⚛️ quantum physics

Reducing the Cost of Qudit State Preparation by Learning a Sparse Structure for SNAP Gates

This paper proposes a method that uses a differentiable mask with sparsity penalties and temperature annealing to simultaneously learn the optimal values and sparse locations of SNAP gate phases, thereby reducing the control complexity of qudit state preparation while maintaining competitive fidelity-resource trade-offs compared to fully parameterized or predefined sparse approaches.

Original authors: Marco Pennazio, Leonardo Ignazio Pagliochini, Riccardo Pellini, Maurizio Ferrari Dacrema, Paolo Cremonesi

Published 2026-10-16

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CC BY 4.0

Original authors: Marco Pennazio, Leonardo Ignazio Pagliochini, Riccardo Pellini, Maurizio Ferrari Dacrema, Paolo Cremonesi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet, super-cooled laboratories where the next generation of computers is being built, scientists are trying to solve a problem of scale. Traditional computers use bits, which are like tiny switches that can be either off or on. But quantum computers use quantum bits, or qubits, which can exist in a mix of both states at once, allowing them to process information in ways that classical machines cannot. To make these machines even more powerful, researchers are looking at a variation called a qudit. Instead of just two states, a qudit can hold many different levels of energy at the same time, much like a piano key that can be pressed to produce a single note, but in this case, the "key" is a container that can hold a specific number of particles called photons. By using these multi-level containers, scientists hope to pack more information into fewer physical devices, making quantum circuits simpler and more efficient.

However, preparing these complex quantum states is incredibly difficult. To set a qudit into the exact configuration needed for a calculation, researchers must apply a sequence of precise operations. One of the most common methods involves a two-step dance of moving the energy around and then tweaking the timing of the waves that make up the state. The first step, called a displacement, shifts the energy levels. The second step uses a specialized tool known as a SNAP gate to apply a unique time delay, or phase, to each energy level. The challenge is that for a qudit with many levels, the number of these time delays needed to get the state just right can explode. If a researcher tries to control every single level independently, the system becomes so complex that the control signals required are long, difficult to calibrate, and prone to errors. It is like trying to tune a massive orchestra where every musician needs a unique, constantly changing instruction; the more musicians you have, the harder it is to keep them all in sync without making mistakes.

To solve this, a team of researchers at the Politecnico di Milano in Italy has developed a new way to teach the computer which instructions are actually necessary. Instead of trying to control every possible time delay or sticking to a rigid, pre-made pattern of controls, they created a system that learns. They started with a known, efficient pattern of controls and then added a layer of "learning" that acts like a smart filter. This filter decides, during the training process, which specific time delays are useful for the task at hand and which ones can be ignored. The system uses a technique called temperature annealing, which is similar to slowly cooling molten metal to let it settle into a strong, solid shape. At first, the system is "hot" and flexible, allowing it to explore many possibilities. As it cools down, it makes firm decisions, turning the flexible filter into a strict on-or-off switch that keeps only the essential controls.

The researchers tested this learning method against older, fixed patterns using two different types of target states: one that is highly structured and regular, and another that is completely random. They simulated the process on computers to see how many time delays were needed to reach a high level of accuracy. The results showed that for the structured, regular states, the learning method was significantly more efficient. In some cases, it reduced the number of required controls by nearly half compared to the best existing methods, without sacrificing the accuracy of the final state. For example, when working with a system that had 32 levels, the new method needed only 81 controls to reach a specific high-precision goal, whereas the best previous method required 152. This reduction is crucial because fewer controls mean shorter, simpler signals that are less likely to fail in a real machine.

However, the advantage was not universal. When the researchers tested the method on the random, chaotic states, the results were more mixed. While the learning method still performed well in some cases, it did not consistently beat the older, fixed patterns across all scenarios. This suggests that the benefit of learning which controls to keep depends heavily on the specific type of information being processed. The study confirms that while a rigid, pre-defined pattern of controls is often good enough, a system that can adapt and learn to prune away unnecessary complexity offers a clear path forward for building more efficient quantum computers. The work does not claim to have solved the problem of quantum control entirely, but it demonstrates that teaching the machine to be selective about its own instructions can lead to substantial savings in resources, bringing the dream of practical quantum computing one step closer to reality.

Technical Summary: Reducing the Cost of Qudit State Preparation by Learning a Sparse Structure for SNAP Gates

Problem Statement
In bosonic quantum computing, preparing arbitrary quantum states in $d$ -level qudits (encoded in superconducting cavities) typically utilizes a sequence of Selective Number-dependent Arbitrary Phase (SNAP) gates and displacement operations. A fully parameterized ansatz assigns an independent phase to every Fock level in every SNAP gate. For a sequence of $B$ SNAP gates acting on a $d$ -dimensional qudit, this results in $B \times d$ independent phase parameters. This steep growth in parameters presents two critical challenges for near-term systems:

Control Complexity: Controlling many phases requires complex control pulses with closely spaced frequency components, making gates longer and harder to calibrate.
Error Accumulation: Increased control complexity leads to higher susceptibility to control errors and noise.

