Interferometric Classical Shadows

• Interferometric classical shadows are efficient measurement protocols that use randomized passive linear-optical unitaries and photon-number measurements to reconstruct photonic quantum states.
• The method employs a block-diagonal measurement channel and its inversion along photon-number sectors, facilitating unbiased Monte Carlo estimation of observables.
• It offers practical scaling with sample complexity O(m^(2d) log T/ε²) and has been experimentally demonstrated on integrated photonic quantum processors.

Interferometric classical shadows constitute an efficient measurement and reconstruction protocol for extracting information about photonic quantum states by using randomized passive linear-optical unitaries and photon-number measurements. The method extends the classical-shadows framework, originally developed for qubit systems, into the domain of photonic Fock spaces, enabling scalable estimation of a wide class of observables relevant to bosonic and linear-optical quantum computing platforms. This approach is particularly aligned with natural photonic hardware and demonstrates favorable scaling properties for polynomials of creation and annihilation operators, alongside experimentally demonstrated feasibility on integrated photonic quantum processors (Thomas et al., 8 Oct 2025).

1. Random Passive Linear-Optical Measurements

The protocol begins with an unknown $m$-mode photonic quantum state $\rho \in L(F_m)$, where $F_m = \bigoplus_{n \ge 0} H_m^n$ is the full Fock space, and $H_m^n$ refers to the $n$-photon subspace. In each protocol round, a random passive linear-optical unitary $U \sim \mathrm{Haar}(U(m))$ is sampled and implemented as the interferometer $\varphi_m(U)$. Subsequently, a measurement is performed in the photon-number basis, yielding an outcome $\bm s = (s_1, ..., s_m)$, with $\sum_i s_i = n$.

In the Heisenberg picture, the corresponding positive-operator valued measure (POVM) element is expressed as:

$$M_{U, \bm s} = U^\dagger |\bm s\rangle \langle \bm s| U,$$

where $\rho \in L(F_m)$0 is the Fock basis state of occupation numbers $\rho \in L(F_m)$1.

2. Measurement Channel and Inverse Reconstruction

Averaging over unitaries $\rho \in L(F_m)$2 and outcomes $\rho \in L(F_m)$3 defines a measurement channel $\rho \in L(F_m)$4:

$\rho \in L(F_m)$5

which is block-diagonal over fixed photon-number sectors:

$\rho \in L(F_m)$6

with $\rho \in L(F_m)$7 projecting onto the $\rho \in L(F_m)$8-photon subspace. Each $\rho \in L(F_m)$9 is invertible, yielding the full inverse map:

$F_m = \bigoplus_{n \ge 0} H_m^n$0

It is important to note that $F_m = \bigoplus_{n \ge 0} H_m^n$1 is not completely positive and trace-preserving.

A single classical "shadow" (or snapshot) is then defined as

$F_m = \bigoplus_{n \ge 0} H_m^n$2

representing an estimator for the quantum state conditioned on the sampled unitary and measurement outcome. In practice, computational effort is focused on evaluating expectation values rather than reconstructing the entire state operator.

3. Monte Carlo Estimation of Observables

Estimation centers on block-diagonal observables $F_m = \bigoplus_{n \ge 0} H_m^n$3, with the empirical estimator for expectation value:

$F_m = \bigoplus_{n \ge 0} H_m^n$4

where $F_m = \bigoplus_{n \ge 0} H_m^n$5 are the sampled unitary-measurement pairs. The estimator is unbiased:

$F_m = \bigoplus_{n \ge 0} H_m^n$6

This protocol enables efficient and scalable learning of such observables, with practical implementation relying on Monte Carlo approaches for large systems.

4. Sample Complexity Bounds and Scaling

The required number of samples for estimating $F_m = \bigoplus_{n \ge 0} H_m^n$7 observables $F_m = \bigoplus_{n \ge 0} H_m^n$8 to additive error $F_m = \bigoplus_{n \ge 0} H_m^n$9 with constant failure probability is characterized by the photonic-shadow-norm

$H_m^n$0

The sample complexity satisfies

$H_m^n$1

For low-degree bosonic observables (degree $H_m^n$2 polynomials in creation/annihilation operators),

$H_m^n$3

leading to overall sample complexity

$H_m^n$4

This scaling is favorable for observables of practical interest, particularly as the number of modes, $H_m^n$5, increases.

5. Comparison with Qubit-Based Classical Shadows

The photonic protocol differs significantly from classical shadows for qubits (e.g., the protocols of Huang et al.). In the qubit context, Haar-random or local Clifford unitaries are applied to $H_m^n$6, followed by computational-basis measurement, resulting in a diagonal-depolarizing inverse channel and efficient classical post-processing. In contrast, the photonic protocol uses Haar-random $H_m^n$7 unitaries acting on the full Fock space, producing an inverse map that is block-diagonal over photon-number sectors. Key technical differences and features include:

Feature	Qubit Shadows	Photonic Shadows
Random Unitary Ensemble	Pauli/Clifford	$H_m^n$8 on Fock space
Measurement Basis	Computational	Photon-number
Inverse Channel	Diagonal depolarizing	Block-diagonal by photon number
Hardware Match	Qubit systems	Linear-optical (passive interferometer)
Coherence	Preserved	Lost between photon-number sectors

Advantages of the photonic approach include alignment with linear-optical hardware and suitability for fixed-photon-number states (e.g., boson sampling, dual-rail encoding). Limitations arise from the loss of coherence between different total-photon sectors and scaling of the inverse map norm in high-photon sectors, which can increase $H_m^n$9.

6. Experimental Implementation and Error Analysis

The protocol has been experimentally realized using a 12-mode integrated interferometer ("Ascella") equipped with a 3-photon quantum dot source and on-chip pseudo–PNR (photon-number-resolving) threshold detectors. The pseudo–PNR detection is achieved via Fourier networks that infer photon numbers from click patterns, replacing true photon-number-resolution.

Principal sources of experimental error include:

• Partial distinguishability of photons, which limits true Haar interference.
• Interferometer drift and calibration errors over long data-acquisition times.
• Detector dark counts and non-unity quantum efficiency.

Error mitigation strategies involve bootstrap sampling for error bar estimation and increasing the number of classical snapshots $n$0 to suppress statistical noise.

7. Prospects and Limitations

Future directions involve incorporating error-mitigation techniques adapted to the linear-optical shadows context, such as loss inversion and symmetrization. Extension to continuous-variable states utilizing squeezers and homodyne detection is also anticipated, with the objective of expanding learning capacity beyond the fixed-photon-number subspaces. A plausible implication is that protocols developed for finite Fock sectors could inform strategies for continuous-variable quantum systems as photonic technology and detector capabilities advance.

The protocol’s ability to efficiently estimate a broad class of observables using photonic hardware positions interferometric classical shadows as a valuable tool for quantum state characterization in emerging linear-optics quantum technologies (Thomas et al., 8 Oct 2025).