Number-Conserving Fermionic-Shadow Tomography

• Number-conserving fermionic-shadow tomography is a protocol that uses random orbital rotations preserving particle number to generate classical shadows of fermionic states.
• It enables simultaneous estimation of k-body correlations with sample complexity scaling as O(η^k/ε²), independent of the ambient mode number.
• The method leverages symmetry-constrained measurement ensembles and efficient inversion techniques, reducing computational overhead in many-body quantum simulations.

Number-conserving fermionic-shadow tomography is a classical-shadow protocol for fixed-particle-number fermionic states that uses random single-particle or orbital rotations commuting with the total-number operator, followed by occupation-basis measurement, to estimate many fermionic observables simultaneously. In the formulation of “Classical shadows of fermions with particle number symmetry” (Low, 2022), the target state is an unknown $\eta$-fermion state $\rho$ on $\wedge^\eta(\mathbb C^n)$, and the protocol is designed to estimate all $k$-reduced density matrices (RDMs) with sample complexity controlled by $\eta$ rather than by the ambient mode number $n$. Subsequent work sharpened this perspective into a framework of number-conserving fermionic-shadow tomography based on random orbital rotations, proving that for every fixed order $k$, all $k$-body fermionic correlations of an $N$-mode $\eta$-particle state can be simultaneously estimated with $\rho$0 samples, independent of $\rho$1, and that this dependence is information-theoretically optimal up to $\rho$2-dependent constants (Koizumi et al., 29 Jun 2026).

1. Conceptual setting and provenance

The central task is the simultaneous estimation of fermionic correlation tensors constrained by particle-number symmetry. For a fixed-$\rho$3 state, the natural observables are $\rho$4-body correlators or, equivalently, matrix elements of the $\rho$5-RDM,

$\rho$6

The defining feature of the number-conserving protocol is that both the measurement ensemble and the reconstruction map remain inside the fixed-particle sector, rather than treating the state as an arbitrary qubit state and only imposing symmetry afterward (Low, 2022).

This line of work emerged from a broader fermionic classical-shadow program. “Fermionic partial tomography via classical shadows” introduced a protocol based on a discrete group of fermionic Gaussian unitaries and proved that estimating all $\rho$7-RDM elements to additive precision $\rho$8 requires

$\rho$9

repeated state preparations. The same work also adapted the method to particle-number symmetry, halving the additional circuit depth at the cost of roughly $\wedge^\eta(\mathbb C^n)$0–$\wedge^\eta(\mathbb C^n)$1 more repetitions in practice (Zhao et al., 2020). The 2022 particle-number-symmetric protocol replaced that symmetry-adapted variant by a scheme whose sample complexity depends on $\wedge^\eta(\mathbb C^n)$2 rather than $\wedge^\eta(\mathbb C^n)$3, yielding a super-exponential improvement when $\wedge^\eta(\mathbb C^n)$4 can be arbitrarily larger than $\wedge^\eta(\mathbb C^n)$5 (Low, 2022).

A later development, “Provably Efficient Learning of Fermionic Correlations under Particle-Number Symmetry,” recast the same theme in terms of all $\wedge^\eta(\mathbb C^n)$6-body fermionic correlations of an $\wedge^\eta(\mathbb C^n)$7-mode $\wedge^\eta(\mathbb C^n)$8-particle state, established a matching lower bound $\wedge^\eta(\mathbb C^n)$9 for any adaptive protocol based on single-copy measurements, and thereby identified the $k$0-dependence as optimal up to constants depending only on $k$1 (Koizumi et al., 29 Jun 2026).

2. Measurement ensemble and shadow formation

The measurement scheme starts from an unknown $k$2-fermion state $k$3 on $k$4. One draws a Haar-random single-particle rotation $k$5, promotes it to the $k$6-particle rotation

$k$7

and uses the fact that $k$8, so total particle number is conserved. After applying $k$9, one measures in the occupational-number basis

$\eta$0

recording the bitstring $\eta$1 with probability

$\eta$2

Each outcome $\eta$3 is one fermionic classical shadow (Low, 2022).

The 2026 formulation expresses the same scheme as a POVM on the $\eta$4-particle sector. With $\eta$5, the induced orbital rotation is

$\eta$6

the single-copy POVM element is

$\eta$7

and the ensemble $\eta$8, with measure $\eta$9, is informationally complete on the $n$0-particle subspace. In that formulation it is also convenient to package the outcome into the one-particle projector

$n$1

on the Grassmannian $n$2 (Koizumi et al., 29 Jun 2026).

