Quantum Imaginary Time Evolution

• Quantum Imaginary Time Evolution (QITE) is a family of algorithms that simulates imaginary-time dynamics via unitary approximations, essential for ground-state preparation.
• It employs local linear-system methods and variational techniques to balance domain size with computation accuracy, managing the trade-off between resource scaling and fidelity.
• QITE’s adaptable framework extends to finite-temperature simulations, optimization problems, and solving PDEs, broadening its applications in quantum computing.

Quantum Imaginary Time Evolution (QITE) is a family of quantum algorithms that approximates the non-unitary imaginary-time propagator $e^{-\beta H}$ by operations that can be executed on quantum hardware. Its canonical use is ground-state preparation: if an initial state $\ket{\psi_0}$ has nonzero overlap with the ground state $\ket{\psi_{\mathrm{gs}}}$, then the normalized state

$$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$

approaches $\ket{\psi_{\mathrm{gs}}}$ as $\beta\to\infty$. Because gate-model hardware implements unitary dynamics rather than $e^{-\beta H}$, QITE replaces each short imaginary-time step by a unitary, a parameter update, or a postselected block-encoding surrogate. In the literature this basic idea has been developed into deterministic local-unitary schemes, McLachlan-projected variational flows, randomized-metric accelerations, coherent double-bracket constructions, fragmented block-encoding algorithms, and specialized methods for finite-temperature observables, combinatorial optimization, molecular electronic structure, and linear PDEs (Andreu et al., 2 Oct 2025, Kolotouros et al., 2024, Kumar et al., 2024).

1. Mathematical basis and geometric interpretation

The foundational observation is that imaginary-time evolution suppresses excited-state components relative to the ground-state component. In one common normalized form, the evolution satisfies

$$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$

while a finite-time estimate for approximate preparation is

$$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$

with $\Delta\le |E_1-E_0|$ a lower bound on the gap and $\ket{\psi_0}$0. This makes explicit that overlap and spectral gap jointly control the useful imaginary-time scale (Kaicher et al., 2024).

A complementary formulation treats imaginary-time evolution as a geometric flow. Writing the pure state as the projector $\ket{\psi_0}$1, one has

$\ket{\psi_0}$2

In this representation the flow is the Riemannian gradient flow of the cost

$\ket{\psi_0}$3

on the unitary orbit $\ket{\psi_0}$4, with

$\ket{\psi_0}$5

For the QITE specialization $\ket{\psi_0}$6, $\ket{\psi_0}$7, the cost is energy-equivalent: $\ket{\psi_0}$8 and the energy obeys

$\ket{\psi_0}$9

This identifies energy variance as the instantaneous speed of cooling and makes clear why eigenstates are stationary points of the flow (Zander et al., 1 Apr 2025).

2. Original QITE algorithm

The original QITE construction discretizes imaginary time into steps of size $\ket{\psi_{\mathrm{gs}}}$0 and seeks, at each step, a Hermitian generator $\ket{\psi_{\mathrm{gs}}}$1 such that

$\ket{\psi_{\mathrm{gs}}}$2

Expanding

$\ket{\psi_{\mathrm{gs}}}$3

in a Pauli basis and linearizing $\ket{\psi_{\mathrm{gs}}}$4 yields a quadratic least-squares objective

$\ket{\psi_{\mathrm{gs}}}$5

with

$\ket{\psi_{\mathrm{gs}}}$6

Minimization gives the linear system

$\ket{\psi_{\mathrm{gs}}}$7

This is the computational core of tomography-based QITE: measure state-dependent correlators, solve a classical linear system, build $\ket{\psi_{\mathrm{gs}}}$8, and implement the corresponding unitary (Andreu et al., 2 Oct 2025).

Scalability is obtained by exploiting locality. If

$\ket{\psi_{\mathrm{gs}}}$9

with $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$0 $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$1-local, then one Trotterizes

$$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$2

with Trotter error $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$3, and replaces each local non-unitary factor by a unitary acting on a domain of $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$4 qubits around the support of $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$5. Larger $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$6 captures more entanglement and correlations and generally improves fidelity, but the basis size grows roughly as $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$7. In broader many-body and molecular settings, domain choice can be guided by Manhattan distance in lattice systems or by mutual-information-selected orbitals in electronic structure problems, and the distinction between exact global-domain QITE and inexact small-domain QITE becomes practically decisive (Andreu et al., 2 Oct 2025, Kaicher et al., 2024).

