Quantum Algorithms

Variational Quantum Imaginary-Time Evolution

Following the imaginary-time ground-state trajectory within a parameterized ansatz by evolving its parameters.

Combining two ideas

Variational quantum imaginary-time evolution (VarQITE) merges the ground-state-finding power of imaginary-time evolution with the shallow circuits of variational methods. Instead of finding a fresh unitary each step, it fixes a parameterized ansatz |psi(theta)> and moves the parameters theta so that the state follows the imaginary-time trajectory as closely as the ansatz allows.

McLachlan's variational principle

Kronos motion — confinement time

The imaginary-time equation of motion is d/dtau |psi> = -(H - )|psi>. Projecting this onto the tangent space of the ansatz via McLachlan's principle yields a linear system A theta_dot = C for the parameter velocities theta_dot. Here A is the quantum geometric (Fubini-Study) metric of the ansatz and C is a gradient-like vector; both are built from measurable quantities. Integrating theta_dot over tau traces the ground-state path.

Ingredients

Compute the metric matrix A from overlaps of parameter derivatives of the state.
Compute the vector C from derivatives of the energy expectation.
Solve A theta_dot = C for the parameter update.
Step theta forward in imaginary time and repeat.

Advantages

VarQITE keeps circuits shallow (the ansatz is fixed) and, because it follows a physical trajectory rather than optimizing a landscape, it avoids some of the local-minimum and barren-plateau difficulties of direct energy minimization. It also naturally yields information used to estimate quantities like the partition function and Gibbs states, extending it beyond ground states to thermal state preparation.

Costs and limits

The method requires measuring the quantum geometric metric, whose size scales with the square of the parameter count, and solving a linear system that can be ill-conditioned, needing regularization. Its accuracy is bounded by ansatz expressibility: if the true trajectory leaves the ansatz manifold, the state can only track a projection of it. VarQITE complements the VQE workflow as an alternative route to the same ground states.