Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Optimal Haar random fermionic linear optics circuits

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proves that Haar-random fermionic linear optics can be sampled directly on a fixed brick-wall matchgate circuit with linear depth, quadratic gate count, and quadratic classical overhead.

desk verdict A genuinely useful construction for Haar random FLO with explicit angle distributions and optimal gate count, but the load-bearing compression proof is a sketch and the depth-optimality claim is unproven. read the letter →

arxiv 2505.24212 v1 pith:KFYJVYEU submitted 2025-05-30 quant-ph

Paolo Braccia , N. L. Diaz , Martin Larocca , M. Cerezo , Diego García-Martín This is my paper
classification quant-ph MSC 81P6860B20
keywords fermioniclinearopticsHaarmeasurematchgatecircuitsrandomquantumbrick-wallarchitectureCliffordgroupunitarydesigns
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

This paper aims to make Haar-random sampling of fermionic linear optics (FLO) circuits as simple as running a fixed circuit and drawing numbers. The authors prove that a nearest-neighbor brick-wall matchgate architecture on $n$ qubits, with $2n$ layers and $n(2n-1)$ gates, produces Haar-random active (non-particle-preserving) FLO when every gate angle is sampled independently from an explicitly given product-of-sines distribution; a parallel construction with $n$ layers and $n^2$ parameters produces Haar-random passive (particle-preserving) FLO. The circuit depth is linear in $n$, the gate count matches the dimension of the relevant Lie algebra, and the classical overhead is only the cost of sampling $\Theta(n^2)$ parameters, avoiding the $\mathcal{O}(n^3)$ matrix compilation used before. The paper also gives a direct algorithm for uniformly sampling Clifford FLO circuits with an optimal average of $n^2/2$ two-qubit gates, and shows Clifford FLO is not a 4-design over FLO. These results matter because FLO sampling underpins randomized benchmarking of continuous gate sets, fermionic classical shadows, and fermion-sampling experiments.

What carries the argument

The load-bearing mechanism is the exact pullback of the Haar measure through a sequence of gate rearrangements. The starting point is a classical angle-coordinate factorization of $\mathrm{SO}(2n)$ and $\mathrm{U}(n)$, whose Haar measures are products of powers of sines. A Lie-algebra isomorphism sends the infinitesimal plane rotations $L_{jk}$ to fermionic bilinears $c_jc_k/2$, which in the qubit representation become the matchgate generators $iZ_q$ and $iX_qX_{q+1}$, so the factorization becomes a triangular matchgate circuit. The paper then uses the turnover identity $U_{R_j}(\alpha)U_{R_{j+1}}(\beta)U_{R_j}(\gamma) = U_{R_{j+1}}(a)U_{R_j}(b)U_{R_{j+1}}(c)$ to reorder gates into a brick wall, and proves (Lemma 4) that under this reordering the unnormalized density $\sin^r(\alpha)\sin^{r+r'+1}(\beta)\sin^{r'}(\gamma)$ maps to $\sin^{r'}(a)\sin^{r+r'+1}(b)\sin^r(c)$: the middle gate's exponent is inherited, while the two outer exponents are exchanged. Bookkeeping with antidiagonals and swap operators yields the final exponent functions $f_n$ and $g_n$, so that the product measure is literally the Haar measure, not an approximation.

What would settle it

Take $n=4$, sample $10^6$ circuits using the distributions in Theorems 1 and 2, compute the empirical frame potentials $\operatorname{Tr}[\tau_{\Lambda}^{(t)}]$ for $t=2$ and $t=3$, and compare them with the exact Haar values, namely the $t$-fold commutant dimensions computable by the SI G method. Since the second moment already distinguishes the Haar measure from any other independent-parameter product measure, a deviation beyond sampling error would show the exponent pattern is wrong; agreement for all $t$ up to at least 3 would corroborate the claim.

Watch

Extended reading notes

Core claim

The central claim is stated as Theorem 1: every FLO circuit can be written, up to a global sign, as $U = L_{2n} \cdots L_1$, where odd layers are products of nearest-neighbor $e^{i\alpha_{jk} X_j X_{j+1}}$ rotations and even layers are products of single-qubit $e^{i\beta_{jk} Z_j}$ rotations. With the angles in their stated ranges, the normalized Haar measure on the adjoint representation is exactly the product measure $d\mu(U) = N \prod_{j,k} \sin(\alpha_{jk})^{f_n(2j,2k-1)} \sin(\beta_{jk})^{f_n(2j-1,2k)}\, d\alpha\, d\beta$, with integer exponents $f_n(u,v)$ given by the min-formula in Eq. (6). Sampling each angle independently from its sine-power distribution therefore yields Haar-random active FLO on a fixed, depth-optimal circuit, with no classical compilation. Theorem 2 is the analogous statement for passive FLO, with $n$ layers and exponent functions $g_n$; Theorem 3 says the Clifford FLO subgroup is not a 4-design over FLO, while earlier results had established it as a 3-design. Algorithm 1 samples Clifford FLO uniformly by choosing angles from $\{0,\pi/2,\pi,3\pi/2\}$ in a triangular matchgate layout, achieving the optimal average two-qubit gate count of $n^2/2$.

Load-bearing premise

The result rests on the assumption that the antidiagonal swap bookkeeping in Eq. (A48) exactly describes how the sine-power exponents change at every gate turnover, including the boundary cases, and that the passive-FLO proof really is completely analogous to the active one; if the tracking is wrong at any gate, the sampled circuits will not be Haar random.

