Measurement-Dressed Imaginary-Time Evolution

• Measurement-Dressed Imaginary-Time Evolution (MDITE) is a quantum algorithm framework that deterministically simulates imaginary-time dynamics using weak QND measurements, adaptive unitaries, and nonlinear backaction.
• The protocol iterates measurement and feedback steps to project any initial state onto a target energy eigenstate with polynomial resource scaling and high fidelity.
• MDITE enables deterministic preparation of complex multi-qubit states, like four-qubit cluster states, overcoming the exponential post-selection challenges of traditional methods.

Measurement-Dressed Imaginary-Time Evolution (MDITE) is a class of quantum algorithms that deterministically simulate non-unitary imaginary-time dynamics by leveraging weak quantum measurements, adaptive feedback unitaries, and nonlinear measurement backactions. MDITE has been developed to circumvent both the exponential post-selection barrier inherent in trajectory-based quantum simulations and the implementation challenge of non-unitary evolutions on quantum hardware. MDITE protocols have demonstrated deterministic ground-state preparation, efficient trajectory engineering, and polynomial resource scaling in scenarios where conventional approaches fail. They are realized through repeated rounds of weak quantum nondemolition (QND) measurements and stroboscopic updates, often tailored to the structure of the target Hamiltonian. The following sections detail the principles, measurement mechanics, feedback rules, multi-qubit effects, stroboscopic implementation, and prominent applications of MDITE.

1. Algorithmic Foundations and Stepwise Protocol

MDITE operates on an N-qubit system with Hamiltonian $H$ and arbitrary initial state $|\psi_0\rangle$. The goal is to project $|\psi_0\rangle$ deterministically onto a selected energy eigenstate $|E_{\text{tgt}}\rangle$. The protocol is iterative:

1. Weak QND Measurement: Perform a QND measurement of $H$ via a weak probe, yielding photon-count outcomes $(n_c,n_d)$. The system state is updated by applying the Kraus operator

$$M_{n_c n_d}(\tau) = \sum_{n} C_{n_c n_d}(E_n \tau) |E_n\rangle\langle E_n|,$$

resulting in state $$|\tilde\psi_{n_c n_d}\rangle = M_{n_c n_d}(\tau) |\psi_{t-1}\rangle$$.

2. Energy Estimation: The peak position of $C_{n_c n_d}(\chi)$ gives a read-out energy estimate within the regime $0 \leq E_n\tau \leq \pi/2$:

$$E_{\text{est}}(n_c, n_d) = \frac{1}{2\tau}\arccos\left(\frac{n_c-n_d}{n_c+n_d}\right).$$

3. Adaptive Unitary Feedback: Based on $E_\text{est}$, apply a feedback unitary:

$$U_{n_c,n_d} = \begin{cases} I, & |E_{\text{est}} - E_{\text{tgt}}| < \delta_{\text{tgt}}, \ U_C, & \text{otherwise} \end{cases}$$

Here, $U_C$ is chosen to transfer amplitude from non-target eigenstates to the target subspace. A diagonal phase-correction $V_{n_c,n_d}$ may also be included:

$V_{n_c n_d} = \exp[i((n_c+n_d)\bmod 2) H\tau].$

4. State Renormalization:

$$|\psi_t\rangle = \frac{U_{n_c, n_d} M_{n_c n_d}(\tau) |\psi_{t-1}\rangle}{\sqrt{\langle\psi_{t-1}|M_{n_c n_d}^\dagger M_{n_c n_d}|\psi_{t-1}\rangle}}.$$

5. Iteration and Convergence: Steps (i)-(iv) are iterated. The Gaussian envelope of the modulation function shrinks around the targeted energy, and only $|E_\text{tgt}\rangle$ remains as a fixed point.

This protocol ensures that, irrespective of stochastic measurement results, the only stable fixed point is the desired energy eigenstate, thus establishing deterministic imaginary-time propagation (Kondappan et al., 2022).

2. QND Measurement Forms and Nonlinear Backaction

The QND measurement interaction is engineered via coupling: $H_{\text{int}} = \hbar\Omega H \otimes a^\dagger a$ where mode $a$ is typically realized within a Mach–Zehnder interferometer. Coherent probe fields yield Kraus operators after photon detection: $$M_{n_c n_d}(\tau) = \sum_{n=0}^{d-1} C_{n_c n_d}(E_n\tau) |E_n\rangle\langle E_n|$$ with modulation function

$$C_{n_c n_d}(\chi) = \frac{\alpha^{n_c+n_d} e^{-|\alpha|^2/2}}{\sqrt{n_c! n_d!}} \cos^{n_c}(\chi) \sin^{n_d}(\chi).$$

For sufficiently large probe photon number, Stirling’s approximation models $C_{n_c n_d}(\chi)$ as a signed Gaussian near its peak: $$C_{n_c n_d}(\chi) \approx s_{n_c n_d}(\chi) A_{n_c n_d} \exp\left[ -\frac{(\cos 2\chi - \frac{n_c-n_d}{n_c+n_d})^2}{8 \tilde\sigma^2_{n_c n_d} } \right].$$ The nonlinear character of this measurement allows for cross-eigenstate backaction, which is essential for enabling multi-qubit interactions and nontrivial imaginary-time evolution (Kondappan et al., 2022).

