Efficiently learning fermionic unitaries with few non-Gaussian gates

Alexey Gorshkov; Mauro E.S. Morales; Sharoon Austin

Efficiently learning fermionic unitaries with few non-Gaussian gates

T0 review · 0 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A fermionic circuit built from Gaussian gates plus a constant number of non-Gaussian gates can be learned efficiently from black-box queries, in both fermionic and qubit implementations.

desk verdict Closes a real open problem with a careful constructive proof; the qubit-case overlap with a concurrent paper is the only caveat. read the letter →

arxiv 2504.15356 v1 pith:CWM4EFGM submitted 2025-04-21 quant-ph

Sharoon Austin , Mauro E.S. Morales , Alexey Gorshkov This is my paper · ORCID

classification quant-ph

keywords fermionicGaussianunitariesnon-GaussiangatesquantumcircuitlearningshadowtomographyMajoranaoperatorsmatchgatehierarchydiamonddistanceJordan-Wignertransformation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that an unknown n-mode fermionic circuit containing t parity-preserving non-Gaussian gates, where t and the gate weight κ are constants, can be learned from finitely many black-box queries in time polynomial in n and 1/ε, producing a channel within ε in diamond distance with high probability. This matters because such circuits are universal building blocks for fermionic quantum computation, and no previous algorithm could learn the unitary itself; only the output states were known to be learnable. The argument works by showing the promised circuit can be rewritten, up to Gaussian transformations, so that all non-Gaussian action is confined to a constant number of Majorana operators, which are then learned by shadow tomography. The same guarantee covers circuits implemented on qubits through the Jordan-Wigner mapping.

What carries the argument

The load-bearing object is the decomposition U_t = G_A u_t G_B (Lemma 2): for any circuit of the promised form, Gaussian unitaries G_A and G_B exist so that u_t is generated by even-weight Majorana strings supported only on the first M = κt Majorana operators. The algorithm's first stage learns the matrix c^(1)_{jk} = (1/d) tr[U_t^† γ_k U_t γ_j] by shadow tomography, and its singular-value decomposition gives approximate Gaussian decouplers G_a, G_b such that W_t = G_a^† U_t G_b^† almost commutes with γ_i for i > M. This Majorana decoupling localizes W_t (or its transformed version for qubits) to m = M/2 modes, where it is learned as a reduced quantum channel via Choi-state shadow tomography, projected onto a valid completely positive trace-preserving map, and realized by a Stinespring dilation V_S; the final description is $U_t^{{(ℓ)}}$ = G_a V_S G_b.

What would settle it

Take a parity-preserving n-mode circuit with two non-Gaussian gates each of the promised form (a single even-weight Majorana exponential, weight at most κ) and compute the matrix c^(1) of Eq. (17); if it does not have at least 2n−κt singular values equal to 1 up to the tomography precision, then Lemma 2 fails for a circuit inside the promised class and the algorithm's SVD step cannot succeed.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: for a unitary U_t promised to have the form U_t = G_t K_t ... G_1 K_1 G_0, with t constant and each non-Gaussian K_i an exponential of an even-weight Majorana product of constant weight κ, there is a learning algorithm that accesses U_t only O(poly(n, $ε^{{-1}}$, log $δ^{{-1}}$)) times, uses the same order of classical processing time, and outputs a description of a quantum channel $U_t^{{(ℓ)}}$ with D⋄($U_t^{{(ℓ)}}$, U_t) ≤ ε and success probability at least 1−δ. The same result holds when the unitary is implemented on qubits via matchgates plus X_1. The authors also show that a simple two-non-Gaussian-gate example lies outside every finite level of the matchgate hierarchy, so the new algorithm covers a class that previously known hierarchy-based learning methods cannot.

Load-bearing premise

The algorithm's success presupposes that every circuit of the promised form can be rewritten, with Gaussian unitaries on both sides, so that the remaining unitary acts only on the first κt Majorana operators—if any promised circuit escapes this normal form, the learning procedure cannot even start.

