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An initialization strategy for addressing barren plateaus in parametrized quantum circuits
Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, Marcello Benedetti
TL;DR
Randomly initialized parametrized quantum circuits can develop barren plateaus with exponentially small gradients, limiting scalable gradient-based optimization. The paper initializes circuits through shallow identity blocks, and reports successful gradient-based training of VQEs and QNNs. The strategy’s scope depends on input states and encoding, with problem-dependent trade-offs remaining.
Problem
Random initialization can produce exponentially vanishing gradients, threatening the scalability of variational quantum eigensolvers and quantum neural networks.
Method
The strategy randomly selects some parameters and chooses the rest so the circuit forms shallow blocks that evaluate to the identity.
Results
The identity block strategy enabled gradient-based training in experiments on variational quantum eigensolvers and quantum neural networks.
Takeaways & Limitations
Limiting effective depth during the first parameter update can avoid initialization-time barren plateaus while retaining existing circuit ansätze.
Takeaways & Limitations
The strategy’s impact depends on input states and data encoding, and VQEs require a fixed shallow entangling layer to initially generate entanglement.
Abstract
from arXiv · showhide
Parametrized quantum circuits initialized with random initial parameter values are characterized by barren plateaus where the gradient becomes exponentially small in the number of qubits. In this technical note we theoretically motivate and empirically validate an initialization strategy which can resolve the barren plateau problem for practical applications. The technique involves randomly selecting some of the initial parameter values, then choosing the remaining values so that the circuit is a sequence of shallow blocks that each evaluates to the identity. This initialization limits the effective depth of the circuits used to calculate the first parameter update so that they cannot be stuck in a barren plateau at the start of training. In turn, this makes some of the most compact ansätze usable in practice, which was not possible before even for rather basic problems. We show empirically that variational quantum eigensolvers and quantum neural networks initialized using this strategy can be trained using a gradient based method.
1 Introduction
Barren plateaus make gradients exponentially small and can cause random initialization to fail, threatening scalable VQE and QNN optimization. The paper instead uses identity-valued shallow blocks to initialize existing circuits.
- Gradients in parametrized quantum circuits can vanish exponentially with the number of qubits, producing barren plateaus.
- Random initialization can make gradient-based optimization fail for a wide class of circuits.
- Barren plateaus affect traditional ansätze such as unitary coupled cluster even with few orbitals when initialized randomly.
- The proposed strategy initializes circuits as sequences of shallow unitary blocks that each evaluate to the identity.
- By limiting effective depth during the first gradient calculation, the strategy aims to enable efficient training of varied parametrized circuits.
2 A quick recap of the barren plateau problem
The barren plateau analysis connects exponentially difficult gradient estimation to circuits whose surrounding operations approximate Haar-random designs. It motivates parameterization choices that avoid such behavior during optimization.
- A parametrized circuit consists of unitary operations with parameterized gates Ul(θl) = exp(−iθlVl) and fixed unitaries Wl.
- The variational objective is the expectation of an observable represented by a Hermitian operator H.
- If either surrounding circuit segment matches the Haar distribution through its second moment, gradient estimation requires exponentially many samples in system size.
- The generalized barren plateau result shows that gradients vanish in expectation for any tangent-space direction of the unitary group.
- The gradient variance also becomes exponentially small as the number of qubits increases.
- Because parametrization strongly affects variational optimization, the paper presents a method intended to resolve barren plateaus for practical use cases.
3 Initializing a circuit as a sequence of blocks of identity operators
The strategy initializes parametrized circuits as identity-evaluating shallow blocks, preserving non-barren gradients for the first update while allowing blocks to support later entanglement. Its effectiveness depends on balancing block depth: shallow enough to avoid 2-design behavior, yet deep enough for entanglement.
- Identity-block construction: Randomly selected parameters are paired with chosen values so each block evaluates to the identity, making the whole initialized circuit an identity.Each second-layer operation is chosen as the adjoint of its corresponding first-layer operation.
