Fermionic-Adapted Shadow Tomography (FAST)

• FAST is a family of classical shadow protocols tailored for fermionic systems, using Majorana strings, Gaussian unitaries, and matchgate circuits to directly estimate k-body reduced density matrices.
• It exploits the inherent fermionic structure—such as parity and Majorana operator bases—to reduce shadow norms and optimize sample complexity compared to qubit-based methods.
• FAST spans diverse strategies from inversion-based classical shadows to two-copy and adaptive-depth protocols, enabling robust, noise-mitigated estimation in quantum many-body simulations.

Searching arXiv for recent FAST-related papers and core fermionic shadow tomography references. Fermionic-Adapted Shadow Tomography (FAST) is the family of “classical shadow” protocols that are tailored to fermionic structure rather than to qubits or generic Pauli operators. In place of qubit-first randomization, FAST uses fermionic observables, Majorana operators, fermionic Gaussian unitaries, matchgate circuits, particle-number symmetry, or other fermion-aware constructions so that the measurement channel is aligned with the algebraic structure of reduced density matrices, Hamiltonians, and correlation functions. Across the literature, FAST encompasses fermionic partial tomography for $k$-RDMs, fermionic classical shadows (FCS), adaptive-depth fermionic classical shadows (ADFCS), triply efficient two-copy schemes for local fermionic observables, and dynamical-correlation-function protocols reformulated for shadow-style estimation (Zhao et al., 2020, Bian et al., 16 Jan 2025, King et al., 2024, Ko et al., 5 Aug 2025).

1. Conceptual basis and fermionic operator structure

The central motivation for FAST is that standard Pauli-based classical shadows can be badly matched to fermionic observables. In the original qubit framework, one samples a random unitary $U$, measures in the computational basis, and inverts the corresponding shadow channel. For Clifford unitaries, the shadow norm is controlled by the Pauli expansion of the observable. For fermionic quantities, this can be unfavorable because Majorana strings become highly nonlocal Pauli strings under Jordan–Wigner-type encodings, and some natural fermionic observables acquire exponentially large shadow norms under Clifford twirling (Bian et al., 16 Jan 2025, Zhao et al., 2020).

FAST instead works directly with fermionic objects. A common starting point is the Majorana representation

$$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$

or an equivalent indexing convention. Fermionic observables are expanded in Majorana strings,

$H = \sum_S \alpha_S \gamma_S,$

while $k$-body reduced density matrices are encoded by degree-$2k$ Majorana operators such as

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$

This basis is Hermitian, self-inverse, and Hilbert–Schmidt orthogonal, and it is preserved by fermionic Gaussian unitaries (Bian et al., 16 Jan 2025, Zhao et al., 2020).

A second structural feature is parity. Fermionic shadow channels typically act naturally on the even operator subspace,

$$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$

and odd-parity operators are either excluded by parity superselection or annihilated by the shadow channel. This is not an incidental technicality: it is one of the main reasons fermionic shadow protocols differ qualitatively from generic qubit shadows (Bian et al., 16 Jan 2025, Zhao et al., 2020).

A representative cross-section of FAST protocols is summarized below.

Protocol	Measurement ensemble	Main guarantee
FGU/FCS for $k$-RDMs (Zhao et al., 2020)	Fermionic Gaussian Clifford unitaries	$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$
ADFCS (Bian et al., 16 Jan 2025)	Depth-$U$0 brickwork matchgates	$U$1
Triply efficient fermionic shadows (King et al., 2024)	Two-copy Bell measurements + Clifford stage	$U$2
Dynamical FAST (Ko et al., 5 Aug 2025)	One- and two-copy shadow-style circuits	Efficient estimation of many commutator and anti-commutator correlators
Constrained shadow 2-RDM (Avdic et al., 12 Nov 2025)	Fermionic Gaussian shadows + SDP reconstruction	$U$3-representable, noise-robust 2-RDM reconstruction

2. Canonical FAST protocols for $U$4-RDMs and Gaussian observables

The first fully developed fermionic shadow protocol for partial tomography uses a discrete ensemble of fermionic Gaussian Clifford unitaries generated by permutations of Majorana modes. In that construction, the measurement channel is diagonal in the Majorana basis,

$U$5

for all degree-$U$6 strings. This yields a shadow norm

$U$7

and therefore the sample complexity

$U$8

for estimating all fermionic $U$9-RDM elements to additive precision $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$0, which was stated to be optimal up to the logarithmic factor (Zhao et al., 2020).