Editorial extensions

If this is right

Fermionic circuits of the promised form become efficiently benchmarkable and verifiable from black-box queries, even though the gate decomposition is unknown.
The same learning guarantees transfer to qubit implementations, so experiments on qubit chains using the Jordan-Wigner mapping can certify such unitaries.
The class of circuits considered complements known classical simulation algorithms: these circuits are simulable, and now also learnable without knowing their internal structure.
Because a two-non-Gaussian-gate example lies outside every finite matchgate-hierarchy level, the new decoupling approach is not subsumed by previous hierarchy-based learning algorithms.
The learned channel description can be implemented with only a constant number of ancillas: 2m+1 fermionic modes or 2m qubits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same strategy suggests a template for learning other unitaries with a few non-Gaussian resources: learn the low-weight response matrix, find decouplers via SVD, reduce to a constant-size core, then learn the core by tomography. If this generalizes, constant-non-Gaussian learning is not a fermionic accident.
The singular-value test on c^(1) gives a natural property-testing primitive: an unknown unitary can be certified to be close to this class by checking that 2n−M singular values of the learned matrix are close to 1, without running the full learning algorithm.
Because the output is a channel that must be dilated with 2m+1 ancillary modes or 2m qubits, the current guarantee is sample-efficient but not ancilla-optimal; reducing the dilation overhead is a direct engineering target.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read on arXiv:2504.15356. It solves a real open problem—turning the state-learning result of Mele and Herasymenko into unitary learning for fermionic circuits with a constant number of non-Gaussian gates—and the main proof survives a careful pass. The only caveat worth flagging is that the qubit-implementation half overlaps with a concurrent paper (Ref [38]), which the authors themselves disclose in a note added.

The construction is genuinely good. The two-step strategy is to learn approximate Gaussian dressings that confine the circuit to a constant number of Majorana modes, then learn the remaining constant-size channel via shadow tomography of Pauli observables and Choi-state projection. The key lemmas—Lemma 2's decomposition, Lemma 5's singular-value perturbation, the Majorana and Pauli decoupling lemmas (6 and 18), Lemma 9's reduced-channel approximation, and Lemma 20's CPTP projection—are all supplied with proofs, and the error propagation is explicit. I agree with the stress-test note that there is no load-bearing flaw. The matchgate hierarchy separation in Lemma 13 is a nice extra.

Soft spots are minor. The proof leans on external shadow tomography sample-complexity results, so it's long and not machine-checked; that's normal for this subfield but means a careful referee is needed. Lemma 2 is the bridge between the circuit promise and the constant-size structure, and its proof in Appendix A is the right place to focus scrutiny. There's a slightly conservative index in Lemma 6 (i > m+1 rather than i > m), but the same argument covers the omitted case, so it doesn't affect Theorem 1. The significance is moderate: it's a valuable contribution to fermionic quantum information and benchmarking, not a paradigm shift.

The paper is aimed at quantum learning theorists and people studying fermionic or matchgate circuits. I'd bring it to a reading group and cite it if I worked in that area. It deserves a serious peer review; with minor revisions (tightening the exposition and fixing the small index point) it should be accepted.

Referee Report

0 major / 5 minor

Summary. The paper studies the problem of learning an n-mode fermionic unitary U_t promised to have the form U_t = G_t K_t ... G_1 K_1 G_0, where each G_i is a fermionic Gaussian unitary, each K_i is a parity-preserving non-Gaussian gate generated by an even-weight Majorana product of constant weight κ, and t is constant. The main result (Theorem 1) is an algorithm that, given black-box access to U_t, outputs a classical description of a quantum channel U_t^(ℓ) such that D⋄(U_t^(ℓ), U_t) ≤ ε, using O(poly(n, ε^{-1}, log δ^{-1})) queries and classical processing time, with probability at least 1−δ. The algorithm first constructs the one-body fermionic correlation matrix c^(1) by shadow tomography, performs a singular-value decomposition to obtain Gaussian unitaries G_a and G_b so that W_t = G_a† U_t G_b† approximately commutes with all but M = κt Majorana operators, and then learns the constant-size reduced quantum channel via Choi-state shadow tomography and Stinespring dilation. The result is established for both the fermionic implementation (Gaussian gates in SO(2n), parity-preserving inputs) and the qubit implementation (Gaussian gates in O(2n)). A side result (Lemma 13) claims that certain unitaries with two non-Gaussian gates lie outside every finite level of the matchgate hierarchy.

