Sequential minimal optimization for quantum-classical hybrid algorithms

Ken M. Nakanishi; Keisuke Fujii; Synge Todo

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Sequential minimal optimization for quantum-classical hybrid algorithms

Ken M. Nakanishi, Keisuke Fujii, Synge Todo

arXiv:1903.12166v1Mar 28, 2019quant-phphysics.comp-ph

TL;DR

Quantum-classical hybrid algorithms need efficient optimization of parameterized circuits, but existing gradient-free and gradient-based methods are not specialized to their structure. The paper proposes sequential minimal optimization that solves parameter subsets using trigonometric cost-function structure. Simulations show substantially faster, initialization-independent convergence and robustness to statistical error, including fidelity above 0.98 after 8192 steps and above 0.95 after 512 steps in reported tasks.

Problem
Quantum-classical hybrid algorithms repeatedly optimize parameterized quantum circuits, making convergence speed important, while better circuit-specialized optimization remains underexplored.
Method
The method divides circuit optimization into subproblems over selected parameters and exactly minimizes their sinusoidal or multivariate trigonometric cost functions.
Results
The proposed method converges faster and with less dependence on initialization than compared methods, achieving fidelity above 0.98 after 8192 steps in one task and above 0.95 after 512 steps in VQE for LiH.
Takeaways & Limitations
The method is presented as a hyperparameter-free optimizer that can substantially accelerate quantum-classical hybrid algorithms and parameter tuning for NISQ devices.

Abstract

from arXiv · show

We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, is robust against statistical error, and is hyperparameter-free. Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters. In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period $2π$, and hence we can exactly minimize with respect to the chosen parameter. Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer. By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible. We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms. We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters. This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.

I. INTRODUCTION

Quantum-classical hybrid algorithms optimize parameterized quantum circuits using classical processing of quantum measurements, making optimization convergence central to performance. The paper proposes a circuit-specialized sequential minimal optimization method and reports faster, more initialization-independent convergence than existing approaches.

Motivation: Quantum-classical hybrid algorithms repeatedly optimize parameters of quantum circuits to minimize an observed expectation-value cost function.Their performance therefore depends strongly on optimization convergence speed.
Related work: Existing approaches include both gradient-free and gradient-based optimization, but circuit-specialized alternatives remain underexplored.Sequential minimal optimization exactly minimizes selected parameters in support vector machines by exploiting problem structure.
Contribution: The proposed method exactly minimizes the cost function over selected circuit parameters by exploiting its sinusoidal dependence on each parameter.For gates exp(iθA) with A^2 = I, the dependence is a sine curve with period 2π and its minimum can be found from three evaluations.
Contribution: The method is hyperparameter-free, converges faster, depends less on initialization, and is robust against statistical error.The update is deterministic once the parameter order is specified.
Evaluation: Numerical simulations compare the proposed optimizer with existing gradient-free and gradient-based methods using a benchmark designed to separate optimization from ansatz representation power.The experiments include a parameterized-circuit benchmark and VQE for lithium hydride.

A. Preconditions

The method applies to parameterized quantum circuits whose parameters are independent, whose gates are fixed or suitable rotation gates, and whose cost is a weighted sum of expectation values. Most hybrid algorithms using hardware-efficient ansätze satisfy these requirements.

Conditions: The circuit parameters should be independent, although this condition can be relaxed by extending the method.The paper identifies this as the first required condition.
Conditions: The circuit should consist only of fixed unitary gates and rotation gates generated by operators satisfying the stated condition.Examples include Hadamard and control-Z fixed gates, and Z- and X-rotation gates.
Conditions: The cost function should be a weighted sum of expectation values of Hermitian operators over specified input states.The paper denotes this cost function by L.
Scope: Most quantum-classical hybrid algorithms with parameterized circuits, including hardware-efficient ansätze, satisfy these requirements.

B. Our method

The method updates one parameter at a time by fitting the cost function’s sinusoidal dependence from three evaluations and selecting its exact minimum. Updates are repeated until convergence, with direct minimum estimation used periodically to limit accumulated statistical error.

Single-parameter update: At each update, the method selects a parameter index sequentially or randomly and evaluates the cost function at specified parameter values using a quantum device.The remaining parameters are held fixed during the single-parameter subproblem.
Single-parameter update: With all other parameters fixed, the cost function becomes a simple trigonometric function of the selected parameter.Its coefficients are independent of that parameter and can be determined from three independent evaluations.
Single-parameter update: The selected parameter is updated to the argument that minimizes the fitted cost function, while the other parameters remain unchanged.The procedure then continues with the next update.
Statistical error: Because the minimum is calculated from the fitted sine curve rather than estimated directly, statistical error can accumulate across updates.The paper proposes directly estimating the minimum periodically to avoid this accumulation.
Iteration: The algorithm repeats parameter selection, evaluation, and updating until the cost function converges.

C. Generalization of our method

The method extends single-parameter updates to subsets of parameters by expressing the restricted cost function as a multivariate trigonometric form. Coefficients are recovered from 3^|M| evaluations, enabling classical minimization over the selected subset.

