Analyzing the Impact of Statistical Noise on Quantum Imaginary-Time Evolution for Infinite 1D Systems

We analyze the impact of statistical noise from finite measurement shots on a quantum imaginary-time evolution (ITE) algorithm designed for infinite one-dimensional systems. The algorithm utilizes a uniform matrix product state (uMPS) ansatz and employs an ancilla qubit to evaluate the cost function. We identify the statistical estimation of this cost function—obtained by sampling the probability of the all-zero state—as the primary barrier to energy convergence on quantum devices.

Our statistical analysis confirms that while the cost value for a fixed circuit follows a normal distribution, the noise model is more complex in the dynamic case. When circuit parameters are iteratively updated using these noisy cost values, the standard deviation of the cost function is found to be √2 times larger than the static shot-noise limit, which we attribute to the propagation of measurement noise through the optimization steps.

Furthermore, we perform repeated ITE simulations and find that the resulting ground-state energy distribution is well-approximated by a Maxwell-Boltzmann-like form. This noise-induced lack of convergence can be overcome by applying the QLanczos algorithm as a post-processing step, which successfully mitigates the statistical effects and yields an accurate ground-state energy.