That optimality claim is tied to a fermionic benchmark. The same work emphasizes that among all degree-$$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$1 Majoranas, at most $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$2 can be measured simultaneously, while there are $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$3 independent $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$4-RDM elements. This implies that any optimal tomographic scheme must use $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$5 distinct measurement settings. FAST therefore pursues the best possible scaling relative to fermionic reduced-density-matrix structure, not relative to the $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$6 criterion used in generic shadow tomography (Zhao et al., 2020).

A number-conserving variant replaces the full fermionic Gaussian Clifford ensemble by number-conserving Gaussian permutations together with local single-qubit Cliffords: $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$7 Its sample complexity remains

$$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$8

for all $$\gamma_{2j-1} = a_j + a_j^\dagger,\qquad \gamma_{2j} = -i(a_j - a_j^\dagger),$$9-RDM elements, while the Gaussian part has depth at most $H = \sum_S \alpha_S \gamma_S,$0, roughly half the depth of the fully Gaussian scheme, at the cost of about $H = \sum_S \alpha_S \gamma_S,$1–$H = \sum_S \alpha_S \gamma_S,$2 more repetitions (Zhao et al., 2020).

A later combinatorial analysis refined the same fermionic-shadow channel, corrected the general second-moment expression, and showed how the affine-matchgate ensemble can be used not only for $H = \sum_S \alpha_S \gamma_S,$3-RDMs but also for fidelity estimation with pure fermionic Gaussian states and for learning Slater determinants. In that setting, the channel remains diagonal on even Majorana sectors, and an $H = \sum_S \alpha_S \gamma_S,$4-electron, $H = \sum_S \alpha_S \gamma_S,$5-mode Slater determinant can be learned to within $H = \sum_S \alpha_S \gamma_S,$6 fidelity given

$H = \sum_S \alpha_S \gamma_S,$7

samples of the Slater determinant (2207.14787). The same analysis also constructs much smaller ensembles of measurement bases that yield the exact same quantum channel, which may help with compilation (2207.14787).

3. Two-copy, joint, and triply efficient fermionic shadows

A separate line of work recasts FAST in the language of triply efficient shadow tomography. For the set

$H = \sum_S \alpha_S \gamma_S,$8

that is, degree-$H = \sum_S \alpha_S \gamma_S,$9 even Majorana monomials, there is no sample-efficient single-copy protocol in the sense of $k$0. More precisely, any single-copy protocol that learns all $k$1 to precision $k$2 with constant success probability requires

$k$3

copies. This does not contradict the $k$4-type bounds of fermionic partial tomography; rather, it shows that triply efficient shadow tomography for local fermionic observables requires entangling measurements across copies (King et al., 2024).

The two-copy remedy uses an initial round of Bell measurements to estimate the magnitudes of all relevant observables and to identify an $k$5-support set $k$6 whose commutation graph has bounded clique number. The remainder of the task becomes a fractional-coloring problem on an induced subgraph of the anticommutation graph. For local fermionic observables, the resulting protocol is triply efficient and uses only two-copy Clifford measurements, with sample complexity

$k$7

where $k$8, $k$9, and, in general,

$2k$0

This is the first triply efficient shadow tomography scheme for the set of local fermionic observables, and the use of two-copy measurements is necessary because sample-efficient schemes are provably impossible using only single-copy measurements (King et al., 2024).

