both raise final cost functions above optimal values.
- Non-Experts: Noise in quantum computers can make algorithms untrainable, but this research shows some noise types are less harmful than others, offering hope for better quantum optimization.
- Critical View: While the paper identifies noise types that avoid NIBPs, real-world quantum hardware may have mixed noise models, and NILS implies even "trainable" circuits may fail to converge to optimal solutions.
Application:
Problem Addressed: Variational quantum algorithms (VQAs)—used for tasks like drug discovery, optimization, and machine learning—suffer from barren plateaus (BPs), where gradients vanish exponentially with circuit size, making training impossible. Noise in quantum hardware exacerbates this issue, causing noise-induced barren plateaus (NIBPs).
Solution: The paper investigates how different noise types affect VQAs:
Unital noise (e.g., depolarizing) causes NIBPs in shallow circuits.
HS-contractive non-unital noise (e.g., amplitude damping) avoids NIBPs but traps cost functions in a noise-induced limit set (NILS), a fixed interval.
Unexpected Findings:
- Non-Unital Noise Avoids NIBPs: Surprisingly, HS-contractive non-unital noise (e.g., amplitude damping) does not cause exponential gradient decay, unlike unital noise. This contradicts the assumption that all noise equally harms trainability.
- Rationale: Non-unital noise contracts the coherence vector but introduces a non-zero shift, preventing gradients from vanishing.
- NILS Emergence: Cost functions under non-unital noise concentrate in a noise-dependent interval (NILS), not a fixed point. This implies optimization may stall near suboptimal values.
- Rationale: Noise-induced shifts accumulate, creating a "limit set" that grows with noise strength.
- Circuit Depth Matters: Unital noise causes NIBPs even in shallow circuits (logarithmic depth), while non-unital noise requires deeper circuits (linear depth) to exhibit similar effects.
- Rationale: Non-unital noise’s contraction factor decays more slowly with depth.
Key Terms:
- Variational Quantum Algorithms (VQAs): Hybrid quantum-classical algorithms that optimize parameterized quantum circuits to solve problems like drug discovery or portfolio optimization.
- Barren Plateau (BP): A phenomenon where gradients of cost functions vanish exponentially with circuit size, making training infeasible.
- Noise-Induced Barren Plateau (NIBP): BP caused by noise in quantum hardware, making gradients vanish even in small circuits.
- Unital Noise: Noise that preserves the identity matrix (e.g., depolarizing noise). It contracts the coherence vector but does not shift it.
- HS-Contractive Non-Unital Noise: Noise that both contracts and shifts the coherence vector (e.g., amplitude damping). It is "Hilbert-Schmidt contractive" if its matrix norm is <1.
- Noise-Induced Limit Set (NILS): A noise-dependent interval where cost functions concentrate for deep circuits, preventing convergence to optimal values.
- Coherence Vector: A classical vector representing a quantum state’s deviation from the maximally mixed state. It is used to analyze noise effects mathematically.
- Gradient: A measure of how cost functions change with parameter updates. Vanishing gradients make training impossible.
- Trainability: The ability of an algorithm to learn optimal parameters. VQAs require non-vanishing gradients to be trainable.
- Hilbert-Schmidt Norm: A measure of the "size" of a matrix, used to quantify noise contraction.
Approach:
Methodology:
- Parameter Shift Rule Extension: Adapted the parameter shift rule (a gradient estimation technique) to noisy settings.
- Noise Analysis: Studied cost function concentration under unital and HS-contractive non-unital noise using coherence vectors and matrix norms.
- Bounds Derivation: Proved bounds on gradient decay and NILS width using mathematical techniques like Levy’s lemma and matrix contraction.
- Simulations: Tested theoretical results with Qiskit simulations on 3–9 qubit systems under depolarizing and amplitude-damping noise.
Problem-Solving Techniques:
- Coherence Vector Transformations: Tracked noise effects through circuits using coherence vectors, which represent quantum states classically.
- Matrix Norm Analysis: Used Hilbert-Schmidt norms to quantify noise contraction and derive bounds on gradient decay.
- Levy’s Lemma: Argued that overlaps between random vectors (e.g., gradients) scale as $1/\sqrt{D}$, where $D$ is the effective dimension.
Results and Evaluation:
Key Findings:
- Unital Noise: Causes NIBPs in circuits with logarithmic depth $L = \omega(\log n)$.
- HS-Contractive Noise: Avoids NIBPs but confines cost functions to NILS.
- Simulations: Depolarizing noise gradients decay exponentially, while amplitude-damping gradients remain finite but cost functions rise with noise.
