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Reducing the Cost of Qudit State Preparation by Learning a Sparse Structure for SNAP Gates
Marco Pennazio, Leonardo Ignazio Pagliochini, Riccardo Pellini, Maurizio Ferrari Dacrema, Paolo Cremonesi
TL;DR
Fully parameterized SNAP–Displacement state preparation assigns a phase to every Fock level in every gate, while fixed sparse patterns may not suit a particular target. The paper learns which phases to retain alongside their values, and finds competitive or improved fidelity–resource trade-offs whose advantage depends on target state and qudit dimension.
Problem
Fully parameterized SNAP ansatzes require B d independent phases, while predefined sparse patterns may retain unnecessary phases or exclude target-useful ones.
Source: §1 · p. 2Method
The Learnable method adds trainable masks to the Diagonal (adaptive) ansatz and jointly optimizes mask coefficients and phase values, using sparsity regularization and temperature annealing.
Source: §1 · p. 2, §6 · p. 14Results
Learnable improves the phase-count trade-off most clearly for Fourier-5, especially at d=16 and d=32, while Haar-random results are mixed and Diagonal (adaptive) performs better in some regions.
Source: §1 · p. 2, §6 · p. 15Takeaways & Limitations
Learning which phases to retain can improve predefined sparse ansatzes, but the benefit depends on the target state and infidelity range.
Source: §6 · p. 15The method can only remove phases from the initial Diagonal (adaptive) structure; broader phase-location learning and tests under realistic noise and hardware constraints remain future work.
Source: §6 · p. 15
Abstract
from arXiv · showhide
Preparing quantum states in bosonic qudits with SNAP--Displacement sequences requires both optimizing the gate parameters and deciding which Fock levels should be controlled by each SNAP gate. A fully parameterized ansatz assigns an independent phase to every Fock level in every gate, leading to a number of phases that grows steeply. This constitutes a problem for short to medium term systems as controlling many phases requires complex control pulses which are subject to increased control errors. Previous work reduced this cost by restricting the trainable phases to predefined sparse patterns, such as a diagonal or a grid, which however may not be optimal for a specific target state. In this work, we address this limitation by introducing a method that learns which phases are needed together with their values, using a differentiable mask. A sparsity penalty encourages the removal of unnecessary phases, while temperature annealing gradually turns the masks into binary decisions. We compare the proposed method with fully parameterized and predefined sparse ansatzes on Fourier and Haar-random target states. The evaluation considers both the minimum number of phases required to reach different infidelity thresholds and the overall trade-off between fidelity and phases required, measured through Pareto frontiers and hypervolume. The results show that learning the location of the phases together with their values provides competitive or improved fidelity--resource trade-offs, although the advantage depends on the target state and qudit dimension.
1. Introduction
The introduction frames SNAP phase count as a resource challenge for variational bosonic-qudit state preparation and motivates learning phase locations instead of fixing them in advance. It reports that the learned approach has competitive or improved trade-offs, with results varying by target state.
1. Introduction
- Bosonic qudits encode information in multiple cavity Fock levels, and alternating displacements with SNAP gates provides universal control.Cavities offer many accessible levels and long coherence times; the introduction notes implementations controlling qudits up to d=20.Source: §1 · p. 1
- A fully parameterized ansatz assigns one independent phase to every Fock level in every SNAP gate, yielding B d phase parameters for B gates.Constructive compilation can also require rapidly increasing gate counts as qudit dimension and target fidelity rise.Source: §1 · p. 2
- Fixed sparse patterns can retain unnecessary phases or exclude phases useful for a specific target, despite reducing the number of active phases.Many independently controlled phases also require complex pulses that can be longer, harder to calibrate, and prone to accumulating control errors.Source: §1 · p. 2
- The Learnable method jointly optimizes trainable masks and phase values, using regularization to remove phases and temperature annealing to approach binary selections.It starts from the Diagonal (adaptive) ansatz and learns which of its allowed phases to retain for a target state.Source: §1 · p. 2
- Learnable reduces SNAP phases in several configurations, with its clearest advantage for Fourier-5 at intermediate infidelities and more mixed results for Haar-random states.The comparison includes Full, Diagonal (adaptive), Diagonal (multiple), and Grid ansatzes, using phase counts, Pareto frontiers, and hypervolume.Source: §1 · p. 2
2. Background
The background defines single-qudit state preparation and introduces the alternating SNAP–Displacement protocol. It then describes heuristic sparse masks that reduce trainable phases but fix their locations before training.