3. Adaptive Feedback: Energy Estimation and Correction

At each round, the energy estimate $E_{\rm est}(n_c,n_d)$ is compared to the target $E_{\text{tgt}}$ within a specified tolerance $\delta_{\text{tgt}}$. Sequence counters accumulate only if the system is within the energetic window: $$m_c^{(t+1)} = \begin{cases} m_c^{(t)} + n_c, & |E_{\rm est}(m_c^{(t)}+n_c,m_d^{(t)}+n_d)-E_{\rm tgt}| < \delta_{\rm tgt} \ 0, & \text{otherwise} \end{cases}$$ The unitary $U_C$ is selected to induce transitions (e.g., single-qubit flips, Hadamards) that re-populate the target subspace. When phase corrections are required (in long-time and multi-qubit regimes), $V_{n_c n_d}$ compensates for sign ambiguities in the effective Kraus action.

Only the ground state (or targeted eigenstate) remains invariant after repeated measurements and feedback, with all other components being exponentially suppressed (Kondappan et al., 2022).

4. Stroboscopic Approximation to Imaginary-Time Evolution

The cumulative effect of repeated measurement-feedback cycles is a narrowing envelope around the target energy: $$\prod_{t=1}^T M_{n_c^{(t)}n_d^{(t)}} \approx A \exp\left[-\frac{(\cos 2\chi - \frac{n_c^{\rm tot}-n_d^{\rm tot}}{n_c^{\rm tot}+n_d^{\rm tot}})^2}{8 \sigma^2}\right].$$ This effectively yields

$$[M(E_{\rm tgt})]^k \propto \sum_n \exp\left(-\frac{k(E_n - E_{\rm tgt})^2}{2\sigma^2}\right) |E_n\rangle\langle E_n| \xrightarrow{k \to \infty} |E_{\rm tgt}\rangle\langle E_{\rm tgt}|$$

representing a projective non-unitary dynamics, stroboscopically approximating $e^{-\tau H}$.

For multi-term Hamiltonians, Trotterization—alternating MDITE applications for each term—enables scalable application to general interacting systems.

5. Generation of Effective Multi-Qubit Interactions

MDITE leverages the nonlinear dependence of weak QND measurements so that only single-qubit light couplings are required to induce multi-qubit imaginary-time operators. For "long-time" interactions ($\tau = \pi/4$): $$\sin\left(\frac{\pi}{2} \sum_{n=1}^N \sigma^z_n\right) = \cos\left(\frac{\pi}{2}(N+1)\right) \prod_{n=1}^N \sigma^z_n$$ and the measurement-induced operator

$$M_{n_c n_d}(\pi/4) \propto \exp\left[\frac{n_d - n_c}{n_c + n_d} \frac{\prod_{n=1}^N \sigma^z_n}{4 \sigma_{n_c n_d}^2}\right]$$

resembles an effective N-body imaginary-time interaction, thereby generalizing the protocol for arbitrary degrees of entanglement and nonlocality (Kondappan et al., 2022).

6. Four-Qubit Cluster-State Preparation: Explicit Construction and Performance

An explicit demonstration of MDITE is the deterministic preparation of the four-qubit cluster state

$$|C_4\rangle = \frac{1}{2}(|0000\rangle + |0011\rangle + |1101\rangle + |1110\rangle)$$

which is the ground state of

$$H_{C_4} = -\sigma^z_1\sigma^z_2 - \sigma^x_3\sigma^x_4 - \sigma^x_1\sigma^x_2\sigma^x_3 - \sigma^z_2\sigma^z_3\sigma^z_4.$$

Each term is handled in a series of MDITE segments, each with appropriate QND Hamiltonian, measurement time $\tau$, feedback unitary $U_C$, and phase correction $V$. For example, to realize $e^{\sigma^z_1\sigma^z_2 t}$, one employs QND coupling to $-(σ^z_1 - σ^z_2)$ and adaptive feedback on $σ^x_1$.

Numerical results show each MDITE segment converges in approximately 10 rounds to unit fidelity within its stabilizer subspace. The final cluster-state fidelity

$F_C = |\langle C_4 | \psi \rangle |^2$

reaches unity. The process is deterministic with nearly unity success probability, obviating the need for post-selection. Error analysis indicates robust performance for detector efficiencies above $\sim 70\%$ in short-time segments and above $95\%$ for phase-sensitive steps.

This construction demonstrates both deterministic preparation of complex entangled states and the flexibility of MDITE in multi-qubit settings (Kondappan et al., 2022).

7. Implementation Considerations and Limitations

Primary sources of imperfection include finite photon number, detector inefficiency (affecting phase-sensitive multi-qubit operations), and spontaneous-emission-induced dephasing. The ratio $(n_c - n_d)/(n_c + n_d)$ is insensitive to photon loss for short-time segments, but high detector efficiency ($>95\%$) is crucial for robust phase correction in processes involving photon parity measurements.

The protocol is fully measurement-based, using only single-qubit QND couplings to cavities or optical modes, and feedback unitaries composed of local gates. The scaling of required rounds and photon resources is moderate for stabilizer Hamiltonians, but may be more demanding for generic noncommuting terms.

Deterministic convergence is ensured by phase-corrected feedback rules, such that only the targeted state survives the iterated measurement-backaction map.

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In summary, measurement-dressed imaginary-time evolution synthesizes nonlinear QND measurements, adaptive feedback unitaries, and stroboscopic update protocols to deterministically emulate non-unitary ground-state and trajectory preparation. It yields robust, efficient, and flexible algorithms that are compatible with modern quantum hardware, and demonstrably enables deterministic preparation of highly entangled multi-qubit states, overcoming the fundamental limitations of post-selection-based quantum simulation (Kondappan et al., 2022).