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Avoiding barren plateaus using classical shadows

Stefan H. Sack, Raimel A. Medina, Alexios A. Michailidis, Richard Kueng, Maksym Serbyn

arXiv:2201.08194v2quant-ph

TL;DR

Barren plateaus cause exponentially vanishing gradients that obstruct variational quantum optimization. The paper defines weak barren plateaus using local Rényi entropies and uses classical shadows to detect them during initialization and optimization. Entropy monitoring with learning-rate reduction provides a procedure for avoiding barren plateaus on NISQ devices.

  • Problem

    Barren plateaus make expressive variational ansätze exponentially hard to optimize because cost-function gradients vanish exponentially with system size.

  • Method

    The paper defines weak barren plateaus through local second Rényi entropies and uses classical shadows to monitor them while adapting the learning rate.

  • Results

    Avoiding weak barren plateaus is sufficient to avoid conventional barren plateaus, and smaller learning rates are less likely to encounter them during optimization.

  • Takeaways & Limitations

    Local entropy tracking offers an efficiently implementable route to barren-plateau-free initialization and optimization on near-term devices.

  • Takeaways & Limitations

    The proposed shadow and purity estimators are stable under small finite noise, but noise remains an important issue for NISQ algorithms.

Abstract

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Variational quantum algorithms are promising algorithms for achieving quantum advantage on near-term devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and throughout the optimization. To this end we define a notion of weak barren plateaus (WBP) based on the entropies of local reduced density matrices. The presence of WBPs can be efficiently quantified using recently introduced shadow tomography of the quantum state with a classical computer. We demonstrate that avoidance of WBPs suffices to ensure sizable gradients in the initialization. In addition, we demonstrate that decreasing the gradient step size, guided by the entropies allows to avoid WBPs during the optimization process. This paves the way for efficient barren plateau free optimization on near-term devices.

I. INTRODUCTION

Variational quantum eigensolvers use quantum circuits to prepare trial states and classical optimization to minimize Hamiltonian energies, but barren plateaus can make expressive ansätze exponentially hard to optimize. This work introduces entropy-based diagnostics and a classical-shadow modification to avoid them during initialization and optimization.

  • Motivation: Barren plateaus make expressive variational ansätze exponentially hard to optimize because cost-function gradients vanish exponentially with system size.Classical optimization is also generally NP-hard and contains many local minima.
  • Contribution: The proposed method introduces weak barren plateaus, diagnosed through local entanglement, as an efficiently detectable route to preventing conventional barren plateaus.Classical shadows estimate local second Rényi entropies alongside energies and gradients.
  • Contribution: The algorithm monitors local second Rényi entropy during initialization and optimization, restarting with a smaller learning rate when a subsystem crosses the WBP threshold.This extends barren-plateau mitigation beyond initialization to the optimization process.
  • Variational quantum eigensolver: VQE prepares a parametrized wave function with a quantum circuit, while a classical computer updates parameters to minimize the Hamiltonian expectation value.The studied hardware-efficient ansatz alternates single-qubit rotations with nearest-neighbor CZ entangling layers.
  • Algorithm: The modified hybrid algorithm adds shadow tomography to estimate cost functions, gradients, and local entanglement without significant overhead.Measurements provide the quantities needed by the classical optimizer and the WBP monitor.

B. Barren plateaus and entanglement

Barren plateaus are linked to circuit scrambling and entanglement, but their experimental diagnosis is difficult because gradients and global entanglement can require exponentially costly measurements. Local-cost barren plateaus arise later with circuit depth, motivating efficient mitigation during optimization.

