Imaginary-Time Quantum Dynamical Emulation

• ITQDE is a quantum simulation method that emulates imaginary-time evolution using ensembles of unitary operations to reveal ground state and thermal properties.
• It employs weighted unitary ensembles and quadrature-based filtering to reconstruct non-unitary propagation with controlled bias and variance.
• The approach enables efficient coarse-grained estimation of spectral and thermodynamic observables on near-term quantum devices with minimal circuit overhead.

Imaginary-Time Quantum Dynamical Emulation (ITQDE) refers to a class of methodologies and quantum algorithms that aim to emulate non-unitary imaginary-time evolution using sequences, ensembles, or combinations of physical (unitary) quantum operations. ITQDE is applicable to a wide range of quantum systems and underpins ground-state preparation, thermal state sampling, quantum criticality studies, and non-equilibrium response estimation, all within regimes relevant to quantum simulation, condensed matter theory, quantum information, and quantum hardware implementation. The central challenge addressed by ITQDE is accessing the dissipative, filtering, or relaxation effects of imaginary-time propagation—exemplified by $e^{-\tau H}$—on quantum computers that can natively perform only unitary operations.

1. Underlying Principles of ITQDE

Imaginary-time evolution transforms a state $|\psi_0\rangle$ via non-unitary propagation $$|\psi(\tau)\rangle \propto e^{-\tau H}|\psi_0\rangle$$, suppressing higher-energy components and revealing ground-state and thermal properties. Since $e^{-\tau H}$ is not directly implementable as a physical process, ITQDE seeks to “emulate” it using quantum hardware-accessible operations.

A foundational approach establishes a quantum dynamical emulation (QDE) map: a superoperator written as an explicit weighted sum (or integral) over unitaries,

$$\mathcal{L}[\rho] = \sum_{ij} k_{ij} \mathcal{U}_i \rho \mathcal{U}_j,$$

where $\mathcal{U}_i$ are physically implementable unitary operators and $k_{ij} \in \mathbb{R}$. Through careful parameter choices, the ensemble or averages over such evolutions recover the effective imaginary-time propagation, up to normalization and truncation errors (Leamer et al., 5 Mar 2024, McCaul, 4 Sep 2025).

In the continuous limit, a central identity employed is a Hubbard-Stratonovich-type equality:

$$e^{-\tau H^2} = \frac{1}{\sqrt{\pi}} \int_{-\infty}^\infty e^{-x^2} e^{-2i\sqrt{\tau} x H} dx,$$

which maps the dissipative filter to an ensemble of real-time unitary evolutions, thus connecting non-unitary and unitary dynamics via Gaussian quadrature (McCaul, 4 Sep 2025). More generally, weighted binomial or probabilistic mixtures of forward and backward real-time evolutions yield the desired non-unitary transformation (Leamer et al., 5 Mar 2024).

2. Algorithmic Strategies and Mathematical Formulation

Various ITQDE approaches are formulated as follows:

• Weighted Unitary Ensembles: Repeatedly apply unitary operators $\mathcal{U} = \exp(-i \sqrt{\Delta\tau/2} H)$ and their adjoints, constructing mixtures such as $$\mathcal{L}[\rho] = \frac{1}{2} (\mathcal{U}\rho \mathcal{U} + \mathcal{U}^\dagger\rho \mathcal{U}^\dagger)$$, which, after many steps, converge to an effective imaginary-time evolution conditioned on overlaps between oppositely evolved states (Leamer et al., 5 Mar 2024).
• Quadrature-Based Filtering: Discretize the Hubbard-Stratonovich integral using Gauss-Hermite quadrature, converting the filter $e^{-\tau(H-\lambda)^2}$ to a finite sum:

$$\langle Z_m^{(\lambda)} \rangle = \sum_{k=1}^m w_k \operatorname{Re}\left[e^{2i\lambda \tau_k} \langle \psi_{-\tau_k} | \psi_{+\tau_k} \rangle\right],$$

where $$|\psi_{\pm \tau_k}\rangle = e^{\mp i \tau_k H}|\psi_0\rangle$$, $w_k$ are quadrature weights, and $\tau_k = \sqrt{\tau} x_k$ with $x_k$ the quadrature nodes (McCaul, 4 Sep 2025). This enables computation of filtered expectation values and spectral densities.