Related measurement designs replace channel inversion by a parent-POVM viewpoint. A joint measurement strategy for quadratic and quartic Majorana monomials uses randomization over products of Majorana operators, a unitary sampled at random from a constant-size set of suitably chosen fermionic Gaussian unitaries, occupation-number measurement, and post-processing. It estimates all quadratic and quartic Majorana monomials to $2k$1 precision using

$2k$2

measurement rounds respectively, matching the performance offered by fermionic classical shadows. On a rectangular lattice of qubits with Jordan–Wigner encoding, this scheme can be implemented in circuit depth

$2k$3

with

$2k$4

two-qubit gates, improving over fermionic and matchgate classical shadows that require depth $2k$5 and $2k$6 two-qubit gates (Majsak et al., 2024).

This suggests that FAST now spans at least two technically distinct paradigms: inversion-based fermionic classical shadows, and joint-measurement constructions that retain the same asymptotic information content while altering the circuit architecture (King et al., 2024, Majsak et al., 2024).

4. Adaptive-depth and hardware-aware FAST

A major bottleneck of full fermionic classical shadows is circuit depth. Haar-random matchgates on a brickwork architecture require depth $2k$7, which is problematic for near-term devices. Adaptive-Depth Fermionic Classical Shadows (ADFCS) address this by replacing global Haar-random matchgates with depth-$2k$8 random brickwork matchgate circuits adapted to the target observable (Bian et al., 16 Jan 2025).

The structural starting point is that the depth-$2k$9 shadow channel remains diagonal on Majorana strings: $$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$0 This implies an unbiased estimator whenever $$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$1, together with the variance bound

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$2

for estimating $$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$3, and therefore

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$4

samples for additive error $$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$5 (Bian et al., 16 Jan 2025).

ADFCS introduces the interaction distance. For a Majorana string $$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$6,

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$7

and for a Hamiltonian

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$8

The depth prescription is then

$$\Gamma_{\boldsymbol{\mu}} = (-i)^k \gamma_{\mu_1}\cdots\gamma_{\mu_{2k}}.$$9

which guarantees $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$0 for every string in the support of $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$1, so the sample complexity remains polynomial and matches the asymptotic order of full FCS (Bian et al., 16 Jan 2025).

The analysis proceeds through a tensor-network representation of the brickwork twirl and, for 2-local strings, an isometric map to a symmetric lazy random walk on a one-dimensional chain. For 2-local $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$2, the required depth becomes

$$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$3

while for constant-$$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$4 strings the same scaling is proposed with $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$5 replacing $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$6 (Bian et al., 16 Jan 2025).

The numerical evidence is fully consistent with this observable-adapted picture. On random 10-qubit states, for small interaction distance $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$7, depth $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$8 already matches FCS accuracy, whereas for larger interaction distance $$\Gamma_{\mathrm{even}} = \mathrm{span}\{\gamma_S : |S|=2j,\ 0\le j\le n\},$$9, depths $k$0 are needed. For the Kitaev chain Hamiltonian

$k$1

with $k$2, $k$3, $k$4, one has $k$5, independent of system size, and depth $k$6 ADFCS gives error comparable to or smaller than full FCS in 10-qubit simulations (Bian et al., 16 Jan 2025).

5. Noise, physical consistency, and constrained reconstruction

FAST on realistic devices is shaped by two distinct issues: noise in the measurement channel, and physical consistency of the reconstructed fermionic object. Error-mitigated fermionic classical shadows address the first issue by assuming gate-independent, time-stationary, and Markovian (GTM) noise. In that setting, the noisy fermionic shadow channel still decomposes by even Majorana sector,

$k$7

and the coefficients $k$8 can be calibrated using the noiseless initial state $k$9 together with matchgate benchmarking ideas. For $$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$0-RDM estimation, the protocol uses

$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$1

state copies and

$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$2

calibration measurements when the relevant average-fidelity parameters remain constant. The method is robust against depolarizing, damping, and $$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$3-rotation noise with constant strengths, but if some sector coefficient vanishes, the noisy shadow channel becomes non-invertible on that sector (Wu et al., 2023).

