Multi-Outcome Circuit Optimization for Enhanced Non-Gaussian State Generation

This paper proposes and demonstrates a multi-outcome optimization strategy for photonic quantum circuits that significantly enhances the success probability of generating non-Gaussian states by leveraging multiple useful measurement patterns and aggregating degenerate outcomes, rather than restricting optimization to a single target outcome.

The Gist

Imagine you are trying to bake the perfect, complex cake (a non-Gaussian quantum state) using a very specific, high-tech oven (a photonic quantum circuit).

In the world of quantum computing, these "cakes" are essential for building powerful, error-proof computers. However, baking them is incredibly tricky. The oven works on probability: you put ingredients in, and sometimes you get the perfect cake, but often you get a burnt cookie or a flat pancake.

The Old Way: The "One-Shot" Gamble

Traditionally, scientists set up their oven to try and bake only one specific type of cake. They would tune the temperature and timing perfectly for, say, a "Chocolate Fudge."

• The Problem: If the oven produces a "Vanilla Sponge" instead, they throw it away. They treat the "Vanilla Sponge" as a failure, even though it might be a delicious cake in its own right.
• The Result: You spend a lot of time and energy, but you only get a perfect cake maybe 1% of the time. The rest is wasted.

The New Idea: The "Multi-Flavor" Strategy

This paper proposes a brilliant new way to think about the oven. Instead of aiming for just one perfect cake, the researchers say: "Let's tune the oven so that any of several different delicious cakes counts as a success!"

They call this Multi-Outcome Optimization. Here is how it works, using two main tricks:

Trick 1: The "Buffet" Approach (Multiplexing)

Imagine you tell the oven: "If you bake a Chocolate Fudge, a Vanilla Sponge, or a Red Velvet, I'll take them all!"

• How it works: The researchers adjust the oven's settings so that different measurement results (different "flavors" coming out of the machine) all produce useful quantum states.
• The Analogy: It's like a restaurant that used to only serve burgers. Now, they realize their kitchen can also make great tacos and salads. Instead of throwing away the taco ingredients because they didn't make a burger, they serve the taco. You get more food (more quantum states) for the same amount of effort.
• The Result: They found that a single machine could produce a whole "hierarchy" of useful states (like different sizes of GKP states or Cat states) just by accepting different outcomes.

Trick 2: The "Group Hug" Approach (Probability Harvesting)

Sometimes, you really only want one specific cake (e.g., the "GKP |0⟩" state). But maybe the oven can make that cake in two different ways:

1. It could happen if you detect 1 photon here and 3 there.
2. Or, it could happen if you detect 2 photons here and 2 there.

• The Old Way: You only accept the "1 and 3" outcome. If the oven does "2 and 2," you throw it away.
• The New Way: You tell the oven, "I'll take the cake if it comes out as '1 and 3' OR '2 and 2'."
• The Analogy: Imagine you are fishing for a specific type of fish. Previously, you only accepted fish caught with a red net. Now, you say, "I'll take the fish if it's caught with a red net or a blue net." You double your catch rate without needing a bigger boat.
• The Result: By "harvesting" all the different ways the machine can accidentally produce the same target state, they significantly increased the success rate.

The Trade-Off: Quality vs. Quantity

Is there a catch? Yes, but it's a small one.
When you accept more outcomes (the "Buffet" or "Group Hug"), the perfect quality of the cake might drop slightly. A "1 and 3" cake might be 99.9% perfect, while a "2 and 2" cake might be 99.5% perfect.

• The Verdict: The paper shows that for many important quantum states, this drop in quality is tiny. You get much more of the state (higher success rate) for a negligible loss in quality. It's like getting 10 slightly imperfect cookies instead of 1 perfect one; for quantum computing, having more resources is often better than having fewer perfect ones.

Why Does This Matter?

Building a quantum computer is hard because these states are so hard to make. Currently, the "success rate" is so low that it's a major bottleneck.

• Before: You might need to run the machine 100 times to get 1 good state.
• After: With this new strategy, you might get 10 good states in those same 100 runs.

The Bottom Line

The authors took a machine that was previously "wasting" half its potential by ignoring alternative results, and they taught it to embrace the chaos. By using smart algorithms to find all the "useful accidents" the machine makes, they turned a low-probability gamble into a much more reliable production line.

They didn't need to build a new, expensive machine; they just changed the recipe to accept more flavors. This makes the path to a scalable, powerful quantum computer much clearer and faster.

Technical Summary

1. Problem Statement

Photonic quantum computing, particularly in the continuous-variable (CV) regime, relies on the generation of non-Gaussian states (e.g., GKP states, Schrödinger cat states) to achieve universal fault-tolerant computation. These states are typically generated using Gaussian Boson Sampling (GBS)-like architectures, where squeezed vacuum states pass through linear interferometers and are conditioned on photon-number-resolving detection (PNRD) on ancillary modes.

