Hamiltonian Expressibility: Variational Quantum Algorithms
Variational Quantum Algorithms Use Hamiltonian Expressibility to Unlock Quantum Potential.
Variable Quantum Algorithms
The rapidly emerging field of quantum computing may use Variational Quantum Algorithms (VQAs) to harness the power of near-term, noisy quantum devices, also known as Noisy Intermediate-Scale Quantum (NISQ) devices. Fusing quantum circuits with classical optimisation, these methods solve complex problems through parameter refinement.
The Variational Quantum Eigensolver (VQE) is designed to find a Hamiltonian operator's ground state. Quantum chemistry uses VQE to determine molecular ground-state energies and combinatorial optimisation using ground-state searches.
The Main Challenge: Selecting a Quantum Circuit In Variational Quantum Algorithms, choosing an effective parametric quantum circuit, or ansatz, is crucial. The ansatz controls quantum computation and affects the quality of the answer and the algorithm's trainability. A good ansatz should work with current quantum hardware, have a short circuit depth to reduce noise, and resist barren plateaus.
The Barren Plateau Issue
Variational Quantum Algorithms struggle with barren plateaus, which are cost function gradients that decrease exponentially with qubit count. This flattening of the optimisation landscape makes it difficult for classical optimisers to recognise meaningful parameter updates, especially for larger quantum systems, preventing convergence.
Even if a circuit needs exploratory power to find the ideal solution, too much power might lead to barren plateaus.
Introduce Hamiltonian Expressibility
Hamiltonian expressibility may help solve the barren plateau problem and ansatz selection. This metric assesses how evenly a circuit explores the problem's energy landscape as defined by a Hamiltonian. The idea is that a circuit's exploration capability should boost its chances of finding good answers. However, its potential to improve solutions has not been completely realised.
Researchers have a method to evaluate Hamiltonian expressibility for a given ansatz and Hamiltonian. VQE solves optimisation issues. A correlation analysis between the two metrics will determine if Hamiltonian expressibility improves Machine Learning for ansatz design.
Key Results: Ideal and Quiet
The study found several noteworthy things under ideal or extremely low noise conditions:
Ansatz Depth and Expressivity
Until saturation, a circuit's Hamiltonian expressibility improves with layer depth. Once saturation is attained, the expressibility value tends to swing inside a “maximally expressive zone” and more layers are no longer needed.
Problem-Specific Expressibility
Hamiltonian expressibility targets issues. Circuits often have higher expressibility than non-diagonal Hamiltonians for problems like Maximum Cut, Minimum Vertex Cover, Maximum Clique, and Random Diagonal. Because diagonal Hamiltonians have basis states as their best solutions, their energy space is naturally “narrower” and easier to explore.
Small-Scale Solution Quality Association
Small-scale problems (like four-qubit ones) are linked.
High Expressibility for Superposition Solutions: Ansätze with high Hamiltonian expressibility perform better for superposition-state and non-diagonal Hamiltonian problems. Classes like Heisenberg Superposition State and Random Non-Diagonal problems showed strong negative associations, meaning lower expressibility values lead to higher accuracy.
Basis-State Solutions Have Limited Expressibility: Circuits with low expressibility are better for problems having basis states, such as diagonal Hamiltonians. Extreme expressibility may backfire in some instances.
The link between expressibility and solution quality weakens as qubits increase, such as to eight. This evidence supports theoretical findings that barren plateaus, which can hinder trainability, have high expressibility.
Key Findings: Noise
Realistic quantum noise sources like decoherence and gate failures substantially modify the relationship between expressibility and solution quality, especially for small-scale settings (4 qubits).
Diagonal Hamiltonians and basis-state issues benefit more from low Hamiltonian expressibility than noiseless problems. This is because simpler, less expressive circuits are shallower and less noisy.
This relationship becomes more complicated for issues with superposition-state solutions and non-diagonal Hamiltonians. An intermediate level of expressibility may yield the best results for some superposition-state circumstances, such as the Heisenberg XXZ model, even if more expressive circuits are more noisy due to their complexity.
This requires balancing a circuit's exploration and noise resistance.
The study demonstrated a bell-shaped pattern in solution quality for some non-diagonal problems under noise: expressibility grows, peaks at an intermediate level, and then falls. This bell-shaped pattern appears when noise levels climb.
Conclusion and Future Plans
The results show that Hamiltonian expressibility can guide VQE protocol ansatz selection for small-scale challenges in ideal and noisy contexts. Expressibility metrics-based circuit selection processes are better for systems with more complex solutions (superposition states).