• Expectation Value Estimation: The central observable—e.g., energy or partition function—is then evaluated as

$$F(\lambda) = \frac{\sum_k w_k \operatorname{Re}\left[e^{2i\lambda\tau_k} \langle \psi_{-\tau_k}|H|\psi_{+\tau_k}\rangle\right]}{\sum_k w_k \operatorname{Re}\left[e^{2i\lambda\tau_k} \langle \psi_{-\tau_k}|\psi_{+\tau_k}\rangle\right]}.$$

• Adaptive Smoothing and Control: Gaussian convolution windows $W_{\delta\lambda}(\lambda-\lambda')$ can be applied to smooth the outputs, suppress high-frequency noise, and trade resolution for sampling variance by adjusting the effective filter parameter $\tau_{\text{eff}} = \tau/(1 + \tau\delta\lambda^2)$ (McCaul, 4 Sep 2025).
• Sampling and Stochastic Trace Estimation: For system-wide estimation, random input states drawn from unitary 2-designs are used to stochastically estimate traces, ensuring an unbiased estimator with variance scaling as $1/\sqrt{K}$ with the number of samples $K$.

3. Complexity, Resolution, and Controlled Bias–Variance Tradeoffs

A central strength of ITQDE is its scaling regime for extracting coarse-grained spectral information. The total number of required overlap evaluations (unary depth) to approximate filtered expectation values to precision $\epsilon$ grows only linearly with the imaginary-time duration $\tau$ and the width of the filter (McCaul, 4 Sep 2025):

• For targeted bandwidth $\kappa_{\text{loc}} = \Delta_{\max}/\Delta_{\min}$ (max-min resolved gap within a spectral window), the sampling budget $K$ must satisfy

$$K \gtrsim \exp\left[c(\Delta_{\max}/\Delta_{\min})^2\right]$$

to resolve gaps $\Delta_{\min}$ in regions where the spectrum is locally dense. However, for “bulk” or coarse-grained questions (e.g., partition function, density of states, spectral plateaux), $\tau$ can remain small, and $K$ is polynomial in system size (McCaul, 4 Sep 2025).

Bias–variance tradeoff can be tuned via the degree of smoothing:

• Larger filter width (smaller $\tau$) increases bias but reduces variance, enabling practical computation of thermodynamic observables.
• For fine-grained eigenvalue resolution (small gaps), exponential growth of the sampling budget is unavoidable; ITQDE does not alter QMA-completeness of exact ground-state estimation.
• Quadrature Discretization: Truncation of the quadrature sum to the dominant nodes is both permissible and scaling-optimal due to exponential decay of high-order weights, minimizing circuit depth without incurring significant bias until resolution approaches the local gap scale.

4. Applications and Experimental Regimes

ITQDE enables various tasks with polynomial resources in hardware, depth, and post-processing:

• Ground-state energy estimation through “spectral staircase” analysis, revealing plateaux corresponding to energy levels via filtered expectation values as a function of probe parameter $\lambda$ (Leamer et al., 5 Mar 2024, McCaul, 4 Sep 2025).
• Partition function and thermodynamic observable computation: integration over the filter yields thermal averages accessible for moderate $\beta$.
• Spectral density analysis by resolving the approximate density of states, with logistic profile observed across spectral gaps.
• Gap estimation and extraction of spectral transitions between eigenstates via fits to logistic forms around transition regions.
• Coarse-grained measurement of “bulk” or aggregate spectral properties far more efficiently than exact spectral estimation would allow.