Editorial extensions

If this is right

  • Randomized benchmarking of continuous matchgate gate sets can run on a fixed hardware-native brick-wall circuit with linear depth, making the protocol practical on near-term devices.
  • Fermionic classical shadow protocols can sample their random unitaries directly from gate parameters, removing the cubic compilation bottleneck that limited previous implementations.
  • Since most sampled angles approach $\pi/2$ for large $n$, Haar-random FLO circuits are mostly Clifford-like, which can be exploited in implementations that prefer Clifford gates.
  • Algorithm 1 provides uniform Clifford FLO sampling with the provably optimal average number $n^2/2$ of two-qubit gates, giving a cheaper way to form a 3-design over FLO for applications where the third moment suffices.
  • The negative result that Clifford FLO is not a 4-design means fourth-moment quantities, such as certain shadow variances, require true Haar FLO or a different design.

Reading between the lines

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

  • One can test the same turnover calculus on other matchgate connectivities, such as periodic boundary conditions, two-dimensional grids, or long-range couplings, by re-deriving the exponent functions $f_n$ and $g_n$ and certifying the result with a frame-potential test; the paper does not do this, but nothing in the method is restricted to open chains.
  • Because most angles concentrate near $\pi/2$ as $n$ grows, truncating the peaked sine distributions or rounding angles to Clifford values should produce approximate, rather than exact, $t$-designs for small $t$; the paper gives the exact distributions but does not quantify the error of such truncations.
  • The exact commutant-dimension computation used to verify the frame potentials could serve as a general certificate that any proposed random-circuit ensemble is Haar-like; the paper presents it as a verification tool rather than as a standalone sampling method.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents circuit architectures for sampling Haar-random active and passive fermionic linear optics (FLO) unitaries directly from gate parameters, avoiding the usual O(n^3) classical matrix-sampling and compilation step. The central results are Theorem 1 and Theorem 2, which give brick-wall decompositions with n(2n-1) and n^2 parameters, respectively, and explicit product-of-sine probability densities (Eqs. (4)-(6) and (11)-(13)). The proofs start from Hurwitz's SO(2n)/U(n) decompositions, map them to matchgate circuits via the Majorana representation, and then use turnover gate identities to compress triangular circuits into brick-wall form. The paper also gives an algorithm for sampling Clifford FLO circuits with O(n^2) average two-qubit gate count, proves that Clifford FLO do not form a 4-design over FLO, and reports numerical frame-potential checks for n=4 that support the low-order moment structure.

Significance. If the central claims are valid, this is a useful advance for quantum information practice: it removes the O(n^3) classical compilation step of previous direct sampling methods and provides circuits with simultaneous Theta(n) depth and Theta(n^2) gate count, which is relevant for randomized benchmarking, fermionic classical shadows, and fermionic random circuit sampling. The explicit formulas are concrete and falsifiable, the local lemmas (Lemmas 1-4 and 6) are derived in detail, and the numerical frame-potential comparisons give nontrivial evidence for the low moments. The main gap is the global compression proof, which is presented as a diagrammatic sketch rather than a formal induction; this gap also affects the passive-FLO theorem, whose proof is delegated as 'completely analogous'.

major comments (3)
  1. [SI A.4, Eqs. (A47)-(A49)] The global compression step that converts the triangular circuit into the brick-wall architecture is the load-bearing step for Theorem 1, but it is asserted rather than proven. The text states that the ordering captured by Sigma_k and the initial assignment 'ensures' the turnover condition, and then lists the final anti-diagonal vectors in Eq. (A48) without a formal induction. Please provide a complete proof, ideally a rigorous induction, that (i) the sequence of turnovers is executable in the stated order, (ii) the middle-gate exponent condition r+r'+1 is satisfied at every turnover, and (iii) the boundary anti-diagonals d_1 and d_{2n-1}, including the k'=n-1 case, yield exactly Eq. (A48) and hence Eq. (6). A hidden swap-ordering or boundary error here would change the sampled distribution and invalidate the main theorem.
  2. [SI B.4] The passive-FLO proof is delegated with the statements 'this optimization is completely identical' and 'completely analogous' to the active case. This is not sufficient: the passive case has n anti-diagonals rather than 2n-1, a different turnover lemma (Lemma 6), and additional phase parameters. A boundary or ordering error in the passive compression would break Theorem 2 just as in the active case. The SI should present the passive analogue of the Sigma_k argument explicitly, including the modified exponent vector definitions and boundary handling, rather than relying on analogy.
  3. [II A, depth claim] The sentence 'This is optimal as one cannot further compress the n(2n-1) gates' is a gate-count lower bound, not a depth lower bound. To justify the advertised 'optimal down-to-the-constant-factor Theta(n) depth', the paper needs an explicit depth lower bound for any nearest-neighbor matchgate circuit sampling the full FLO group, together with a matching analysis of the depth of the 2n layers in Eq. (3), including the intra-layer scheduling of non-commuting XX gates on adjacent edges.
minor comments (4)
  1. [SI B.4, Eq. (B26)] There are typos in the intermediate factors of the Lemma 6 proof: 'sin(t1)' should be 'sin(t3)', and '(sin theta3 sin theta3)' should be '(sin theta2 sin theta3)'. The final formula is correct, but these errors make verification harder.
  2. [Eq. (3)] The stated range 'for 1<k<n' conflicts with the layers L_1 and L_{2n} appearing in Eq. (2). The range should be stated consistently, e.g., 1<=k<=n, with the angle domains specified accordingly.
  3. [Fig. 8] The text states that the sampled frame potentials match the exact commutant dimensions 'despite the statistical uncertainty' but does not quantify that uncertainty. Adding error bars or a table of the numerical values used for n=4 and t=1,...,5 would make the verification reproducible.
  4. [Algorithm 1 and SI F] The notation [k] is used without a definition; the proof assumes l is chosen from {1,...,k}. Please define the interval notation explicitly in the main text and use it consistently in the algorithm and its proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Haar-measure derivation is anchored in the external Hurwitz decomposition, the turnover measure transformation is proven with an explicit Jacobian, and the numerical benchmarks are checked against independently computed commutant dimensions.