Previous work addressed this by restricting trainable phases to predefined sparse patterns (e.g., Diagonal or Grid structures). However, these heuristic patterns are fixed before training. Consequently, they may retain unnecessary phases or exclude phases that are critical for specific target states, potentially limiting the fidelity-resource trade-off.

Methodology
The authors propose a "Learnable" method that dynamically determines both the values and the locations of necessary SNAP phases. The approach integrates machine learning techniques for differentiable masking into the variational quantum state preparation framework.

Differentiable Masking: The method begins with the structural mask of the "Diagonal (adaptive)" ansatz (a predefined band of learnable phases). A second, continuous "learned mask" ( $M_{learn}$ ) is introduced, where each entry corresponds to a trainable logit $z_{b,f}$ . These logits are mapped to the interval $[0, 1]$ via a sigmoid function with a temperature parameter $\tau$ .
Joint Optimization: The gate parameters (displacement amplitudes/phases and SNAP phases) and the mask logits are optimized jointly using gradient-based methods (Adam). The objective function $\mathcal{J}$ maximizes state fidelity while penalizing the number of retained phases and large gate parameters:
$\mathcal{J} = F(\vec{\alpha}, \tilde{\Theta}) - R_{gate} - R_{mask}$
The mask regularization term $R_{mask}$ includes an $\ell_1$ term (approximating the count of active phases) and an entropy term (encouraging mask values to be binary, i.e., 0 or 1, rather than intermediate).
Temperature Annealing: To convert the continuous soft mask into a discrete binary decision, a geometric annealing schedule gradually reduces the temperature $\tau$ from 1 to 0.05 over 10,000 training steps. This sharpens the sigmoid function, forcing mask entries toward 0 or 1.
Post-Processing: After training, the soft mask is thresholded to a binary hard mask. A short fine-tuning phase (200 steps) follows to optimize gate parameters under the fixed binary mask, ensuring the final configuration is physically realizable.

Key Contributions

Adaptive Phase Selection: Unlike previous sparse ansatzes with fixed structural masks, this method learns which phases within a starting heuristic structure are necessary for a specific target state.
Differentiable Sparsity: The introduction of a differentiable mask with temperature annealing allows for the end-to-end optimization of phase sparsity without discrete, non-differentiable operations during the main training loop.
Comprehensive Evaluation: The method is evaluated against fully parameterized and predefined sparse ansatzes (Diagonal adaptive/multiple, Grid) using Fourier-5 and Haar-random target states across qudit dimensions $d \in \{8, 16, 32\}$ .

Results
The evaluation focuses on the trade-off between infidelity ( $I$ ) and the number of active SNAP phases, measured via minimum phase counts at specific thresholds, Pareto frontiers, and hypervolume indicators.

Phase Reduction: The Learnable method consistently reduces the number of required SNAP phases compared to baselines in several configurations.
- For Fourier-5 states, the method shows significant gains, particularly at larger dimensions ( $d=16, 32$ ) and intermediate infidelities ( $10^{-2}$ to $10^{-4}$ ). For instance, at $d=32$ and $I \le 10^{-3}$ , Learnable required 81 phases compared to 152 for the best baseline (a ~47% reduction).
- For Haar-random states, results are more mixed. Learnable achieved strict improvements at $d=8$ and $d=16$ but was less competitive at $d=32$ , where it did not achieve the minimum phase count at any reported threshold.
Pareto Frontiers & Hypervolume:
- For Fourier-5 states, Learnable provides a clear advantage in the intermediate infidelity region ( $10^{-2}$ to $10^{-4}$ ), offering better trade-offs than the Diagonal (adaptive) baseline.
- In hypervolume analysis restricted to $I \le 10^{-3}$ , Learnable achieved the highest mean hypervolume for Fourier-5 states across all dimensions.
- For Haar-random states, the Diagonal (adaptive) baseline often maintained a higher hypervolume over the complete range, though Learnable remained competitive in specific regions.
Duration: Reduced phase counts did not always translate to shorter gate durations, as duration depends on the distribution of phases across blocks. However, in several favorable cases (e.g., Fourier-5 at $d=32$ ), Learnable achieved both fewer phases and shorter durations.

Significance and Claims
The paper claims that learning the location of phases alongside their values provides competitive or improved fidelity-resource trade-offs compared to static sparse patterns. The authors emphasize that the advantage is state-dependent and dimension-dependent.

The method is most effective for structured states (like Fourier-5) and in the intermediate infidelity regime relevant to near-term applications.
The benefit diminishes for highly random states (Haar) at higher dimensions, suggesting that the initial heuristic structure (Diagonal adaptive) may already be near-optimal for those cases, or that the optimization landscape becomes more difficult for the joint learning of masks and parameters.
The work demonstrates that differentiable masking is a viable strategy for reducing control overhead in bosonic state preparation, though it currently operates by pruning an existing structure rather than discovering a completely new topology from scratch.

The authors conclude that while the method improves upon predefined sparse ansatzes, the magnitude of the benefit relies heavily on the specific target state and the infidelity range considered. Future work is suggested to explore broader initial structures and realistic noise constraints.

Technical Summary: Reducing the Cost of Qudit State Preparation by Learning a Sparse Structure for SNAP Gates

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