On prospects and limitations of variational quantum algorithms/О перспективах и ограничениях вариационных квантовых алгоритмов тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Рабинович Даниил Сергеевич

• Рабинович Даниил Сергеевич
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Рабинович Даниил Сергеевич

Contents

Page

Introduction

Chapter 1. Main concepts in quantum computations

1.1 Notation

1.1.1 Pure quantum states

1.1.2 Projective measurements

1.1.3 Unitary evolution

1.1.4 Density operators

1.1.5 Completely Positive Trace Preserving maps

1.2 Language of quantum computing

1.2.1 Operator basis

1.2.2 Quantum gates

1.2.3 Quantum circuits

1.2.4 Standard noise models

Chapter 2. Variational Quantum Algorithms

2.1 Variational Quantum Algorithms

2.1.1 Variational ansatz

2.2 Variational quantum eigensolver

2.3 Quantum Approximate Optimization Algorithm

2.3.1 Combinatorial optimization problems

2.3.2 Energy vs ground state overlap

2.4 Parameter Concentration

2.5 Layerwise training saturation

2.6 Discussion

Chapter 3. Gate error robustness of Variational Quantum

Algorithms

3.1 Parameter perturbation noise

3.2 Perturbative analysis in presence of stochastic parameter perturbation

3.3 Numerical simulations of stochastic perturbations

Page

3.3.1 VQE for Ising Hamiltonian

3.3.2 QAOA for MAX-3-SAT, MAX-CUT and state preparation

3.3.3 Shift of optimal parameters in the presence of noise

3.4 Perturbations to the individual parameters

3.5 Discussion

Chapter 4. Mitigating quantum gate errors using hardware

inspired Zero Noise Extrapolation

4.1 Error mitigation techniques

4.2 Qubit permutations as a mean of varying the strength of the noise . 64 4.2.1 Zero noise extrapolation with permutation fit

4.3 Numerical results for ZNE with permutation fit

4.3.1 Problems under consideration

4.3.2 Exact extrapolation with all permutations

4.3.3 Scaling with a noise strength and a problem size

4.3.4 ZNE with different noise models

4.3.5 Comparison with existing ZNE techniques

4.4 Conclusions

Chapter 5. Hardware native variational ansatz for quantum

approximate optimization

5.1 QAOA implementations

5.2 Hardware specific modification to QAOA ansatz

5.2.1 Ion-based quantum computers

5.2.2 Ion native QAOA ansatz

5.3 Symmetry protection

5.4 Benchmarking the ansatz

5.4.1 Sherrington-Kirkpatrick Hamiltonians

5.4.2 Exhaustive solution of all instances for n = 6 qubits

5.4.3 A large system size

5.5 Discussion and Conclusion

List of symbols and abbreviations

Bibliography

Page

List of Figures

List of Tables

Appendix A. Zero Noise Extrapolation

A.1 Relative deviation from linear trend

Appendix B. Ion native QAOA

B.1 Structure of symmetric state space V

B.2 Exhaustive benchmarking of ion native QAOA ansatz

Введение диссертации (часть автореферата) на тему «On prospects and limitations of variational quantum algorithms/О перспективах и ограничениях вариационных квантовых алгоритмов»

Introduction Relevance of the work

The concept of quantum computation hinges on the clever manipulation of amplitudes of quantum states in an attempt to prepare a state with desired properties. The use of quantum hardware is the key here, as quantum systems naturally exploit properties such as superposition and entanglement, allowing for the manipulation of an exponential number of degrees of freedom. Indeed, a general n-qubit quantum state possesses 2n degrees of freedom, quickly making large problems intractable for classical devices [1; 2]. This has led researchers to believe that a universal quantum computer would be inherently more powerful than any classical device [3].

The first celebrated quantum algorithms, such as Deutsch-Josza's [4], Shor's [5], and Grover's [6], indeed promised significant speedup—exponential in the case of Deutsch-Josza's and Shor's, and quadratic in Grover's case. However, several decades after their original proposals, the implementation of practically relevant algorithms on real hardware remains limited to the smallest sizes [7; 8]. Indeed, in the current era of Noisy Intermediate-Scale Quantum (NISQ) devices, the size of feasible quantum circuits is limited by the precision of individual operations and the coherence times of qubits [9]. The successful implementation of large-scale quantum algorithms on such devices cannot be achieved without error correction protocols [10; 11], which are not fully available yet, given the current operation imprecision.