full rationale

The paper's central claim, Theorem 1, derives the brick-wall Haar measure from three ingredients: (i) Hurwitz's external SO(2n) decomposition and measure (Lemma 1, cited to Ref [37], a historical external source), (ii) a proven Lie-algebra isomorphism between so(2n) and the matchgate dynamical algebra (Lemma 2, proven by explicit commutator and trace computations), and (iii) a turnover transformation that is an exact three-gate identity cited from external works [27,41] whose authors do not overlap with the present paper. Critically, the measure transformation under the turnover is not cited but proven in Lemma 4 via a Jacobian computation using the unique Haar measure of SU(2) and the matrix-element relations sin(alpha)sin(beta)=sin(b)sin(c), sin(beta)sin(gamma)=sin(a)sin(b); the algebra in Eq. (A46) goes through explicitly and yields the claimed exponent-swap pattern. The function f_n in Eq. (6) is the output of the anti-diagonal compression procedure in SI A.4 (Eqs. (A47)-(A49)), not an input assumed in advance: it is read off from the transformed anti-diagonal vectors that follow from applying the turnover rules to the Hurwitz exponents. No fitted parameters appear anywhere; the sampling distributions are parameter-free and derived. The numerical section provides genuine external validation: frame potentials are compared against t-fold commutant dimensions computed by a general representation-theoretic method (SI G, Weyl character ring) that is independent of the specific FLO Haar measure claim, and for t>=2 a uniform-parameter alternative clearly deviates from the Haar values, demonstrating that the derived exponent pattern is what makes the sampling correct. The self-citations present (Refs [23], [57], [72]) support standard background facts (irrep decomposition of operators under matchgate action, frame-potential formalism, Weingarten calculus) and are not load-bearing for the central derivation. The weakest step is the global compression in SI A.4, which is presented as a diagrammatic/sketch argument asserting that the swap ordering ensures the r+r'+1 condition throughout; the paper itself flags the caveat that Lemma 4's transformation is 'only valid when the original power of the sinusoidal function associated with the middle gate equals the sum of the powers of the two external gates plus one'.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; all angle distributions are derived from the Haar measure. The construction relies on standard Lie theory, the known turnover identity, and the spinor representation of SO(2n); none of these are invented for this paper.

assumptions (5)
  • standard math Hurwitz decomposition of SO(d) and U(d) with the associated left/right-invariant Haar measures (Ref [37])
    Starting point for all circuit constructions; used in Lemma 1 and SI B.1 to obtain the initial triangular Haar random FLO circuits.
  • domain assumption Adjoint action of FLO unitaries is isomorphic to the standard representation of SO(2n) (Eq. (23)/(A32))
    Bridges matrix decompositions to matchgate circuits; relies on the spinor representation of SO(2n) and the mapping of Majorana operators, cited to Refs [17,23,28].
  • domain assumption Turnover identity URj(α)URj+1(β)URj(γ) = URj+1(a)URj(b)URj+1(c) from Refs [27,41]
    The circuit compression from triangular to brick-wall uses this algebraic identity. The paper proves how the probability densities transform under it (Lemma 4), but takes the existence of the angle mapping from [27,41] as given.
  • domain assumption The gate generators {Z_q, X_q X_{q+1}} (active) and {Z_q, X_qY_{q+1}-Y_qX_{q+1}} (passive) generate the full dynamical Lie algebra so(2n) and u(n) respectively
    Needed for the circuit to cover all of FLO; proven via the isomorphism in Lemma 2 and standard Lie closure arguments cited to [17,23].
  • standard math Schur's lemma and Weyl character ring computations for exact commutant dimensions (SI G)
    Used to compute the exact frame potential values that the numerical experiments compare against.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Optimal Haar random fermionic linear optics circuits." pith.science (2026). https://pith.science/paper/KFYJVYEU

@misc{pith2026250524212,
  author       = {Pith},
  title        = {Pith review of: Optimal Haar random fermionic linear optics circuits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KFYJVYEU}},
  note         = {Machine review of arXiv:2505.24212}
}
abstract

Sampling unitary Fermionic Linear Optics (FLO), or matchgate circuits, has become a fundamental tool in quantum information. Such capability enables a large number of applications ranging from randomized benchmarking of continuous gate sets, to fermionic classical shadows. In this work, we introduce optimal algorithms to sample over the non-particle-preserving (active) and particle-preserving (passive) FLO Haar measures. In particular, we provide appropriate distributions for the gates of $n$-qubit parametrized circuits which produce random active and passive FLO. In contrast to previous approaches, which either incur classical $\mathcal{O}(n^3)$ compilation costs or have suboptimal depths, our methods directly output circuits which simultaneously achieve an optimal down-to-the-constant-factor $\Theta(n)$ depth and $\Theta(n^2)$ gate count; with only a $\Theta(n^2)$ classical overhead. Finally, we also provide quantum circuits to sample Clifford FLO with an optimal $\Theta(n^2)$ gate count.