Significance. If the claims hold, the paper solves an open problem from Ref. [17], extending efficient learnability of fermionic Gaussian states with a constant number of non-Gaussian gates to learnability of the corresponding unitaries, in both fermionic and qubit implementations. The proof is constructive and unusually detailed: Lemma 2 supplies the Gaussian-dressing decomposition, Lemma 5 provides a singular-value perturbation bound for the decoupling step, Lemmas 6 and 18 give Pauli-decoupling statements, Lemma 9 bounds the reduced-channel approximation, and Lemma 20 gives a CPTP projection guarantee for learned Choi states. The algorithm is explicit (Algorithms 1 and 2), with sample-complexity estimates for the shadow-tomography steps. The matchgate-hierarchy result, if corrected as noted below, is a valuable complementary contribution.

minor comments (5)

[Theorem 1 and Section III] The quantity m = M/2 is not well-defined when M = κt is odd. Since the non-Gaussian gates have even weight, the support can always be placed on an even number of Majorana modes, and the statement should define m = ceil(M/2) or explicitly replace M by an even upper bound such as M' = 2⌊κt/2⌋. The subsequent ancilla counts (2m+1 and 2m) remain constant under this change, so the theorem is unaffected, but the present wording is ill-defined for odd κt.
[Lemma 6 and its use in Lemma 9] Lemma 6 states the Pauli-decoupling bound for i > m+1, but the proof in Appendix D establishes the stronger statement for i > m, which is the condition needed by Lemma 9 and the final guarantees in Lemmas 22 and 23. The statement should be strengthened to i > m, or the proof should be adjusted to the weaker statement and the later arguments should be modified accordingly; as written, the lemma is weaker than the proof.
[Lemma 13] The claim that U_t = K G(θ) K with θ = π/p and p an odd integer lies outside every finite level of the matchgate hierarchy is false for p = 1, because then sin(2^j θ) = 0 for all j ≥ 1 and F_j reduces to γ_2, which is Gaussian. The statement should require p odd with p > 1, or the proof should treat p = 1 separately.
[Equation (9)] There is a typo in the matchgate matrix: the (4,4) entry should be A22, not A21. In addition, the stated condition det(A) = det(B) = ±1 should presumably be det(A) = det(B), as is standard for matchgates.
[Proofs of Lemmas 22 and 23] The notation "input parameters (ϵ2, δ/2)" is easily misread as ε·2 rather than ε^2. Since the final bound goes through only if Algorithm 1 is run with accuracy ε^2 (because Lemma 5 gives ε0 ≤ T1(n)∥E^(1)∥^{1/2}), the text should be clarified, e.g., by writing (ε^2, δ/2) explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from an externally sourced decomposition proved in Appendix A, with all tomography and SVD steps carrying explicit error bounds.

full rationale

The central claim of Theorem 1 is a constructive learning algorithm, not a repackaged input. The load-bearing structural input is Lemma 2, which is adapted from external Ref. [17] and proved in Appendix A via an explicit G_aux construction mapping the span of transformed non-Gaussian generators into the first M Majorana modes; the proof does not assume the theorem it is used to prove. The subsequent SVD construction in Lemma 5 uses the estimated matrix c(1) only to extract Gaussian dressings G_a and G_b, with the Majorana-decoupling error bounded by ||E(1)||^{1/2}; this is a statistical estimation bound, not a fit renamed as a prediction. The remaining steps (Pauli decoupling, reduced-channel approximation, Choi-state shadow tomography, CPTP projection, and Stinespring dilation) are standard and are each supplied with explicit lemmas and error propagation. The only self-citation in the author list is Ref. [14] (background on learning Gaussian unitaries); it is not used in the proof of Theorem 1 and therefore does not create circularity. No step reduces to its own input by definition, and no load-bearing uniqueness claim is imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters and no new physical entities are introduced. The constants M = kappa t and w = (kappa + 1)t are determined by the circuit promise, not fitted to data. The paper's inputs are the circuit promise, black-box access to the unitary, and standard shadow tomography results.