The second issue is addressed by constrained shadow tomography for molecular simulation. There, shadow measurements target diagonal 2-RDM elements in randomly rotated fermionic bases,

$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$4

and reconstruction proceeds by a bi-objective semidefinite program that enforces DQG $$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$5-representability constraints together with a nuclear-norm regularization on slack matrices: $$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$6 subject to

$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$7

and the shadow constraints

$$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$8

The parameter $$M = O\!\left[\binom{n}{k}\frac{k^{3/2}\log n}{\varepsilon^2}\right]$$9 interpolates between pure variational 2-RDM minimization and strong shadow-data fidelity (Avdic et al., 12 Nov 2025).

Numerically, this shadow-v2RDM formulation significantly improves accuracy, noise resilience, and scalability. For hydrogen chains $U$00, it yields significantly lower energy error and 2-RDM Frobenius error than unconstrained FCS at fixed total shot budget, and the smallest eigenvalue of the reconstructed 2-RDM is exactly zero even when unconstrained FCS produces negative eigenvalues. For $U$01 in cc-pVDZ with a $U$02 active space, the constrained reconstruction tracks CASCI across the potential-energy curve. On IBM’s ibm_fez device for an $U$03 geometry path, the same framework recovers the correct potential-energy shape, and the sv2RDM circuits are up to 2 orders of magnitude shallower than the many-unitary FCS protocol used as comparison (Avdic et al., 12 Nov 2025).

This suggests that in noisy molecular simulation, FAST is increasingly as much a constrained inverse problem as a randomized measurement protocol. The shadow data alone is not the whole estimator; the physically admissible fermionic manifold matters just as much (Wu et al., 2023, Avdic et al., 12 Nov 2025).

6. Dynamical FAST, wave-function ansätze, and comparative scope

FAST has also been extended beyond static tomography. For dynamical correlation functions, two protocol families—FAST 1 for commutators and FAST 2 for anti-commutators—reformulate

$U$04

into expectation values of fermionic one-body or Majorana observables on modified time-evolved states. The resulting circuits require at most two-copy measurements with uncontrolled Hamiltonian simulation. For commutator-type correlators of one-body operators, all correlators at fixed $U$05 can be estimated with

$U$06

For retarded Green’s functions, the corresponding bounds are

$U$07

again depending on the $U$08 versus $U$09 regime. The stated effect is to reduce the number of measurement circuits by an order of one or two with respect to the number of qubits across a range of scenarios (Ko et al., 5 Aug 2025).

A related, but distinct, use of fermion-adapted shadows turns measurement data into an ansatz rather than a reusable classical predictor. In the shadow-CSE construction, one measures classical shadows of the residual of the contracted Schrödinger equation in random orbital bases,

$U$10

and uses the resulting operators to build the shadow ansatz

$U$11

The key statement is that the classical shadows of the CSE vanish if and only if the wave function satisfies the Schrödinger equation, so randomly sampling only the two-electron space yields an exact ansatz regardless of the total number of electrons (Wang et al., 2024). This is not FAST in the narrower reusable-shadow sense, but it belongs to the same fermion-adapted shadow methodology.

A common misconception is that FAST universally dominates alternative many-observable estimators. The current literature does not support that claim. A coherent adaptive-QGE framework for fermionic partial tomography is not a shadow protocol, but it was explicitly compared against fermionic shadow tomography and can outperform it in the high-precision regime, with numerical improvements by a factor of 100 for FeMoco and by a factor of 500 for a 100-site Fermi–Hubbard 2-RDM task (Koizumi et al., 1 May 2025). A plausible implication is that FAST is best understood as one major branch of fermionic measurement theory—especially strong for reusable data, low- to moderate-precision many-observable estimation, and structure-aware circuit design—rather than as a universally optimal solution for every precision regime.

Across these developments, the unifying theme is stable: FAST chooses the measurement ensemble, observable basis, and reconstruction map so that fermionic algebra, parity, locality in Majorana index, particle number, and $U$12-representability are not afterthoughts. They are the organizing principles of the protocol itself (Zhao et al., 2020, Bian et al., 16 Jan 2025, King et al., 2024, Avdic et al., 12 Nov 2025, Ko et al., 5 Aug 2025).