Risk Preference Optimization: A New Approach to Fidelity Control in Noisy Bosonic Quantum State Preparation

When Quantum State Preparation Meets Noise: Average Fidelity Is Not the Only Answer

Executing quantum algorithms on bosonic quantum computers requires state preparation as an indispensable preliminary step—encoding inputs into target quantum states. However, in real physical environments, noise is unavoidable, and the prepared state is actually a probabilistic ensemble of pure states rather than an ideal single state.

According to this study, the core logic of existing state preparation methods is: averaging the variational ansatz over all possible photon loss event sequences, thereby optimizing an output state with a specific average fidelity. This strategy is simple and direct, but it overlooks a critical piece of information: the distribution shape of fidelity within the ensemble.

Does a state with high average fidelity mean that every sampled pure state is sufficiently good? The answer is no. The distribution may simultaneously contain "lucky samples" with extremely high fidelity and "catastrophic samples" with extremely low fidelity, and the average masks this polarization. For downstream applications, being able to obtain high-fidelity states with a certain probability is often more practically meaningful than improving the overall average level.

Bosonic quantum computers are a class of quantum computing platforms that use quantum modes of bosons (typically photons) as information carriers, with typical representatives including photonic linear optical quantum computing and continuous-variable quantum computing. Unlike quantum computers based on two-level systems (such as superconducting transmons or ion traps), bosonic quantum computers have infinite-dimensional Hilbert spaces, where each bosonic mode can in principle accommodate any number of photons. This high-dimensional nature provides richer resources for quantum information encoding, but also brings unique noise challenges: photon loss is the dominant decoherence mechanism in such platforms, where the environment inevitably absorbs or scatters photons, causing quantum states to decay toward the vacuum state. The goal of state preparation is to transform easily accessible input states (such as coherent states) into target states required by algorithms (such as Schrödinger cat states or squeezed states), but under photon loss noise, the actual output cannot remain pure and instead forms a probabilistic mixture of pure states corresponding to different photon loss events.

Fidelity is a measure quantifying the closeness between quantum states, typically defined as the degree of overlap between two state density matrices. For a pure state |ψ⟩ and a target state |φ⟩, fidelity simplifies to |⟨ψ|φ⟩|²; for a mixed state ρ and a target pure state |φ⟩, it is ⟨φ|ρ|φ⟩. Fidelity ranges from 0 to 1, with values closer to 1 indicating that the prepared state is closer to the ideal target. Infidelity is 1 minus fidelity, the complement of fidelity, and in scenarios where infidelity is extremely small, it approximately equals the square of the trace distance between the prepared state and the target state. The reason for using infidelity rather than fidelity to report improvement magnitudes is that when fidelity is already close to 1 (e.g., 0.99), directly comparing changes in fidelity is not sensitive enough; whereas reducing infidelity from 10⁻² to 10⁻⁶ means the error rate is reduced to one-millionth of the original, a relative change that is numerically more significant and better reflects the practical benefits of the optimization strategy.

Introducing Risk-Sensitive Optimization: From Expectation to Distribution

This work borrows the framework of risk-sensitive optimization to redefine the objective function. This framework was first successfully applied in reinforcement learning, with the core idea of optimizing not just the expected value of a distribution, but specific statistical features of the distribution.

Researchers adapted the risk-seeking objective function from risk-sensitive optimization to the state preparation scenario. Unlike standard average fidelity optimization, the risk-seeking strategy actively sacrifices average fidelity in exchange for significant improvements in the fidelity of some pure states within the ensemble.

This is essentially a trade-off problem: when you need to "obtain high-quality states with a certain probability," pursuing average optimality may not actually be the optimal solution. This study explicitly takes this scenario as the optimization objective, rather than the traditional overall average optimum.

Risk-sensitive optimization has its core idea rooted in the concept of risk preferences from decision theory, later introduced into reinforcement learning to handle uncertainty in reward distributions. In standard reinforcement learning, the agent maximizes expected cumulative reward, i.e., averaging rewards over all possible trajectories. However, in many real-world scenarios, optimizing only the expected value leads to neglect of extreme outcomes—the agent may obtain moderate rewards with very high probability while encountering catastrophic failure with low probability. Risk-sensitive optimization changes the optimization objective by introducing risk measures (such as Conditional Value-at-Risk CVaR, variance, Sharpe ratio, etc.): risk-averse strategies tend to avoid the high-loss tail, while risk-seeking strategies actively pursue the high-gain tail, even if this means sacrificing expected value. In this quantum state preparation study, the risk-seeking objective function is designed to directly optimize the fidelity (or infidelity quantile) of the best-performing portion of pure states in the ensemble, rather than the average fidelity of all pure states. Such objective functions are typically formulated mathematically as quantile optimization or tail optimization, and their optimal solutions often lie at the boundary of the feasible region, with corresponding preparation strategies fundamentally different from average-optimal strategies.

Experimental Results: Up to Four Orders of Magnitude Reduction in Infidelity for Some States

Under the experimental conditions set by the researchers, the risk-seeking objective function produced remarkable results: compared with standard average fidelity optimization, the infidelity of some pure states in the prepared ensemble can be reduced by up to four orders of magnitude.

What does a four-order-of-magnitude gap mean? Simply put, if under the standard method the infidelity of certain pure states is 10⁻², the risk-seeking method can push it down to the 10⁻⁶ level. This is not fine-tuning but a qualitative leap.

Of course, this gain does not come without a cost—the average fidelity decreases accordingly. But the researchers emphasize that this trade-off is reasonable and necessary in specific application scenarios: when you only care about "whether you can obtain sufficient states with acceptable probability," a slight degradation in average metrics is entirely acceptable.

Controllable Trade-off Curve: Letting Applications Choose Their Operating Point

Another valuable contribution of this study is demonstrating the controllability of the trade-off. The researchers showed that flexible adjustment is possible between "the probability of obtaining high-fidelity states" and "the minimum infidelity threshold achieved."

This means users can select the most suitable operating point according to specific application requirements. For scenarios with extremely stringent fidelity requirements but tolerance for low success rates, one can push toward extreme risk-seeking; for scenarios with success rate requirements but slightly higher fidelity tolerance, more moderate parameter settings can be chosen. This flexibility transforms risk-sensitive optimization from a theoretical tool into a deployable engineering approach.

Significance for Quantum Computing Engineering

This work reveals a frequently overlooked issue: the evaluation metrics for state preparation themselves need to be re-examined according to application scenarios. While average fidelity is convenient as a general benchmark, it may steer optimization in directions that deviate from actual needs.

The introduction of risk-sensitive optimization provides a more refined control dimension for state preparation on noisy quantum systems. Especially for platforms like bosonic quantum computing where photon loss noise dominates, actively managing the fidelity distribution shape may be more engineering-valuable than single-mindedly pursuing average metrics.

From a broader perspective, this approach may also inspire optimization strategy design for other quantum operations (such as quantum gates and quantum error correction) in noisy environments—when the distribution itself becomes the optimization object, the space for exploration is far larger than expected-value optimization.