assumptions (3)

standard math Weyl's singular value perturbation theorem holds as used in Lemma 4.
Used to show that learned singular values of the noisy matrix c^(1) remain close to 1, which is essential for the Majorana decoupling step.
domain assumption Shadow tomography guarantees for the fermionic Gaussian unitary ensemble [26] and the local Clifford ensemble [27] hold as stated.
Lemmas 11 and 12 use these sample-complexity bounds as black boxes; if they fail, the algorithm's sample complexity is unsupported.
domain assumption Each non-Gaussian gate K_i is an exponential of a single even-weight Majorana product of weight at most kappa.
Stated in Section III; the decomposition Lemma 2 and the bound w = (kappa + 1)t depend on this single-generator form.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Efficiently learning fermionic unitaries with few non-Gaussian gates." pith.science (2026). https://pith.science/paper/CWM4EFGM

@misc{pith2026250415356,
  author       = {Pith},
  title        = {Pith review of: Efficiently learning fermionic unitaries with few non-Gaussian gates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CWM4EFGM}},
  note         = {Machine review of arXiv:2504.15356}
}

abstract

Fermionic Gaussian unitaries are known to be efficiently learnable and simulatable. In this paper, we present a learning algorithm that learns an $n$-mode circuit containing $t$ parity-preserving non-Gaussian gates. While circuits with $t = \textrm{poly}(n)$ are unlikely to be efficiently learnable, for constant $t$, we present a polynomial-time algorithm for learning the description of the unknown fermionic circuit within a small diamond-distance error. Building on work that studies the state-learning version of this problem, our approach relies on learning approximate Gaussian unitaries that transform the circuit into one that acts non-trivially only on a constant number of Majorana operators. Our result also holds for the case where we have a qubit implementation of the fermionic unitary.

Figures

Figures reproduced from arXiv: 2504.15356 by the authors.

**Figure 2.** FIG. 2. This figure shows the implementation of the learned quan [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The matrix [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. The qubit unitary, composed of parity-preserving two-qubit [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Efficient learning of bosonic Gaussian unitaries
quant-ph 2025-10 conditional novelty 7.0 of 10

We present the first provably efficient algorithm, in both query and time complexity, for learning arbitrary multi-mode bosonic Gaussian unitaries under the energy-constrained diamond distance.
Energy-independent tomography of Gaussian states
quant-ph 2025-08 unverdicted novelty 7.0 of 10

A tomography protocol estimates Gaussian states in trace distance with sample complexity independent of energy (up to doubly logarithmic factors), a doubly exponential improvement over prior methods.

Reference graph

Works this paper leans on

46 extracted references · 25 canonical work pages · cited by 2 Pith papers

[38]

Iyer, Mildly-interacting fermionic unitaries are efficiently learnable (2025), arXiv:2504.11318 [quant-ph]

V . Iyer, Mildly-interacting fermionic unitaries are efficiently learnable (2025), arXiv:2504.11318 [quant-ph]

arXiv 2025
[17]

A. A. Mele and Y . Herasymenko, PRX Quantum 6, 010319 (2025)

work page 2025
[1]

I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455–2467 (1997)

work page 1997
[2]

Mohseni, A

M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Phys. Rev. A 77, 032322 (2008)

2008
[3]

G. M. D’Ariano and P. Lo Presti, Phys. Rev. Lett. 86, 4195 (2001)

2001
[4]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V . James, N. K. Langford, T. C. Ralph, and A. G. White, Phys. Rev. Lett. 93, 080502 (2004)

work page 2004
[5]

J. Haah, R. Kothari, R. O’Donnell, and E. Tang, in FOCS (2023)

work page 2023
[6]

Leone, S

L. Leone, S. F. E. Oliviero, S. Lloyd, and A. Hamma, Phys. Rev. A 109, 022429 (2024)

work page 2024

Show all 46 references

[7]

H. Zhao, L. Lewis, I. Kannan, Y . Quek, H.-Y . Huang, and M. C. Caro, PRX Quantum 5, 040306 (2024)

2024
[8]

Huang, Y

H.-Y . Huang, Y . Liu, M. Broughton, I. Kim, A. Anshu, Z. Lan- dau, and J. R. McClean, in STOC (2024) p. 1343–1351

2024
[9]

Jozsa and A

R. Jozsa and A. Miyake, Proc. R. Soc. A: Math464, 3089–3106 (2008)

2008
[10]

B. M. Terhal and D. P. DiVincenzo, Phys. Rev. A 65, 032325 (2002)

2002
[11]

S. B. Bravyi and A. Y . Kitaev, Ann. Phys. (N. Y .).298, 210–226 (2002)

2002
[12]

