Large gradients via correlation in random parameterized quantum circuits

Tyler Volkoff; Patrick J. Coles

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Large gradients via correlation in random parameterized quantum circuits

Tyler Volkoff, Patrick J. Coles

arXiv:2005.12200v2May 25, 2020quant-ph

TL;DR

Uncorrelated random parameterized quantum circuits can suffer exponentially vanishing gradients, limiting gradient-based optimization. The paper proves that spatial or temporal parameter correlations can circumvent this problem, with QAOA-inspired variational Grover circuits transitioning from barren plateaus to trainability near optimal search depth.

Problem
Uncorrelated random parameterized quantum circuits can exhibit exponentially vanishing gradients, creating a challenge for efficient gradient-based optimization.
Method
The paper analyzes random parameterized quantum circuit architectures with spatially or temporally correlated parameters, including QAOA-inspired variational Grover circuits.
Results
Variational Grover circuits exhibit barren plateaus when oracle applications scale as 2^(cn−log2 n) with 0 < c < 1/3, but not when c > 1/2.
Takeaways & Limitations
Parameter correlation can mitigate or avoid barren plateaus, but it reduces the volume of quantum state space accessible to the circuit and creates a tradeoff with algorithm complexity.
Takeaways & Limitations
The conclusions rely on an assumption that the target ground state lies within the RPQC orbit, and the proposed strategy is presented for specific variational algorithms.

Abstract

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Scaling of variational quantum algorithms to large problem sizes requires efficient optimization of random parameterized quantum circuits. For such circuits with uncorrelated parameters, the presence of exponentially vanishing gradients in cost function landscapes is an obstacle to optimization by gradient descent methods. In this work, we prove that reducing the dimensionality of the parameter space by utilizing circuit modules containing spatially or temporally correlated gate layers can allow one to circumvent the vanishing gradient phenomenon. Examples are drawn from random separable circuits and asymptotically optimal variational versions of Grover's algorithm based on the quantum alternating operator ansatz (QAOA). In the latter scenario, our bounds on cost function variation imply a transition between vanishing gradients and efficient trainability as the number of layers is increased toward $\mathcal{O}(2^{n/2})$, the optimal oracle complexity of quantum unstructured search.

1. Introduction

Variational quantum algorithms use random parameterized quantum circuits optimized through measurement-driven classical updates, but barren plateau landscapes make gradient descent difficult. The paper proposes correlated circuit parameters as a way to avoid barren plateaus, including for global cost functions and QAOA-inspired Grover search.

Variational quantum algorithms couple a random parameterized quantum circuit to a classical module that optimizes a measured cost function.
Barren plateau landscapes challenge gradient descent because cost-function gradients can become exponentially small.
Correlated parameters can avoid barren plateaus in selected algorithms even when the cost function is a global projection onto a pure register state.
The paper studies spatially correlated single-qubit layers, Haar-random multiqubit gates, and QAOA-inspired circuits.
Grover-inspired variational circuits transition from barren plateaus at low depth to trainability near depths associated with high search success.

2. Background

Barren plateau analysis studies gradients of cost functions over parameterized quantum states and identifies exponentially concentrated gradients as the defining failure mode. Spatial or temporal parameter correlations are presented as a strategy for avoiding this problem, including with global measurements and as an initialization scheme.

The framework represents n-qubit states as a parameterized family indexed by parameters in a compact domain with probability density p.
A barren plateau occurs when the probability of a gradient exceeding any fixed threshold decreases exponentially with n.
When the analyzed region contains global minima, a barren plateau precludes efficient gradient-descent trainability.
Spatially or temporally correlated layers are motivated by the repeated structure of layers used in many quantum algorithms.
The paper treats correlated random parameterized quantum circuits as an initialization strategy that can circumvent barren plateaus even for globally measured costs.

3. Large gradients in separable circuits

Correlating parameters in separable random circuits can prevent barren plateaus for global and local cost functions, but trainability depends on circuit structure and input-state purity. These benefits contrast with exponentially vanishing gradients in uncorrelated circuits and can weaken for mixed inputs or insufficient depth.

