Expressive Power and Limitations of Multi-photon Quantum Neural Networks

Technical Summary: Expressive Power and Limitations of Multi-photon Quantum Neural Networks

Problem Statement
Quantum Neural Networks (QNNs), particularly those utilizing data re-uploading (DRQNNs), have demonstrated universal approximation capabilities. While linear optical networks with multi-photon inputs (Boson sampling) have shown quantum advantages in computational tasks, the specific mechanism by which increasing the photon number enhances the expressivity of QNNs remains unexplored. Specifically, it is unclear whether the expressivity of Multi-photon Quantum Neural Networks (MPQNNs) can be infinitely enhanced by increasing the photon number or if there exist fundamental limitations. This paper addresses the quantitative characterization of MPQNN expressivity and the impact of photon number ($n$), mode number ($m$), and layer number ($L$) on approximation error.

Methodology
The authors propose a framework for MPQNNs implemented on linear optical networks, extending the DRQNN architecture to the multi-photon Fock space.

1. Model Definition: The model consists of a circuit part with data re-uploading (alternating encoding blocks via phase shifters and trainable blocks via universal unitary linear optical networks) and an observable part. The input is a Fock state $|n, 0, \dots, 0\rangle$ representing $n$ identical photons.
2. Hypothesis Space Characterization: The authors derive the hypothesis space of MPQNNs. They establish that the output function $h(x)$ is a composition of a real polynomial $g$ (determined by the observable) and trigonometric polynomials $y_j$ (determined by the trainable circuit parameters).
3. Error Bound Derivation: The study analyzes two distinct cases based on the nature of the measured observable:
   • Fixed Observable: The weights of the photon number partitions are predetermined.
   • Trainable Observable: The weights are optimized alongside the circuit parameters during training.
     Using Jackson's inequality, the authors derive upper bounds on the approximation error for both cases, relating the error to the smoothness of the target function and the parameters $n, m, L$.
4. Numerical Simulation: To validate theoretical findings, the authors developed a dynamic programming algorithm to efficiently simulate the output photon number distribution of linear optical networks driven by Fock states. This approach improves upon Ryser's algorithm for calculating permanents, reducing runtime complexity.

Key Contributions and Results
The paper provides a quantitative analysis of how photon number affects MPQNN expressivity, yielding contrasting results for fixed versus trainable observables:

• Fixed Observable Case:

  • The approximation error is bounded by $O((dL)^{-K})$, where $d = \min\{n, m-2\}$.
  • Threshold Effect: A critical threshold exists at $n = m-2$. Below this threshold, increasing the photon number polynomially enhances expressivity.
  • Limitation: Once $n$ exceeds $m-2$, further increases in photon number do not improve expressivity. This is because the number of trainable parameters in the linear optical network (dependent only on $m$ and $L$) becomes insufficient to span the higher-dimensional function space generated by the increased photon number.

• Trainable Observable Case:

  • The approximation error is bounded by $O(d^{-K})$, where $d = \min\{nL, \max\{(m-2)L, n\lfloor\frac{m-1}{2}\rfloor\}\}$.
  • Continuous Enhancement: Unlike the fixed case, increasing the photon number always enhances the expressivity of the MPQNN, even without altering the linear optical network structure. As $n \to \infty$, the error bound tends to zero.
  • Cost: This enhanced expressivity comes at the cost of introducing $\binom{n+m-1}{m-1}$ additional trainable parameters in the observable, which grows exponentially with $n$.

• Numerical Validation: Simulations on univariate trigonometric polynomial approximation tasks confirm that the mean test loss decreases as both the photon number and layer number increase, aligning with the theoretical bounds.

Significance
The authors claim this work elucidates the performance enhancement and inherent limitations of multi-photon quantum features in QNNs. By quantitatively characterizing the relationship between photon number and expressivity, the paper offers guidance for implementing QNNs on photonic platforms. Specifically, it clarifies that while multi-photon inputs offer advantages, their utility is constrained by the architecture of the trainable blocks in the fixed observable scenario. The findings suggest that to fully leverage multi-photon advantages, either the observable must be trainable or the network structure must be adjusted to accommodate the increased degrees of freedom. The work serves as a reference for understanding the trade-offs between photon number, parameter count, and approximation capability in quantum machine learning.