L. G. Valiant, SIAM J. Comput. 31, 1229 (2002)

2002
[13]

K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Com- mun. Math. Phys. 404, 629–700 (2023)

2023
[14]

Oszmaniec, N

M. Oszmaniec, N. Dangniam, M. E. Morales, and Z. Zimbor´as, PRX Quantum 3, 020328 (2022)

2022
[15]

Cudby and S

J. Cudby and S. Strelchuk, Learning gaussian operations and the matchgate hierarchy (2024), arXiv:2407.12649 [quant-ph]

2024 arXiv
[16]

D. J. Brod and E. F. Galv ˜ao, Phys. Rev. A 84, 022310 (2011)

2011
[18]

Gonz ´alez-Cuadra, D

D. Gonz ´alez-Cuadra, D. Bluvstein, M. Kalinowski, R. Kaubruegger, N. Maskara, P. Naldesi, T. V . Zache, A. M. Kaufman, M. D. Lukin, H. Pichler, B. Vermersch, J. Ye, and P. Zoller, Proc. Natl. Acad. Sci.120 (2023)

2023
[19]

R. Ott, D. Gonz ´alez-Cuadra, T. V . Zache, P. Zoller, A. M. Kauf- man, and H. Pichler, Error-corrected fermionic quantum pro- cessors with neutral atoms (2024), arXiv:2412.16081 [quant- ph]

2024 arXiv
[20]

Schuckert, E

A. Schuckert, E. Crane, A. V . Gorshkov, M. Hafezi, and M. J. Gullans, Fermion-qubit fault-tolerant quantum computing (2024), arXiv:2411.08955 [quant-ph]

2024 arXiv
[21]

Dias and R

B. Dias and R. Koenig, Quantum 8, 1350 (2024)

2024
[22]

Reardon-Smith, M

O. Reardon-Smith, M. Oszmaniec, and K. Korzekwa, Quantum 8, 1549 (2024)

2024
[23]

Mocherla, L

A. Mocherla, L. Lao, and D. E. Browne, Extending match- gate simulation methods to universal quantum circuits (2024), arXiv:2302.02654 [quant-ph]

2024 arXiv
[24]

Bampounis, R

A. Bampounis, R. S. Barbosa, and N. de Silva, Matchgate hier- archy: A clifford-like hierarchy for deterministic gate teleporta- tion in matchgate circuits (2024), arXiv:2410.01887 [quant-ph]

2024 arXiv
[25]

Castaneda and N

J. Castaneda and N. Wiebe, Hamiltonian learning via shadow 11 tomography of pseudo-choi states (2023), arXiv:2308.13020 [quant-ph]

2023 arXiv
[26]

A. Zhao, N. C. Rubin, and A. Miyake, Phys. Rev. Lett. 127, 110504 (2021)

2021
[27]

Huang and R

H.-Y . Huang and R. Kueng, Predicting features of quantum sys- tems from very few measurements (2019), arXiv:1908.08909 [quant-ph]

2019 arXiv
[28]

R. Levy, D. Luo, and B. K. Clark, Phys. Rev. Res. 6, 013029 (2024)

2024
[29]

Kempe, D

J. Kempe, D. Bacon, D. P. DiVincenzo, and K. B. Whaley, En- coded universality from a single physical interaction (2001), arXiv:quant-ph/0112013 [quant-ph]

2001 arXiv
[30]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

2010
[31]

Watrous, The Theory of Quantum Information , 1st ed

J. Watrous, The Theory of Quantum Information , 1st ed. (Cam- bridge University Press, USA, 2018)

2018
[32]

Surawy-Stepney, J

T. Surawy-Stepney, J. Kahn, R. Kueng, and M. Guta, Quantum 6, 844 (2022)

2022
[33]

Cudby and S

J. Cudby and S. Strelchuk, Gaussian decomposition of magic states for matchgate computations (2023), arXiv:2307.12654 [quant-ph]

2023 arXiv
[34]

Bravyi and R

S. Bravyi and R. Koenig, Classical simulation of dissipative fermionic linear optics (2011), arXiv:1112.2184 [quant-ph]

2011 arXiv
[35]

Bittel, A

L. Bittel, A. A. Mele, J. Eisert, and L. Leone, Optimal trace- distance bounds for free-fermionic states: Testing and improved tomography (2025), arXiv:2409.17953 [quant-ph]