Regime of practicality: The so-called “free snack” region defines a bandwidth domain (in energy or $\lambda$) where coarse spectral information is accessible efficiently, while fine eigenvalue resolution and exponentially small energy gaps remain prohibitively costly (McCaul, 4 Sep 2025).

The required physical operations involve basic unitary evolutions and overlap measurements (e.g., via the Hadamard test or direct measurement); advanced state preparation or ancilla technologies are unnecessary. The only significant non-unitary operation is in classical post-processing, sidestepping major practical barriers to quantum simulation.

5. Methodological Variants and Generalizations

Multiple ITQDE variants exist:

• QDE/ITQDE via forward/backward time evolution: Binomial-weighted mixtures of real-time unitary evolutions and their adjoints, with overlaps reconstructing the imaginary-time filter (Leamer et al., 5 Mar 2024).
• Quadrature-based ITQDE: Direct mapping of non-unitary filters to ensembles of variable-duration unitaries via discrete quadrature, as outlined above (McCaul, 4 Sep 2025).
• Measurement-based and probabilistic schemes: (See related, but not identical, approaches in (Xie et al., 2022, Mao et al., 2022)) combine weak measurements, conditional unitaries, and post-selection to emulate imaginary-time dynamics with manageable overhead, though with varying success probability and sampling cost.
• Smoothing/Filtering: Explicit bias–variance control via convolving ITQDE estimators with symmetric windows enables user control of spectral resolution and sampling stability.
• Generalization to operator-valued filtering: Allows Gaussian or other analytic filters beyond the simple $e^{-\tau(H-\lambda)^2}$.

These approaches differ in details of circuit structure, normalization, applicability to exact eigenvalue estimation, and resource scaling; all share the core idea of avoiding explicit non-unitary propagation by using quantum-accessible unitary processings and classical recombination.

6. Impact on Quantum Complexity and Near-Term Quantum Simulation

ITQDE does not evade the hardness of the local Hamiltonian problem: exact resolution of exponentially small gaps or ground-state energies remains QMA-hard. However, it provides a “middle ground” where

• Bulk spectral properties and thermal averages become tractable at polynomial sampling and circuit cost,
• Fine-grained spectral features remain complex and resource-intensive but are explicitly linked to sampling bandwidth and filter parameters,
• Minimal hardware requirements (no ancillas, depth-efficient circuits, no explicit state preparation) make ITQDE highly suitable for NISQ and early fault-tolerant quantum devices,
• Bias–variance management is principled, transparent, and tunable, allowing algorithmic adjustment to available hardware and experimental tolerances.

These features enable meaningful “coarse-grained” quantum simulations for applications in condensed matter physics, quantum chemistry, statistical mechanics, and quantum information even before scalable quantum error correction.

7. Outlook, Open Issues, and Limitations

• The quadrature approach to ITQDE achieves exponentially decreasing discretization error at linear growth of required quantum resources with respect to the filtering parameter for fixed resolution (McCaul, 4 Sep 2025).
• The method is robust to coherent control imperfections and does not require amplitude amplification or complex state preparation (Leamer et al., 5 Mar 2024, McCaul, 4 Sep 2025).
• The necessary shot (sampling) overhead becomes prohibitive only when attempting to resolve bandwidths much smaller than the typical spectral gap; thus, ITQDE is best applied to bulk property estimation.
• There is an inherent limitation: as the desired energy resolution approaches local spectral gaps, sampling costs increase exponentially, preserving the fundamental QMA-hardness of exact eigenvalue estimation.
• Extensions to dynamical response and open system settings, as well as connections to thermodynamic fluctuation theorems, have been articulated (Leamer et al., 5 Mar 2024).

The ITQDE methodology thus delineates a “free snack” regime in quantum complexity, allowing efficient access to meaningful spectral information and thermodynamic properties on near-term hardware without violating computational hardness results for exact ground-state problems.