2. Background
- State preparation seeks gates that transform an initial state into a target state, with fidelity and infidelity measuring preparation quality.The paper focuses on states defined over a single qudit.Source: §2 · p. 2
2.1. The SNAP–Displacement Protocol
- The SNAP–Displacement protocol alternates B SNAP gates with B+1 displacement gates, optimizing displacements and phases to maximize target-state fidelity.Displacements change Fock-level populations, while SNAP gates apply independent phases; simulations include bumper states to account for leakage above the qudit levels.Source: §2.1 · pp. 2–3
- The fully parameterized protocol independently optimizes B d SNAP phases, and the paper counts nonzero phases as its main resource measure.Source: §2.1 · p. 3
2.2. Predefined Sparse Phase Ansatzes
- Predefined sparse ansatzes use fixed phase-location patterns, including diagonal bands, multiple diagonals, and diagonal-plus-antidiagonal grids.Their structural masks reduce trainable phases, but optimization cannot change which phase locations are allowed; Learnable instead selects within the Diagonal (adaptive) mask.Source: §2.2 · pp. 3–4
3. Learnable Phase Locations
The Learnable method adds a trainable mask to the Diagonal (adaptive) structural mask so it can select useful SNAP phases alongside optimizing gate parameters. A differentiable soft mask and joint fidelity–sparsity objective produce a binary selection, followed by refinement and compression.
3. Learnable Phase Locations
- The method combines the Diagonal (adaptive) structural mask with a learned mask that selects which allowed SNAP phases to retain for a target state.The structural mask sets candidate locations; the learned mask selects among them.Source: §3 · p. 4
3.1. Differentiable Learned Mask
- Trainable logits map through a sigmoid to continuous mask values, allowing gradient-based optimization of phase retention.The learned mask is combined element-wise with the structural mask.Source: §3.1 · p. 4, §3 · p. 4
- Temperature annealing sharpens the mask from smooth values toward zero or one, while the joint objective decides whether each phase is retained or suppressed.The temperature decreases from 1 to 0.05 over 10 000 training steps, then remains fixed.Source: §3.1 · p. 5
3.2. Joint Parameter Optimization
- The objective jointly optimizes displacement parameters, SNAP phases, and mask logits to maximize fidelity while penalizing gate parameters and retained mask entries.The Learnable experiments set the gate-regularization weights λ1 and λ2 to zero, so sparsity is controlled only through the learned mask.Source: §3.2 · pp. 5–6
- Mask regularization favors suppressing unnecessary phases, while entropy regularization discourages intermediate mask values and the fidelity term protects phases needed for preparation.Mask regularization controls whether a phase remains learnable; ℓ1 gate regularization instead acts on its numerical value and may set it to zero.Source: §3.2 · p. 6
- After training, the hard effective mask fixes suppressed phases to zero; gate parameters are fine-tuned before greedy phase removal and structural compression.Greedy refinement accepts a phase removal only when the target infidelity remains satisfied; structural compression removes all-zero SNAP gates and combines adjacent displacements.Source: §3.2 · p. 6
4. Methodology
The evaluation compares ansatzes on Fourier-5 and Haar-random states at several qudit dimensions, using phase counts, infidelity, Pareto frontiers, and hypervolume. Training and hyperparameter searches are repeated across configurations, with distinct procedures for the Learnable ansatz and the baselines.
