Lecture 5: Variational Quantum Eigensolver

Summary

The lecture introduces the Variational Quantum Eigensolver (VQE), a hybrid quantum-classical algorithm designed to address challenges in quantum phase estimation. Unlike quantum phase estimation, which requires accurate wave functions and complex unitary transformations, VQE optimizes unitary transformations to minimize energy. The algorithm iteratively updates quantum states based on classical calculations to find the ground state energy. However, VQE faces three main challenges: finding the correct state, measuring Hamiltonians efficiently, and navigating the large Hilbert space. The lecture also discusses early experimental implementations and methods to address symmetry-breaking issues in molecular simulations.

Takeaways

• 🔍 The quantum phase estimation method requires an approximate wave function as input, but it has issues with accuracy and implementability on current quantum computers.
• ⚙️ Variational Quantum Eigensolver (VQE) was proposed to address these issues by optimizing unitary transformations to minimize energy, avoiding long gate sequences.
• 🖥️ VQE is a hybrid algorithm where part of the calculation is done on a quantum computer, creating trial wave functions, and the rest on a classical computer, optimizing the parameters.
• 📉 The classical computer helps by iteratively providing new parameters for the quantum computer to improve the energy minimization.
• 🌐 One challenge with VQE is finding the correct unitary transformation, which is computationally complex due to the large space of possibilities.
• 🔬 The Hamiltonian in VQE must be partitioned for measurement since it cannot be measured all at once, presenting another challenge.
• 💡 The third challenge is dealing with the large Hilbert space, which includes various electronic states and symmetries, making the search for the ground state difficult.
• 🧬 Early implementations of VQE by IBM in 2017 showed discrepancies between classical simulations and quantum hardware results, possibly due to symmetry breaking.
• 🔄 Adding constraints to enforce symmetries in the VQE can help improve accuracy and reduce noise in quantum hardware results.
• 📊 Symmetry breaking leads to state transitions that cause 'kinks' in potential energy surfaces, which can be mitigated by enforcing symmetry constraints.

Q & A

What is the main motivation behind developing the Variational Quantum Eigensolver (VQE)?

The VQE was developed to address two major problems with the Quantum Phase Estimation (QPE) method: the need for an accurate approximate wave function and the complexity of implementing the required unitary transformations on current quantum computers. VQE aims to optimize the unitary transformation directly to minimize energy without breaking it down into a long sequence of elementary gates.

How does the VQE algorithm work?

VQE is a hybrid algorithm where energy minimization is split between quantum and classical computers. The quantum computer sets up a trial wave function by rotating and entangling qubits, measures the Hamiltonian expectation value, and passes the result to a classical computer. The classical computer then adjusts the parameters for the quantum computer to create a new wave function, and this process is repeated until the lowest energy is found.

What are the three main challenges of the VQE approach?

The challenges include: (1) finding the right unitary transformation in a large space, (2) partitioning the Hamiltonian into measurable pieces, and (3) navigating the large Hilbert space of qubits, which corresponds to the entire Fock space of the original problem and includes various electronic states and spin configurations.

How does VQE differ from Quantum Phase Estimation (QPE) in terms of wave function preparation?

Unlike QPE, which requires an accurate approximate wave function to achieve correct energy estimates, VQE directly optimizes the wave function and unitary transformation to minimize the energy, making it more flexible and feasible for near-term quantum computers.

What was the key issue observed in the early implementation of VQE on real quantum hardware?

In early experiments, such as those conducted by IBM in 2017, a kink-like structure appeared in the potential energy surface when using VQE on hardware. This was suspected to be caused by symmetry breaking, where different quantum states with distinct symmetries switch roles, leading to incorrect energy estimates.

What solution was proposed to address the symmetry-breaking issue in VQE results?

The solution involved adding a penalty term to the VQE functional to enforce symmetry constraints. By monitoring symmetries such as electron number and spin, and imposing penalties for breaking these symmetries, the accuracy of the VQE results could be improved.

How did the researchers confirm that symmetry-breaking caused the kink in the energy surface?

Researchers investigated the hydrogen molecule (H2) and observed that the kink in the potential energy surface coincided with a change in spin symmetry between singlet and triplet states. This confirmed that the kink was due to symmetry-breaking rather than a lack of correlation.

What role does the classical computer play in the VQE algorithm?

The classical computer in VQE adjusts the parameters (such as the angles of unitary rotations) for the quantum computer's trial wave function based on the energy estimates obtained from the quantum measurements. This iterative process continues until the lowest energy is found.

What are some practical benefits of imposing symmetry constraints in VQE?

Imposing symmetry constraints not only improves the accuracy of the energy estimates but also helps reduce noise in quantum hardware, such as depolarization errors. Symmetry constraints can purify results and enhance the overall performance of the VQE algorithm.

What are the ongoing challenges in improving VQE for future applications?

Current challenges include efficiently parameterizing the unitary transformations, finding optimal measurement schemes for the Hamiltonian, and navigating the large qubit space in quantum simulations. These issues are critical for making VQE more scalable and effective on quantum hardware.