Quantum Imaginary-Time Evolution with Polynomial Resources in Time

TL;DR

This work resolves a key challenge in quantum simulation by introducing a quantum algorithm for normalized imaginary-time evolution with polynomial resource scaling in the imaginary-time duration $\tau$ and, under a reasonable ground-state overlap assumption, in the system size $n$. Central to the approach is an adaptive normalization within a quantum signal processing (QSP) framework, enabling stable post-selection probabilities and robust state preparation for long imaginary times. The authors prove rigorous polynomial-time guarantees, provide detailed resource analyses, and demonstrate the method on ground-state tasks and open-system (Lindbladian) dynamics, including numerical experiments up to $\tau=50$ and system sizes up to a handful of qubits. The results offer a path to shallower circuits and practical imaginary-time algorithms for early fault-tolerant quantum computing, with potential impact on ground-state energy estimation and dissipative quantum dynamics across quantum chemistry and many-body physics.

Abstract

Imaginary-time evolution is fundamental for 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 a stable success probability over long imaginary-time intervals. Our algorithm approximates the target state with error polynomially small in the inverse imaginary time using a polynomial number of elementary quantum gates and a single ancilla qubit, with success probability close to one. When the initial state has reasonable overlap with the ground state, this algorithm also achieves polynomial resource cost in the system size. Numerical experiments validate our theoretical analysis for evolution time up to 50, demonstrating the algorithm's effectiveness for long-time evolution. Building on this technique, we further develop imaginary-time-evolution-based algorithms for ground-state-related problems and for simulating open quantum systems. These algorithms reduce circuit depth compared with existing methods and illustrate the effectiveness of imaginary-time evolution in early fault-tolerant quantum computing.

Figures (3)

• Figure 1: The performance of the ITE circuit $V_{{f_{\tau, \lambda}}}^{\epsilon}(U_H)$ with different choices of $\lambda$ (horizontal axis) and $\tau = 20$. The blue line shows the state infidelity between the ITE state $|{\phi(\tau)}\rangle$ and the output state $|\widetilde{\phi}(\tau)\rangle$. The orange line shows the success probability of obtaining the output state. The vertical axis is scaled by a logarithm of 10 for better visibility.
• Figure 2: Experiment results for applying our ITE-based algorithms to ground-state problems for antiferromagnetic Heisenberg (AFM) chains. (a) The expectation value of output state with respect to the Hamiltonian as $\tau$ increases. The inset plot shows success probability of obtaining the imaginary-evolution state, with the red dashed line as the theoretical lower bound. The inset plot and main plot share the same x-axis label. (b) The list of estimated energy recorded in numerical simulations of Algorithm \ref{['alg:ground']} for 3-, 4-, and 5-qubit AFM instances. (c) The logarithm of the difference between the measured energy and the ground state energy for the 5-qubit AFM instance, plotted against accumulated resource consumption, where each circuit uses the same number of shots.
• Figure 3: Numerical performance of Algorithm \ref{['alg:lindbladian']} for Lindbladian simulation. (a,b) Infidelity and resource cost as a function of evolution time $t = 1, \ldots, 20$ with $N = t^3$ steps. Panel (a) shows results for four 4-qubit TFIM instances with jump operators from ding2024simulatingpeng2025quantumyu2025lindbladianhuang2025robust. Panel (b) shows an antiferromagnetic Heisenberg chain in Equation \ref{['eqn:hamiltonian']} with five random sets of jump operators. In each panel, the blue curve (log scale) is the state infidelity between the normalized $|\rho(t)\rangle\!\rangle$ and the algorithm output; the orange curve (log scale) is the total number of queries to $U_H$ and $U_{H_c}$ required, including repetitions due to post-selection. (c) Trade-off between infidelity and average resource cost for the Heisenberg model of (b) at final time $t = 20$, varying the number of steps $N$. Both axes are in logarithmic scale.

Theorems & Definitions (41)

• Lemma 1
• Theorem 2
• Theorem 3
• Corollary 4
• Lemma 5
• Theorem 6
• Theorem 7
• Theorem 8
• Definition S1: Eigenphase transformation
• Definition S2