Shedding light on classical shadows: learning photonic quantum states

1. Problem Statement

Characterizing unknown quantum states is a fundamental challenge in quantum information science. Traditional Quantum State Tomography (QST) requires an exponential number of state copies relative to the number of subsystems to reconstruct the full density matrix, rendering it intractable for large-scale systems. While Classical Shadow Tomography (introduced by Huang et al.) offers an efficient alternative for qubit-based systems by using randomized measurements to predict properties, extending this framework to photonic platforms presents unique challenges:

• Infinite-dimensional Hilbert Space: Photonic systems are described by Fock states in an infinite-dimensional space, unlike finite-dimensional qubits.
• Measurement Constraints: Photonic experiments typically use Photon-Number Resolving (PNR) detectors or threshold detectors, which project states onto fixed photon-number subspaces, erasing coherence between different photon-number sectors.
• Computational Hardness: Simulating linear optical transformations generally involves computing matrix permanents, which is #P-hard, making standard classical post-processing inefficient.
• Hardware Limitations: Existing photonic devices (e.g., integrated interferometers) often lack the ability to perform arbitrary unitary evolutions required by standard shadow protocols (like Clifford groups) and are restricted to passive linear optics.

The paper addresses the gap in efficiently learning properties of photonic quantum states using only passive linear optical transformations and photon-number detection.

2. Methodology

The authors propose a Photonic Classical Shadow Protocol tailored to the constraints of linear optics. The protocol consists of three stages:

A. Data Collection (Quantum Phase)

1. Input: An unknown $m$-mode photonic state $\rho$ (potentially a superposition of different photon numbers).
2. Random Evolution: A random unitary matrix $U$ is drawn from the Haar measure over the unitary group $U(m)$. This corresponds to a random passive linear optical network (interferometer).
3. Measurement: The evolved state is measured using Photon-Number Resolving (PNR) detectors in the computational basis (Fock basis).
4. Snapshot: The outcome is a pair $(U, s)$, where $s$ is the measurement outcome (a tuple of photon numbers per mode). The "classical shadow" is the collection of $N$ such pairs: $S(\rho, N) = \{(U_1, s_1), \dots, (U_N, s_N)\}$.
   • Note: Unlike qubit shadows where one stores the reconstructed snapshot $\hat{\rho}$, here the raw pairs $(U, s)$ are stored because reconstructing $\hat{\rho}$ explicitly is computationally expensive.

B. Theoretical Framework (Channel Inversion)

The protocol relies on defining a quantum channel $\mathcal{M}$ that maps the input state to the average of the measurement outcomes.

• Visible Space: Due to PNR detection, the channel $\mathcal{M}$ is not tomographically complete for the entire infinite-dimensional space; it is block-diagonal, preserving coherence only within fixed photon-number subspaces ($n$-photon sectors).
• Representation Theory: The authors utilize the representation theory of the unitary group $U(m)$ acting on Fock spaces. The channel $\mathcal{M}$ decomposes into irreducible representations (irreps) of $U(m)$.
• Inverse Channel: They derive a closed-form expression for the inverse channel $\mathcal{M}^{-1}$ using Schur's Lemma and Clebsch-Gordan coefficients. This allows the reconstruction of expectation values without explicitly computing the full density matrix.

C. Classical Post-Processing

To estimate the expectation value $\langle O \rangle_\rho$ of an observable $O$:

1. Shadow-Norm: They introduce a photonic shadow-norm ($\|\cdot\|_P$) which bounds the sample complexity. They prove that for observables that are polynomials of degree $d$ in creation/annihilation operators, the shadow-norm scales as $O(\|O\|_\infty^2 m^3 d)$.
2. Efficient Estimation: Instead of computing matrix permanents (which is exponential), they develop an algorithm to compute the estimator $\langle O \rangle_{\hat{\rho}}$ directly from $(U, s)$.
   • For observables of constant degree $d$, the expectation value can be computed in time polynomial in the number of modes ($m$) and photons ($n$), specifically $O(\text{poly}(N, n, m^d))$.
   • This is achieved by decomposing the observable into monomials and using Ryser's algorithm to compute the necessary matrix permanents of size $d \times d$ (where $d$ is small).

D. Experimental Mitigation (Pseudo-PNR)

Since true PNR detectors are difficult to scale, the experiments utilize Pseudo-PNR measurements. This involves passing each output mode through a small Fourier interferometer connected to threshold detectors. The authors provide a statistical mitigation technique to correct the bias introduced by this setup, allowing the recovery of the true distribution.

3. Key Contributions

1. First Photonic Shadow Protocol: The introduction of a rigorous classical shadow protocol specifically designed for passive linear optical systems with PNR detection.
2. Theoretical Guarantees:
   • Proof of sample efficiency: The number of samples required scales polynomially with the number of modes and the degree of the observable.
   • Proof of time efficiency: A novel algorithm allows for the computation of expectation values in polynomial time for low-degree observables, bypassing the #P-hardness of general linear optics simulation.
3. Experimental Demonstration: Successful implementation on two integrated photonic quantum processing units (QPUs):
   • Ascella (QPU A): 12 modes.
   • Belenos (QPU B): 24 modes.
4. Versatility: Demonstration of the protocol across five distinct applications, ranging from correlation functions to learning complex quantum states.

4. Experimental Results

The authors validated the protocol on the Quandela cloud platforms (Ascella and Belenos) using up to 4 photons.

• Application A.1 (Low-order Correlators): Accurately estimated two-mode correlation matrices with an average absolute error of 0.03.
• Application A.2 (Lie-Algebraic Invariants): Measured linear-optical invariants (quantities constant under linear evolution) with a relative error of 0.67%, demonstrating the ability to detect nonlinear effects or characterize the processor.
• Application A.3 (Binned Probabilities): Recovered binned probability distributions for mode partitions with a Total Variation Distance (TVD) of ~0.02, comparable to state-of-the-art benchmarks.
• Application B.1 (Hamiltonian Energy Estimation): Estimated the ground state energy of the Bose-Hubbard Hamiltonian.
  • In the superfluid regime ($U=0$), the state is prepared via linear optics.
  • Crucially, they numerically demonstrated the protocol works even in the interacting regime ($U>0$), where the ground state cannot be prepared by linear optics alone, proving the protocol's ability to learn properties of states generated by non-linear processes using only linear optical measurements.
• Application B.2 (Learning Boson Sampling States): Successfully "learned" a Boson Sampling state. Given copies of an unknown state $\phi_m(W)|n\rangle$, the protocol reconstructed a unitary $V$ such that the fidelity between the learned state and the true state was 0.96.

5. Significance

• Scalability: This work bridges the gap between theoretical quantum learning and practical photonic hardware. It proves that scalable characterization is possible without full tomography, even on large-scale integrated photonic chips.
• Hardware Agnosticism: The protocol works for generic photonic states, including those produced by non-linear processes (e.g., measurement-based state preparation or Kerr non-linearities), making it a universal tool for characterizing future fault-tolerant photonic quantum computers.
• Resource Efficiency: By leveraging the structure of linear optics (representation theory), the authors reduced the computational bottleneck from exponential to polynomial for relevant observables, making the post-processing feasible on classical computers.
• Future Applications: The framework opens the door to applying classical shadows in photonic quantum machine learning, error mitigation, and benchmarking of large-scale quantum advantage experiments (like Boson Sampling).

In summary, this paper establishes a rigorous, efficient, and experimentally verified framework for learning properties of photonic quantum states, overcoming the specific challenges of infinite-dimensional spaces and measurement constraints inherent to photonics.