Figures

Figures reproduced from arXiv: 2505.24212 by the authors.

Figure 1
Figure 1. Schematic representation of our results. Our goal is to find optimal circuits which implement Haar random active, passive, or Clifford FLO unitaries. Previous methods are based on parametrizing the entries of matrices from the SO(2n), U(n) and B2n ∩SO(2n), respectively, to later sample 2n × 2n matrices according to the Haar measure, and then compile these into qubit-based circuits. In this work we pro￾pose a more ef… view at source ↗
Figure 2
Figure 2. Optimal Haar random active FLO circuits. These quantum circuits produce Haar random active FLO on n = 3 (a) and n = 4 qubits (b), respectively, when the parameters are sampled according to Eq. (4), (5), and (6). Gates with the same color belong to the same “anti-diagonal”, which can be used to understand the arrangement of the probability density functions within the circuits. Namely, how the powers of the sine func… view at source ↗
Figure 4
Figure 4. Structure of Haar random active FLO cir￾cuits. Distribution of the powers of sine functions (f50 in Eq. (6)) that have to be sampled in a Haar random active FLO circuit with n = 50 qubits, according to the architecture in Theorem 1. Each gate is shown as a rectangle, with larger blocks corresponding to Rxx rotations and smaller ones to Rz rotations. Colors encode the values of f50; the horizontal axis denotes the ci… view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: Optimal Haar random passive FLO circuits. These quantum circuits produce Haar random passive FLO on n = 4 (a) and n = 5 qubits (c), respectively, when the parameters are sampled according to Eq. (11), (12), and (13). Gates with the same color belong to the same “anti-d…
Figure 6
Figure 6. Figure 6: Decomposition of the Rxy−yx(θ) passive FLO gate. We show how to implement an Rxy−yx(θ) rotation in terms of Rz and Rxx gates (up to an unobservable global sign). This decomposition was obtained following Eqs. (16) and (18) as indicated by the colors in the ladder-like …
Figure 7
Figure 7. Figure 7: Number of two-qubit gates of random Clif￾ford FLO circuits. We compare the average number of two￾qubit gates obtained by sampling 104 random Clifford FLO circuits for each circuit width n, according to Algorithm 1 (blue diamonds) or the one proposed in Ref. [35] (orang…
Figure 9
Figure 9. Figure 9: Bringing the FLO Haar measure to optimal depth. We graphically show the set of rules that allow us to find optimal Haar random active FLO circuits. In par￾ticular, these rules specify how the Haar measure transforms under certain “turnovers” of the gates, according to …

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Emergence of Generic Entanglement Structure in Doped Matchgate Circuits

    quant-ph 2025-07 conditional novelty 6.0 of 10

    Non-Gaussian doping of matchgate circuits restores generic entanglement growth in unitary evolution and, at extensive per-time injection rates, stabilizes a volume-law phase under measurements.

Reference graph

Works this paper leans on

97 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [1]

    Here, we note that while there exist sev- eral parametrizations for these Lie groups, not all of them are well suited for implementation as quan- tum circuits

    We start with known parametrizations of the standard representations of the classical compact Lie groupsSO(2n) andU(n), found by Hurwitz, together with the corresponding invariant mea- sures. Here, we note that while there exist sev- eral parametrizations for these Lie groups, not all of them are well suited for implementation as quan- tum circuits. For i...

  2. [2]

    We identify isomorphisms between the Lie algebras so(2n) oru(n), and the dynamical Lie algebra of the respective FLO circuits, active or passive

  3. [3]

    We use these isomorphisms to translate Hurwitz’s decompositions into suboptimal quantum circuits with a triangular shape

  4. [4]

    turnovers

    We derive a set of rules that allow us to track how the Haar measure transforms under certain rearrangements of the circuits’ gates known as turnovers[27, 41]. This allows us to bring the pre- vious triangular circuits into their optimal shape. We will now delve into more details for the case of active FLO (the reasoning for passive FLO is completely anal...

  5. [5]

    L. G. Valiant, Quantum computers that can be simu- lated classically in polynomial time, inProceedings of the thirty-third annual ACM symposium on Theory of com- puting(2001) pp. 114–123

    2001

  6. [6]

    Knill, Fermionic linear optics and matchgates, arXiv preprint arXiv:quant-ph/0108033 (2001)

    E. Knill, Fermionic linear optics and matchgates, arXiv preprint arXiv:quant-ph/0108033 (2001)

    arXiv 2001

  7. [7]

    B. M. Terhal and D. P. DiVincenzo, Classical simulation of noninteracting-fermion quantum circuits, Physical Re- view A65, 032325 (2002)

    2002

  8. [8]

    D. P. DiVincenzo and B. M. Terhal, Fermionic linear op- tics revisited, Foundations of Physics35, 1967 (2005)

    2005

Show all 97 references
  1. [9]

    I. D. Kivlichan, J. McClean, N. Wiebe, C. Gidney, A. Aspuru-Guzik, G. K.-L. Chan, and R. Babbush, Quantum simulation of electronic structure with lin- ear depth and connectivity, Physical review letters120, 110501 (2018)

