Steering the Fidelity in State Preparation on Noisy Bosonic Systems with Risk-Seeking Optimization

Technical Summary: Steering the Fidelity in State Preparation on Noisy Bosonic Systems with Risk-Seeking Optimization

Problem Statement
In bosonic quantum computing, executing algorithms requires encoding inputs into specific target quantum states via a process known as state preparation. In realistic, noisy environments, photon loss within physical cavities causes the system's evolution to be non-deterministic. Consequently, the prepared state is not a single pure state but a mixed state, represented as a probabilistic ensemble of pure states (quantum trajectories), each corresponding to a specific sequence of photon loss events.

Current state preparation methods, such as the SNAP-Displacement (SD) protocol, typically optimize a variational ansatz by maximizing the average fidelity between the output density matrix and the target state. This approach effectively averages over all possible noise trajectories. The authors argue that this standard optimization fails to account for the distribution of fidelities across the ensemble. Specifically, it offers no direct control over the probability of obtaining high-fidelity states, often resulting in a scenario where the average fidelity is maximized, but the probability of achieving very low infidelity (high precision) in any single experimental run remains negligible.

Methodology
To address this limitation, the authors adapt the framework of risk-sensitive optimization, originally developed for Reinforcement Learning and Finance, to the domain of noisy quantum state preparation. Instead of optimizing the expectation value (average fidelity), they propose optimizing specific characteristics of the fidelity distribution to favor high-fidelity outcomes.

The core methodology involves:

1. Trajectory-Wise Sampling: Rather than evolving a density matrix, the authors simulate the system by sampling individual quantum trajectories. In each step of the SD-ansatz (a sequence of Displacement and SNAP gates), a Kraus operator corresponding to a specific number of photon losses is sampled according to its probability. This yields a pure state for that specific trajectory.
2. Risk-Seeking Objectives: The authors introduce two objective functions based on the entropic risk measure:
   • Fidelity Entropic Risk: Applies the risk measure directly to the pure-state fidelity ($F$).
   • Log-Infidelity Entropic Risk: Applies the risk measure to the negative logarithm of the pure-state infidelity ($-\log(1-F)$).
     Both objectives utilize a risk coefficient $\beta > 0$. A positive $\beta$ induces risk-seeking behavior, weighting high-fidelity trajectories more heavily than low-fidelity ones. As $\beta$ increases, the optimization prioritizes trajectories with fidelities closer to 1, even if this reduces the overall average fidelity.
3. Gradient Estimation: Since the number of possible trajectories grows combinatorially, exact gradient computation is infeasible. The authors employ Stochastic Computation Graphs (SCGs) to construct an unbiased Monte Carlo gradient estimator. This estimator accounts for the dual dependence of the objective on the model parameters: the probability of sampling a specific trajectory and the value of the fidelity for that trajectory.

Key Contributions

• Reframing State Preparation: The paper shifts the optimization paradigm from maximizing average fidelity to shaping the fidelity distribution, explicitly targeting the probability of obtaining high-fidelity states.
• Adaptation of Risk-Sensitive Optimization: The work successfully adapts the entropic risk measure to the SNAP-Displacement protocol, introducing trajectory-wise objective functions that directly influence the distribution of outcomes.
• Controlled Trade-offs: The study demonstrates that the risk coefficient $\beta$ serves as a tunable parameter, allowing researchers to navigate a trade-off between the average fidelity of the ensemble and the probability of achieving a specific low-infidelity threshold.

Results
The authors evaluated their approach using a simulated bosonic system with an 8-level qudit and an SD-ansatz of 4 blocks, subject to photon loss ($T_1 = 1$ ms). They compared the risk-seeking objectives against the standard average fidelity baseline across 100 independent optimization runs.

• Infidelity Reduction: Using risk-seeking objectives, the minimum achievable pure-state infidelity was reduced by up to four orders of magnitude compared to the standard average fidelity approach. For instance, while the average fidelity baseline struggled to reach infidelities below $10^{-2}$ in most runs, risk-seeking objectives successfully reached thresholds of $10^{-6}$.
• Performance of Objectives:
  • The Fidelity Entropic Risk objective achieved the highest $\epsilon$-probabilities (probability of sampling a trajectory below a threshold $\epsilon$) for a given infidelity, particularly at higher $\beta$ values. However, it required very large $\beta$ values to reach the lowest thresholds, and the success rate (percentage of runs reaching the threshold) remained low for extreme targets.
  • The Log-Infidelity Entropic Risk objective demonstrated a more favorable balance. It achieved higher Success Rates (SR) across various thresholds compared to the other methods. For example, it achieved an 80% success rate for an infidelity threshold of $10^{-4}$ with $\beta=1.0$. However, it generally yielded slightly lower $\epsilon$-probabilities than the Fidelity Entropic Risk.
• Trade-off Dynamics: Increasing $\beta$ consistently lowered the minimum infidelity threshold but generally decreased the average fidelity and increased its standard deviation. Furthermore, while higher $\beta$ allowed for lower thresholds, it often reduced the $\epsilon$-probability for larger thresholds and eventually saturated the success rate, indicating a limit to the benefits of increasing risk-seeking behavior.

Significance and Claims
The paper claims that risk-seeking optimization provides a flexible framework for tailoring state preparation to specific application requirements. The primary significance lies in the ability to control the trade-off between the reliability of the optimization process (Success Rate), the probability of obtaining a high-fidelity state ($\epsilon$-probability), and the strictness of the fidelity requirement (infidelity threshold).

The authors emphasize that while standard optimization seeks a "compromise" across all noise trajectories, risk-seeking optimization can prioritize a subset of trajectories where fidelities can be maximized, effectively "steering" the system toward high-fidelity outcomes at the cost of the ensemble average. The work suggests that for applications requiring high-fidelity states with a certain probability (rather than high average fidelity), risk-sensitive objectives offer a superior operating point. The authors conclude modestly, noting that the observed behaviors should be further investigated across different target states, larger Fock-space dimensions, and different noise models to confirm generalizability.