Quantitative Results:
- Gradient Decay: For unital noise, $|\partial C/\partial \theta| \propto \exp(-rL)$, where $r$ is the noise contraction factor.
- NILS Width: $\Lambda_\infty = \frac{|h|}{1-p}\sqrt{1 - 1/d}$, where $p$ is the noise contraction factor.
- Simulation Trends: Amplitude-damping noise preserves gradients better than depolarizing noise, but final cost functions rise with noise probability $p$.
Practical Deployment and Usability:
Real-World Applicability: Results guide noise mitigation strategies. For example, amplitude-damping-resistant circuits could improve VQE algorithms for molecular energy estimation.
Ease of Use: The findings require understanding noise models and circuit depth, which may be challenging for non-experts. Tools like Qiskit can simulate these effects, but hardware implementation depends on noise characterization.
Examples:
Quantum Chemistry: Use amplitude-damping-resistant circuits in VQE to estimate molecular ground states.
Optimization: Apply HS-contractive noise models in QAOA to solve combinatorial problems like MaxCut.
Limitations, Assumptions, and Caveats:
- Assumes noise models are either unital or HS-contractive; real hardware may have mixed noise types.
- Theoretical results hold for circuits with specific depths (e.g., logarithmic for unital noise).
- NILS bounds depend on noise parameters ($p$, $r$), which must be empirically measured.
- Simulations are limited to small qubit systems (≤9 qubits), which may not capture large-scale behavior.
Promises and Horizons:
Future Benefits: Insights could lead to noise-aware VQA designs, such as circuits that exploit non-unital noise properties.
New Research Areas: Characterizing HS-contractive maps in real quantum hardware, exploring intermediate noise regimes, and developing error mitigation tailored to non-unital noise.
Evolution: NILS may inspire new optimization techniques that navigate noise-dependent cost function landscapes.
Conflict of Interest:
The research was funded by the U.S. Army Research Office (ARO) and DARPA, which may influence the focus on quantum computing applications relevant to defense and national security. The authors acknowledge support from these agencies but do not report direct conflicts of interest.
Addendum: Prior Work McClean and Biamonte
McClean et al. (2018) first identified barren plateaus (BPs) in noise-free variational quantum algorithms (VQAs), showing that gradients vanish exponentially with circuit size for random parameterized circuits, rendering training infeasible. Their work focused on noise-free scenarios, but noted that noise could exacerbate BPs by further suppressing gradients.
This paper extends McClean’s framework to noise-induced barren plateaus (NIBPs), proving that unital noise (e.g., depolarizing) causes exponential gradient decay even in shallow circuits (logarithmic depth), while HS-contractive non-unital noise (e.g., amplitude damping) avoids NIBPs but confines cost functions to a noise-induced limit set (NILS)—a noise-dependent interval
McClean’s BP analysis applies to noise-free circuits, whereas this paper focuses on open-system noise effects. While McClean showed gradients vanish in noise-free random circuits, this paper demonstrates that non-unital noise can preserve gradients but introduces NILS, a new phenomenon where cost functions plateau near noise-specific values.
McClean’s work implies trainability challenges in large noise-free circuits; this paper shows that unital noise worsens these challenges, while non-unital noise may allow deeper circuits but risks convergence to suboptimal solutions.
Biamonte (2021) proved that VQAs can achieve universal quantum computation under specific ansatz conditions, establishing their theoretical power. That work focuses on the expressibility of VQAs, showing they can approximate any quantum operation if the ansatz is sufficiently expressive.
This paper complements Biamonte’s theoretical results by addressing practical noise challenges. While Biamonte’s universality implies VQAs can solve any quantum problem in principle, this paper highlights how noise—particularly unital noise—practically limits trainability via NIBPs.
Biamonte assumes idealized noise-free circuits, whereas this paper explicitly models noise effects (unital vs. non-unital). This paper shows that noise mitigation strategies (e.g., avoiding unital noise) are critical for practical deployment.
Synthesis: Bridging Theory and Practice
We can discern a continuum of trainability challenges—from noise-free BPs to noise-induced pathologies: While McClean’s work motivates the need for structured ansatzes to avoid BPs, this paper shows that noise-aware design (e.g., leveraging HS-contractive noise) is equally critical.
Biamonte’s proof of VQA universality assumes noise-free circuits, but this paper’s NILS results suggest that even if a VQA is theoretically universal, noise may trap it in suboptimal solutions. This underscores the need for noise-resilient Ansätze or hybrid error mitigation strategies.