2025
[36]

Lyu and K

X. Lyu and K. Bu, Fermionic gaussian testing and non-gaussian measures via convolution (2024), arXiv:2409.08180 [quant- ph]

2024 arXiv
[37]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, J. Phys. A: Math. Theor. 53, 023001 (2020)

2020
[39]

Stewart, Linear Algebra and its Applications 28, 213 (1979)

G. Stewart, Linear Algebra and its Applications 28, 213 (1979)

1979
[40]

Kliesch and I

M. Kliesch and I. Roth, PRX Quantum 2, 010201 (2021)

2021
[41]

A. Hahn, D. Burgarth, and K. Yuasa, New J. Phys. 24, 063027 (2022). Appendix A: Details of Algorithm 1 In this Appendix, we present proofs for several lemmas re- lated to Algorithm 1. The following lemma describes the de- composition result (Theorem 4 in Ref. [17]) with minor ...

2022
[42]

The entries of the matrix fαβ defined as fαβ = 1 2n tr S†( ¯Pβ⊗IB)S( ¯Pα⊗IB) , (57) where S = ¯Wt from Eq

Learning the matrix fαβ Lemma 12 (Learning Pauli observables with shadow tomog- raphy for the qubit implementation). The entries of the matrix fαβ defined as fαβ = 1 2n tr S†( ¯Pβ⊗IB)S( ¯Pα⊗IB) , (57) where S = ¯Wt from Eq. (36), ¯Pα ∈ {I,X,Y,Z }⊗m are Pauli strings supported ...
[43]

The results presented here apply for channels corresponding to the uni- tariesWt (for the fermionic implementation) and ¯Wt (for the qubit implementation)

Technical lemmas for Choi states In this subsection, we state and prove key technical lemmas regarding the Choi state learned in Algorithm 2. The results presented here apply for channels corresponding to the uni- tariesWt (for the fermionic implementation) and ¯Wt (for the qu...
[44]

The projected Choi state Jp satisfies the bound ∥J(E)−Jp∥1≤ϵl, (C19) whereϵl =C0ϵ1 (withC0 being a constant). Proof. The projection, with respect to the Frobenius norm (defined as∥A∥2 = p tr[A†A]), to a trace-preserving map is defined as ProjTP[X] = argminX′∥X−X′∥2 (C20) s.t. ...
[45]

For the fermionic imple- mentation, we modify the states and observables as follows

Fermionic implementation: states and observables for Algorithm 1 In Lemma 10, we perform state tomography on some qubit state|ψj⟩ (where the index j corresponds to the bitstring x with weight 1 and xj = 1 ) to estimate physical observables O+ k and construct the matrix c(1). F...
[46]

We define states on the fermionic modesA1,A2, 1,..., 2n, whereA1,A2 are ancilla modes

Fermionic implementation: states and observables for Algorithm 2 We now modify the states and observables in Algorithm 2 used to construct the Choi stateJ( ˆE) for the reduced quantum channel corresponding to the unitary Wt. We define states on the fermionic modesA1,A2, 1,...,...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [38]

Iyer, Mildly-interacting fermionic unitaries are efficiently learnable (2025), arXiv:2504.11318 [quant-ph]

V . Iyer, Mildly-interacting fermionic unitaries are efficiently learnable (2025), arXiv:2504.11318 [quant-ph]

arXiv 2025

[2] [17]

A. A. Mele and Y . Herasymenko, PRX Quantum 6, 010319 (2025)

work page 2025

[3] [1]

I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455–2467 (1997)

work page 1997

[4] [2]

Mohseni, A

M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Phys. Rev. A 77, 032322 (2008)

2008

[5] [3]

G. M. D’Ariano and P. Lo Presti, Phys. Rev. Lett. 86, 4195 (2001)

2001

[6] [4]

J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V . James, N. K. Langford, T. C. Ralph, and A. G. White, Phys. Rev. Lett. 93, 080502 (2004)

work page 2004

[7] [5]

J. Haah, R. Kothari, R. O’Donnell, and E. Tang, in FOCS (2023)

work page 2023

[8] [6]

Leone, S

L. Leone, S. F. E. Oliviero, S. Lloyd, and A. Hamma, Phys. Rev. A 109, 022429 (2024)

work page 2024