Uncorrelated circuits: The variance of derivatives for uncorrelated random circuit parameters vanishes exponentially with qubit number, concentrating gradients near zero.This is the defining barren plateau behavior for the global cost function considered.
Spatial correlations: Perfectly correlating angles across layers reduces the parameter space to a permutation-invariant circuit that can avoid barren plateaus for global costs.The correlated circuit can retain descriptive power for compiling the ground state while using only an L-dimensional parameter space.
Input-state dependence: Polynomially increasing derivative variance eliminates barren plateaus for the correlated variational compiling task, but this scaling holds only for pure input states.For mixed inputs, the analysis instead finds exponentially vanishing gradient behavior under suitable noise levels.
Input-state dependence: For input impurity δ = 0.01, gradient variance shows a crossover, whereas δ = 0.10 clearly exhibits barren plateaus across n = 1, . . . , 60 qubits.The Monte Carlo results show that correlated parameters alone do not guarantee trainability when the input register is insufficiently pure.
Spatial and temporal correlations: Spatially correlated separable circuits with noncommuting layers can avoid barren plateaus, and temporal correlations further increase derivative magnitudes.For the asymptotic cost function, the derivative is nonzero and independent of n as n →∞.
Local cost functions: Faithful local cost functions avoid barren plateaus for correlated circuits, whereas global costs for some approximate-optimization circuits may require exponentially deep, cross-layer-correlated ansätze.This makes local costs preferable when they are available for training circuits with otherwise uncorrelated layers.

4. Large gradients in ξ-separable circuits

ξ-separable circuits group n qubits into ξ registers of m qubits and use permutation-invariant tensor products of Haar-distributed unitaries. The analysis shows that correlated circuits can avoid barren plateaus for pure inputs, while mixed inputs may still exhibit them.

Circuit construction: n = ξm qubits are grouped into ξ registers of m qubits, with the RPQC formed from permutation-invariant tensor products of Haar-distributed m-qubit unitaries.This extends correlated single-qubit circuits to separable circuits containing multiqubit gates.
Application setting: The translation-invariant RPQC uses repeated copies of an m-qubit unitary, W(θ)^⊗ξ, for estimating the energy of a sum of local observables.The unitary structure is motivated by uniformly random operations on an m-qubit register.
Analytical method: The variance calculation uses Haar integration, Egorychev’s contour-integral method, and the Harish-Chandra-Itzykson-Zuber integral.The resulting expression depends on eigenvalue data, including a Vandermonde determinant.
Trainability: For pure input states, the BPL phenomenon is avoided, whereas nonzero input mixedness produces BPL behavior.In the single-qubit example, the input is ρ = diag(1 − δ, δ), with 0 < δ < 1/2; maintaining constant variance as ξ grows requires δ to scale logarithmically with ξ.

5. Trainability of variational algorithms for unstructured search

Variational Grover circuits use alternating oracle and rotation layers, but uncorrelated parameters can cause barren plateaus. Correlating parameters across structurally identical layers reduces the parameter space and yields a depth-dependent transition from vanishing gradients to trainability.

Circuit setup: Alternating Grover circuits contain oracle and local-rotation layers related to the quantum alternating operator ansatz.Their efficient optimization is nontrivial despite containing an optimal Grover circuit.
Correlated parameterization: Correlating parameters across layers with the same structure reduces the variational parameter space to two dimensions and can avoid BPL at sufficient depth.Without this correlation, the circuit necessarily exhibits BPL during optimization.
Implications: The trainable parameter submanifold is relevant because variational Grover algorithms may target optimal total depth complexity even though original Grover search minimizes oracle applications.The less efficient version still improves over classical unstructured search, with success rate O(n^−1/2) and iteration complexity compared against O(2^n/2).
Low-depth regime: For L ∼ 2^(cn−log2 n) with 0 < c < 1/3, the less efficient variational Grover cost function exhibits BPL.The derivative-related quantity decays exponentially with n in this regime.
Trainable regime: For L ∼ 2^(cn−1/2 log2 n) with c ≥ 1/2, the correlated variational Grover circuit can be efficiently optimized by gradient descent.This depth is comparable to that required for successful algorithm performance.
Numerical support: Monte Carlo results support an asymptotic crossover from BPL at low depth to its absence at high depth.Additional fits identify layer and problem-size scaling consistent with this transition, including approximately L^5 and 2^−1.8n behavior.

