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Higher Order Derivatives of Quantum Neural Networks with Barren Plateaus
M. Cerezo, Patrick J. Coles
TL;DR
Barren plateaus make QNN gradients exponentially small, raising the question of whether Hessians or other higher-order derivatives can enable escape. The paper derives general quantum-hardware formulas for these derivatives and shows that they too are exponentially suppressed, so higher-order optimization does not avoid barren-plateau scaling.
Problem
Barren plateaus cause QNN cost gradients to vanish exponentially with system size, and it was unclear whether higher-order derivative information could help escape them.
Method
The paper uses iterative parameter-shift formulas, organized through a Pascal tree, to analyze Hessians and arbitrary higher-order partial derivatives.
Results
Hessian elements and higher-order partial derivatives are exponentially vanishing in barren plateaus, making their estimation exponentially hard.
Takeaways & Limitations
Optimization methods using Hessians or other higher-order derivative information do not circumvent the exponential scaling caused by barren plateaus.
Abstract
from arXiv · showhide
Quantum neural networks (QNNs) offer a powerful paradigm for programming near-term quantum computers and have the potential to speedup applications ranging from data science to chemistry to materials science. However, a possible obstacle to realizing that speedup is the Barren Plateau (BP) phenomenon, whereby the gradient vanishes exponentially in the system size $n$ for certain QNN architectures. The question of whether high-order derivative information such as the Hessian could help escape a BP was recently posed in the literature. Here we show that the elements of the Hessian are exponentially suppressed in a BP, so estimating the Hessian in this situation would require a precision that scales exponentially with $n$. Hence, Hessian-based approaches do not circumvent the exponential scaling associated with BPs. We also show the exponential suppression of higher order derivatives. Hence, BPs will impact optimization strategies that go beyond (first-order) gradient descent. In deriving our results, we prove novel, general formulas that can be used to analytically evaluate any high-order partial derivative on quantum hardware. These formulas will likely have independent interest and use for training quantum neural networks (outside of the context of BPs).
I. Introduction
Barren plateaus can make QNN optimization exponentially difficult, motivating analysis of whether higher-order derivatives can help. This work shows that Hessians and higher-order derivatives remain exponentially suppressed, while introducing general formulas for evaluating them on quantum hardware.
- Parameterized quantum circuits adapt to NISQ constraints and support both VQAs and QNNs through quantum cost or gradient evaluation paired with classical optimization.
- Barren plateaus cause cost gradients to vanish exponentially with system size, potentially destroying quantum speedup by requiring exponential precision.
- Hessian elements are exponentially vanishing in a barren plateau, so estimating them requires exponentially increasing precision.
- Higher-order derivative information therefore does not circumvent barren-plateau scaling in optimization methods beyond gradient descent.
- The paper derives general higher-order derivative formulas, introduced through the Pascal tree, that can be evaluated on quantum hardware.These formulas are also applicable to training parameterized quantum circuits and other uses of higher-order information.
II. Preliminaries
The paper introduces background on cost functions, parameter-shift differentiation, and barren plateaus before presenting its main results.
- The preliminaries establish the cost function, parameter shift rule, and barren plateaus needed for the subsequent analysis.
A. Cost function
The cost is modeled as a sum of expectation values over input states passed through a parameterized quantum circuit, covering both QNNs and VQAs.
- The cost function sums expectation values over N input states processed by the parameterized circuit.For efficient computation, N must grow at most polynomially with the qubit count n.
- In QNNs, the input states represent training data points, while VQAs typically use one input state.This makes the cost formulation general enough for both paradigms.
B. Parameter shift rule
The parameter shift rule computes derivatives of parameterized quantum-circuit costs by evaluating the cost at shifted parameter values, exactly rather than through finite differences.
- For a parameterized unitary e^(-iθ_iσ_i/2), the derivative with respect to θ_i is obtained using the parameter shift rule.
- The rule uses indices other than the differentiated parameter as unchanged parameters in the shifted evaluations.
- Parameter shifting exactly expresses a first-order derivative as a difference of cost values at two parameter settings, not as a finite-difference approximation.
C. Barren plateaus
Barren plateaus make cost-function derivatives exponentially small across the landscape, requiring exponential precision to identify a minimizing direction. The paper treats this consequence independently of the mechanism producing the plateau.
- C. Barren plateaus: Exponential suppression of cost derivatives means estimating gradients and finding a minimizing direction requires exponential precision.Chebyshev-based analysis shows derivatives are exponentially small on average across the landscape.
