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Fast simulation of fermions with reconfigurable qubits

T0 review · 3 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Fermionic circuits can be simulated on reconfigurable qubits with logarithmic overhead, and with constant overhead for structured circuits such as the fermionic fast Fourier transform.

desk verdict Fresh interleave gadget gives O(log N) fermionic permutations, but the cost model hides feedforward and reconfiguration latency. read the letter →

arxiv 2509.08898 v1 pith:AYCT6WUW submitted 2025-09-10 quant-ph cond-mat.str-elphysics.atom-ph

Nishad Maskara , Marcin Kalinowski , Daniel Gonzalez-Cuadra , Mikhail D. Lukin This is my paper
classification quant-phcond-mat.str-elphysics.atom-ph MSC 81P68
keywords fermion-to-qubitmappingJordan-WignerencodingdynamicalreconfigurablequbitsfermionicfastFouriertransformSachdev-Ye-Kitaevmodelmid-circuitmeasurementquantumsimulation
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 establishes a compilation method that turns any fermionic circuit into a qubit circuit with dramatically less overhead than previously known. Standard fermion-to-qubit mappings pay a cost that grows linearly with the number of fermionic modes. Here, by switching the Jordan-Wigner ordering dynamically and implementing the switch with measurement and feedforward, the cost drops to O(log N) in the worst case and O(1) for important structured subroutines. If correct, this means fermionic statistics need not be an asymptotic bottleneck for quantum simulation on qubit hardware, and algorithms for chemistry, materials, and the Sachdev-Ye-Kitaev model become practical at much larger sizes.

What carries the argument

The interleave operation: a fermionic permutation that preserves order within two contiguous blocks A and B. The circuit applies a CNOT cascade to block B, permutes qubits, applies one layer of CZ gates from A to the nearest left B-mode, then reverses; the CNOT cascade is compiled to constant depth using ancillas, mid-circuit measurement, and feedforward. This broadcasts O(N) CZ gates into O(N^2) effective crossings. Recursive mergesort-style decomposition of arbitrary permutations into log N interleaves converts this into the O(log N) depth bound.

What would settle it

Measure, on a reconfigurable qubit platform or a simulator with realistic timing, the wall-clock latency of one interleave layer including measurement, classical parity computation, feedforward, and qubit movement as a function of N. If this latency grows faster than O(log N) (e.g., linearly), the claimed constant-depth interleave is not realized in that hardware model, and the O(log N) permutation bound fails. A more targeted check: for a specific permutation with N=8 or 16, simulate the full CNOT cascade with classically computed corrections and compare the output to the exact fermionic SWAP

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: an arbitrary fermionic circuit on N modes with T layers of two-mode gates can be simulated by a qubit circuit of O(N) qubits and depth O(T log N). The simulation works by dynamically changing which Jordan-Wigner encoding is in use: before each layer, a fermionic permutation reorders the modes so that every tunneling gate acts between adjacent qubits. The paper's technical core is a fast implementation of these fermionic permutations. An interleave—a permutation that only crosses two contiguous blocks—can be done in constant depth with O(N) Clifford gates and ancillas, because a CNOT cascade broadcasts a single layer of CZ gates into all the crossings of an O(N

Load-bearing premise

The paper assumes that all classical computation needed to compute feedforward corrections between a measurement and the conditional Pauli gate—up to O(N) logic operations—and any physical qubit reconfiguration takes no more time than O(log N); if feedforward latency or move time grows with N, the depth bounds shrink to linear.

Editorial extensions

If this is right

  • Any parallel fermionic circuit of depth T can be compiled to a qubit circuit of depth O(T log N), so fermionic statistics add only logarithmic overhead even in the worst case.
  • The fermionic fast Fourier transform can be implemented in depth O(log N) with O(1) overhead per qubit, a factor of N/log N improvement over swap-network encodings.
  • Sparse Sachdev-Ye-Kitaev Hamiltonians and translation-invariant materials models can be Trotterized or block-encoded with O(log N) depth per cycle, reducing gate counts by orders of magnitude at hundreds of modes.
  • Translation-invariant free-fermion states, including chiral topological insulators, p/d-wave superconductors, and critical states, can be prepared exactly in O(log L) depth on an L^D lattice.
  • Because the encoding circuits are Clifford and use only Pauli preparation and measurement, they are compatible with error-corrected, fault-tolerant implementations at modest cost.

