Quantum Imaginary-Time Evolution with Polynomial Resources in Time

Abstract: Imaginary-time evolution is fundamental to analyzing quantum many-body systems, yet classical simulation requires exponentially growing resources in both system size and evolution time. While quantum approaches reduce the system-size scaling, existing methods rely on heuristic techniques with measurement precision or success probability that deteriorates as evolution time increases. We present a quantum algorithm that prepares normalized imaginary-time evolved states using an adaptive normalization factor to maintain stable success probability over large imaginary times. Our algorithm approximates the target state to polynomially small errors in inverse imaginary time using polynomially many elementary quantum gates and a single ancilla qubit, achieving success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial resource complexity in the system size. We extend this approach to ground-state preparation and ground-state energy estimation, achieving reduced circuit depth compared to existing methods. Numerical experiments validate our theoretical results for evolution time up to 50, demonstrating the algorithm's effectiveness for long-time evolution and its potential applications for early fault-tolerant quantum computing.