What Problem Does A Variational Quantum Eigensolver Solve?

You don’t need to be a physicist to understand it

Do you want to get started with Quantum Machine Learning? Have a look at Hands-On Quantum Machine Learning With Python.

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm. It aims to find an upper bound of the lowest eigenvalue of a given Hamiltonian.

Don’t get scared, even if you’re not a physicist. This post provides a high-level overview. The Hamiltonian is a quantum operator that corresponds to the total energy of a physical system. It is the central part of Schrödinger’s equation.

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Schrödinger’s equation says that the energy (𝐸) — a scalar value corresponds to 𝐻 — an operator. A quantum operator is a linear mapper that turns one quantum state vector into another one — it is a matrix.

So, of course, a scalar value is not a matrix. In fact, we’re not saying this. Because 𝐻̂ is an operator on Ψ, it doesn’t work without it. And, so, 𝐸 is the wave’s energy — it is a characteristic of the wave. It doesn’t mean anything without it. For that reason, we can’t cancel Ψ from the equation.

So, when 𝐸 and 𝐻̂ are not equal, how do they relate?

The answer results from the spectral theory — that is, the theory about eigenvalues and eigenvectors of linear operators.

Unless you speak German or studied math or physics, the words eigensolver, eigenvalue, and eigenvector may seem strange. Eigen is a German word meaning “own,” “innate,” or “typical.” And, indeed the eigenvalues and eigenvectors are unique properties that embody the spirit and nature of the matrix.

So, what is it? The fundamental equation is pretty simple: 𝐴𝑣=𝜆𝑣

𝐴 is a matrix, 𝑣 is a vector — the eigenvector of the matrix, and 𝜆 is an eigenvalue of the matrix.

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For instance, the matrix [[1 4][3 2]] has the eigenvector [1 1] with the eigenvalue 5. Let’s check this.