Reading between the lines

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

  • If classical feedforward latency can be hidden under gate execution or pipelined, the practical overhead might be even smaller than the paper's asymptotic bound suggests; a hardware-aware cost model would test this.
  • The deformation machinery (deletions and duplications of O(1)-overhead permutations) suggests a general design principle: algorithms should be co-designed around a small toolbox of structured fermionic permutations, not compiled from arbitrary permutations afterward.
  • The equivalence proven here tightens the computational gap between fermionic and qubit models; one testable consequence is that fermionic advantage in quantum simulation, if any, must come from something other than statistics alone, such as memory or gate locality.
  • On architectures without native non-local connectivity or with slow feedforward, the overhead could revert to linear; the practical crossover size depends on the device and is a concrete benchmark to measure.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a dynamical Jordan-Wigner encoding for simulating arbitrary fermionic circuits on qubit processors. The central claim (Theorem 1) is that any T-layer, N-mode fermionic circuit can be simulated by O(N) qubits in depth O(T log N), and that structured circuits such as the fermionic fast Fourier transform can be simulated with O(1) overhead per layer. The key technical ingredient is a new compilation of fermionic permutation operators F_p: an arbitrary permutation is decomposed into O(log N) interleaves, each implemented by O(N) Clifford gates, O(N) ancillas, mid-circuit measurement, and classical feedforward, compressing the O(N^2) FSWAP network into O(N) gates. Applications to sparse SYK models, translation-invariant state preparation, and fault-tolerant compilation are discussed. The algebraic proofs of Lemmas 1 and the interleave construction are given in the Supplemental Material, with an extended toolbox of O(1)-overhead reflections and deformations.

Significance. If the cost model is accepted, the result is significant: it gives a constructive, parameter-free fermion-to-qubit compilation whose per-operation overhead is logarithmic for arbitrary circuits and constant for a useful class of structured permutations. This would substantially improve over the linear overhead of swap-network approaches and would make several simulation subroutines, including the FFFT and translation-invariant free-fermion state preparation, asymptotically faster. The paper's proofs are algebraic and detailed in the SI, and the numerical evidence for the interleave-SYK construction is suggestive. The strength of the contribution is, however, tied to the physical interpretation of 'depth'; the idealized cost model in Definition 2 is the main point of fragility.

major comments (3)
  1. [Methods, Definition 2; Lemma 1; Discussion] The above is the core concern.
  2. [Lemma 2, proof after Eq. (10)] This is a rigor issue in the central derivation, not a conceptual error.
  3. [SI Appendix A, Eq. (A10) and subsequent edge-case discussion] The SI proof is convincing in outline, but the omitted details are load-bearing for Lemma 1.
minor comments (6)
  1. [Abstract and Section 'Fast Fermion Permutations'] The abstract states O(1) overhead for 'circuits with additional structure, including important subroutines like the fermionic fast Fourier transform.' As written, the 1D FFFT has depth O(log N), not O(1); the intended meaning is O(1) overhead per FFFT layer. Please state this explicitly to avoid confusion.
  2. [Eq. (7) and surrounding text] The notation L(i) Delta L(j) is used for the symmetric difference of intervals. This is defined in the text, but the reader must constantly re-derive that A Delta B means (A \ B) union (B \ A). A short sentence reminding the reader, or a diagram, would improve clarity.
  3. [Methods, Definition 2] Definition 2 permits 'any classical computation involving O(N) classical logic operations' to be free, but does not state whether this is O(N) sequential operations, O(N) parallel processors, or O(N) work with O(log N) depth. The distinction matters for the physical interpretation of the theorem. Please specify the parallel classical complexity model.
  4. [Figure 2 caption and SI, Fig. 14] The text says the CNOT cascade can be compiled into a 'depth-2 CNOT circuit' using one ancilla per CNOT. This is a standard measurement-based construction, but the number of ancillas, the measurement pattern, and the exact feedforward rule are not shown. A short derivation or a reference to the specific result in [49, 92] would help.
  5. [SYK numerical results, Fig. 4(c) and Fig. 7] The paper states that numerical optimization is used for random permutations and for the interleave-SYK model, but no code, data, or statistical details are provided. In a theory paper this is not fatal, but adding a reproducibility statement would strengthen the claims.
  6. [Throughout] There are several typos: 'pratical' in the Introduction, 'signficant' in the Fig. 1 caption, and 'analagous' in the Methods. A careful proofread is recommended.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: central compilation proof is constructive and self-contained; only caveats are the Definition 2 cost model that makes classical feedforward free and a minor non-load-bearing self-citation for hardware reconfiguration.