- Depth trade-off: Block depth must be sufficient for entanglement during training but shallow enough that an isolated block does not approach a 2-design and produce impractically small gradients.The initialization uses M blocks of depth L, with L chosen sufficiently small to prevent 2-design behavior.
- First-update gradients: Constant-depth blocks keep the initialized circuit from entering a barren plateau, enabling efficient gradient estimation for the initial learning iteration.The gradient variance otherwise decreases exponentially with qubit number and approaches a fixed lower value as circuit depth increases.
- First-update gradients: Changing one parameter leaves the other blocks acting as identities, so the circuit used for the first gradient update has shallow effective depth.This protection is stated to hold only for the first training parameter update.
- Depth trade-off: Gradients are expected to be larger for gates near block boundaries because the corresponding estimation circuits are shallower and farther from 2-designs.The same intuition applies to parameters near the beginning or end of a block.
- Residual zero-gradient cases: Gradients can still vanish when the Hamiltonian commutes with a gate, but adding a shallow deterministic entangling layer can resolve such cases.The added layer is written as B before the sequence of identity blocks and is also used for VQE training.
4 Experimental results
Experiments show that identity-block initialization avoids exponentially vanishing initial gradients and supports gradient-based training for both MNIST quantum neural networks and a VQE.
- 4.1 Initializing a parametrized quantum circuit: For |0⟩^⊗n inputs, random initialization produced exponentially decreasing gradient variance, whereas identity-block initialization kept variance invariant with system size.Each point used 200 circuits.
- 4.1 Initializing a parametrized quantum circuit: For amplitude-encoded MNIST inputs, random initialization caused gradient variance to vanish with system size, while identity blocks avoided exponential decay at initialization.Each figure point was computed from 200 circuits.
- 4.2 Training a quantum neural network classifier: The identity-block MNIST classifier trained successfully in all trials without entering a barren plateau, while all-zero initialization trained less efficiently.Gradient variance decreased only as the model converged to an objective minimum.
- 4.3 Training a variational quantum eigensolver: At VQE initialization, identity blocks yielded mostly larger gradient variances than random initialization, with variance increasing away from each block’s center.The comparison used 200 trials and equal total circuit depth.
- 4.3 Training a variational quantum eigensolver: The identity-block VQE avoided a plateau during training, with gradient variance approaching zero as the energy converged to the ground-state energy.The experiment used a 7-qubit Heisenberg model and a fixed shallow entangling input layer.
5 Conclusion
The identity block strategy addressed barren plateaus in the reported VQE and QNN experiments, while its applicability depends on input states, encoding methods, and initialization alone.
- The identity block strategy enabled good performance in the reported variational quantum eigensolver and quantum neural network experiments.
- Input states and data encoding methods require further study because their effects differ between VQEs and QNNs.VQE initialization does not initially generate entanglement, whereas encoded QNN inputs can already be highly entangled.
- The approach addresses barren plateaus solely through parameter initialization, leaving layer-wise training, regularization, and structural constraints for comparison.The relative merits of these alternatives remain a topic for future work.
7 Appendix
The appendix derives vanishing gradients and gradient variance over the unitary group, then shows why identity initialization can retain nonzero expected gradients while variance still scales exponentially.
- 7.1 Vanishing gradient: For E(U) = ⟨0|U†HU|0⟩, the appendix evaluates directional derivatives along tangent-space directions Z = iUM.Here H is Hermitian, U is unitary, and M is Hermitian.
- 7.1 Vanishing gradient: Identity initialization makes the initial circuit constant, and the expected gradient is zero only when the Hamiltonian commutes with the observable.The appendix states that this condition is generally not the case and that the result also applies to block initialization.
- 7.2 Vanishing variance: The variance calculation uses Haar-unitary second moments and matrix-entry identities to obtain an expression involving H, M, and N.The derivation evaluates terms individually before combining them into the variance.
- 7.2 Vanishing variance: The resulting gradient variance decreases exponentially with the number of qubits.The appendix defines H_kl as matrix entries and M_00, (M^2)_00 through expectation values.