    2018

  2. [10]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell,et al., Hartree-fock on a superconducting qubit quantum computer, Science369, 1084 (2020)

    2020

  3. [11]

    J. M. Arrazola, O. Di Matteo, N. Quesada, S. Jahangiri, A. Delgado, and N. Killoran, Universal quantum circuits for quantum chemistry, Quantum6, 742 (2022)

    2022

  4. [12]

    Verstraete, J

    F. Verstraete, J. I. Cirac, and J. I. Latorre, Quantum cir- cuits for strongly correlated quantum systems, Physical Review A79, 032316 (2009)

    2009

  5. [13]

    Kraus, Compressed quantum simulation of the ising model, Physical review letters107, 250503 (2011)

    B. Kraus, Compressed quantum simulation of the ising model, Physical review letters107, 250503 (2011)

    2011

  6. [14]

    Cervera-Lierta, Exact ising model simulation on a quantum computer, Quantum2, 114 (2018)

    A. Cervera-Lierta, Exact ising model simulation on a quantum computer, Quantum2, 114 (2018)

    2018

  7. [15]

    Jiang, K

    Z. Jiang, K. J. Sung, K. Kechedzhi, V. N. Smelyanskiy, and S. Boixo, Quantum algorithms to simulate many- body physics of correlated fermions, Physical Review Ap- plied9, 044036 (2018)

    2018

  8. [16]

    Dallaire-Demers, J

    P.-L. Dallaire-Demers, J. Romero, L. Veis, S. Sim, and A. Aspuru-Guzik, Low-depth circuit ansatz for prepar- ing correlated fermionic states on a quantum computer, Quantum Science and Technology4, 045005 (2019)

    2019

  9. [17]

    Sopena, M

    A. Sopena, M. H. Gordon, D. Garc ´ ıa-Mart ´ ın, G. Sierra, and E. L´ opez, Algebraic Bethe Circuits, Quantum6, 796 (2022)

    2022

  10. [18]

    R. Ruiz, A. Sopena, M. H. Gordon, G. Sierra, and E. L´ opez, The bethe ansatz as a quantum circuit, Quan- tum8, 1356 (2024)

    2024

  11. [19]

    R. Ruiz, A. Sopena, B. Pozsgay, and E. L´ opez, Ef- ficient eigenstate preparation in an integrable model with hilbert space fragmentation, arXiv preprint arXiv:2411.15132 (2024)

    2024

  12. [20]

    Bravyi, Lagrangian representation for fermionic linear optics, Quantum Info

    S. Bravyi, Lagrangian representation for fermionic linear optics, Quantum Info. Comput.5, 216–238 (2005)

    2005

  13. [21]

    Jozsa and A

    R. Jozsa and A. Miyake, Matchgates and classical simu- lation of quantum circuits, Proceedings of the Royal Soci- ety A: Mathematical, Physical and Engineering Sciences 464, 3089 (2008)

    2008

  14. [22]

    D. J. Brod and E. F. Galvao, Extending matchgates into universal quantum computation, Physical Review A—Atomic, Molecular, and Optical Physics84, 022310 (2011)

    2011

  15. [23]

    D. J. Brod and A. M. Childs, The computational power of matchgates and the xy interaction on arbitrary graphs, Quantum Information and Computation14, 901 (2014)

    2014

  16. [24]

    D. J. Brod, Efficient classical simulation of matchgate cir- cuits with generalized inputs and measurements, Physical Review A93, 062332 (2016)

    2016

  17. [25]

    Helsen, S

    J. Helsen, S. Nezami, M. Reagor, and M. Walter, Match- gate benchmarking: Scalable benchmarking of a continu- ous family of many-qubit gates, Quantum6, 657 (2022)

    2022

  18. [26]

    K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Commu- nications in Mathematical Physics404, 629 (2023)

    2023

  19. [27]

    N. L. Diaz, D. Garc ´ ıa-Mart ´ ın, S. Kazi, M. Larocca, and M. Cerezo, Showcasing a barren plateau the- ory beyond the dynamical lie algebra, arXiv preprint arXiv:2310.11505 (2023)

    2023 arXiv

  20. [28]

    N. L. Diaz, P. Braccia, M. Larocca, J. M. Matera, R. Rossignoli, and M. Cerezo, Parallel-in-time quantum 11 simulation via page and wootters quantum time, arXiv preprint arXiv:2308.12944 (2023)

    2023

  21. [29]

    A. A. Mele and Y. Herasymenko, Efficient learning of quantum states prepared with few fermionic non- gaussian gates, PRX Quantum6, 010319 (2025)

    2025

  22. [30]

    K¨ okc¨ u, T

    E. K¨ okc¨ u, T. Steckmann, Y. Wang, J. Freericks, E. F. Dumitrescu, and A. F. Kemper, Fixed depth hamilto- nian simulation via cartan decomposition, Physical Re- view Letters129, 070501 (2022)

    2022

  23. [31]

    K¨ okc¨ u, D

    E. K¨ okc¨ u, D. Camps, L. B. Oftelie, J. K. Freericks, W. A. de Jong, R. Van Beeumen, and A. F. Kemper, Algebraic compression of quantum circuits for hamiltonian evolu- tion, Physical Review A105, 032420 (2022)

    2022

  24. [32]

    Guaita, L

    T. Guaita, L. Hackl, and T. Quella, Representation theory of gaussian unitary transformations for bosonic and fermionic systems, arXiv preprint arXiv:2409.11628 (2024)