  • Barren plateaus: Barren plateaus are parameter regions where gradient variance and typical gradients vanish exponentially in the number of qubits.This behavior prevents efficient optimization and can eliminate potential quantum advantage.
  • Barren plateaus: For local cost functions, barren plateaus occur at circuit depths scaling polynomially with system size, whereas global costs can exhibit them at modest constant depths.The depth dependence differs substantially between local and global objectives.
  • Entanglement connection: Circuits approaching unitary 2-designs develop nearly maximal volume-law entanglement, making entanglement-induced and local-cost barren plateaus equivalent in this work.The paper relates scrambling, entanglement growth, and vanishing local gradients.
  • Experimental challenge: Diagnosing barren plateaus directly requires exponentially many measurements for exponentially small gradients and must distinguish them from convergence at a local minimum.Checking volume-law entanglement scaling on a device is also described as formidable.
  • Motivation: Most existing mitigation approaches target initialization, leaving a need for efficient strategies that also address barren plateaus arising during optimization.The paper uses this gap to motivate its entropy-guided procedure.

C. Weak barren plateaus and improved algorithm

Weak barren plateaus are defined using local second Rényi entropies relative to the Page value, enabling practical diagnosis with classical shadows. The proposed optimizer restarts with a smaller learning rate whenever local entropy reaches the WBP threshold.

  • Scope: The entropy restriction limits local entanglement but does not imply that the entire circuit is classically simulable.Classical simulatability depends on how entanglement entropy scales with system size.
  • Diagnosis: Classical shadows estimate local second Rényi entropies together with the cost and gradients, making WBP monitoring practical on NISQ devices.Gradients require 2pN additional tomographies when computed by the parameter-shift rule.
  • Algorithm: The WBP-free algorithm updates parameters only when S2 remains below the Page-value threshold; otherwise it restarts with a smaller learning rate.The same entropy test is applied at initialization and throughout the optimization loop.

III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE

The paper connects 2-design scrambling, local Rényi entropy, and barren plateaus, then defines WBPs as an efficiently testable precursor. Avoiding a WBP is sufficient to avoid a conventional barren plateau, with local-subsystem testing feasible for fixed k.

  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: If a unitary ensemble forms a 2-design, typical k-qubit reduced states have second Rényi entropy concentrated near the Page value.This links random-circuit scrambling to near-maximal local entanglement.
  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: Because 2-design behavior also produces exponentially vanishing local-gradient variance, entanglement-induced barren plateaus coincide with barren plateaus for local cost functions.The paper uses this equivalence to motivate entropy-based diagnosis.
  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: A weak barren plateau is defined by S2 of a k-qubit reduced state reaching at least α times the corresponding Page entropy.The threshold is intended as a computationally efficient modification of the conventional barren-plateau condition.
  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: Avoiding a WBP is sufficient to avoid a conventional barren plateau, although a WBP is necessary rather than sufficient for a conventional plateau.The practical implication is that finding one suitable non-WBP subregion certifies average avoidance of exponentially small gradient variance.
  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: For fixed k, classical shadows can test local purity with system-size-independent accuracy and only logarithmic overhead when checking all size-k subregions.The implementation is therefore efficient on NISQ devices when k does not scale with N.
  • III. WEAK BARREN PLATEAUS AND INITIALIZATION OF VQE: The scrambling region expands through entangling layers, reaching a WBP when it extends beyond k qubits and a conventional plateau after spreading across the system.Numerical results illustrate corresponding entropy growth and gradient-variance decay with circuit depth.

WBP BP

Weak barren plateaus arise when local subsystems become sufficiently entangled, preceding full barren plateaus and providing an efficiently measurable warning signal. Controlling initialization angles and learning-rate-induced entropy growth can preserve gradients and avoid this regime.