full rationale

The core result (Theorem 1) is derived from explicit constructions, not from fitted inputs or self-citations. Lemma 3 characterizes any fermionic permutation F_p as exactly the CZ circuit over mode pairs that change relative order under the Jordan-Wigner encoding; Lemma 1 shows the interleave realizes that CZ circuit via CNOT/CZ identities, compressing the CNOT cascade with measurement-based techniques from Ref. [92], an external reference. Lemma 2 gives a constructive mergesort-style decomposition of an arbitrary permutation into log N interleaves, proven in the text. No parameter is fit to data and then renamed a prediction, no uniqueness theorem is imported from the authors, and no ansatz is smuggled in by citation. The FFFT and state-preparation applications simply reuse this proven toolbox. Two caveats are noted without being circular: (1) Definition 2 declares 'any classical computation involving O(N) classical logic operations' to be free between measurement and conditional Pauli gates, so all depth claims are logical-layer depths under that cost model; if physical feedforward latency must be counted, the O(1)-depth interleave and hence Theorem 1's logarithmic bound are conditional on that model. (2) In the Discussion, Ref. [89] (co-authored by M. D. Lukin) is cited for O(log N) physical reconfiguration overhead on neutral-atom arrays; this supports the hardware claim but is not load-bearing for the mathematical theorem. The internal algebra is sound, so there is no circularity; score 1 reflects the absence of circularity with a cost-model caveat.

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

The central claim requires no fitted constants. It leans on a hardware model (reconfigurable connectivity, fast classical feedforward) and on known measurement-based Clifford compilation, which are listed as axioms. No new physical entities are introduced.

assumptions (5)
  • domain assumption The computation model permits non-local connectivity, mid-circuit measurement, and O(N)-scale classical feedforward that is 'fast' (does not contribute to reported depth). Definition 2.
    All depth bounds (Theorem 1, Lemma 1) count the interleave as O(1) because the CNOT cascade is compiled using measurement and feedforward instantiated by this model.
  • standard math Any Clifford circuit with O(1) gates per qubit can be parallelized into an O(1)-depth circuit with O(1) ancillas per qubit using measurement and feedforward.
    Invoked in 'Fast Fermion Permutations' and Fig 2c; referenced to [49, 63, 92], not re-derived in this paper.
  • domain assumption The fermionic circuit model (Definition 1), with parallel layers of two-mode gates, faithfully represents the simulation tasks of interest.
    The compilation theorem is stated for this circuit family; real Hamiltonians must first be decomposed into this model.
  • domain assumption Physical qubit permutations in reconfigurable hardware can be performed with O(log N) asymptotic overhead.
    The discussion cites [89] for neutral-atom arrays; if the move cost were O(N), the overall scaling would change.
  • standard math Jordan-Wigner mapping properties (Eqs. 5-8), including the CZ-circuit equivalence for changing orderings (Lemma 3).
    Used to define F_p and in the interleave correctness proof.

