Exploring entanglement and optimization within the Hamiltonian Variational Ansatz (2008.02941v2)

Abstract: Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail. In this paper, we focus on a special family of quantum circuits called the Hamiltonian Variational Ansatz (HVA), which takes inspiration from the quantum approximation optimization algorithm and adiabatic quantum computation. Through the study of its entanglement spectrum and energy gradient statistics, we find that HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied "hardware-efficient ansatz." We also numerically observe that the optimization landscape of HVA becomes almost trap free when the ansatz is over-parameterized. We observe a size-dependent "computational phase transition" as the number of layers in the HVA circuit is increased where the optimization crosses over from a hard to an easy region in terms of the quality of the approximations and speed of convergence to a good solution. In contrast with the analogous transitions observed in the learning of random unitaries which occur at a number of layers that grows exponentially with the number of qubits, our Variational Quantum Eigensolver experiments suggest that the threshold to achieve the over-parameterization phenomenon scales at most polynomially in the number of qubits for the transverse field Ising and XXZ models. Lastly, as a demonstration of its entangling power and effectiveness, we show that HVA can find accurate approximations to the ground states of a modified Haldane-Shastry Hamiltonian on a ring, which has long-range interactions and has a power-law entanglement scaling.

• The paper analyzes the entanglement spectrum of HVA circuits, showing they explore a more structured state space than random circuits.
• It reveals a computational phase transition where increased circuit depth leads to polynomial scaling in optimization efficiency, contrasting exponential trends.
• The study demonstrates that initializing near the identity operator mitigates vanishing gradients, enhancing performance for quantum many-body models.

Hamiltonian Variational Ansatz: Insights into Entanglement and Optimization

This paper addresses significant challenges and explores potential solutions in the field of quantum variational algorithms, with a focus on the Hamiltonian Variational Ansatz (HVA). As quantum computing advances into the Noisy Intermediate-Scale Quantum (NISQ) era, optimizing these algorithms to perform efficiently on available quantum hardware is pivotal. The paper explores two fundamental aspects: the entanglement properties and the optimization landscape of HVA.

Quantum variational algorithms, such as the Variational Quantum Eigensolver (VQE), are designed to approximate ground states of quantum many-body systems. A critical factor for their success is the ansatz, the parameterized quantum circuit that should ideally be expressive enough to capture the desired quantum states. The HVA emerges as a promising candidate due to its problem-specific structure, which is less prone to the notorious barren plateaus that hinder optimization in randomly initialized circuits.

Key Findings

1. Entanglement Spectrum and State Space: The authors provide an analysis of the entanglement spectrum of HVA circuits when applied to the transverse field Ising model and the XXZ model. The paper establishes that HVA circuits navigate a restricted state space compared to typical random circuits, which aids in avoiding barren plateaus. For the transverse field Ising model, the HVA state space is notably more structured, suggesting an inherent suitability for the model.
2. Computational Phase Transitions and Over-Parameterization: The paper reveals a "computational phase transition" that occurs with increasing circuit depth, transitioning from a hard to easy optimization landscape. Critically, the threshold depth to achieve effective optimization scales polynomially with the number of qubits, contrasting the exponential scaling in learning random unitaries. This discovery underlines the efficiency of HVA for certain quantum models, indicating that increasing circuit depth can ameliorate optimization challenges.
3. Initialization and Optimization Strategies: Analysis of different initialization strategies indicates that starting near the identity operator can bypass vanishing gradient issues, which are prevalent in random circuit approaches. For the XXZ model, experiments suggest an “identity start” reduces entanglement pathway variances and improves optimization outcomes.
4. Entangling Power and Model Scalability: The research demonstrates HVA's capability in tackling models with power-law entanglement scaling, such as the modified Haldane-Shastry model. This points to HVA’s robustness in finding ground states of complex models characterized by significant entanglement, potentially extending its applicability to more challenging 2D systems.

Practical and Theoretical Implications

• Optimization without Barren Plateaus: The results indicate that appropriately designed HVA circuits can significantly mitigate the barren plateau problem, a major hurdle in quantum algorithm scalability.
• Implications for Quantum Hardware: The insights into over-parameterization and initialization strategies suggest pathways to optimize circuit performance within the coherence and error margins of current quantum devices.
• Relation to Machine Learning: The parallels between over-parameterization in deep neural networks and HVA suggest a shared optimization landscape framework, warranting further exploration into algorithmic similarities and cross-pollination of strategies.

Future Research Directions

The findings encourage the exploration of HVA for larger, more complex lattice models, particularly in two dimensions where classical methods struggle. Additional research could focus on hybrid strategies combining HVA with other optimization techniques, potentially harnessing their individual strengths. Understanding over-parameterization in the presence of quantum noise is another promising avenue, as is the extension of these strategies to hybrid algorithms that leverage classical computational resources.

Overall, the HVA offers a structured path forward for tackling challenging quantum many-body problems, with both immediate and long-term impacts on the field of quantum computing. Through deeper comprehension of its entanglement properties and optimization potential, researchers can craft more effective quantum algorithms, pushing the boundaries of what can be achieved with near-term quantum devices.