A ground-state preparation method that projects non-unitary imaginary-time evolution onto a parameterized quantum circuit.
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Imaginary time suppresses higher-energy components exponentially. Quantum imaginary-time evolution replaces the non-unitary operator with a sequence of measurable parameter updates, avoiding the need to implement e^(−τH) directly.
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Takes A Hermitian reachable as a sum of terms, as sparse-access oracles or as a block-encoding; a way to prepare trial states, and — for the methods that need it — a trial state whose overlap with the ground state is not negligible; a target additive error and a confidence . Returns A scalar estimate of the lowest eigenvalue with a stated additive-error guarantee, plus the run or query budget it consumed. Whether that estimate is also a rigorous upper bound is a property of the method and is not promised by the slot.
Imaginary-time evolution obeys . Because the exponential is non-unitary, a circuit implementation measures a metric and force vector and solves a classical linear system for the parameter velocity. The record therefore treats energy descent as small-instance evidence only: it does not turn a variational proxy into a proof of the exact ground state, and it calls out conditioning and ansatz expressivity as first-class limitations.
import numpy as np
# One QITE step. The two measurement functions are the parts that come
# from a device or simulator; they are stood in for here with the
# analytic values of a single-qubit RY ansatz under H = -Z, so the whole
# file runs as written. Replace them with real measurements to use it.
def measure_metric(theta):
return np.array([[0.25]]) # Fubini-Study metric of RY(theta) on one qubit
def measure_imaginary_time_force(theta):
# -d<H>/dtheta / 2 for H = -Z, where <H> = -cos(theta), so the step
# descends toward theta = 0 (the ground state) rather than away from it.
return np.array([-0.5 * np.sin(theta[0])])
theta = np.array([0.3])
dt = 0.05
metric = measure_metric(theta)
force = measure_imaginary_time_force(theta)
velocity = np.linalg.solve(metric + 1e-8 * np.eye(len(theta)), force)
theta = theta + dt * velocity
print('theta after one imaginary-time step:', theta)
RESULT = {'theta': [float(v) for v in theta], 'velocity': [float(v) for v in velocity]}Classical imaginary-time methods can store a full vector or tensor network but their memory and contraction cost grow with system size and entanglement.
QITE trades non-unitary state storage for repeated expectation-value measurements and a classical linear solve; it is not automatically cheaper on shallow noisy hardware.
Report energy, overlap with a trusted small-system ground state, metric conditioning, and measurement budget together.
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Introduces quantum imaginary-time evolution and discusses ground, excited, and thermal-state preparation.
arxiv.org/abs/1901.07653 ↗