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Cite this review

Pith. "Pith review of Fast simulation of fermions with reconfigurable qubits." pith.science (2026). https://pith.science/paper/AYCT6WUW

@misc{pith2026250908898,
  author       = {Pith},
  title        = {Pith review of: Fast simulation of fermions with reconfigurable qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYCT6WUW}},
  note         = {Machine review of arXiv:2509.08898}
}
read the original abstract

Performing large-scale, accurate quantum simulations of many-fermion systems is a central challenge in quantum science, with applications in chemistry, materials, and high-energy physics. Despite significant progress, realizing generic fermionic algorithms with qubit systems incurs significant space-time overhead, scaling as O(N) for N fermionic modes. Here we present a method for faster fermionic simulation with asymptotic space-time overhead of O(log(N)) in the worst case, and O(1) for circuits with additional structure, including important subroutines like the fermionic fast Fourier transform. This exponential reduction is achieved by using reconfigurable quantum systems with non-local connectivity, mid-circuit measurement, and classical feedforward, to generate dynamical fermion-to-qubit mappings. We apply this technique to achieve efficient compilation for key simulation tasks, including Hamiltonian simulation of the sparse Sachdev-Ye-Kitaev model and periodic materials, as well as free-fermion state-preparation. Moreover, we show that the algorithms themselves can be adapted to use only the O(1)-overhead structures to further reduce resource overhead. These techniques can lower gate counts by orders of magnitude for practical system sizes and are natively compatible with error corrected computation, making them ideal for early fault-tolerant quantum devices. Our results tightly bound the computational gap between fermionic and qubit models and open new directions in quantum simulation algorithm design and implementation.

Figures

Figures reproduced from arXiv: 2509.08898 by the authors.

Figure 1
Figure 1. Simulating arbitrary fermionic circuits with qubits. (a) A fermionic circuit consisting of arbitrary non￾local tunneling and interaction gates. (b) Such a circuit can be implemented on a quantum computer with natively fermionic degrees of freedom, such as a reconfigurable array of fermionic atoms. (c) Our key technical result is an algorithm to compile an arbitrary permutation of fermionic modes, with at most O(log(… view at source ↗
Figure 2
Figure 2. Low-overhead fermionic permutations with qubits. (a) A key ingredient in our approach to simulating fermionic circuits with qubits is implementing an encoded fermionic permutation Fp. Here, we show how Fp can be implemented with O(N) ancilla qubits and depth O(log(N)), by applying multiple layers of interleave operations. (b) An interleave Ip is defined as a fermionic permutation, where modes are partitioned into tw… view at source ↗
Figure 3
Figure 3. Toolbox of efficient fermionic permutations. (a) There are a variety of structured fermionic permutations that can be performed with O(1)-overhead, in addition to the interleave. (b) These operations include a 1D reflection, where the order of the fermionic modes is reversed. (c) It also includes a more intricate 2D reflection, which switches from a row-wise ordering where the index of the mode at position (r, c) is… view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Compiling non-local SYK models. (a) We consider non-local models where each mode is involved in d distinct four-majorana interactions, described by a constant￾degree and d-colorable interaction graph. (b) A single Trot￾ter cycle or block encoding of these Hamiltonians …
Figure 5
Figure 5. Figure 5: FFFT and low-depth state preparation: (a) The 1D FFFT on L fermionic modes can be implemented in log(L) steps of O(1) depth, each of them consisting of parallel interleave operations (see Fig. 1a) and one layer of entangling gates [21, 68]. (b) The 2D FFFT on a L × L l…
Figure 6
Figure 6. Figure 6: Majorana permutation protocol. Consider a system of N qubits simulating fermions via the Jordan-Wigner encoding. There are 2N majorana operators that form the fermionic operations. Here, our goal is to realize a majorana permutation Mp that non-locally shuffles the maj…
Figure 7
Figure 7. Figure 7: Interleave-SYK Hamiltonians and their properties. (a) Spectral form factor (SFF) of SYK Hamiltonians (N=14) for the sparse, complete, and interleave models. (b) Minimum values (dips) of I-SYK form factors as a function of interleave permutation rounds. The asymptotic v…
Figure 8
Figure 8. Figure 8: 1D reflection protocol. (a) In the more general version of the protocol, we start by introducing an ancilla register of N qubits, initialized in |0⟩. Then, we apply CNOT circuits to compress the target FSWAP, such that the targets are always acting on ancilla qubits. T…
Figure 9
Figure 9. Figure 9: 2D Reflection Protocol. (1) A 2D reflection is defined as follows. Consider fermionic modes laid out in a 2D Lr × Lc grid, and initially encoded in a Jordan-Wigner encoding, with m0(r, c) = rLc + c and r = 0, ..., Lr − 1, c = 0, ..., Lc − 1. The 2D reflection switches …
Figure 10
Figure 10. Figure 10: Protocol for deformed permutations. Here, we illustrate the deformation gadgets (deletion and duplication) by showing how to implement a deformation of a 2D reflection. The key idea is to introduce an ancilla register on which we will apply the standard (i.e. undeform…
Figure 11
Figure 11. Figure 11: State preparation circuits for translationally-invariant Slater determinants. Translational-invariant Slater determinants, corresponding to ground states of particle-number conserving non-interacting Hamiltonians, can be prepared at any filling using a O(log L)-depth …
Figure 12
Figure 12. Figure 12: State preparation circuit for translationally-invariant d-wave superconductors. (1) The state is first prepared in momentum space by applying a two-mode circuit corresponding to a Bogoliubov transformation for every pair of modes k ↑ (yellow) and −k ↓ (blue). (2) Thes…
Figure 13
Figure 13. Figure 13: State preparation circuit for translationally-invariant p-wave superconductors. (1) The state is first prepared in momentum space, where two-modes circuits (gray rectangles) are applied in parallel to create local superpositions of 0 and 2 fermions shared between k (y…
Figure 14
Figure 14. Figure 14: Encoding fermionic permutations into error-correcting codes. (a) System qubits, which are long-lived and persist through the computation, should be stored in an error-correcting code, such as the standard surface code. (b) In contrast, the temporary ancilla qubits do …
Figure 15
Figure 15. Figure 15: Compilation of the FFFT. (a) The FFFT can be constructed recursively [68]. Consider a 1D array with N = 8 fermions (orange circles). The recursion starts by separating the even and odd index sites via a permutation. Then two FFFT on N/2 modes each is performed. Finall…