  • WBP BP: Before a full barren plateau, the second Rényi entropy of a two-qubit region reaches its maximum while gradients remain well behaved and do not decrease exponentially with system size.The two-qubit entropy saturates at a significantly lower circuit depth than the full-system bipartite entropy, illustrating that WBP precedes BP.
  • WBP-free initialization: Restricting rotation angles with ϵθ ∈ [0, 1) delays both weak and full barren plateaus, with ϵθ = 0 remaining WBP-free for all circuit depths.At ϵθ = 0, diagonal CZ entanglers create no entanglement from the computational-basis zero state.
  • WBP-free initialization: Avoiding a weak barren plateau is sufficient for avoiding a full barren plateau and yields gradients that vanish at most polynomially with system size during initialization.Figure 3 relates delayed entropy growth to the absence of the gradient-variance signature of a BP.
  • Bounding entanglement increase at a single optimization step: Large learning rates produce larger changes in trace distance and purity, making the optimization more prone to entering a weak barren plateau.The continuity-bound behavior is illustrated numerically from a BP-free initialization with small qubit rotations.
  • Bounding entanglement increase at a single optimization step: A sufficiently small learning rate limits purity changes between gradient-descent steps and can preserve WBP avoidance when the preceding state is WBP-free.The bound uses trace distance and the QFIM, with purity related to second Rényi entropy by e^-S2 = tr ρA^2.
  • Bounding entanglement increase at a single optimization step: The QFIM and gradients can be estimated on NISQ hardware, including with classical shadows, enabling practical learning-rate selection from the continuity bound.The resulting expression is intended to be evaluated on a real device during optimization.

B. Optimization performance with learning rate

Learning-rate adaptation helps the Heisenberg-model optimization avoid high-entanglement regions, but overly large or overly small rates both impair performance. The best tested behavior occurs at an intermediate rate selected to avoid WBPs.

  • B. Optimization performance with learning rate: η = 1 rapidly converges to an energy far from the target ground state while the second Rényi entropy spikes into a WBP region.The algorithm responds by restarting with a smaller learning rate.
  • B. Optimization performance with learning rate: η = 0.1 avoids an initial WBP but becomes trapped in a local minimum with large entanglement entropy, motivating a smaller α threshold and restart.With α = 0.5, this trajectory satisfies the WBP condition and is restarted at an even smaller learning rate.
  • B. Optimization performance with learning rate: η = 0.01 avoids the large-entanglement region and converges very close to the true ground-state energy in the Heisenberg-model optimization.Its gradient norm is the smallest among the tested learning rates, whereas η = 0.001 performs worse.
  • B. Optimization performance with learning rate: Further reducing the learning rate to η = 0.001 degrades performance because convergence becomes slower and the final energy expectation value is larger.This supports choosing the highest learning rate that still avoids a WBP.
  • B. Optimization performance with learning rate: An optimization strategy that adapts the learning rate at every step is proposed as a promising direction, but testing that strategy is beyond this work’s scope.The authors present this as speculation rather than an evaluated result.

C. Classical simulatability and performance comparison

The method constrains local second Ŕenyi entropy to avoid weak barren plateaus without requiring classical simulability, and numerical studies show that learning-rate control supports optimization across challenging models.

  • Classical simulatability: The WBP criterion can remain compatible with volume-law entanglement, so avoiding WBPs does not imply that the circuit is efficiently classically simulable.Classical simulation is efficient only under stronger poly-logarithmic entanglement scaling, whereas the WBP condition can allow volume-law states.
  • Performance comparison: In the random-graph Heisenberg model, layerwise optimization reaches a WBP for both tested learning rates, while small-angle initialization avoids it.The graph geometry induces nonlocal interactions and volume-law entanglement, motivating α = 1 without prior knowledge of the target entanglement.
  • Performance comparison: Good convergence with small-angle initialization is achieved only at η = 0.01; η = 0.1 fails to reach a local minimum and retains a large gradient norm after 500 iterations.This shows that avoiding a WBP is insufficient by itself: the learning rate must also permit convergence into the local-minimum basin.
  • Performance comparison: The method also prevents barren-plateau occurrence and finds the ground state in the volume-law-entangled SYK model.
  • Performance comparison: Smaller learning rates are less likely to encounter BPs, but overly small rates degrade gradient-descent performance, motivating selection of the largest WBP-free rate.The authors suggest dynamically adapting the learning rate during optimization, but systematic study of entanglement in local minima remains incomplete.
  • Classical simulatability: Choosing α requires prior knowledge of the target state's entanglement structure when using entanglement-based postselection to avoid higher-entanglement local minima.The entanglement structure may be inferred from the Hamiltonian or from small problem instances.
  • Performance comparison: The proposed shadow-based diagnosis and observable or purity estimation are reported to remain stable under small but finite noise.
  • Performance comparison: Although shadows can be implemented on near-term devices, realizing sufficiently deep entangling circuits that produce a BP remains unclear on current NISQ hardware.