    2024 arXiv

  25. [33]

    Oszmaniec, N

    M. Oszmaniec, N. Dangniam, M. E. Morales, and Z. Zim- bor´ as, Fermion sampling: a robust quantum computa- tional advantage scheme using fermionic linear optics and magic input states, PRX Quantum3, 020328 (2022)

    2022

  26. [34]

    Somma, H

    R. Somma, H. Barnum, G. Ortiz, and E. Knill, Efficient solvability of Hamiltonians and limits on the power of some quantum computational models, Physical Review Letters97, 190501 (2006)

    2006

  27. [35]

    M. L. Goh, M. Larocca, L. Cincio, M. Cerezo, and F. Sauvage, Lie-algebraic classical simulations for quan- tum computing, arXiv preprint arXiv:2308.01432 (2023)

    2023

  28. [36]

    Miller, Z

    A. Miller, Z. Holmes, ¨O. Salehi, R. Chakraborty, A. Nyk¨ anen, Z. Zimbor´ as, A. Glos, and G. Garc ´ ıa-P´ erez, Simulation of fermionic circuits using majorana propaga- tion, arXiv preprint arXiv:2503.18939 (2025)

    2025

  29. [37]

    Oszmaniec and Z

    M. Oszmaniec and Z. Zimbor´ as, Universal extensions of restricted classes of quantum operations, Physical review letters119, 220502 (2017)

    2017

  30. [38]

    A. Zhao, N. C. Rubin, and A. Miyake, Fermionic par- tial tomography via classical shadows, Physical Review Letters127, 110504 (2021)

    2021

  31. [39]

    Heyraud, H

    V. Heyraud, H. Chomet, and J. Tilly, Unified frame- work for matchgate classical shadows, arXiv preprint arXiv:2409.03836 10.48550/arXiv.2409.03836 (2024)

    2024 doi arXiv

  32. [40]

    Mezzadri, How to generate random matrices from the classical compact groups, Notices of the American Math- ematical Society54, 592 (2007)

    F. Mezzadri, How to generate random matrices from the classical compact groups, Notices of the American Math- ematical Society54, 592 (2007)

    2007

  33. [41]

    Diaconis and P

    P. Diaconis and P. J. Forrester, Hurwitz and the origins of random matrix theory in mathematics, Random Ma- trices: Theory and Applications6, 1730001 (2017)

    2017

  34. [42]

    Zyczkowski and M

    K. Zyczkowski and M. Kus, Random unitary matrices, Journal of Physics A: Mathematical and General27, 4235 (1994)

    1994

  35. [43]

    Gottesman, The heisenberg representation of quan- tum computers, arXiv preprint quant-ph/9807006 (1998)

    D. Gottesman, The heisenberg representation of quan- tum computers, arXiv preprint quant-ph/9807006 (1998)

    1998 arXiv

  36. [44]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Physical Review A70, 052328 (2004)

    2004

  37. [45]

    Camps, E

    D. Camps, E. K¨ okc¨ u, L. Bassman Oftelie, W. A. De Jong, A. F. Kemper, and R. Van Beeumen, An algebraic quan- tum circuit compression algorithm for hamiltonian sim- ulation, SIAM Journal on Matrix Analysis and Applica- tions43, 1084 (2022)

    2022

  38. [46]

    Foxen, C

    B. Foxen, C. Neill, A. Dunsworth, P. Roushan, B. Chiaro, A. Megrant, J. Kelly, Z. Chen, K. Satzinger, R. Barends, et al., Demonstrating a continuous set of two-qubit gates for near-term quantum algorithms, Physical Review Let- ters125, 120504 (2020)

    2020

  39. [47]

    D. C. McKay, C. J. Wood, S. Sheldon, J. M. Chow, and J. M. Gambetta, Efficientzgates for quantum comput- ing, Phys. Rev. A96, 022330 (2017)

    2017

  40. [48]

    C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Physical Review A54, 3824 (1996)

    1996

  41. [49]

    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. Sloane, Quantum error correction and orthogonal geom- etry, Physical Review Letters78, 405 (1997)

    1997

  42. [50]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics16, 1050 (2020)

    2020

  43. [51]

    M. West, A. A. Mele, M. Larocca, and M. Cerezo, Real classical shadows, arXiv preprint arXiv:2410.23481 (2024)

    2024 arXiv

  44. [52]

    Webb, The clifford group forms a unitary 3-design, Quantum Information and Computation16, 1379 (2016)

    Z. Webb, The clifford group forms a unitary 3-design, Quantum Information and Computation16, 1379 (2016)

    2016

  45. [53]

    Zhu, Multiqubit clifford groups are unitary 3-designs, Physical Review A96, 062336 (2017)

    H. Zhu, Multiqubit clifford groups are unitary 3-designs, Physical Review A96, 062336 (2017)

    2017

  46. [54]

    Kueng and D

    R. Kueng and D. Gross, Qubit stabilizer states are complex projective 3-designs, arXiv preprint arXiv:1510.02767 (2015)

    2015 arXiv

  47. [55]

    H. Zhu, R. Kueng, M. Grassl, and D. Gross, The clif- ford group fails gracefully to be a unitary 4-design, arXiv preprint arXiv:1609.08172 (2016)

    2016 arXiv

  48. [56]

    Hashagen, S

    A. Hashagen, S. Flammia, D. Gross, and J. Wallman, Real randomized benchmarking, Quantum2, 85 (2018)

    2018

  49. [57]