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Forward citations

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Reference graph

Works this paper leans on

45 extracted references · 8 linked inside Pith · cited by 11 Pith papers

  1. [1]

    The ordering of modes at timetis described bym t(i)

    Switching Jordan-Wigner encodings We start by defining in detail the operation that is required to switch between two Jordan-Wigner encodings. The ordering of modes at timetis described bym t(i). For eachiandt, letL t(i) ={j|m t(j)< mt(i), j= 1, ..., N}denote the set of all modes to the left ofiduring stept. The Jordan-Wigner encoding of the majorana ferm...

  2. [2]

    2b implements a correct fermionic permutationF p

    Proof of Correctness of Interleave Circuit Next, we prove Lemma 1, by showing that the interleave circuit depicted in Figure. 2b implements a correct fermionic permutationF p. In particular, the approach we take here is to directly show equivalence with the CZ circuit defined in Lemma 3 using simple circuit identities. Note that a more general framework i...

  3. [3]

    Diagonal entries are defined up to multiples of 2, so can be mapped to to{0,1}

    Composition of two circuits is described by matrix addition,C Z(A1) +CZ(A2) =C Z(A1 +A 2). Diagonal entries are defined up to multiples of 2, so can be mapped to to{0,1}. Off-diagonal entries are defined up to integers, and can be mapped to{0,1/2}. Definition 5.AnN-qubit circuitC X (P)composed entirely of CNOT gates is specified by a matrixP∈F N×N 2 . Two...

  4. [4]

    adjacent

    Deformations of structured permutations The examples above are specific operations that can be performed withO(1) overhead. In practice, to compile generic permutations, and more complex structures, it is helpful to consider permutations which are simple deforma- tions of these structures. To enable this, we define two kinds of deformations: deletions and...