C. Generalization of our method: For a subset M of parameters, the method fixes all other parameters and defines a restricted cost function over the selected variables.M identifies the indices on which minimization is performed.
C. Generalization of our method: The restricted cost function is represented as a trigonometric expression involving sine and cosine terms for the selected parameters.The coefficient vector has dimension 3^|M|.
C. Generalization of our method: The multi-parameter update uses 3^|M| independent evaluations to determine the trigonometric coefficients before minimizing the restricted cost function.The update procedure repeats until convergence, with direct minimum estimation periodically used to avoid error accumulation.
C. Generalization of our method: The algorithm chooses a subset of indices and evaluates the cost function at parameter settings encoded by α ∈ {0, ±1}^|M| \ {0}^|M|.These choices allow the coefficients to be determined efficiently with a discrete Fourier transform.

D. Special case of our method

When multiple rotation gates share one parameter, the cost function becomes a finite trigonometric sum whose coefficients can be determined from sampled evaluations and minimized classically.

Shared parameters arise when several rotation gates use the same parameter, such as in symmetry-constrained ansatz states.
For a parameter used S_j times, the cost function contains harmonics through s=S_j and therefore has 2S_j+1 coefficients.
The trigonometric coefficients are independent of θ_j and can be determined from values of the cost function at selected parameter settings.
The update procedure follows the single-parameter method after determining the coefficients for the shared-parameter cost function.

A. Numerical setups

The simulations evaluate the method on a controlled parameter-optimization benchmark and a four-qubit LiH VQE task, using randomized initialization and finite-sample cost estimates.

Task 1: The benchmark minimizes a fidelity-related cost whose exact minimum is −1, enabling optimization comparisons independent of circuit representation power.The target parameters are randomly sampled from [0, 2π), and the cost is estimated from computational-basis samples.
Task 2: The second task is VQE for lithium hydride at a bond distance using four qubits and the molecular Hamiltonian from Kandala et al.
Optimization procedures: The proposed method uses single-parameter minimization at each update and is compared with Powell, Nelder-Mead, CG, BFGS, and SPSA.
Circuit specification: Figure 1 supplies the simulated parameterized circuit, whose parameters θ are optimized to minimize L, with D denoting circuit repetition depth.
Sampling and initialization: Circuit parameters are initialized uniformly in [0, 2π), cost estimates use 1024 samples except in Fig. 3, and each simulation runs from 100 initializations.

B. Numerical results

Across the benchmark and LiH VQE tasks, the proposed method converges faster, remains less dependent on initialization, and is particularly tolerant of statistical error.

Task 1: Fidelity exceeded 0.98 after 8192 steps in task 1 for the proposed method, independently of the initial parameters.Other methods reached much lower fidelity for some initializations; gradient-based methods generally outperformed gradient-free methods.
Statistical-error tolerance: Fidelity exceeded 0.9x with 256 accumulations for almost all initial parameters, and the advantage increased as statistical error grew.
Task 2: Fidelity exceeded 0.95 after 512 steps in task 2 for almost all initial parameter choices, near the circuit’s representation-power limit.
Mechanistic comparison: Gradient-based BFGS and CG require cost-function estimates for all parameter gradients, which can disadvantage them when step counts measure total estimates.SPSA adds noise through finite differences, potentially interrupting fast convergence.

IV. CONCLUSION

The method exploits the trigonometric dependence of parameterized-circuit cost functions by dividing optimization into solvable subproblems, achieving faster convergence and potential robustness to statistical error.

Method: The cost function’s dependence on an individual circuit parameter is a sine curve with period 2π, enabling exact minimization over that parameter.The method divides the parameterized-circuit optimization problem into subproblems involving subsets of parameters.
Properties: The method is reported to be hyperparameter-free, less dependent on initial parameters, and robust against statistical error.The paper also expects robustness to noise in NISQ devices because the parameter–cost relation may remain robust against noise.
Numerical results: The proposed method converges to better solutions much faster than existing methods, especially when statistical error is large.This result is demonstrated through numerical simulations comparing optimization algorithms.
Extensions: The approach may also be applied to classical variational methods, including MERA optimization using variational unitary matrices inspired by parameterized quantum circuits.Neural networks composed of rotation matrices are mentioned as another possible candidate.

Appendix A: Derivation of Eq. (6)

The appendix derives the single-parameter cost-function form by rewriting a rotation gate and proving the resulting statement for circuits with arbitrary parameter counts.

Single-parameter derivation: The j-th rotation gate is rewritten using Eq. (2) to analyze how the cost function depends on θ_j.The derivation separates parameter-dependent and parameter-independent circuit components.
Cost-function form: The cost function is expressed as a finite combination of sine and cosine terms involving θ_j and coefficients indexed by the optimization step.The displayed terms include squared trigonometric factors and linear sine and cosine terms.
Inductive proof: Mathematical induction establishes the statement P(m) for all natural numbers m, beginning with a single-parameter case and extending from P(t) to P(t + 1).The induction assumes the parameterized circuit, Hamiltonian, and quantum state satisfy the stated conditions.
Generalization: The proof concludes after verifying both the base case and inductive step, covering circuits with any natural number of parameters.A separate assumption considers circuits with S rotation gates sharing the same parameter.