The practical significance of this ensemble is that it preserves the physically relevant $n$3 symmetry at the measurement level. The protocol therefore does not spend samples learning amplitudes or coherences outside the fixed-$n$4 manifold. A plausible implication is that the observed improvement in sample complexity is structurally tied to restricting both the quantum channel and the inverse reconstruction map to the symmetry sector, rather than merely post-selecting number-conserving outcomes.

3. Channel inversion and unbiased $n$5-RDM estimators

The reconstruction problem is organized through the measurement channel

$n$6

For fixed particle number, this channel is invertible on $n$7, giving the single-shot state estimator

$n$8

The resulting single-shot estimator of a $n$9-RDM matrix element is unbiased: $k$0 where $k$1, $k$2, and $k$3 is a diagonal estimation matrix of size $k$4 whose $k$5th diagonal entry depends only on $k$6 (Low, 2022).

In compact form,

$k$7

This diagonal structure is the algebraic core of the protocol: the inverse channel is explicit, yet its action is organized by the overlap pattern between an index set $k$8 and the occupied reference set $k$9.

The 2026 work rewrites the same inversion problem using contraction and extension maps,

$k$0

with

$k$1

and gives the one-shot estimator

$k$2

Equivalently, centering $k$3 by $k$4,

$k$5

with coefficients $k$6 given explicitly, and unbiasedness takes the form

$k$7

This reformulation makes the Grassmannian geometry and the dependence on centered minors explicit (Koizumi et al., 29 Jun 2026).

4. Variance bounds, sample complexity, and optimality

For the 2022 protocol, the average variance over all $k$8 satisfies

$k$9

A combinatorial bound gives

$N$0

Hence, to drive the average variance below $N$1, it suffices that

$N$2

The dependence is independent of $N$3 except via the mild factor $N$4, and this is a super-exponential improvement over previous $N$5 scalings when $N$6 (Low, 2022).

The same work emphasizes two regimes. First, when $N$7 can be arbitrarily larger than $N$8, the gain over mode-number-dependent approaches is strongest. Second, even in the worst-case of half-filling, the protocol still provides a factor of $N$9 advantage in sample complexity. It also estimates all $\eta$0-reduced density matrices, applicable to estimating overlaps with all single Slater determinants, with at most $\eta$1 samples. In the detailed variance analysis this appears as

$\eta$2

so only $\eta$3 samples suffice on average to estimate all $\eta$4-RDM elements, independent of $\eta$5 or $\eta$6 (Low, 2022).

The 2026 analysis gives a pointwise variance upper bound for each entry,

$\eta$7

and therefore $\eta$8 shots to achieve mean-squared error at most $\eta$9 for each entry. With median-of-means to boost success probability $\rho$00, the total copies are

$\rho$01

The same paper proves a matching lower bound $\rho$02 for any adaptive single-copy measurement protocol, using a two-hypothesis distinguishing framework and optimizing a block size $\rho$03 (Koizumi et al., 29 Jun 2026).

A common misunderstanding is to read the $\rho$04 statement as a uniform bound for arbitrary observables. In the cited results, that scaling is stated for estimating all $\rho$05-RDM elements on average, whereas the general fixed-$\rho$06 guarantees scale as $\rho$07 or through the explicit $\rho$08-dependent bound above (Low, 2022).

5. Computational structure and implementation

The protocol’s computational content is not limited to its sample complexity. A naïve evaluation of

$\rho$09

would require $\rho$10 time. The 2022 construction instead exploits the block-diagonal form of $\rho$11 and the fact that $\rho$12 is a matchgate unitary. This reduces each matrix element to the evaluation of $\rho$13-by-$\rho$14 submatrices of $\rho$15 and to Pfaffians of $\rho$16 matrices. By rewriting number operators in the Majorana basis and using fermionic linear optics Pfaffian formulas, one can compute all overlaps for fixed $\rho$17 in $\rho$18 time (Low, 2022).

At the quantum level, the cost per shot is one depth-$\rho$19 circuit implementing $\rho$20 plus an $\rho$21-particle measurement. The summary scaling is:

• quantum cost per shot: one depth-$\rho$22 circuit implementing $\rho$23 plus an $\rho$24-particle measurement;
• sample complexity to reach mean-squared error $\rho$25 on all $\rho$26-RDMs: $\rho$27
• post-processing per $\rho$28-RDM from one shot: $\rho$29;
• for $\rho$30, one recovers $\rho$31 sample complexity even for the largest RDM (Low, 2022).