In response to these challenges, a new paradigm of quantum computation has emerged. In this paradigm, called Variational Quantum Algorithms (VQAs) [12; 13], quantum hardware works in tandem with a classical computer. In a process reminiscent of machine learning [14; 15], a short parameterized quantum circuit (called an ansatz) is trained using feedback loop from the classical co-processor. In this loop the classical computer optimizes the parameters of the circuit to minimize a specific objective function. Typical examples of VQAs include Variational Quantum Eigensolvers (VQEs) [16—20]—algorithms designed to find the ground state and energy of physically inspired problems, Quantum Approximate Optimization Algorithm (QAOA) [21—34], designed to approximately solve combinatorial optimization problems, quantum machine learning [14; 35—40] and beyond.

Despite some advantages, such as a degree of noise resilience [41—43] and robustness against the systematic limitations of NISQ devices [44—47], certain drawbacks of these algorithms have been identified. These include trainability limitations [48; 49] and the infamous barren plateau effect [50—55] which can also be induced by noise [56]. From the practical standpoint, the performance of variational algorithms is still known to deteriorate in the presence of gate errors [12; 57—60], which strongly restricts the number of operations which can be used in the circuit [45].

In this regard, it remains crucial to study the limitations of variational quantum algorithms to understand their potential for practical realizations. By understanding these limitations, it becomes possible to develop strategies that would enhance the performance of these algorithms on real platforms, ultimately pushing NISQ devices to their maximum potential.

Dissertation goals

This thesis aims to investigate the limitations and prospects of variational quantum algorithms. Specifically, we theoretically study the performance of quantum algorithms within the limitations, induced by imperfect NISQ devices. In particular, we:

1. Study the structure of optimal parameters for the Quantum Approximate Optimization Algorithm and analyze the effects of suboptimal training strategies.

2. Examine the impact of parameter perturbations on the quality of solutions found by Variational Quantum Eigensolvers and the Quantum Approximate Optimization Algorithm. Analytically investigate the scaling of perturbations to the expectation values with respect to the strength of the noise.

3. Investigate the sensitivity of different layers of Quantum Approximate Optimization circuit to the parameter perturbations and explore the potential for the reduction of the algorithm execution time.

4. Analyze the role of inhomogeneously distributed errors in quantum computers. Study how the measured expectation value depends on the specific abstract-to-physical qubit mapping used for circuit execution.

5. Explore the effects of inhomogeneously distributed errors on the ability to perform zero noise extrapolation—a strategy that recovers noiseless expec-

tations from noisy expectation values. Identify if provable guarantees for protocol performance can be established.

6. Conduct numerical experiments across a range of problem Hamiltonians, noise models, and error distributions to validate the proposed zero noise extrapolation protocol.

7. Assess the role of the compilation problem in the execution of the Quantum Approximate Optimization Algorithm, which limits its implementation on noisy devices.

8. Propose modifications to the ansatz that eliminate the need for a compilation step, thereby bypassing the gate-based model. Numerically benchmark the proposed ansatz, assuming a trapped-ion-based quantum computer.

Statements defended

1. For the QAOA circuit with the n qubit problem Hamiltonian H = 1 — \t)(t\, where t G {0,1}xn is an arbitrary bitstring, we

a) Prove a linear relation 7 + 2^ = tt between the optimal parameters for p =1 depth circuit with an arbitrary problem size n [32].

b) Numerically validate the linear relation 7p + 2/3p = tt between optimal parameters of the final QAOA layer for circuits of up to n = 17 qubits and depth p = 5 [32].

2. For the QAOA circuit with the n qubit problem Hamiltonian H = 1 — \t)(t\ for an arbitrary bitstring t G {0,1}xn, trained layerwise, we

a) Prove the existence of nontrainable quantum states whose overlap with the target state \t) can not be improved [31].

b) Formulate and prove necessary conditions for the onset of layerwise training saturations [31]. These conditions explain properties of quantum states, for which training saturations has been numerically confirmed by E. Campos in the joint work [31].