The Core Challenge:

• Low Success Rates: The generation of specific non-Gaussian states is inherently probabilistic. Conventional optimization strategies focus on maximizing the success probability for a single, specific measurement outcome (heralding pattern).
• Wasted Resources: This single-outcome approach ignores other measurement patterns generated by the same physical circuit that might also produce useful, high-fidelity non-Gaussian states. These alternative outcomes are typically discarded as "waste," leading to suboptimal resource utilization and low overall acceptance probabilities.

2. Methodology

The authors propose a Multi-Outcome Optimization Strategy that treats the generation of non-Gaussian states as a multi-objective problem. Instead of targeting one outcome, the framework optimizes circuit parameters to maximize the aggregate utility across a diverse set of measurement patterns.

Key Technical Components:

• Architecture: A GBS-like device with $N$ modes. Input modes undergo displacement and squeezing, pass through a passive linear unitary ($U$), and are measured on $N-1$ ancillary modes. The remaining mode is the heralded output.
• Optimization Algorithms:
  1. Beam Search (Pattern Discovery): To navigate the exponentially large space of possible Fock basis outcomes, the authors use a beam search strategy. It identifies the top-$B$ most probable measurement patterns dynamically, without relying on prior physical intuition or symmetry assumptions.
  2. Fixed-Pattern Optimization: Once promising patterns are identified, the circuit parameters (squeezing levels, displacements, interferometer phases) are fine-tuned using a global optimization routine (Basin-Hopping with L-BFGS-B).
• Loss Functions:
  • For Fixed Patterns: A linear combination of probability ($p_k$) and fidelity ($F_k$) is maximized.
  • For Beam Search: A non-linear filtration objective is used to suppress low-fidelity patterns and sharpen gradients around high-quality candidates, ensuring the optimizer focuses on viable states.
• Rotation Invariance: The fidelity metric is rotation-invariant ($F = \max_\phi |\langle \phi | e^{i\hat{n}\phi} | \psi \rangle|^2$), allowing the optimizer to find states regardless of their phase-space orientation. This simplifies convergence and identifies patterns that may require only a phase shift correction.
• Simulation Environment: Simulations were performed using the Strawberry Fields framework in the Fock basis with a truncation dimension of $D=30$ (validated against $D=50$).

3. Key Contributions

The paper introduces two distinct mechanisms to enhance state generation:

1. Resource Multiplexing: Configuring a single circuit to produce a hierarchy of different useful non-Gaussian states (e.g., GKP logical $|0\rangle$ and $|1\rangle$ with varying photon numbers, or both even and odd cat states) from different measurement outcomes.
2. Probability Harvesting: Aggregating degenerate outcomes (different measurement patterns that herald the same target state) to significantly boost the production rate of a single specific resource.

4. Results

The framework was benchmarked against four target state families using 2-mode and 3-mode circuits with 12 dB of squeezing:

• GKP States:
  • Multiplexing: A 2-mode circuit successfully generated a hierarchy of logical states ($|1_A\rangle$ for photon counts $n \in \{4, 6, 8, 10\}$) with fidelities $>99\%$ and an aggregated success probability of 11.2%.
  • Harvesting: For the GKP $|0_A^4\rangle$ state, aggregating outcomes $(1,3), (3,1),$ and $(2,2)$ increased the success probability from ~2.1% (single outcome) to 6.5%, with only a minor fidelity trade-off.
• Schrödinger Cat States:
  • The optimizer naturally discovered parity-based clustering. A 2-mode circuit produced both even ($|cat_+\rangle$) and odd ($|cat_-\rangle$) cat states with an aggregated probability of 9.5%.
  • A 3-mode circuit utilizing pattern clusters ($P n_i = 4$ and $P n_i = 5$) achieved a 10.0% success rate.
• Binomial Codes & Cubic Phase States:
  • Binomial codes ($|0_{S=2}\rangle$ and $|0_{S=3}\rangle$) were generated with an aggregated probability of 2.5%, nearly triple the single-outcome rate.
  • Cubic phase states achieved a 6.7% success rate, matching the fidelity of deep reinforcement learning approaches but with a systematic optimization framework.

Robustness to Loss:
Simulations with 1% and 10% photon loss showed that while fidelity degrades (especially for higher-energy states), the multi-outcome strategy remains robust. For probability harvesting, the degradation was uniform across aggregated patterns.

5. Significance

• Turning Waste into Utility: The work demonstrates that "waste" outcomes in Gaussian circuits are not inherently useless but represent a rich, untapped resource landscape.
• Scalability without Hardware Upgrades: By optimizing existing hardware to accept multiple outcomes, the success probability of generating critical non-Gaussian resources can be significantly enhanced without requiring immediate hardware improvements (e.g., higher squeezing or lower loss).
• Practical Implementation: The strategy provides a clear path toward more scalable photonic quantum computing. It suggests that future experimental setups can employ dynamic feed-forward systems (phase shifters) or software-defined tracking to correct for the phase rotations associated with different heralding patterns, thereby utilizing the full capacity of the photonic circuit.
• Algorithmic Foundation: The paper provides the essential algorithmic groundwork and open-source code (via GitHub) for optimizing complex photonic circuits, bridging the gap between theoretical state engineering and experimental realization.