Imaginary Time Propagation Method

• Imaginary Time Propagation is a computational method that applies a Wick rotation to the Schrödinger equation, exponentially damping excited state amplitudes to project the system onto its ground state.
• Modern implementations leverage ancilla-based unitary embeddings, Trotterization, and amplitude amplification to tackle non-unitarity and achieve high-fidelity ground-state convergence.
• This method underpins both classical and quantum algorithms across electronic structure, many-body physics, and finite-temperature simulations, enabling precise and scalable system analyses.

The imaginary time propagation (ITP) method is a central computational tool for accessing ground-state and thermal properties of quantum many-body systems by transforming the real-time evolution dictated by the Schrödinger equation into a diffusion process in imaginary time. This formal analytic continuation underlies a broad class of numerical and quantum algorithms for ground-state preparation, tunneling-rate estimation, statistical-mechanical ensemble construction, and high-precision electronic structure calculations across quantum chemistry, condensed matter, and nuclear physics. ITP has recent realizations in both classical and quantum computational frameworks, with rigorous formulations exploiting unitary-embedding, ancilla-based postselection, measurement-driven stochasticity, and variational circuit optimization.

1. Formal Foundations and Ground-State Projection

The mathematical core of ITP is the Wick rotation from real to imaginary time in the time-dependent Schrödinger equation,

$$i\hbar\,\frac{\partial}{\partial t}\,\ket{\psi(t)} = \hat{H}\,\ket{\psi(t)}$$

by $t \rightarrow -i\tau$, leading to

$$\frac{\partial}{\partial \tau}\,\ket{\psi(\tau)} = -\,\hat{H}\ket{\psi(\tau)},$$

which formally integrates to

$$\ket{\psi(\tau)} = e^{-\hat{H}\tau}\,\ket{\psi(0)}.$$

This evolution exponentially damps excited-state amplitudes, projecting the state onto the ground-state eigenvector for any initial state with nonzero ground-state overlap (Turro et al., 2021).

Normalization is essential because $e^{-\tau\hat{H}}$ is not unitary; this is achieved either by explicit normalization or through a shifted Hamiltonian $\hat{H} - E_T$, producing

$$\ket{\psi(\tau)} = \frac{e^{-\left(\hat{H} - E_T\right)\tau}\ket{\psi(0)}}{\|e^{-(\hat{H} - E_T)\tau}\ket{\psi(0)}\|}.$$

This formalism supports classical evolution, quantum algorithms, and hybrid methods.

2. Unitary Embedding and Quantum Algorithmic Realizations

Quantum implementations must address the non-unitarity of $e^{-\tau\hat{H}}$. A widely adopted strategy is to extend the system Hilbert space by introducing an ancilla qubit, constructing a global unitary operation in $\mathbb{C}^2 \otimes \mathbb{C}^N$ such that projection or postselection on the ancilla reproduces imaginary-time dynamics. The explicit construction (Turro et al., 2021): $$\hat{U}(\tau) = \begin{pmatrix} Q_{\rm ITP}(\tau) & \sqrt{\mathbb{1} - Q_{\rm ITP}(\tau)^2} \ \sqrt{\mathbb{1} - Q_{\rm ITP}(\tau)^2} & -Q_{\rm ITP}(\tau) \ \end{pmatrix},$$ with

$$Q_{\rm ITP}(\tau) = \left[\mathbb{1} + e^{-2(\hat{H}-E_T)\tau}\right]^{-1/2} e^{-(\hat{H}-E_T)\tau}.$$

Ancilla postselection on $\ket{0}_{\rm ancilla}$ yields the desired imaginary-time-propagated system state up to normalization. The success probability is bounded below by overlap and time-dependent terms, $P(0) \ge |c_0|^2 / [1 + e^{2\tau(E_0-E_T)}]$.

Gate-level implementation can be approached either by explicit circuit decomposition (single- and two-qubit gates, e.g., for STO-2G hydrogen on IBM hardware), or via optimal-control pulse engineering (e.g., constructing a single high-fidelity pulse for a transmon qudit). For multi-qubit/k-local Hamiltonians, Trotterization allows scaling to large systems with polynomial gate costs in domain size $k$, at the expense of error scaling with time step and commutator norms (Turro et al., 2021).

3. Measurement-Based, Deterministic, and Stochastic Frameworks

Alternative quantum ITP algorithms include measurement-based deterministic schemes (Mao et al., 2022) and measurement-driven stochastic protocols. In the measurement-based approach, ancilla-coupled weak measurements, post-selected and corrected by conditional unitaries, drive the system toward the ground state with unit (asymptotic) probability. Suzuki–Trotter decomposition of $e^{-i\epsilon H \otimes Y}$, followed by adaptive application of correction unitaries based on measurement sequences, yields a deterministic, convergence-guaranteed evolution for $E_{\rm th}$ below the first excited state. The required number of steps and measurement complexity is system-dependent and can become exponential in the absence of favorable spectral structure.

Stochastic postselection-based approaches, as in probabilistic ITP, require amplitude amplification to address low success probabilities, especially in the deep imaginary-time regime (Turro et al., 2021).

