Certification of linear optical quantum state preparation

1. Problem Statement

Certifying the correct functioning of large-scale quantum devices is a fundamental challenge. In the context of Linear Optical Quantum Computing (LOQC) and photonic quantum advantage demonstrations, the goal is to verify that a device produces a desired output state.

• The Core Difficulty: Standard fidelity witnesses and tomography rely on comparing an experimental state to a specific, fixed target state vector. However, in photonic systems, the computational power of LOQC relies on the indistinguishability of photons.
• The Indistinguishability Issue: Photons possess "internal" degrees of freedom (e.g., frequency, polarization, temporal profile) that are often not directly resolved by standard photon-counting detectors. A "perfect" LOQC experiment requires photons to be indistinguishable in these internal modes.
• The Gap: Standard certification methods fail because:
  1. They assume a fixed internal state, whereas any state within a specific equivalence class (permutation-symmetric states) yields identical LOQC statistics.
  2. They often require homodyne/heterodyne detection (continuous variable), which is difficult to implement in standard pulsed LOQC setups.
  3. They assume perfect indistinguishability, ignoring the reality of partial distinguishability errors which degrade performance.

The paper addresses the need for a certification framework that is operationally meaningful for LOQC (focusing on the relevant equivalence class) and experimentally feasible using standard photon-counting detectors.

2. Methodology and Theoretical Framework

A. Defining LOQC Fidelity

The authors redefine the certification task by introducing an LOQC equivalence class ($C_{LO}$).

• Equivalence: Two states are LOQC-equivalent if they produce identical probability distributions for all linear optical experiments. This occurs if they share the same generalized indistinguishabilities (expectation values of permutation operators).
• Target Class: The target is not a single state vector but the set of all states generated by evolving $n$ indistinguishable photons (plus vacuum) through a unitary $U$.
• New Metric: They define LOQC Fidelity ($F_{LO}$) as the maximum standard fidelity between the experimental state and any state in the target equivalence class. This effectively measures the "closeness" to the operational goal rather than a specific internal mode structure.

B. Decomposition of the Witness

The authors prove (Theorem 1) that the LOQC fidelity can be lower-bounded by measuring two distinct quantities:

1. Photon Reversibility ($p_1$): The probability that the state, when passed through the inverse unitary ($U^\dagger$), returns to the initial single-photon input configuration. This certifies the correct manipulation of external modes.
2. Indistinguishability Weight ($c_n$): A bound on the weight of the fully indistinguishable (symmetric) sector of the input state. This certifies the quality of the internal modes.

The fidelity bound is given by:
$$F_{LO} \ge p_1 + c_n - 1$$

C. Indistinguishability Witnesses

To measure $c_n$, the paper evaluates and compares four existing proposals for quantifying multi-photon indistinguishability, promoting them to fidelity witnesses:

1. Superposed HOM Dips: Based on bunching statistics in a Fourier-transformed network.
2. Two-Mode Correlator: Based on second-order number correlations in a specific interferometer.
3. Cyclic Interferometer: Based on interference fringes in a cyclic network measuring cyclic permutation symmetry.
4. Fourier Transform Witness: Based on the total probability of "forbidden" output events (zero-transmission laws) in a Discrete Fourier Transform (DFT) interferometer.

D. Assumptions and Robustness

The authors analyze the sample complexity, tightness, and semi-device-independence (robustness to errors in the trusted interferometer) of each method.

• Partition Representation: Many witnesses assume the input state has a "positive partition representation" (an incoherent mixture of distinguishability sectors).
• Key Insight: The Fourier Transform Witness is unique because it is invariant under permutation twirling. This means it provides a valid bound for any input state (even those with negative partition representations or coherent distinguishability errors) without requiring the "positive partition" assumption.

3. Key Contributions

1. Operational Redefinition of Fidelity: The paper formalizes the concept of LOQC fidelity, shifting from certifying a specific internal state to certifying membership in an equivalence class of operationally equivalent states.
2. Unified Framework: It provides a general method (Theorem 2 & 3) to convert any indistinguishability witness into an LOQC fidelity witness by combining it with a photon reversibility measurement.
3. Comparative Analysis: A rigorous theoretical and numerical comparison of four indistinguishability witnesses reveals:
   • Cyclic Interferometer: Tight but not semi-device-independent (errors can overestimate fidelity) and has exponential sample complexity ($O(2^n)$).
   • Superposed HOM & Two-Mode Correlator: Efficient but rely on strong assumptions (positive partition representation) and are not tight in general regimes.
   • Fourier Transform Witness: Offers the best balance. It is tight, semi-device-independent (numerically supported), requires no structural assumptions on the input state (due to twirling invariance), and has constant sample complexity ($O(1)$) for specific noise models.
4. Experimental Demonstration: The authors experimentally implemented all four witnesses on a programmable silicon nitride photonic chip using three photons generated via Spontaneous Parametric Down-Conversion (SPDC).

4. Experimental Results

• Setup: A 12-mode integrated photonic processor was used to implement the four different interferometer configurations. Three photons were injected with tunable temporal delays to control their mutual distinguishability.
• Indistinguishability Comparison:
  • The Fourier and Cyclic witnesses provided tight bounds on the indistinguishability parameter ($c_n$).
  • The Superposed HOM and Two-Mode Correlator provided looser bounds.
  • Crucially, in regimes with specific time-delay noise models (which create "negative partition" states), the Superposed HOM witness failed, overestimating indistinguishability (falsely certifying high quality), while the Fourier witness remained robust and accurate.
• Full Fidelity Certification: The authors successfully certified the fidelity of a 3-photon, 4-mode state preparation device against 12 Haar-random target unitaries. By combining the measured reversibility and the Fourier-based indistinguishability bound, they demonstrated a rigorous lower bound on the LOQC fidelity.

5. Significance and Conclusion

This work provides a practical, scalable, and rigorous protocol for certifying photonic quantum devices.

• Scalability: By identifying the Fourier transform witness as having $O(1)$ sample complexity (for relevant noise models) and minimal assumptions, the paper removes a major bottleneck for certifying large-scale photonic quantum computers.
• Robustness: The demonstration that the Fourier method is invariant under permutation twirling makes it the most reliable choice for real-world experiments where the exact nature of distinguishability errors (coherent vs. incoherent) is unknown.
• Impact: The proposed framework allows researchers to verify the "usefulness" of a quantum state for LOQC tasks without needing full state tomography or complex continuous-variable detection, paving the way for the benchmarking of future fault-tolerant photonic quantum computers.

In summary, the paper establishes that certifying linear optical quantum state preparation is best achieved by measuring photon reversibility and using a Fourier-transform-based indistinguishability witness, which offers the optimal trade-off between tightness, robustness, and experimental efficiency.