Quantum Algorithms

Quantum Imaginary-Time Evolution

Approximating the projection onto a Hamiltonian's ground state by simulating evolution in imaginary time.

Imaginary time and ground states

Real-time evolution applies exp(-i H t), which preserves energy. Replacing t with -i*tau gives imaginary-time evolution exp(-H tau), which is not unitary: it damps high-energy components faster than low-energy ones. As tau grows, any starting state with nonzero ground-state overlap converges to the ground state after normalization. This is the quantum analog of the power method for finding the lowest eigenvector.

The non-unitary problem

Kronos motion — confinement time

Quantum computers apply unitary operations, but exp(-H tau) is not unitary. Quantum imaginary-time evolution (QITE) works around this by finding, at each small step, a unitary that reproduces the action of the imaginary-time step on the current state. It solves a small linear system, built from measured expectation values, whose solution gives the parameters of that local unitary update.

Per-step recipe

Approximate exp(-H delta_tau) acting on the current state by a unitary exp(-i A delta_tau).
Determine A by measuring correlation functions and solving a linear least-squares system.
Apply the unitary and proceed to the next step.
Repeat until the energy converges to the ground-state value.

Cost and locality

The unitary that mimics a local imaginary-time step can act on a growing region as correlations spread, so the operator A may involve increasingly many qubits, raising measurement cost. For systems with limited correlation length, the support of A stays bounded and QITE remains efficient. The method needs no variational optimization loop, avoiding barren plateaus, but pays in measurement overhead per step.

Variational relative

A lighter-weight alternative, variational quantum imaginary-time evolution, restricts the dynamics to a parameterized ansatz and evolves its parameters along the imaginary-time trajectory, trading exactness for shallower circuits. Both target the same goal as the VQE workflow: the ground state and its energy.