3. Variational and metric-based formulations

Variational QITE (varQITE) replaces the explicit construction of a fresh $$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$8 at every step by a parameterized circuit

$$\ket{\psi(\beta)}= \frac{e^{-\beta H}\ket{\psi_0}} {\sqrt{\langle \psi_0|e^{-2\beta H}|\psi_0\rangle}}$$9

and projects imaginary-time dynamics onto the ansatz manifold through the McLachlan variational principle,

$\ket{\psi_{\mathrm{gs}}}$0

This yields

$\ket{\psi_{\mathrm{gs}}}$1

with

$\ket{\psi_{\mathrm{gs}}}$2

and

$\ket{\psi_{\mathrm{gs}}}$3

In discrete form,

$\ket{\psi_{\mathrm{gs}}}$4

Operationally, original QITE is deterministic and systematically improvable through the domain size $\ket{\psi_{\mathrm{gs}}}$5, whereas varQITE is easier to deploy on hardware but depends strongly on ansatz expressivity, initialization, and trainability; barren plateaus and convergence to local minima are explicit concerns (Andreu et al., 2 Oct 2025).

A closely related formulation writes the variational flow in terms of the quantum Fisher information matrix (QFIM),

$\ket{\psi_{\mathrm{gs}}}$6

so that the main cost of standard variational QITE is the $\ket{\psi_{\mathrm{gs}}}$7 metric estimation, which requires $\ket{\psi_{\mathrm{gs}}}$8 state preparations for $\ket{\psi_{\mathrm{gs}}}$9 parameters. Random-measurement imaginary-time evolution (RMITE) accelerates this by estimating the QFIM from random-basis measurements. For an exact unitary $\beta\to\infty$0-design $\beta\to\infty$1,

$\beta\to\infty$2

and one sample costs at most $\beta\to\infty$3 state preparations, so $\beta\to\infty$4 samples cost $\beta\to\infty$5. A more aggressive family replaces the QFIM by averaged classical Fisher information matrices (CFIMs); these do not guarantee exact McLachlan geometry, but they do satisfy an energy-descent theorem. The relation

$\beta\to\infty$6

is presented as a conjecture rather than a theorem (Kolotouros et al., 2024).

4. Resource scaling, stability, and approximation regimes

For original QITE, the brute-force Pauli basis on $\beta\to\infty$7 qubits requires $\beta\to\infty$8 expectation values, while a local-domain implementation reduces the dominant linear system to roughly $\beta\to\infty$9. In practical code, the main bottleneck for large $e^{-\beta H}$0 is solving a linear system of $e^{-\beta H}$1 equations, typically via least squares. The matrix $e^{-\beta H}$2 is often singular, so one uses a pseudo-inverse or least-squares solve; analogous ill-conditioning can arise in varQITE whenever $e^{-\beta H}$3 is required (Andreu et al., 2 Oct 2025).

The approximation regime is highly problem dependent. In heuristic many-body benchmarks, QITE with a short-range domain can track Trotterized ITE well for some local systems, but long-range spin models, strongly correlated molecular active spaces, and long imaginary-time horizons often destabilize small-domain QITE. The same benchmark study concludes that success is strongly contingent on initial-state quality and on whether the physics can be captured by small local domains (Kaicher et al., 2024).

Norm handling is also not a peripheral issue. Standard ground-state QITE uses normalized trajectories, but excited-state and Krylov-style extensions require accurate norm reconstruction. This has motivated dedicated improvements to the QITE equations and norm estimation procedures in molecular settings, including quantum Lanczos excited-state calculations and folded-spectrum QITE for general excited states (Tsuchimochi et al., 2022).

5. Algorithmic variants and hardware-oriented refinements

Several later variants modify either the update rule or the circuit realization while preserving the basic imaginary-time objective. One line replaces the full local unitary compilation by a qDRIFT-style randomized implementation of the QITE generator. In time-dependent drifting QITE,

$e^{-\beta H}$4

is not implemented as a full product formula; instead one samples one Pauli term with probability $e^{-\beta H}$5 and applies

$e^{-\beta H}$6

This removes depth dependence on the operator-pool size, gives inverse-linear convergence in the number of steps, and admits a cross-step measurement-reduction protocol whose total cost for an observable scales as

$e^{-\beta H}$7

independent of the number of time steps $e^{-\beta H}$8 (Huang et al., 2022).