  5. [32]

    Our approach is to represent CZ circuits as quadratic polynomials [93], whose coefficients are encoded into matrices

    F ormalism for working with CNOT + CZ circuits To start, we introduce some formalism and definitions, which we will use to generate efficient algorithms and protocols for constructing CZ circuits relevant for implementing fast fermionic permutations. Our approach is to represent CZ circuits as quadratic polynomials [93], whose coefficients are encoded int...

  6. [33]

    2.C Z(A)acts on a computational basis state|x⟩=|x 1⟩|x2⟩...|xn⟩by applying a phaseC Z(A)|x⟩= (−1) fA(x)|x⟩ wheref A(x) =x T Ax= P ij xiAijxj

    We often decomposeA=A 0 + 1 2 (B+B T ), whereA 0 ∈F N 2 is a diagonal matrix andB∈F N(N−1)/2 2 is a lower-triangular matrix. 2.C Z(A)acts on a computational basis state|x⟩=|x 1⟩|x2⟩...|xn⟩by applying a phaseC Z(A)|x⟩= (−1) fA(x)|x⟩ wheref A(x) =x T Ax= P ij xiAijxj

  7. [35]

    In particular,C X (P)X j = Q i(Xi)Pij CX (P)

    The matrixPcaptures howX-operators transform. In particular,C X (P)X j = Q i(Xi)Pij CX (P)

  8. [36]

    In particular,C X (P)|x⟩=|x ′⟩where(x ′)i =P j Pijxj, or equivalentlyx ′ =P x

    Similarly,Pcaptures how computational basis states transform. In particular,C X (P)|x⟩=|x ′⟩where(x ′)i =P j Pijxj, or equivalentlyx ′ =P x

Show all 45 references
  1. [37]

    Lemma 4.A CZ circuitC Z(A)conjugating by a CNOT circuitC X (P)is equivalent to a CZ circuitC Z(A)where A=P T APis defined below

    Composition of twoC X circuits is equivalent to matrix multiplication,C X (P1)CX (P2) =C X (P1P2). Lemma 4.A CZ circuitC Z(A)conjugating by a CNOT circuitC X (P)is equivalent to a CZ circuitC Z(A)where A=P T APis defined below. 19 Proof.Letxbe a bit-string. By definition, the ...

  2. [38]

    2D reflection

    General purpose algorithm for compiling a fermionic permutation Our key technical result is a way of compressing the CZ circuits arising from fermionic permutation, into more efficient ones constructed by conjugating CZ circuits with CNOT circuits. In the main text, we present...

  3. [40]

    Particle-number conserving TIFF states Consider first the case of a particle-number conserving TIFF state, i.e., a Slater determinant, with a unit cell composed of a finite number of modes that repeat periodically in space. Such a state corresponds, e.g., to the ground state o...

  4. [41]

    These states involve pairing of modes in momentum space, related by reflectionk,-k

    Superconducting TIFF states With a slight modification, our protocol can be also used to prepare TI superconducting states inO(logL) depth. These states involve pairing of modes in momentum space, related by reflectionk,-k. To see this, let as first consider the mean-field Ham...

  5. [42]

    (D17) Here,c † R,α andc R,α are the fermionic creation and annihilation operators, respectively, at unit cellRand internal indexα∈ {1,

    Generic TIFF state Finally, we consider the most general quadratic fermionic Hamiltonian in real space, given by H= X R,R′ X α,β tαβ(R−R ′)c† R,αcR′,β + 1 2 ∆αβ(R−R ′)c† R,αc† R′,β + H.c. .(D17) Here,c † R,α andc R,α are the fermionic creation and annihilation operators, respe...

  6. [43]

    It turns out that fermionic permutationsF p areClifford circuits, which are among the least-expensive operations in standard fault-tolerant architectures [71, 95]

    Encoding fermionic permutations into error-correcting codes When working with encoded information, not all single and two-qubit operations are equal in terms of resources [57, 90]. It turns out that fermionic permutationsF p areClifford circuits, which are among the least-expe...