3. We consider a parametrized quantum circuit of q gates of the form Uk(Ok) = e %Akdk ,A2k = 1. Under the weak parameter perturbation assumption, when every parameter of the circuit receives a stochastic perturbation of typical scale a ^ 1 we prove that the expectation value of any observable is perturbed by a term

proportional to a2 [61]. Applying this to the VQE circuits, we show that the noise induced energy error is bounded by the term, proportional to qa2.

4. Under the inhomogeneous error distribution assumption, when the gate errors between distinct pairs of qubits are different, a method of zero noise extrapolation is proposed. In this method, the effective strength of the noise is varied by considering different logical-to-physical qubit mappings.

a) The method is proven to recover exact noiseless value (up to terms quadratic in the strength of the noise) for circuits of regular graph topology when all qubit permutations are considered [62].

b) The numerical simulations of the proposed method are conducted using up to 100 permutations. The simulations demonstrate that the method allows recovering the VQE noiseless energy for transverse field Ising model and water molecule with accuracy of 10-2 (and beyond) for up to 12 qubits [62].

5. QAOA ansatz can be modified to utilize native system interactions, allowing to bypass the gate-based model [34]. For the example of ion based quantum computer, this allows to modify QAOA to solve arbitrary instances of the Sherrington-Kirk-patrick Hamiltonian of n = 6 qubits, guaranteeing more than 62.5% overlap with the ground state, using no more than 6 layers of the developed ansatz [34].

Scientific novelty

1. Previous works have demonstrated the effect of parameter concentrations [24; 63—65], and in the joint work [30] we have analytically proven the scaling of optimal parameters with respect to system size for a problem of state preparation. In this thesis, we focus on the structure of optimal circuit parameters for the same problem and prove, as well as numerically validate, a linear relation between the parameters of individual layers. This allows truncating the space of variational parameters, thereby simplifying circuit optimization.

2. Layerwise optimization—a suboptimal training strategy in which circuit parameters are optimized layer by layer—is known to simplify optimization [24; 36]. However, in the joint work [31], this training strategy has

been shown to saturate, i.e. not being able to improve performance beyond a certain level. In this thesis, we explain the onset of training saturation by demonstrating the existence of non-trainable states with layerwise QAOA and establishing necessary conditions for saturation. Thus, violating these conditions removes training saturation and improves algorithmic performance.

3. The effect of parameter miscalibration has been studied before (see for instance [66—68]), yet its impact on the algorithmic performance remains underexplored. In this thesis, we prove that in the presence of parameter perturbation, the expectation value of any observable is perturbed by a term proportional to a2, where a is the strength of the perturbation. This study allows us to quantify the robustness of VQAs against stochastic perturbations and propose strategies for the algorithm execution time reduction.

4. Existing methods of zero noise extrapolation typically require enlarging the circuit [68—72], which introduces new sources of noise, e.g. noise caused by limited coherence times. In this thesis, we propose a hardware-inspired method of zero noise extrapolation, leveraging the inhomogeneous distribution of errors in NISQ devices and allowing the recovery of noiseless expectation values from noisy circuits.

5. Previous realizations of hardware-native interactions in QAOA have typically been limited to minimizing the system native Hamiltonian [45]. Our proposal for an ion-compatible circuit for QAOA allows solving more general, non-native problem Hamiltonians, bypassing the gate-based model and simplifying algorithmic implementations.

Theoretical and practical significance

The results of the thesis can be used to assist the optimization and simplify the execution of variational quantum algorithms. The obtained structure of the QAOA optimal parameters and the proven necessary conditions for layerwise training saturation open the pathway towards efficient implementation of variational algorithms. The proposed techniques allow (i) quantifying the effect of noise on the performance

of variational algorithms, and (ii) partially alleviate the effect of noise either by performing error mitigation, or by bypassing the main source of noise.

The validity of the work is supported by numerical experiments and mathematical proofs, where applicable.

