Quantum Imaginary-Time Evolution (QITE)

• Quantum Imaginary-Time Evolution (QITE) is a method that approximates the non-unitary cooling process through sequences of localized unitary operations.
• It achieves polynomial resource scaling per iteration, significantly reducing computational cost compared to classical methods that require exponential resources.
• QITE enables efficient preparation of ground, excited, and thermal states on near-term devices by avoiding deep circuits, ancilla overhead, and extensive optimization.

Quantum Imaginary-Time Evolution (QITE) is a quantum algorithmic paradigm that enables the efficient computation of ground, excited, and thermal states of quantum Hamiltonians through iterative quantum operations that mimic the cooling process of classical imaginary-time evolution. QITE achieves this by systematically replacing non-unitary imaginary-time operations, which cannot be directly realized on quantum hardware, with a sequence of unitary transformations acting on localized domains of the quantum register. This approach circumvents several bottlenecks of classical algorithms and alternative quantum methods, offering polynomial resource scaling per iteration under realistic physical assumptions, and is compatible with near-term quantum devices due to its low circuit depth and absence of ancillary overhead.

1. Principles and Mathematical Formulation

QITE is founded on the classical imaginary-time evolution method, where a trial state $|\Phi(0)\rangle$ is evolved according to the differential equation

$$-\partial_\beta |\Phi(\beta)\rangle = H |\Phi(\beta)\rangle$$

such that, in the limit $\beta \to \infty$, the normalized state converges to the ground state,

$$|\Psi\rangle = \lim_{\beta \to \infty} \frac{|\Phi(\beta)\rangle}{\|\Phi(\beta)\|}$$

as long as $\langle\Phi(0)|\Psi\rangle \neq 0$.

Since direct implementation of the non-unitary operator $e^{-\beta H}$ is not possible on a quantum computer, QITE achieves an effective approximation by representing each small imaginary-time step $e^{-\Delta\tau h_m}$ (with $h_m$ a local Hamiltonian term) via a locally acting unitary $e^{-i A_m}$, where $A_m = \sum_I a[m]_I \sigma_I$ is an expansion in a local operator basis (e.g., tensor products of Pauli operators). The coefficients $a[m]_I$ are determined dynamically by minimizing the distance between the true imaginary-time evolved and the unitary-approximated wavefunction using measurements and a linear solve over the local domain.

This process is systematically Trotterized over the full Hamiltonian: $$e^{-\beta H} \approx \left( e^{-\Delta\tau h_1}\, e^{-\Delta\tau h_2}\, ...\, e^{-\Delta\tau h_K} \right)^{\beta/\Delta\tau}$$ with each $e^{-\Delta\tau h_m}$ approximately replaced by $e^{-iA_m}$ acting only on a domain whose size is determined by the system's correlation length. This approach is justified by Uhlmann’s theorem, which ensures that, for finite correlation lengths, local unitaries suffice to approximate imaginary-time evolution on geometrically local Hamiltonians.

2. Quantum Resource Efficiency and Classical Comparison

Classically, imaginary-time methods and Lanczos algorithms for Hamiltonians on $N$ qubits require exponential space and time per iteration due to the necessity of storing and manipulating exponentially large vectors (Krylov subspace vectors: $|\Phi\rangle,H|\Phi\rangle,H^2|\Phi\rangle,...$). Specifically, the cost scales as $\exp(\mathcal{O}(N))$ or $\exp(\mathcal{O}(C^d))$ for $d$-dimensional systems with correlation length $C$.

QITE, in contrast, reduces the required resources per iteration exponentially. Provided the state’s correlations are localized within a finite domain ($D\ll N$), it suffices to update local regions, resulting in polynomial scaling with system size for each Trotterized update. The quantum version of the Lanczos algorithm further leverages this by building a Krylov subspace from the locally updated QITE states and similarly achieves polynomial cost under the same domain-size assumptions.

The absence of deep circuits, ancillae, and high-dimensional classical optimization steps—required for phase estimation and variational quantum eigensolvers (VQEs)—makes QITE particularly suited for current and near-term hardware.