6. Conclusions

Correlating RPQC parameters can mitigate or avoid barren plateau landscapes, but this introduces a tradeoff between algorithm complexity and trainability. The results broaden strategies for designing efficient variational quantum algorithms, while motivating further study of correlation as pre-training.

Conclusions: Spatial or temporal parameter correlation can mitigate or avoid barren plateau landscapes in specific variational quantum algorithms.Correlation reduces the volume of quantum state space accessible to the RPQC.
Conclusions: Parameter correlation creates an expected tradeoff between algorithm complexity and trainability.The paper identifies expressibility as one way to quantify this tradeoff.
Conclusions: The results broaden the available strategies for defining efficient variational quantum algorithms for near-term quantum processors.
Conclusions: Future work could investigate correlated parameters as a pre-training approach followed by training with relaxed correlation.

Appendix A. Proof of (5)

The proof transforms the parameter integral through a linear change of variables that preserves both the measure and integration domain. The resulting expression yields a central-binomial-coefficient result independent of the number of layers.

Appendix A. Proof of (5): A linear change of variables preserves the measure and integration domain while rewriting the parameter integral.The variables are transformed modulo 2π across the L parameters.
Appendix A. Proof of (5): The resulting integral is evaluated using the central binomial coefficient.
Appendix A. Proof of (5): The final result is independent of L.

Appendix B. Proof of (20) and (22)

The appendix derives the stated expressions by combining SU(2) coherent-state identities, rotation formulas, and small-γ expansions of the random parameterized circuit. These intermediate formulas are then used to obtain equations (20) and (22).

Appendix B. Proof of (20) and (22): The proof uses SU(2) coherent states in the spin j = n/2 representation.The coherent-state construction is introduced alongside the collective lowering operator J−.
Appendix B. Proof of (20) and (22): A rotation formula for real a supplies the transformation identity used in the derivation.
Appendix B. Proof of (20) and (22): For small γ, the RPQC is expanded before the appendix evaluates the resulting expressions.
Appendix B. Proof of (20) and (22): The appendix bounds the relevant trigonometric expressions using critical-point and denominator properties over their periods.

Appendix C. Avoiding BPL in the ring of disagrees algorithm

For the ring of disagrees, correlated QAOA layers provide a low-dimensional setting without barren plateaus for the local cost function, whereas the global cost function still encounters them at exponential depth. The correlated approach is therefore trainable in the local setting but can be suboptimal for the optimization problem.

Appendix C. Avoiding BPL in the ring of disagrees algorithm: The QAOA circuit alternates driver and mixer layers with parameters βj and γj.
Appendix C. Avoiding BPL in the ring of disagrees algorithm: For the 2-local ring-of-disagrees cost function, E remains O(n) for every mixer parameter βj, so no BPL occurs.This permits efficient training of the mixer angles when driver layers are short-time evolutions generated by C.
Appendix C. Avoiding BPL in the ring of disagrees algorithm: With correlated driver and mixer parameters, the global cost function encounters BPL when L ∼ 2^(cn−2).
Appendix C. Avoiding BPL in the ring of disagrees algorithm: The Dirichlet-kernel bound implies that no BPL is encountered if L ln L scales as 2^(n−log2 n).
Appendix C. Avoiding BPL in the ring of disagrees algorithm: For the ring of disagrees problem, layer-correlated QAOA is low-dimensional but suboptimal because its maximum cost increases with L.