    Mitsuhashi and N

    Y. Mitsuhashi and N. Yoshioka, Clifford group and uni- tary designs under symmetry, PRX Quantum4, 040331 (2023)

    2023

  50. [58]

    Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

    D. Gottesman,Stabilizer codes and quantum error cor- rection(California Institute of Technology, 1997)

    1997

  51. [59]

    A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Physical Review A86, 032324 (2012)

    2012

  52. [60]

    A. A. Mele, Introduction to haar measure tools in quan- tum information: A beginner’s tutorial, Quantum8, 1340 (2024)

    2024

  53. [61]

    Ragone, Q

    M. Ragone, Q. T. Nguyen, L. Schatzki, P. Braccia, M. Larocca, F. Sauvage, P. J. Coles, and M. Cerezo, Representation theory for geometric quantum machine learning, arXiv preprint arXiv:2210.07980 (2022)

    2022 arXiv

  54. [62]

    Wiersema, E

    R. Wiersema, E. K¨ okc¨ u, A. F. Kemper, and B. N. Bakalov, Classification of dynamical lie algebras of 2- local spin systems on linear, circular and fully connected topologies, npj Quantum Information10, 110 (2024)

    2024

  55. [63]

    K¨ okc¨ u, R

    E. K¨ okc¨ u, R. Wiersema, A. F. Kemper, and B. N. Bakalov, Classification of dynamical lie algebras gener- ated by spin interactions on undirected graphs, arXiv preprint arXiv:2409.19797 10.48550/arXiv.2409.19797 (2024)

    2024 doi arXiv

  56. [64]

    Aguilar, S

    G. Aguilar, S. Cichy, J. Eisert, and L. Bittel, Full classification of pauli lie algebras, arXiv preprint arXiv:2408.00081 (2024)

    2024 arXiv

  57. [65]

    Schatzki, M

    L. Schatzki, M. Larocca, Q. T. Nguyen, F. Sauvage, and M. Cerezo, Theoretical guarantees for permutation- equivariant quantum neural networks, npj Quantum In- formation10, 12 (2024)

    2024

  58. [66]

    S. Kazi, M. Larocca, and M. Cerezo, On the universality ofs n-equivariantk-body gates, New Journal of Physics 26, 053030 (2024). 12

    2024

  59. [67]

    Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nature Physics18, 283 (2022)

    I. Marvian, Restrictions on realizable unitary operations imposed by symmetry and locality, Nature Physics18, 283 (2022)

    2022

  60. [68]

    Spengler, M

    C. Spengler, M. Huber, and B. C. Hiesmayr, Composite parameterization and haar measure for all unitary and special unitary groups, Journal of Mathematical Physics 53, 10.1063/1.3672064 (2012)

    2012 doi

  61. [69]

    Zeier and T

    R. Zeier and T. Schulte-Herbr¨ uggen, Symmetry princi- ples in quantum systems theory, Journal of mathematical physics52, 113510 (2011)

    2011

  62. [70]

    B. C. Hall,Lie groups, Lie algebras, and representations (Springer, 2013)

    2013

  63. [71]

    P. A. Knight, Fast rectangular matrix multiplication and qr decomposition, Linear algebra and its applications 221, 69 (1995)

    1995

  64. [72]

    Strassen, Gaussian elimination is not optimal, Nu- merische mathematik13, 354 (1969)

    V. Strassen, Gaussian elimination is not optimal, Nu- merische mathematik13, 354 (1969)

    1969

  65. [73]

    Le Gall, Powers of tensors and fast matrix multipli- cation, Proceedings of the 39th international symposium on symbolic and algebraic computation , 296 (2014)

    F. Le Gall, Powers of tensors and fast matrix multipli- cation, Proceedings of the 39th international symposium on symbolic and algebraic computation , 296 (2014)

    2014

  66. [74]

    A. V. Aho,The Design and Analysis of Computer Algo- rithms(Reading/Addison-Wesley, Reading, 1974)

    1974

  67. [75]

    Ginibre, Statistical ensembles of complex, quaternion, and real matrices, Journal of Mathematical Physics6, 440 (1965)

    J. Ginibre, Statistical ensembles of complex, quaternion, and real matrices, Journal of Mathematical Physics6, 440 (1965)

    1965

  68. [76]

    Garc ´ ıa-Mart ´ ın, M

    D. Garc ´ ıa-Mart ´ ın, M. Larocca, and M. Cerezo, Deep quantum neural networks form gaussian processes, arXiv preprint arXiv:2305.09957 (2023)

    2023 arXiv

  69. [77]

    Fulton and J

    W. Fulton and J. Harris,Representation Theory: A First Course(Springer, 1991)

    1991

  70. [78]

    J. E. Humphreys,Introduction to Lie algebras and rep- resentation theory, Vol. 9 (Springer Science & Business Media, 2012)

    2012

  71. [79]

    Racah, Group theory and spectroscopy, Springer Tracts in Modern Physics, Volume 37 , 28 (2006)

    G. Racah, Group theory and spectroscopy, Springer Tracts in Modern Physics, Volume 37 , 28 (2006)

    2006

  72. [80]

    The Sage Developers, Sagemath, the sage mathemat- ics software system,https://www.sagemath.org(2021), version 9.3, 2021

    2021

  73. [81]

    A. W. Knapp,Lie Groups Beyond an Introduction, Vol. 140 (Springer Science & Business Media, 2013)

    2013

  74. [82]

    Nazarov, O

    A. Nazarov, O. Postnova, and T. Scrimshaw, tatistical ensembles of complex, quaternion, and real matricess, Journal of the London Mathematical Society109, e12813 (2024). Supp. Info. A: Proof of Theorem 1 In this section we provide the detailed proof of Theorem 1, which we here...