A second line changes the operator manifold itself. For NISQ implementation, nonlocal approximation (NLA) and extended local approximation (eLA) relax the original locality constraint on $e^{-\beta H}$9. In a 10-vertex 3-regular max-cut instance, the paper reports a one-step circuit depth of $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$0 for LA-D6 versus $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$1 for NLA-D2, while NLA-D3 reached $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$2 on a problem whose ground-state energy is $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$3 (Nishi et al., 2020). Related work on dense classical optimization Hamiltonians introduces a reduced-parameter QITE ansatz in which entanglement is mediated by a small set of pivot qubits, enabling each compressed QITE layer to be implemented with constant two-qubit depth using dynamic fan-out circuits. On current IBM hardware the semi-classical adaptive variant performs favorably to the unitary implementation, whereas the fully dynamic construction exposes the trade-off between entangling-depth reduction and the overhead of mid-circuit measurement and feed-forward. Using a fidelity threshold of $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$4 relative to the noiseless QITE ansatz, the paper estimates that dynamic fan-out QITE would outperform unitary implementations when the measurement and two-qubit gate errors are reduced by $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$5 and the feedback latency is halved (Lund et al., 5 Mar 2026).

Other variants alter the time structure itself. Multiple-Time QITE (MT-QITE) assigns different imaginary times to different Hamiltonian partitions within one Trotter layer. Because all partition terms are treated from the same reference state, measurements can be reused across time-step scans, updates become parallelizable, and benchmark fidelity improved by one to two orders of magnitude after 10 Trotter steps while using about one order of magnitude fewer measurements (Castillo et al., 11 Dec 2025). Double-bracket QITE (DB-QITE) instead realizes imaginary-time descent coherently through the group-commutator approximation to Brockett’s double-bracket flow, with discrete cooling law

$$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$6

but it inherits intrinsic saddle points near excited eigenstates where the variance nearly vanishes (Zander et al., 1 Apr 2025). In the block-encoding setting, fragmented imaginary-time evolution factorizes

$$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$7

and reruns short probabilistic fragments sequentially, reducing wasted depth on failed runs. One primitive uses one single ancillary qubit throughout, and another saturates an imaginary-time no-fast-forwarding bound in the regime $$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$8 (Silva et al., 2021).

6. Applications and empirical behavior

Ground-state preparation in spin and molecular systems remains the central application. A review benchmark on the transverse-field Ising model used

$$\frac{\partial}{\partial \tau}\ket{\psi(\tau)}=-(H-E_\tau)\ket{\psi(\tau)}, \qquad E_\tau=\langle\psi(\tau)|H|\psi(\tau)\rangle,$$9

with $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$0, QITE domain sizes $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$1, and a two-repetition hardware-efficient varQITE ansatz. Energy converged on a similar timescale for exact ITE, QITE, and varQITE, while larger $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$2 improved QITE fidelity and energy accuracy (Andreu et al., 2 Oct 2025). On superconducting hardware, variational QITE was demonstrated for $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$3 and $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$4, with convergence within 4 iterations. For $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$5, the fidelity improved from about $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$6 to about $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$7; for LiH, a cluster-mean-field plus hardware-efficient implementation reached about $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$8, and a direct 4-qubit UCC implementation reached about $$\tau_\eta=\mathcal O\!\left(\frac{1}{\Delta}\log\left(\frac{1}{\gamma_{\text{init}}\eta}\right)\right),$$9 (Zong et al., 2023).

Molecular applications also show that initial-state design can dominate performance. For dissociating and open-shell systems, replacing Hartree–Fock by a broken-symmetry reference and adding a spin penalty,

$\Delta\le |E_1-E_0|$0

can accelerate convergence. In stretched $\Delta\le |E_1-E_0|$1 at $\Delta\le |E_1-E_0|$2 Å with diradical character $\Delta\le |E_1-E_0|$3, broken-symmetry QITE reached chemical precision in about $\Delta\le |E_1-E_0|$4 iterations versus about $\Delta\le |E_1-E_0|$5 from HF. In square $\Delta\le |E_1-E_0|$6, broken-symmetry QITE required about $\Delta\le |E_1-E_0|$7 iterations versus about $\Delta\le |E_1-E_0|$8 for HF. For $\Delta\le |E_1-E_0|$9 dissociation, the paper reports initial fidelities of about $\ket{\psi_0}$00 for RHF, about $\ket{\psi_0}$01 for BS2, and about $\ket{\psi_0}$02 for BS3 at $\ket{\psi_0}$03 Å, with the practical crossover for $\ket{\psi_0}$04 discussed near $\ket{\psi_0}$05 and the advantage especially pronounced by $\ket{\psi_0}$06 (Poudel et al., 25 Apr 2025).