- C. Barren plateaus: Barren plateaus can arise from deep unstructured circuits, global cost functions, entanglement, hardware noise, and other mechanisms.The cited literature links the phenomenon to several circuit, cost-function, and hardware properties.
- C. Barren plateaus: The results are formulated independently of the mechanism that produces a barren plateau, so they apply across these mechanisms.
III. Hessian matrix elements
The paper bounds Hessian elements using repeated parameter-shift evaluations and proves that they vanish exponentially with high probability in a barren plateau. Consequently, Hessian-based algorithms require precision that grows exponentially with system size.
- III. Hessian matrix elements: The Hessian is the square matrix of second derivatives Hij = ∂i∂jC(θ) of the cost function.
- III. Hessian matrix elements: Repeated application of the parameter-shift rule expresses second derivatives as sums of cost-function evaluations at up to four points.
- III. Hessian matrix elements: Proposition 1 derives a probability bound for Hessian elements under the assumption that the mean first derivative satisfies ⟨∂iC⟩θ = 0.
- III. Hessian matrix elements: The proof bounds Hessian-element magnitudes through parameter-shift relations and controls derivative deviations using Chebyshev’s inequality and a union bound.
- III. Hessian matrix elements: Hessian matrix elements are exponentially vanishing with high probability in a barren plateau, requiring exponentially growing precision for their estimation.Corollary 1 combines the Hessian bound with the barren-plateau variance condition.
IV. Higher order partial derivatives
The section derives general expressions for higher-order cost derivatives by repeatedly applying the parameter-shift rule, then shows that efficiently computable derivatives vanish exponentially in barren plateaus.
- IV. Higher order partial derivatives: The coefficients in these derivative formulas are indexed by frequency shifts and parameter multiplicities, with allowed shifts determined by whether each multiplicity is even or odd.For even multiplicities, ω_l ∈ {0, ±1}; for odd multiplicities, ω_l ∈ {±1/2, ±3/2}.
- IV. Higher order partial derivatives: Higher-order partial derivatives are expressed as sums of up to 2^|α| cost-function evaluations, so restricting |α| to O(log(n)) keeps the evaluation overhead polynomial.The derivative order may include repeated differentiation with respect to the same parameter.
- IV. Higher order partial derivatives: The Pascal tree generates the signed coefficients used in the higher-order derivative formulas, with its restricted width accounting for periodic parameter shifts.The tree is obtained by modifying Pascal-triangle addition when entries exceed the maximum width.
- IV. Higher order partial derivatives: For a barren plateau, any efficiently computable higher-order partial derivative is exponentially vanishing in n with high probability.The bound has the form G(n) ∈ O(1/q^n) for some q > 1 when |α| ∈ O(log(n)).
V. Discussion
The discussion concludes that barren plateaus suppress the Hessian and higher-order derivatives, making higher-order estimation exponentially hard despite its potential usefulness for analyzing and optimizing quantum landscapes.
- V. Discussion: When a barren plateau exists, the Hessian and other high-order partial derivatives of the cost vanish exponentially in n with high probability.This result applies to the cost function considered for both Variational Quantum Algorithms and Quantum Neural Networks.
- V. Discussion: Iterative applications of the parameter-shift rule reduce the higher-order result to the vanishing of first-order derivatives and motivate the Pascal tree construction.The Pascal tree is introduced as a mathematical construct arising from these repeated applications.
- V. Discussion: Estimating higher-order partial derivatives in a barren plateau is exponentially hard, limiting their use as a route around barren-plateau scaling.The conclusion concerns higher-order derivative information rather than only first-order gradient descent.
Appendix A Explicit description of d(ωl,Nk)
The appendix explains how repeated parameter shifts generate the Pascal tree and provides an analytical procedure for its coefficients.
- Appendix A Explicit description of d(ωl,Nk): Repeated application of the parameter-shift rule produces the first rows of the Pascal tree, which initially coincide with the corresponding rows of Pascal’s triangle.The construction changes once periodicity restricts the tree width.
- Appendix A Explicit description of d(ωl,Nk): Parameter periodicity restricts the Pascal-tree width because shifts differing by 2π produce the same unitary up to an unobservable global phase.This restriction is what causes out-of-range entries to be combined into the central entry.
- Appendix A Explicit description of d(ωl,Nk): The coefficient recursion begins with explicit low-order values and then analytically determines d(ωl,Nk) for higher Nk.The listed base cases include d(±1,0) = 0, d(0,0) = 1, and d(±1/2,1) = ±1.