  7. [44]

    This is in sharp contrast to an architecture based on local FSW AP networks, which requires depthN/2 andO(N 2) total Clifford gates

    Compiling the fermionic fast fourier transform (FFFT) into a fault-tolerant gateset Our approach to implementing fermionic permutations enables an asymptotic reduction in the Clifford cost of implementing the FFFT, by enabling the interleave fermionic permutation ofNmodes to b...

    2016

  8. [45]

    In Figure

    Considerations arising from 2D FFFT The advantage of the dynamical JW approach over FSW AP networks becomes even more dramatic when considering more complex applications of the FFFT, for example to perform higher-dimensional FFFT. In Figure. 4, we discuss the 2D FFFT, and expa...

  9. [85]

    & Williamson, D

    Ren, Y., Tantivasadakarn, N. & Williamson, D. J. Effi- cient preparation of solvable anyons with adaptive quan- tum circuits.Physical Review X15(2025)

    2025

  10. [86]

    Christos, M.et al.A model of d-wave superconductiv- ity, antiferromagnetism, and charge order on the square lattice.Proc. Natl. Acad. Sci. U.S.A.120, e2302701120 (2023)

    2023

  11. [87]

    P., Berg, E., Kivelson, S

    Arovas, D. P., Berg, E., Kivelson, S. A. & Raghu, S. The hubbard model.Annu. Rev. Condens. Matter Phys.13, 239–274 (2022)

    2022

  12. [88]

    & Gull, E

    Qin, M., Sch¨ afer, T., Andergassen, S., Corboz, P. & Gull, E. The hubbard model: A computational perspec- tive.Annu. Rev. Condens. Matter Phys.13, 275–302 (2022)

    2022

  13. [89]

    B., Bluvstein, D., Lukin, M

    Tan, D. B., Bluvstein, D., Lukin, M. D. & Cong, J. Compiling quantum circuits for dynamically field- programmable neutral atoms array processors.Quan- tum8, 1281 (2024)

    2024

  14. [90]

    Preprint at https://arxiv.org/abs/2506.20661 (2025)

    Bluvstein, D.et al.Architectural mechanisms of a uni- versal fault-tolerant quantum computer. Preprint at https://arxiv.org/abs/2506.20661 (2025)

    2025

  15. [91]

    M., Su, Y., Tran, M

    Childs, A. M., Su, Y., Tran, M. C., Wiebe, N. & Zhu, S. Theory of trotter error with commutator scaling.Phys. Rev. X11, 011020 (2021)

    2021

  16. [92]

    & Woerner, S

    B¨ aumer, E. & Woerner, S. Measurement-based long- range entangling gates in constant depth.Phys. Rev. Res.7, 023120 (2025)

    2025

  17. [93]

    Quantum circuits and low-degree poly- nomials overF 2.J

    Montanaro, A. Quantum circuits and low-degree poly- nomials overF 2.J. Phys. A50, 084002 (2017)

    2017

  18. [94]

    Haldane, F. D. M. Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parity anomaly”.Phys. Rev. Lett.61, 2015–2018 (1988)

    2015

  19. [95]

    A game of surface codes: Large-scale quan- tum computing with lattice surgery.Quantum3, 128 (2019)

    Litinski, D. A game of surface codes: Large-scale quan- tum computing with lattice surgery.Quantum3, 128 (2019)

    2019

  20. [96]

    D.et al.Improved Fault-Tolerant Quan- tum Simulation of Condensed-Phase Correlated Elec- trons via Trotterization.Quantum4, 296 (2020)

    Kivlichan, I. D.et al.Improved Fault-Tolerant Quan- tum Simulation of Condensed-Phase Correlated Elec- trons via Trotterization.Quantum4, 296 (2020)

    2020

  21. [97]

    & Fowler, A

    Gidney, C. & Fowler, A. G. Efficient magic state facto- ries with a catalyzed ccz to 2t transformation.Quantum 3, 135 (2019)