QITE has also been used as an optimization algorithm. For PUBO problems such as weighted MaxCut and LABS, a separable linear ansatz

$\ket{\psi_0}$07

can already perform strongly. On weighted MaxCut, QITE with a separable ansatz often outperforms the Goemans–Williamson algorithm on graphs up to 150 vertices, with average approximation ratio around $\ket{\psi_0}$08 for $\ket{\psi_0}$09. On LABS, linear-QITE achieved average ground-state probability comparable with $\ket{\psi_0}$10 QAOA, and the tested entangling quadratic ansatz showed no significant advantage for sizes up to $\ket{\psi_0}$11 (Bauer et al., 2023).

Beyond zero-temperature ground states, QITE has been used for finite-temperature observables and dynamics. On IBM five-qubit devices, finite-temperature energies, static correlations, dynamical correlators, and excitation spectra were computed for spin Hamiltonians up to four sites. For two-site TFIM observables, the mean absolute percentage error over the tested $\ket{\psi_0}$12 range lay between $\ket{\psi_0}$13 and $\ket{\psi_0}$14, and the extracted spectral peaks at $\ket{\psi_0}$15 were close to exact values $\ket{\psi_0}$16 (Sun et al., 2020). A different generalization reinterprets QITE as a solver for linear PDEs by tracking not only the normalized trajectory but also the changing scale of the state vector; in numerical simulations of the heat equation, 1D and 2D solutions on six and ten qubits achieved perfect fidelity and zero mean squared error at $\ket{\psi_0}$17 in the reported exact-support cases (Kumar et al., 2024).

7. Limitations, misconceptions, and open problems

A recurrent misconception is to treat QITE as a single algorithm. The literature instead uses the term for several operationally distinct families: local linear-system QITE, McLachlan/VarQITE, randomized-metric approximations such as RMITE, coherent double-bracket schemes, fragmented operator-function methods, and problem-structured low-depth variants. What unifies them is the attempt to approximate the same normalized imaginary-time trajectory, not a single fixed circuit architecture (Andreu et al., 2 Oct 2025, Kolotouros et al., 2024, Silva et al., 2021).

The main limitations are equally recurrent. Original QITE is deterministic and ansatz-independent in the narrow sense used by the literature, but its cost depends strongly on domain size $\ket{\psi_0}$18, Hamiltonian partitioning, and repeated measurement of large correlation matrices. VarQITE can be easier to implement on hardware, yet its performance depends heavily on ansatz expressivity and initialization and may suffer from barren plateaus or convergence to local minima. In many realistic long-range or strongly correlated systems, small-domain QITE ceases to track ITE reliably, so exact behavior would require essentially global domains (Andreu et al., 2 Oct 2025, Kaicher et al., 2024).

Several open technical questions remain unsettled. For RMITE, the sample complexity $\ket{\psi_0}$19 needed for useful metric estimates and the efficient handling of measurement outcomes remain open, and the Haar-average identity for averaged CFIMs is still a conjecture rather than a theorem (Kolotouros et al., 2024). For DB-QITE, saddle points are intrinsic to the geometry rather than pathologies of implementation, but they can create low-variance bottlenecks that are practically severe (Zander et al., 1 Apr 2025). For MT-QITE, gains depend on partition quality and candidate-time optimization can grow combinatorially with the number of partitions (Castillo et al., 11 Dec 2025). For dynamic-circuit QITE, constant entangling depth does not by itself guarantee better hardware performance, because measurement error, feed-forward latency, and ancillary overhead can dominate on current devices (Lund et al., 5 Mar 2026).

Taken together, these results suggest that QITE is best understood not as a monolithic ground-state routine but as a broader imaginary-time methodology. Its central abstraction—the replacement of $\ket{\psi_0}$20 by implementable surrogates that preserve the normalized trajectory—has proved adaptable across ground states, excited states, thermal observables, optimization, and PDEs, but the decisive questions remain structural: overlap with the target state, the geometry of the chosen update manifold, the measurement budget, and the hardware cost of realizing the surrogate dynamics.