Presentations and validation of the results

The results of the thesis have been presented at the following conferences and seminars:

1. Scientific Seminar of Steklov Mathematical Institute of Russian Academy of Sciences (December 3, 2024, Moscow, Russia)

2. 8th International Conference on Quantum Techniques in Machine Learning (November 25-29, 2024, Melbourne, Australia)

3. Scientific Seminar of Russian Quantum Center (October 25, 2024, Moscow, Russia)

4. 24th Asian Quantum Information Science Conference (August 26-30, 2024, Sapporo, Japan)

5. VII International Conference on Quantum Technologies (July 9-12, 2023, Moscow, Russia)

6. VI International Conference on Quantum Technologies (July 12-16, 2021, Moscow, Russia)

Structure of the dissertation

The dissertation consists of introduction, five chapters, conclusions, bibliography, list of symbols and abbreviations, list of figures, and supplementary material.

Publications

The work in this thesis is based on the following publications:

1. Progress towards analytically optimal angles in quantum approximate optimisation / D. Rabinovich [et al.] // Mathematics. 2022. Vol. 10, no. 15. P. 2601

2. Training saturation in layerwise quantum approximate optimization / E. Campos [et al.] // Physical Review A. 2021. Sept. 15. Vol. 104, no. 3. P. L030401

3. Robustness of variational quantum algorithms against stochastic parameter perturbation / D. Rabinovich [et al.] // Phys. Rev. A. 2024. Apr. Vol. 109, issue 4. P. 042426. URL: https : / /link. aps . org/doi/10 . 1103/ PhysRevA.109.042426

4. Mitigating quantum gate errors for variational eigensolvers using hardware-inspired zero-noise extrapolation / A. Uvarov [et al.] // Phys. Rev. A. 2024. July. Vol. 110, issue 1. P. 012404. URL: https://link.aps. org/doi/10.1103/PhysRevA.110.012404

5. Ion-native variational ansatz for quantum approximate optimization / D. Rabinovich [et al.] // Physical Review A. 2022. Vol. 106, no. 3. P. 032418

Acknowledgments

I would like to express my gratitude to my supervisor Professor Jacob Bia-monte for assisting me over the course of my PhD studies and on the challenging path toward completing this dissertation. I also thank all my colleagues and collaborators from Deep Quantum Laboratory, especially Andrey Kardashin, Alexey Uvarov, and Akshay Vishwanathan, for the fruitful joint work and endless scientific discussions. Special gratitude goes to Ernesto Campos and Soumik Adhikary for the continuous scientific collaboration, which greatly helped me throughout this challenging four-year journey.

I would also like to thank Professors Vladimir Palyulin, Irina Bobkova and Vladimir Voikov for their support in my career and wise advice when I needed it the most.

Finally, I would like to thank my parents, brother, and most importantly my wife, Elena Rabinovich, for the unwavering care, support, and encouragement in both my scientific career and personal life.

Заключение диссертации по теме «Другие cпециальности», Рабинович Даниил Сергеевич

Results

Overall the thesis presents the following results.

1. Depth p =1 QAOA optimal parameters for state preparation relate as 7 + 2^ = w, regardless of the system size. This behavior is numerically validated for the last layer of the QAOA circuit up to system size n = 17 and circuit depth p = 5.

2. The existence of non-improvable states, i.e. those for which training would saturate, is demonstrated for layerwise QAOA for the problem of state preparation. Necessary conditions for the onset of layerwise training saturation are established.

3. In the presence of parameter perturbation noise, expectation values of any observable are proven to receive perturbations quadratic in the typical scale of parameter perturbation. For applications in VQE and QAOA, this allows us to establish the noise threshold that can still be tolerated by the algorithm. Additionally, a strategy to reduce the execution time of variational algorithms is proposed.

4. We propose a novel zero noise extrapolation strategy, which utilizes the inhomogeneity of errors in NISQ devices. The protocol is proven to recover the perfect noiseless expectation value for circuits of regular graph topology when all permutations are considered. In numerical simulations for up to n = 12 qubits, the proposed protocol demonstrates orders of magnitude improvement in the energy of noisy VQE.

5. A hardware-specific modification to QAOA is proposed. The modified circuit is able to minimize all instances of n = 6 qubit graph optimization problems with weights Kij = ±1, and shows potential for solving problems for at least up to 20 qubits. For n = 6 qubits required circuit depth is shown to be smaller than that of the unmodified QAOA.

Список литературы диссертационного исследования кандидат наук Рабинович Даниил Сергеевич, 2025 год

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