Quantum Imaginary Time Evolution (QITE)

Overview

Imaginary-time evolution is a mathematical trick from statistical physics. If you evolve a quantum state under the operator e to the minus tau times H, where H is the Hamiltonian and tau is a kind of imaginary time, every component of the state that is not in the ground state decays, and the weaker the energy, the slower the decay. Run long enough, and only the ground state is left. The catch is that e to the minus tau H is not unitary, so a quantum computer cannot apply it directly.

Quantum imaginary-time evolution, or QITE, approximates this non-unitary evolution with a sequence of short unitary steps chosen by solving small classical linear equations. It was proposed as an alternative to VQE for finding ground states and thermal states on near-term hardware, with the advantage that it does not need a classical optimizer loop over circuit parameters.

The problem it solves

Finding the lowest-energy (ground) state of a quantum system, similar to VQE's goal, but using a different mathematical approach that can sometimes avoid VQE's optimization difficulties.

History and background

Imaginary-time evolution has long been used in classical simulations, such as the power method for finding ground states and quantum Monte Carlo. Motta, Sun, Tan, and coauthors from Caltech and Google posted the QITE algorithm in 2019, and it appeared in Nature Physics in 2020. In parallel, McArdle and coauthors proposed a variational ansatz-based version in npj Quantum Information in 2019, and Yuan and coauthors published a general theory of variational quantum simulation the same year.

The original paper demonstrated QITE on a superconducting qubit device for small Hamiltonians, and later work extended it to thermal states and dynamical properties. Sun and coauthors in 2021 used QITE to compute finite-temperature properties of spin systems, and Motta and coauthors proposed the quantum Lanczos method for faster convergence. Experiments on cloud devices have been used to find ground states of small molecules and spin models.

How it works

Simulates evolution in 'imaginary time' (a mathematical trick from statistical mechanics) which naturally drives a quantum system toward its ground state, implemented through a sequence of small quantum circuit updates.

Start with a state that has some overlap with the ground state. Apply e to the minus tau H. Each energy eigencomponent is multiplied by e to the minus tau times its energy, so components with higher energy shrink relative to the ground state. After normalizing, the state moves closer to the ground state, and the process converges as tau grows, at a rate set by the energy gap.

A quantum computer can only apply unitary operations. QITE breaks the evolution into small steps of size delta tau. For each step, it asks which unitary operation, generated by a local Hermitian operator A, produces the same change in the state as the non-unitary step, to first order. This is a small linear system, whose coefficients are expectation values measured on the quantum device, and whose solution gives the coefficients of A.

The unitary found is then added to the circuit. Because each step is determined by measurements on the current state, the circuit grows with each step, and the method does not use a parameterized ansatz or a classical optimizer. The price is that the circuit depth increases linearly with the number of steps, and the size of the local operators needed grows with the correlation length of the state.

Step by step

1. Prepare an initial state, such as a product state or Hartree-Fock state, with nonzero overlap with the ground state.
2. Choose a small time step delta tau and a set of local operators from which the update generator will be built.
3. Measure the expectation values needed to build the linear system: the matrix of overlaps between the derivatives generated by the operators, and the vector of the energy-weighted terms.
4. Solve the linear system classically for the coefficients of the update generator.
5. Append the corresponding unitary, for example a product of Pauli rotations, to the circuit.
6. Repeat for the desired number of steps, then measure the energy and other observables of the final state.

A worked example

Consider two spins with Hamiltonian H equal to the Pauli Z Z interaction plus a transverse field on each spin. The ground state is a particular entangled state. Starting from the product state with both spins along the field, a few QITE steps with delta tau of 0.1 move the state toward the ground state, with the energy decreasing monotonically from the initial value toward the exact minimum.

For a molecule with many terms in its Hamiltonian, each step needs local operators acting on a number of qubits that depends on the correlation length, and the number of measurements grows quickly. Early experiments used four or fewer qubits for this reason.

Complexity and resources

What hardware it needs

• A near-term quantum device with enough coherence to run circuits whose depth grows with the number of steps.
• Many measurements per step to build the linear system.
• A classical processor to solve the small linear systems and manage the circuit.

Applications

• Finding ground states of molecules and spin models on near-term devices.
• Preparing thermal states and computing finite-temperature properties.
• Initial-state preparation for quantum phase estimation.
• Studying lattice models where correlations are short-ranged.

Limitations and caveats

• Circuit depth grows with each step, which is hard for noisy devices.
• For states with long-range correlations, the required operators become large and the classical cost explodes.
• The method needs an initial state with nonzero overlap with the ground state.
• Measurement noise in the linear system can make the update unstable.

Common misconceptions

• Imaginary time is not real time. It is a mathematical device that makes the evolution non-unitary.
• QITE does not need a parameterized ansatz, but its circuit is built from measured data and can be deep.
• It does not remove all barriers to solving molecules on near-term hardware.

How it compares

• VQE uses a fixed ansatz and an outer optimization loop, which can suffer from barren plateaus.
• Quantum phase estimation gives guaranteed accuracy but needs fault-tolerant hardware.
• Variational imaginary-time evolution uses a parameterized circuit and the same metric as quantum natural gradient.

Real-world impact

An active alternative to VQE for near-term quantum chemistry simulation, particularly useful in cases where VQE's classical optimization step struggles to converge.

Why it matters

QITE showed that ground-state preparation can be done without the optimization loop that makes VQE difficult. It also introduced a useful technique, converting non-unitary operations into unitary ones using local measurements.

The method's limits, especially its dependence on correlation length, illustrate the general lesson that near-term quantum algorithms tend to work best for weakly correlated states that classical methods also handle.

Frequently asked questions

Keep learning

• Related algorithm: Variational Quantum Eigensolver (VQE)
• Related algorithm: Quantum Natural Gradient Descent
• Related algorithm: Quantum Gibbs Sampling
• Related algorithm: Quantum Phase Estimation (QPE)

This page is written for educational purposes. Complexity statements describe the best known results at the time of review (September 30, 2026), and research in this area moves quickly, so check recent papers before relying on a specific bound.

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