    2024

  75. [83]

    Decomposition ofSO(d)matrices We first show that any matrixO∈SO(d) can be decomposed as O=O 1O2 · · ·Od−1 ,(A6) for Oj =R j(θj,j+1)· · ·R1(θ1,j+1),(A7) with Euler anglesθ 1,j+1 ∈[0,2π), andθ j′,j+1 ∈[0, π] forj ′ >1. Here, the Givens rotations are defined by Rj(θ) =e θLjj+1 = ...

  76. [84]

    Isomorphism between the Lie algebras ofSO(2n)andSPIN(2n) We now provide an explicit isomorphism between the dynamical Lie algebra of FLO circuits andso(2n). Lemma 2.The isomorphismφconnectinggin Eq.(21)andso(2n)is given by the linear map φ(Ljk ) = cjck 2 .(A17) Proof.To show t...

  77. [85]

    turnovers

    F romSO(2n)matrices to FLO circuits We here explain how to transform the decomposition ofSO(2n) matrices in Eqs. (A6) and (A7) into FLO circuits, by simply using the isomorphism in Eq. (A17). This results in Haar random active FLO circuits consisting of gates arranged in “ladd...

  78. [86]

    turnover

    Optimal Haar random FLO circuits We now make use of the “turnover” property of the gatesU Rj andU Rj+1 defined in Eq. (A26), to bring the circuits in Lemma 3 to optimal depth. This property was explained in Refs. [27, 41], and corresponds to the following transformation URj (α...

  79. [87]

    We recall that the standard representation of the unitary groupU(d) consists of the unitary matrices of sized×d, acting irreducibly onC d

    Decomposition ofU(d)matrices The derivation for the unitary groupU(d) is completely analogous to that forSO(d). We recall that the standard representation of the unitary groupU(d) consists of the unitary matrices of sized×d, acting irreducibly onC d. These matrices satisfy tha...

  80. [88]

    The isomorphism between the Lie algebras ofU(n)andSO(2n)∩SP(2n,R) We now provide an explicit isomorphism between the Lie algebras of the groupsU(n) andSO(2n)∩SP(2n,R). Analogously to the case of active FLO, we will use this isomorphism (together with the one between the algebr...

  81. [89]

    F romU(n)matrices to passive FLO circuits Using the previous isomorphisms betweenU(n),SO(2n)∩SP(2n,R), and the adjoint action of passive FLO, one can directly translate Hurwitz’s construction in the SI B 1 to obtain passive Haar random FLO circuits with a triangular shape, as ...

  82. [90]

    Optimal Haar random passive FLO circuits We finally proceed to determine how the Haar measure on passive FLO transforms under the turnover property of the gatesU ˜Rj andU ˜Rj+1 . Just as for the case of active FLO circuits, this property can be expressed by the following equal...

  83. [91]

    DecomposingH ⊗t into irreps,

  84. [92]

    Taking the tensor productH ⊗t ⊗(H ∗)⊗t ∼= L(H⊗t),

  85. [93]

    29 Indeed, if H⊗t = M λ mH⊗t λM i=1 Tλ,i,(G2) then, as we will see in Sec

    Applying Schur’s Lemma. 29 Indeed, if H⊗t = M λ mH⊗t λM i=1 Tλ,i,(G2) then, as we will see in Sec. G 2, by Schur’s Lemma dim L(H⊗t)G = X λ mH⊗t λ 2 .(G3) We now detail these steps

  86. [94]

    These multiplicity coefficients can be computed by writing the characters ofV λ andV µ as elements of the weight lattice (more on this in Subsection G 3)

    Irreps in the tensor powers Let us begin with the decomposition of theG-module T ≡ H⊗t = M λ mH λM i=1 Hλ,i ⊗t = M λ1,···,λ t mλM i1,···,i t=1 Hλ1,i1 ⊗ · · · ⊗ Hλt,it .(G4) Given twoG-irreps of typeλandµ, their tensor decomposes as V λ G ⊗V µ G ∼= M ν mλ⊗µ jM j=1 V ν G ,(G5) w...

  87. [95]

    Schur’s Lemma and the decomposition of the space ofG-invariantt-fold tensor operators LetMandNbe irreducibleG-modules. Schur’s Lemma (see, for example, page 7 in Fulton and Harris [73]) states that Hom(M,N) G = ( span{A}ifM ∼= N 0 else ,(G8) 30 whereA:M − → Nis the identity is...

  88. [96]

    Multiplicity computation via the W eyl Character Ring In this section we show how to compute the multiplicity coefficients appearing the the tensor product of irreducible representations (see e.g., Eq. (G5)). LetPbe theweight latticeof some semisimple Lie algebrag 5 P= n rX i=...

  89. [97]

    We now specialize to the cases of active and passive FLO

    Active and Passive FLO commutants We note the method presented here works for essentiallyanyrep of any reductive Lie algebra. We now specialize to the cases of active and passive FLO. LetH ∼= (C2)⊗n be the state space ofnfermionic modes, and letR act :SO(2n)− →U(H) andRpass :U...

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.