APPENDIX

Classical shadows provide a near-term protocol for estimating local observables and subsystem properties without full state tomography. Random single-qubit Pauli measurements generate shadow states whose empirical averages converge to the desired quantities.

  • APPENDIX: Classical shadows replace full state tomography with sequential random single-qubit Pauli measurements that support prediction of local linear and polynomial properties.The measurement budget scales logarithmically with the number of target properties but exponentially with their support size.
  • APPENDIX: Each repetition prepares the variational state, selects random single-qubit Pauli observables, performs one joint measurement, and stores the postmeasurement states.Random single-qubit Clifford gates implement the basis changes needed for x-, y-, and z-basis measurements.
  • APPENDIX: Averaging independent classical shadows reproduces the underlying state in expectation and becomes exact as the number of measurement repetitions tends to infinity.The cited results indicate substantially faster convergence in practice than this limiting statement alone suggests.
  • APPENDIX: Partial traces of global shadows yield subsystem shadows, allowing local density matrices and range-k observables to be estimated directly without constructing global approximations.For local observables, the protocol can jointly estimate many expectation values using a measurement budget with logarithmic dependence on their number.
  • APPENDIX: The appendix presents a formal joint-estimation theorem for L range-k observables with additive accuracy ϵ and failure probability δ.The theorem’s statement is supported by a concentration bound and a union-bound argument over the observables.

2. Estimating subsystem purities

Subsystem purities are estimated from pairwise products of subsystem classical shadows, with convergence controlled by estimator variance. The resulting costs favor smaller purities and local subsystems, while gradient estimation can be more expensive because Hamiltonian-term errors accumulate.

  • 2. Estimating subsystem purities: Distinct pairs of subsystem shadows provide an empirical purity estimator whose variance decreases with the number of measurements.Restricting to distinct pairs makes the expectation values factorize and ensures convergence to the true subsystem purity.
  • 2. Estimating subsystem purities: Smaller subsystem purities require fewer measurements to estimate accurately, whereas WBP-regime purities demand sufficiently small ϵ because they decay exponentially with subsystem size k.The accuracy requirement concerns the subsystem size rather than the full system size.
  • 2. Estimating subsystem purities: Median-of-means can improve the failure-probability dependence from δ to ln(1/δ), but worsens the dependence on ϵ by a constant factor.The trade-off is reported as useful primarily when estimating polynomially many subsystem purities.
  • 2. Estimating subsystem purities: The parameter-shift rule computes each variational single-qubit rotation gradient from the difference between two energy evaluations at θ+ and θ−.The resulting gradients are exact up to finite-sampling errors, and each energy can be decomposed into local Hamiltonian terms.
  • 2. Estimating subsystem purities: Gradient estimation can scale quadratically with system size because the Hamiltonian has at least linearly many terms and the full gradient requires one shifted calculation per parameter.Summing individually accurate term estimates can require rescaling the per-term accuracy to ϵ/L, producing additional measurement cost.

4. Example of error accumulation in an Ising model

An Ising-chain example shows that correlated measurement errors can accumulate when estimating a sum of locally simple Hamiltonian terms. In this worst-case construction, the required measurement budget scales with the number of terms.