4.1. Experimental Setting
- Experiments use Fourier-5 and Haar-random target states at d ∈ {8, 16, 32}, and count phases at infidelity thresholds from 10^-1 to 10^-5.Simulations include bumper states beyond the qudit space, and fidelity is evaluated over the full extended space so leakage reduces target overlap.Source: §4.1 · pp. 6–7
4.2. Evaluation Protocol
- At each target infidelity threshold, the minimum phase count is selected from configurations meeting that threshold; Pareto frontiers and hypervolume summarize broader fidelity–phase trade-offs.Hypervolume is compared only between ansatzes within the same setting and qudit dimension; 95% confidence intervals use 1000 bootstrap resamples.Source: §4.2 · pp. 7–8
4.3. Training Protocol
- Gate parameters are optimized with Adam, using multiple random initializations; the Learnable mask is annealed, hardened, and followed by a 200-step fine-tuning stage.Baseline configurations use 100 initializations and Learnable configurations use 10.Source: §4.3 · p. 8
4.4. Hyperparameter Optimization
- For each target state, qudit dimension, and ε, the Learnable search fixes B = d and selects remaining hyperparameters; it then greedily removes phases and structurally compresses the ansatz.Each run has a maximum wall time of four days; this mainly affects d = 32 experiments.Source: §4.4 · pp. 8–9
4.5. Baseline Hyperparameter Optimization
- Baseline hyperparameters are explored with a scalarized fidelity–phase objective and a Tree-structured Parzen Estimator, using up to 50 configurations per penalty weight.The same scalarization weights as [14] enable direct comparison with the proposed method; this study penalizes phase count rather than duration.Source: §4.5 · pp. 9–10
5. Results and Discussion
The evaluation compares phase counts at fixed infidelity thresholds, Pareto frontiers, and hypervolume across ansatzes. Learnable’s advantage is clearest for Fourier-5 in the intermediate-accuracy region, while hypervolume results vary by target state and analysis range.
5. Results and Discussion
- The evaluation compares minimum phase counts at target infidelities, Pareto frontiers, and hypervolume across the explored range and the region I≤10^-3.A larger hypervolume indicates a better trade-off within the same qudit dimension; the two analysis ranges are computed independently.Source: §5 · p. 10, §5.1 · p. 11, §5.3 · pp. 12–13
5.1. Number of SNAP Phases per Target Infidelity
- Learnable beats all baselines in 7 of 15 Fourier-5 configurations and has five strict improvements and two ties for Haar, with results depending on dimension.For Fourier-5 at d=32, it uses 81 rather than 152 phases at I≤10^-3 and 153 rather than 348 at I≤10^-4; Haar at d=32 is less competitive.Source: §5.1 · p. 11
5.2. Pareto Frontiers
- Fourier-5 frontiers favor Learnable at intermediate infidelities, whereas Haar results are weaker: at d=32 the frontiers largely overlap and no ansatz consistently leads.The Fourier-5 advantage appears approximately between 10^-2 and 10^-4, especially at d=32; differences are less marked at the lowest and highest infidelities.Source: §5.2 · pp. 11–12, Figure 2 · p. 13, Figure 3 · p. 13
5.3. Hypervolume Analysis
- For I≤10^-3, Learnable improves Fourier-5 hypervolume at d=16 and d=32, while Haar results are mixed and Diagonal (adaptive) leads over the complete range.The Fourier-5 improvements have non-overlapping confidence intervals at d=16 and d=32; the Haar comparison at d=32 favors Diagonal (adaptive).Source: §5.3 · pp. 13–14, Table 3 · p. 14, Table 4 · p. 14
6. Conclusions
The Learnable ansatz uses a differentiable mask to select which phases from the Diagonal (adaptive) structure to retain. It can reduce phase counts, with the clearest gains for Fourier-5, while results vary with target state and infidelity range.
6. Conclusions
- The method extends Diagonal (adaptive) with a jointly optimized differentiable mask that learns which allowed phases to retain.A regularization term and temperature annealing encourage phase removal and progressively binary mask selections.Source: §6 · p. 14
- The evaluation covers Fourier-5 and Haar-random targets at qudit dimensions up to d=32, using phase counts, Pareto frontiers, and hypervolume.Hypervolume is assessed across the complete explored range and within I≤10^-3.Source: §6 · p. 14
- Learnable reduces phase requirements most clearly for Fourier-5, while Haar-random results are mixed and Diagonal (adaptive) performs better in some dimensions and frontier regions.For Fourier-5, Learnable has the highest mean hypervolume in the restricted I≤10^-3 region at all considered dimensions.Source: §6 · p. 15
- The method can remove phases only from the initial Diagonal (adaptive) structure; broader phase-location learning and tests under realistic noise and hardware constraints remain future directions.The authors also identify different mask architectures as a possible direction.Source: §6 · p. 15
Declaration on Generative AI
The authors report using GPT-5.6 Sol for paraphrasing, rewording, and grammar and spelling checks, then reviewing and editing the content and taking responsibility for it.
Declaration on Generative AI
- The authors used GPT-5.6 Sol for paraphrasing, rewording, and grammar and spelling checks, then reviewed and edited the content and took responsibility for its publication.Source: Declaration on Generative AI · p. 15