The 2026 pseudocode formulation makes the classical workflow explicit. For shots $\rho$32, one samples $\rho$33, measures after $\rho$34, forms the one-particle projector $\rho$35, computes centered minors

$\rho$36

and updates all entries by

$\rho$37

In that presentation, memory is $\rho$38 and classical time per shot is $\rho$39 (Koizumi et al., 29 Jun 2026).

6. Related protocols, distinctions, and research trajectory

Number-conserving fermionic-shadow tomography sits within a larger family of symmetry-aware shadow methods, but those methods target different objects and use different ensembles.

Work	Core idea	Stated result
(Zhao et al., 2020)	Fermionic Gaussian shadows with a particle-number-symmetry adaptation	Depth halved at the cost of roughly $\rho$40–$\rho$41 more repetitions
(Low, 2022)	Haar-random single-particle bases that conserve particle number	Super-exponential improvement when $\rho$42; $\rho$43 post-processing
(Koizumi et al., 29 Jun 2026)	Random orbital rotations with matching lower bound	$\rho$44 samples and $\rho$45 lower bound
(Hearth et al., 2023)	Local “All-Pairs” protocol with one layer of two-body gates	$\rho$46 samples for arbitrary few-body observables
(Avdic et al., 12 Nov 2025)	Constrained shadow tomography for 2-RDM reconstruction	DQG-constrained SDP with nuclear-norm regularization
(King et al., 2024)	Two-copy triply efficient shadow tomography for local fermionic observables	Single-copy sample-efficient schemes are provably impossible for local fermionic observables

The distinctions are substantive. The orbital-rotation protocols of (Low, 2022) and (Koizumi et al., 29 Jun 2026) are single-copy, fixed-particle-number schemes for simultaneously estimating all $\rho$47-RDM entries or all $\rho$48-body fermionic correlations. By contrast, the “All-Pairs” protocol applies one layer of random two-body gates followed by occupation-basis measurement and yields $\rho$49 samples to reconstruct arbitrary few-body observables, together with an $\rho$50-time post-processing algorithm for fixed few-body weight (Hearth et al., 2023). “Constrained Shadow Tomography for Molecular Simulation on Quantum Devices” uses one-body fermionic Gaussian unitaries to generate shadow data for the 2-RDM and then solves a bi-objective SDP imposing $\rho$51-representability constraints and nuclear-norm regularization, with main costs scaling as $\rho$52 in floating-point operations and $\rho$53 in memory (Avdic et al., 12 Nov 2025).

The topic also intersects with quantum chemistry ansatz construction. “Shadow Ansatz for the Many-Fermion Wave Function in Scalable Molecular Simulations on Quantum Computing Devices” measures classical shadows of the contracted Schrödinger equation residual and builds a product of number-conserving transformations

$\rho$54

where $\rho$55. That work states $\rho$56 measurement settings for the residual and quadratic rather than quartic scaling in orbital count for both measurements and circuit depth (Wang et al., 2024). This is number-conserving and shadow-based, but its primary objective is variational wave-function construction rather than direct $\rho$57-RDM tomography.

An additional nuance concerns impossibility and optimality statements. “Triply efficient shadow tomography” proves that sample-efficient schemes for local fermionic observables are provably impossible using only single-copy measurements, and introduces a two-copy Bell-measurement framework that is triply efficient for local fermionic observables (King et al., 2024). That result concerns a different observable class and protocol model than the fixed-$\rho$58, fixed-particle-number single-copy setting of (Low, 2022) and (Koizumi et al., 29 Jun 2026). Conversely, within the latter setting, the 2026 lower bound shows that the $\rho$59 dependence is already optimal up to $\rho$60-dependent constants (Koizumi et al., 29 Jun 2026).

Numerically, the 2026 paper reports that for estimating all 1-RDM entries with $\rho$61, $\rho$62, $\rho$63, and success at least $\rho$64, the orbital-rotation shadow uses $\rho$65, hence $\rho$66 shots, while a FU-U shadow gives $\rho$67; the stated outcome is a reduction in query count by roughly an order of magnitude compared with state-of-the-art methods for one-body correlation estimation (Koizumi et al., 29 Jun 2026). This suggests that particle-number symmetry is not merely a formal constraint but a quantitatively exploitable structural resource for fermionic observables estimation.