  • 4. Example of error accumulation in an Ising model: Correlated estimators can amplify outlier errors when many Hamiltonian terms are summed, producing an additional L^2 measurement-cost scaling.The example is explicitly presented as a contrived worst-case argument, with a straightforward generalization to the paper’s Heisenberg Hamiltonian.
  • 4. Example of error accumulation in an Ising model: For a 1D Ising chain, computational-basis measurements reduce energy estimation to estimating a biased coin whose two outcomes correspond to the ground and Néel states.The chain has L = N − 1 interaction terms, and the two state energies are −J(N − 1) and +J(N − 1).
  • 4. Example of error accumulation in an Ising model: The estimator outputs the ground-state energy with probability 1 − λ and the highest-excited-state energy with probability λ, yielding a variance governed by the corresponding coin toss.The resulting variance is proportional to L^2 = (N − 1)^2 unless λ equals 0 or 1.
  • 4. Example of error accumulation in an Ising model: In general, estimating the total energy requires a measurement budget that scales with the number L of Hamiltonian terms, even when each term is individually cheap to evaluate.The conclusion follows from the central-limit behavior of the constructed estimator.
  • 4. Example of error accumulation in an Ising model: The appendix’s concentration analysis uses bounded independent centered variables and a union bound to control simultaneous errors across local observables.The argument applies to range-k observables with operator norm at most one.

Appendix B: Unitary t-designs

Unitary t-designs are ensembles whose first t moments match those of the Haar measure, with larger t indicating closer agreement with fully random unitaries. For random pure states, the expected reduced-state purity determines how closely subsystems approach maximal mixing.

  • Appendix B: Unitary t-designs: A unitary t-design is an ensemble whose t-fold channel matches the Haar average for every operator on the t-fold Hilbert space.Approximate t-designs relax this equality by measuring the channel distance with the diamond norm.
  • Appendix B: Unitary t-designs: Larger t-design order captures more Haar moments, and known exact constructions include t = 2 and t = 3.The Pauli group provides a 1-design as an operator-algebra basis.
  • Appendix B: Unitary t-designs: The appendix analyzes N-qubit random pure states generated by an ensemble forming a 2-design and studies reduced states across a bipartition.The subsystem A has dimension d_A, while the full Hilbert space has dimension d = 2^N.
  • Appendix B: Unitary t-designs: The expected purity of a reduced subsystem is obtained from the second moment of the unitary ensemble and controls its deviation from the maximally mixed state.The derivation relates trace-distance bounds to the reduced-state purity.
  • Appendix B: Unitary t-designs: When the complementary subsystem is significantly larger than A, the expected deviation of ρ_A from the maximally mixed state is exponentially small.This establishes the relevant near-maximal-mixing behavior for sufficiently smaller subsystems.

2. Bounding the expected second R´enyi entropy

The expected second Rényi entropy is bounded using purity and von Neumann entropy inequalities, yielding values close to the Page entropy for unitary 2-designs.

  • A lower bound is obtained by applying Jensen’s inequality and analyzing the expected purity of the reduced density matrix.
  • The upper bound follows from S2(ρA) ≤ S(ρA) and the Page-entropy upper bound for the expected von Neumann entropy.
  • For subsystem size k under dA/d¬A = 1/2N−2k ≪ 1, the bounds are k ln 2 − 1/2N−2k ≤ EE(S2) ≤ k ln 2 − 1/2 1/2N−2k.
  • For unitary ensembles forming a 2-design, the expected second Rényi entropy is close to the Page entropy.

Appendix D: Entanglement growth and learning rate

The SYK experiments show that entropy-guided learning-rate control avoids weak barren plateaus during optimization, while convergence depends on choosing a sufficiently small learning rate. Identity-block initialization gives the best reported optimization performance.

  • Appendix E: Algorithm performance for SYK model: The SYK case is a theoretical demonstration because its nonlocal Hamiltonian prevents efficient classical-shadow estimation of the energy expectation value.
  • Appendix E: Algorithm performance for SYK model: Resetting the learning rate avoids WBPs during SYK ground-state optimization for the tested small-angle and identity-block initializations.The experiments use N = 10, k = 2, circuit depth p = 100, and 100 random instances.
  • Appendix E: Algorithm performance for SYK model: η = 0.1 avoids the WBP for both initializations, whereas η = 1 causes both to encounter a WBP during optimization.
  • Appendix E: Algorithm performance for SYK model: The identity-block initialization achieves the best optimization performance, attributed to faster entanglement growth from finely tuned parameters.
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