3. Implementation Strategy and Circuit Structure

QITE’s implementation proceeds by:

1. Decomposition and Trotterization: Fragmenting the Hamiltonian $H$ into local terms $h_m$ and performing a Trotter approximation for the full time evolution.
2. Local Unitary Synthesis: For each $h_m$, measuring local observables to determine the coefficients $a[m]_I$ in $A_m$ by solving a linear system that minimizes the norm $\| |\bar\Psi'\rangle - e^{-iA_m}|\Psi\rangle \|^2$, where $|\bar\Psi'\rangle$ is the normalized output of imaginary-time evolution for $h_m$.
3. Iterative Circuit Construction: Sequentially applying the corresponding $e^{-iA_m}$ gates, each acting on $O(D)$ qubits, leading to shallow circuits even for large $N$ if the correlation length remains small.
4. Domain Size Consideration: For unentangled (product) initial states, $D$ is minimal; as the state entangles during evolution, $D$ may increase but remains bounded for systems with finite correlation length.

Prototyping was performed both on the Rigetti Quantum Virtual Machine (QVM) and the Aspen-1 QPU, with explicit measurement of local Pauli strings for circuit synthesis and Trotter steps corresponding to local mean-field or correlated domains (adjusted by increasing $D$ for accuracy).

4. Applications: Ground, Excited, and Thermal State Preparation

QITE and its extensions are applicable to a wide range of settings:

• Quantum Simulation: Enabling ground state (and, via the quantum Lanczos algorithm, excited state) preparation for arbitrary Hamiltonians, including those relevant to quantum chemistry and materials science.
• Optimization Problems: Solving combinatorial and hard optimization problems mapped to finding Hamiltonian ground states.
• Thermal State Sampling: Employing QITE as a generator for Gibbs/thermal averages via the quantum minimally entangled typical thermal states (QMETTS) protocol. Here, an initial product state is evolved via QITE and measured, then “collapsed” back to a new product state, facilitating sampling thermal traces:

$$\langle \hat O \rangle = \frac{\operatorname{Tr}[\hat O e^{-\beta H}]}{Z}$$

without requiring deep circuits or ancillae, which is valuable for studying finite-temperature phase transitions.

• Hybrid Algorithms: Combining QITE with Krylov subspace methods (quantum Lanczos) enables rapid convergence to ground and excited states without the exponential memory or optimization costs of their classical analogs.

In all these applications, QITE is markedly more resource-efficient than phase estimation (no deep ancilla-based circuits) and VQE (no outer-loop high-dimensional optimization).

5. Experimental Demonstrations and Performance

Experimental and numerical results support the practical feasibility of QITE:

• Classical Emulation: QITE was benchmarked on 1D Heisenberg chains, antiferromagnetic transverse-field Ising models, long-range Heisenberg systems, a Hubbard chain, MAXCUT instances, and the minimal basis H$_2$ molecule. The ground state energy converges consistently as domain size $D$ increases.
• Noise Robustness: In simulated and physical experiments (Rigetti QVM and Aspen-1 QPU), QITE achieved good convergence to ground-state energies even in the presence of realistic noise (decoherence, readout, and gate errors). For example, QITE achieved comparable measurement overhead and energy accuracy to leading VQE protocols on the QVM and robust convergence on the two-qubit QPU.
• Resource-Error Tradeoffs: Using reduced domain size $D$ leads to fewer gates and shorter circuits, with an associated tradeoff between accuracy and circuit depth. Empirically, stopping QITE after initial convergence (before oscillations) yields accurate estimations of ground state energies.

A summary table of observed behaviors (for $D$ choices and system sizes) is:

Domain Size $D$	Accuracy (Ground State Energy)	Circuit Depth
$D=1$	Mean-field	Minimal
$D>1$	Tracks classical (exact) result	Increases
Max $D$	Exact to classical precision	Maximal

6. Impact, Limitations, and Outlook

QITE’s resource efficiency, avoidance of high-depth circuits and ancillae, and independence from complex optimization routines provide a compelling route for scalable quantum simulation, optimization, and eigenstate preparation on near-term quantum devices. The method’s utility is further amplified by its role as a subroutine in more advanced protocols for thermal sampling and excited state preparation.

Key caveats and open directions include:

• Correlation Length Dependency: The assumption of a finite, bounded correlation length is crucial for domain localization and polynomial resource scaling.
• Entanglement Growth: As imaginary-time evolution entangles the initially local state, the effective domain size $D$ may need to be increased, impacting circuit resources.
• Stopping Criteria: With non-maximal $D$, oscillatory behavior in energy convergence may appear. Selecting an appropriate stopping rule is important for practical accuracy.
• Noise and Error: While relatively robust, further error mitigation and adaptive domain management may be required as hardware scales or for more challenging Hamiltonians.

The QITE framework outlined in this work establishes a foundation for advanced quantum algorithms that can tackle physically and computationally complex problems in simulation, optimization, and machine learning using resource-efficient and hardware-compatible strategies.