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arXiv:2310.17996

Symmetry breaking and restoration for many-body problems treated on quantum computers

Ruiz, Andres

Published Oct 27, 2023• 0 citations
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Abstract

This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.

DOI
10.48550/arxiv.2310.17996

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This thesis explores the application of the Symmetry-Breaking/Symmetry-Restoration methodology on quantum computers to better approximate a Hamiltonian's ground state energy within a variational framework in many-body physics. This involves intentionally breaking and restoring the symmetries of the wave function ansatz at different stages of the variational search for the ground state. The Variational Quantum Eigensolver (VQE) is utilized for the variational component together with an ansatz inspired by the Bardeen-Cooper-Schrieffer (BCS) theory. The applications were demonstrated using the pairing and Hubbard Hamiltonians. Two approaches were identified with the VQE method: varying the symmetry-breaking ansatz parameters before or after symmetry restoration, termed Quantum Projection After Variation and Quantum Variation After Projection, respectively. The main contribution of this thesis was the development of a variety of symmetry restoration techniques based on the principles of the Quantum Phase Estimation algorithm, the notion of a Quantum "Oracle," and the Classical Shadow formalism. In the final part, hybrid quantum-classical techniques were introduced to extract an approximation of the low-lying spectrum of a Hamiltonian. Assuming accurate Hamiltonian moment extraction from their generating function with a quantum computer, two methods were presented for spectral analysis: the t-expansion method and the Krylov method, which provides, in particular, information about the evolution of the survival probability. Furthermore, the Quantum Krylov method was introduced, offering similar insights without the need to estimate Hamiltonian moments, a task that can be difficult on near-term quantum computers.
We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional postprocessing. The final goal is to develop configuration interaction techniques based on many-body trial states preoptimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and the quantum Krylov approaches for the postprocessing. The latter method combined with the quantum variation after projection leads to very fast convergence toward the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing Hamiltonian.
We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional post-processing. The final goal is to develop configuration interaction techniques based on many-body trial states pre-optimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and quantum Krylov approaches for the post-processing. The latter method combined with the quantum variation after projection (Q-VAP) leads to very fast convergence towards the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing hamiltonian.
We present a new method to perform variation after projection in many-body systems on quantum computers that does not require performing explicit projection. The technique employs the notion of ``oracle'', generally used in quantum search algorithms. We show how to construct the oracle and the projector associated with a symmetry operator. The procedure is illustrated for the parity, particle number, and total spin symmetries. The oracle is used to restore symmetry by indirect measurements using a single ancillary qubit. An illustration of the technique is made to obtain the approximate ground state energy for the pairing model Hamiltonian.
We present a new method to perform variation after projection in many-body systems on quantum computers that does not require performing explicit projection. The technique employs the notion of ``oracle'', generally used in quantum search algorithms. We show how to construct the oracle and the projector associated with a symmetry operator. The procedure is illustrated for the parity, particle number, and total spin symmetries. The oracle is used to restore symmetry by indirect measurements using a single ancillary qubit. An Illustration of the technique is made to obtain the approximate ground state energy for the pairing model Hamiltonian.
This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.
The generating function of a Hamiltonian HHH is defined as F(t)=⟨e−itH⟩F(t)=\langle e^{-itH}\rangleF(t)=⟨e−itH⟩, where ttt is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian ⟨HK⟩\langle H^{K}\rangle⟨HK⟩ at various orders KKK. With the constraint of minimizing the quantum resources needed on near-term quantum computers, we show that the evaluation of the generating function can be made with only one extra ancillary qubit. Due to the limited stability of actual quantum devices, the function F(t)F(t)F(t) can a priori be estimated on a restricted time interval limiting automatically the orders of the moments that could be computed. Despite of this current limitation and even with a small number of moments, a post-processing on classical computers can be used to predict approximate ground or excited state energies and/or approximate long-time evolutions. This post-processing can be achieved using methods based on the Krylov space and/or on the ttt-expansion approach that is based on imaginary time evolution. Hybrid quantum-classical calculations are illustrated in many-body interacting systems using the pairing and Fermi-Hubbard models. Possible extensions using symmetry breaking and restoration as well as a multi-reference framework are discussed.