4. Classical Imaginary Time Propagation in Electronic Structure and Many-Body Physics

ITP underpins a wide array of classical ground-state algorithms, including:

• Mean-field tunneling: Imaginary-time-dependent Hartree or Hartree–Fock equations eradicate real-time "self-trapping," enabling accurate tunneling-rate calculations via Euclidean-action integration and initial-value methods for complex many-body systems (McGlynn et al., 2020).
• Time-dependent density functional theory (it-TDDFT): Imaginary-time propagation of Kohn–Sham orbitals, subject to orthonormalization at each step, can robustly reach ground states where self-consistent-field (SCF) iterations struggle and is applicable to periodic systems, DFT+U, and non-collinear spinor representations. Adaptive time stepping and state-specific integrators accelerate convergence (McFarland et al., 2020).
• Configuration interaction and multi-configurational methods: High-precision splitting schemes (symmetric/complex-coefficient) enable efficient and stable diffusion for multidimensional grid- and orbital-based methods in quantum chemistry, as exemplified by high-order propagation for 2D systems, arbitrary magnetic fields, and block-based codes (class: itp2d) (Luukko et al., 2013, Bader et al., 2013).
• Tensor contraction and multi-layer approaches: In optimized second quantization (MCTDH-oSQR), imaginary-time evolution is tailored using spectral-gauge minimization, particle-number-conserving contractions, and penalty operators to enforce physical subspace selection, leading to robust convergence for bosonic or fermionic models (Weike et al., 2019).

5. Extensions to Thermal States and Response Functions

ITP naturally generalizes to finite-temperature physics, as in the preparation of Gibbs states via nonunitary evolution $e^{-\beta H}$ from a maximally mixed initial ensemble (Turro, 2023). Quantum algorithms achieve this by diluting the nonunitary propagator into a block-unitary over an enlarged Hilbert space, with ancilla-based projective measurement yielding the correct thermal density matrix. The scaling of success probability, resource requirements, and validation on hardware for (few-qubit) neutron-spin systems is explicitly demonstrated.

In many-body perturbation theory (e.g., GW polarizability calculations), Laplace-transformed energy denominators yield expressions for response functions in terms of imaginary-time propagators. These are efficiently evaluated using energy windowing and Gauss–Laguerre quadrature, reducing scaling from quartic to cubic in system size (Kim et al., 2017).

Radiative-capture reaction rates in nuclear physics are also formulated in imaginary time, with temperature-dependent rates computed via propagation of channel-coupled wavepackets and operator insertions, bypassing explicit summation over scattering states and continuum normalization complexities (Hikota et al., 2015).

6. Variational, Multi-Time, and Hybrid Algorithmic Advancements

Measurement-efficient and high-fidelity ITP can be achieved through variational imaginary-time methods, hybrid quantum-classical iteration, and the partitioning of time slices. In the Multiple-Time QITE (MT-QITE) algorithm (Castillo et al., 11 Dec 2025), allocating independent imaginary-time parameters to each Hamiltonian term and reusing tomography across trial times dramatically reduces measurement budgets and enhances convergence fidelity, while allowing parallel execution across nonlocal partitions.

Alternative quantum algorithms eschew heavy classical postprocessing by building instantaneous orthogonal bases at each time step, minimizing basis size and circuit depth without locality assumptions (Jouzdani et al., 2022). In integrated quantum dynamical emulation (ITQDE) (Leamer et al., 2024), nonunitary propagators such as $e^{-\tau H^2}$ are constructed from weighted superpositions of time-symmetric unitaries, enabling extraction of spectra, ground- and excited-state observables, and even fluctuation theorems from real-time unitary circuits and Hadamard-test overlap measurements.

7. Numerical Performance, Hardware Demonstrations, and Practical Considerations

Quantum simulations using ITP have been validated on superconducting and ion-trap hardware for molecular models (e.g., hydrogen, few-spin nuclear Hamiltonians). Results report high-fidelity ground-state preparation within a few Trotter–Suzuki steps, with further stabilization via amplitude amplification (Turro et al., 2021):

System	Method/Hardware	Fidelity	Comments
Hydrogen atom	IBM Quantum, GRAPE	0.942–0.9978	Amplitude amplification, optimal control
2-neutron system	Transmon simulation	0.9919–0.999999996	Robust to decoherence/noise
Short-step ITP	IBMQ Manila	<20% infidelity	Energetic convergence in few steps

Classical implementations routinely scale to thousands of states in large grid calculations, with high-order operator splitting, efficient orthonormalization, and adaptive step-size leading to rapid, mesh-independent convergence (Luukko et al., 2013, Bader et al., 2013). For multi-layer tensor methods, spectral-gauge and number-sector-conserving contractions minimize computational overhead and guarantee physicality (Weike et al., 2019).

Limitations include exponential resource scaling in certain measurement-driven schemes, gate-depth challenges for simulating large non-stoquastic Hamiltonians, and the need for aggressive mitigation strategies against non-unitary errors and postselection bias on NISQ hardware. Nonetheless, ITP remains foundational for preconditioning, state preparation, and hybrid classical–quantum workflows in the push toward scalable quantum simulation.