Classical simulation and quantum resource theory of non-Gaussian optics

1. Problem Statement

Continuous-variable (CV) quantum computing, particularly in optical systems, offers inherent noise resilience but relies heavily on non-Gaussian resources (e.g., Fock states, GKP states, cat states) to achieve universal quantum computation. While Gaussian states and operations can be efficiently simulated classically using covariance matrix formalism, the introduction of non-Gaussian elements makes classical simulation intractable in general.

Existing simulation methods face significant limitations:

• Wigner Function Negativity: Simulation cost scales with the volume of negative regions in the Wigner function, which is intrinsic to non-Gaussianity but often difficult to quantify for general states.
• Fock Basis Decomposition: Methods relying on truncating the Fock basis (e.g., Stellar Rank) suffer from scaling issues for states requiring infinite or very large Fock expansions (like GKP or cat states).
• Lack of Operational Measures: There is a need for resource measures that directly quantify the computational cost of simulating non-Gaussian circuits.

The authors aim to develop efficient classical simulation algorithms for CV systems with non-Gaussian inputs and to define corresponding resource measures that quantify the "cost" of non-Gaussianity.

2. Methodology

The core methodology involves decomposing arbitrary non-Gaussian states into superpositions of Gaussian states and simulating the evolution using an extended Gaussian formalism.

A. Phase-Sensitive Gaussian Simulator

Standard covariance matrix formalism tracks means and covariances but loses global phase information. To simulate superpositions of Gaussian states, the authors developed a phase-sensitive simulator:

• Reference State Approach: They introduce a fixed Gaussian reference state $|G_0\rangle$.
• Triple Overlap Calculation: The inner product between two Gaussian states $|G_1\rangle$ and $|G_2\rangle$ is computed via the trace of three states: $\text{Tr}(G_0 G_1 G_2) = \langle G_2|G_0\rangle \langle G_1|G_2\rangle \langle G_0|G_1\rangle$.
• Analytical Formula: Using characteristic functions and Gaussian integrals, they derive an exact formula for $\langle G_1|G_2\rangle$ that depends on the covariance matrices ($\sigma$), displacement vectors ($\mu$), and the overlap with the reference state. This allows tracking relative phases in superpositions.

B. Simulation Algorithms

The authors propose two algorithms based on decomposing a non-Gaussian state $|\psi\rangle = \sum c_i |G_i\rangle$:

1. Exact Simulation:

   • Computes the Born probability by evaluating all pairwise overlaps $\langle G_j | x \rangle \langle x | G_i \rangle$.
   • Cost: Scales quadratically with the number of Gaussian states in the decomposition ($O(\chi^2)$), where $\chi$ is the Gaussian Rank.

2. Approximate Simulation (Sparsification):

   • Approximates the state by sampling terms from the decomposition according to a probability distribution $p(i) = |c_i| / \|c\|_1$.
   • Constructs a sparse state $|\Omega\rangle$ with $k$ terms.
   • Fast Norm Estimation: Since sparsification yields an unnormalized state, a naive norm calculation is quadratic. The authors introduce a Fast Norm Estimation algorithm using random coherent state sampling (Gaussian ensemble) to estimate the norm in linear time.
   • Cost: Scales linearly with the $L_1$ norm of the coefficients ($O(\|c\|_1)$), which corresponds to the Gaussian Extent.

3. Key Contributions

A. New Resource Measures

The paper defines two measures of non-Gaussianity within the framework of Quantum Resource Theories:

1. Gaussian Rank ($\chi$): The minimum number of Gaussian states required to exactly represent a state $|\psi\rangle$.
2. Gaussian Extent ($\xi$): The square of the minimum $L_1$ norm of coefficients required to represent $|\psi\rangle$ as a superposition of Gaussian states ($\xi = \inf (\sum |c_i|)^2$).

Properties:

• Operational Meaning: These measures directly quantify the simulation overhead.
• Monotonicity: Both measures are non-increasing under Gaussian operations (unitaries, measurements, and composition with Gaussian states).
• Faithfulness: They equal 1 if and only if the state is a probabilistic mixture of Gaussian states.
• Relation to Robustness: The Gaussian extent is shown to be equal to the lower semicontinuous robustness of non-Gaussianity for pure states. This provides a condition for optimal decompositions: the optimal free states in the decomposition must saturate the optimal witness operator.

B. Non-Multiplicativity

The authors prove that the Gaussian extent is not multiplicative. They provide a counterexample using two Fock states $|1\rangle \otimes |1\rangle$. The optimal decomposition for the tensor product yields a lower cost per copy than the product of individual costs, implying that multi-mode expansions can be more efficient than single-mode expansions for simulation.

C. Optimal Decompositions

Using the connection to robustness, the authors derive optimal decompositions for key CV states:

• Fock State $|1\rangle$: Optimal decomposition involves a superposition of phase-shifted squeezed states with specific parameters ($\alpha = 2/3, \xi = \ln\sqrt{3}$).
• GKP States: Decomposed into a sum of squeezed states, showing exponential suppression of high-order terms.
• Cat States: Analysis of even/odd cat states shows that for large amplitudes, the standard decomposition is optimal.

4. Results and Applications

• Gaussian Boson Sampling (GBS):

  • The authors apply their simulator to GBS. They show that while the simulation cost scales exponentially with the number of Fock state inputs, it allows for efficient simulation of logarithmically many Fock states.
  • They demonstrate that using the Gaussian extent (non-Gaussianity resource) provides tighter bounds on simulation cost compared to using non-classicality resources (which assume only passive symplectic operations).

• Cat State Breeding:

  • They derive a lower bound on the number of cat states required to generate a specific grid state (sensor state) using the monotonicity of the Gaussian extent.
  • Table 1 in the paper provides numerical bounds for different squeezing levels ($\Delta$), showing that the required number of cat states increases as the target squeezing becomes finer. These bounds are independent of the input cat state magnitude.

• Comparison with Other Methods:

  • The proposed method outperforms Fock-basis truncation methods for states with infinite stellar rank (like GKP).
  • It offers tighter bounds than Wigner negativity for multi-copy scenarios due to the sub-multiplicative nature of the Gaussian extent.

5. Significance

This work bridges the gap between practical classical simulation and fundamental resource theory in continuous-variable quantum computing.

1. Algorithmic Efficiency: It provides the first efficient classical simulation algorithms specifically tailored for non-Gaussian optics that scale with the "amount" of non-Gaussianity rather than the dimension of the Hilbert space.
2. Resource Quantification: By defining Gaussian Rank and Extent, it offers a rigorous way to quantify the "magic" (non-Gaussianity) required for quantum advantage in CV systems, analogous to the role of magic states in qubit systems.
3. Design Guidance: The derived bounds for state transformations (e.g., cat state breeding) provide fundamental limits for experimentalists designing fault-tolerant CV quantum computers.
4. Theoretical Insight: The discovery of the non-multiplicativity of the Gaussian extent challenges assumptions about how resources scale in multi-mode systems, suggesting that joint processing of modes can reduce simulation costs.

In summary, the paper establishes that the difficulty of simulating non-Gaussian optics is directly tied to the Gaussian rank and extent, providing both a computational toolset and a theoretical framework to understand the resources necessary for quantum advantage in optical systems.