    2019

  22. [98]

    & Kitaev, A

    Bravyi, S. & Kitaev, A. Universal quantum computa- tion with ideal clifford gates and noisy ancillas.Phys. Rev. A71, 022316 (2005)

    2005

  23. [99]

    Fault-tolerant postselected quantum computation: Threshold analysis

    Knill, E. Fault-tolerant postselected quantum computation: Threshold analysis. Preprint at https://arxiv.org/abs/quant-ph/0404104 (2004)

    2004 arXiv

  24. [100]

    Fault-tolerant postselected quan- tum computation: Schemes

    Knill, E. Fault-tolerant postselected quan- tum computation: Schemes. Preprint at https://arxiv.org/abs/quant-ph/0402171 (2004)

    2004 arXiv

  25. [101]

    Magic state distillation: Not as costly as you think.Quantum3, 205 (2019)

    Litinski, D. Magic state distillation: Not as costly as you think.Quantum3, 205 (2019). 15

    2019

  26. [102]

    & Reichardt, B

    Paetznick, A. & Reichardt, B. W. Universal fault- tolerant quantum computation with only transversal gates and error correction.Phys. Rev. Lett.111, 090505 (2013)

    2013

  27. [103]

    T., Duclos-Cianci, G

    Anderson, J. T., Duclos-Cianci, G. & Poulin, D. Fault- tolerant conversion between the steane and reed-muller quantum codes.Phys. Rev. Lett.113, 080501 (2014)

    2014

  28. [104]

    Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes

    Bombin, H. Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes. Preprint at https://arxiv.org/abs/1311.0879 (2015)

    2015 arXiv

  29. [105]

    Dimensional jump in quantum error cor- rection

    Bombin, H. Dimensional jump in quantum error cor- rection. Preprint at https://arxiv.org/abs/1412.5079 (2016)

    2016 arXiv

  30. [106]

    E., Kubica, A

    Beverland, M. E., Kubica, A. & Svore, K. M. Cost of universality: A comparative study of the overhead of state distillation and code switching with color codes. PRX Quantum2, 020341 (2021)

    2021

  31. [107]

    & Seetharam, K

    Bomb ´ ın, H., Pant, M., Roberts, S. & Seetharam, K. I. Fault-tolerant postselection for low-overhead magic state preparation.PRX Quantum5, 010302 (2024)

    2024

  32. [108]

    & Noh, K

    Chamberland, C. & Noh, K. Very low overhead fault- tolerant magic state preparation using redundant ancilla encoding and flag qubits.npj Quantum Inf.6(2020)

    2020

  33. [109]

    & Jones, C

    Gidney, C., Shutty, N. & Jones, C. Magic state cultiva- tion: growing t states as cheap as cnot gates. Preprint at https://arxiv.org/abs/2409.17595 (2024)

    2024 arXiv

  34. [110]

    & Mosca, M

    Kliuchnikov, V., Maslov, D. & Mosca, M. Fast and efficient exact synthesis of single qubit uni- taries generated by clifford and t gates. Preprint at https://arxiv.org/abs/1206.5236 (2013)

    2013 arXiv

  35. [111]

    Ross, N. J. & Selinger, P. Optimal ancilla-free clif- ford+t approximation of z-rotations. Preprint at https://arxiv.org/abs/1403.2975 (2016)

    2016 arXiv

  36. [112]

    & Svore, K

    Paetznick, A. & Svore, K. M. Repeat-until-success: Non-deterministic decomposition of single-qubit uni- taries. Preprint at https://arxiv.org/abs/1311.1074 (2014)

    2014 arXiv

  37. [113]

    & Svore, K

    Bocharov, A., Roetteler, M. & Svore, K. M. Efficient synthesis of universal repeat-until-success quantum cir- cuits.Phys. Rev. Lett.114(2015). 16 Supplemental Material Appendix A: F ermionic Permutations The core technical result of this paper is a constructive procedure for e...

    2015

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