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Custom fermionic codes for quantum simulation

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Pith's one-line read This paper argues that strict locality is the wrong target for fermionic simulation encodings: quasi-local codes can reduce both qubit count and Pauli weight, and a general graph-based construction lets users tailor the code to the…

desk verdict Useful modular framework for fermionic encodings with a real but fixable gap: path-consistency is assumed, not proven. read the letter →

arxiv 2009.11860 v1 pith:OXLNUWHZ submitted 2020-09-24 quant-ph

Riley W. Chien , James D. Whitfield This is my paper · ORCID
classification quant-ph
keywords fermion-to-qubitmappingsquasi-localencodingsPauliweightreductionSYKmodelgeneralizedJordan-Wignerstringstoriccodequantumsimulationdeviceconnectivity
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 argues that strict locality is the wrong target when encoding fermionic systems into qubits for quantum simulation. The authors generalize a known local fermionic encoding so that the user chooses a system graph—any graph with paths between every pair of modes the Hamiltonian couples—and realizes each coupling by a generalized Jordan-Wigner string running through that graph. They show that relaxing locality this way can lower both the number of qubits and the Pauli weight (the number of qubits an operator acts on nontrivially) of the encoded operators, improving on strictly local encodings in examples and reducing the q=2 SYK model's worst-case Pauli weight from $O(N^3)$ to $O(N^2\log N)$. The construction interpolates between Jordan-Wigner and fully local toric-code-style encodings, and offers a concrete path for designing codes around device connectivity.

What carries the argument

The central object is a pair of graphs: the interaction graph, fixed by the Hamiltonian's two-body (and higher) terms, and the freely chosen system graph into which qubits are placed. Each vertex of degree $d$ holds $\lceil d/2 \rceil$ qubits carrying local Majorana operators; each edge of the system graph comes with an encoded edge operator, a product of two local Majoranas, and each plaquette comes with a stabilizer, a product of edge operators around the boundary. A coupling between two modes not joined by an edge is realized as a generalized Jordan-Wigner string, a product of edge operators along any path in the system graph; the stabilizers are what make different path choices equivalent. The paper also uses Fenwick-tree or ternary-tree local Majorana bases to reduce per-vertex operator weight from $O(d)$ to $O(\log d)$, and interprets blocking as truncating the MERA state-preparation circuit for the toric code state underlying local encodings.

What would settle it

Take a pair of boundary modes in the ternary-MERA or hyperbolic encoding, compute the generalized Jordan-Wigner strings along two distinct geodesics, and check whether their product is a stabilizer; if the two strings differ by an operator that is not a product of plaquette stabilizers, the encoding would not faithfully represent the fermionic Hamiltonian.

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Extended reading notes

Core claim

The central claim is that locality is overrated for fermion-to-qubit encodings: requiring that every Hamiltonian term map to a constant-weight local operator forces extra qubits and restricts the possible code geometries, while a quasi-local encoding—one in which couplings are carried by strings through a chosen system graph—can be cheaper. Because any path between two coupled modes gives an edge operator as a product of local edge operators along that path, the encoded operators are generalized Jordan-Wigner strings; they are valid if different paths between the same modes agree up to plaquette stabilizers. The paper demonstrates this on a square lattice with diagonal couplings, where omitting the diagonal edges from the system graph reduces qubit count and Pauli weight, and on the all-to-all SYK model, where virtual-mode geometries (a central hub, ternary tree, ternary MERA, hyperbolic tiling) bring the worst-case Pauli weight from $O(N^3)$ with a linear Jordan-Wigner chain down to $O(N^2\log N)$. The same framework yields a block-partitioning trade-off between locality and qubit count, and a method for embedding fermionic modes in device-specific lattices such as heavy-hexagon hardware.

Load-bearing premise

The construction assumes that any two routes through the chosen encoding graph between the same pair of fermionic modes give qubit operators that differ only by the graph's built-in consistency checks; the paper verifies this for the square lattice but not for the tree, MERA, or hyperbolic graphs used in its main examples.

Editorial extensions

If this is right

  • For a square lattice with nearest-neighbor and nearest-diagonal-neighbor couplings, omitting the diagonal edges from the system graph reduces both qubit count and the Pauli weight of encoded diagonal couplings, so strictly local encodings are not always optimal.
  • For the all-to-all q=2 SYK model, replacing the linear Jordan-Wigner geometry with a virtual geometry (hub, ternary tree, MERA-like, or hyperbolic tiling) reduces the worst-case Pauli weight from $O(N^3)$ to $O(N^2\log N)$ while keeping qubit count linear in $N$.
  • The number of qubits needed to encode $N$ fermionic modes in a general block or site construction is $N + \sum_i \lceil d(s_i)/2 \rceil$, where $d(s_i)$ is the degree of site $s_i$ in the coarse-grained graph; this interpolates between Jordan-Wigner's $N$ qubits and roughly $2L^2$ for the local square-lattice code.
  • State-preparation circuits for local encodings can be truncated to yield quasi-local codes: each additional level of the MERA circuit adds qubits and makes operators local on a finer lattice, realizing a tunable qubit-versus-locality trade-off.
  • Because the system graph can be chosen freely, the construction can be customized to a device's connectivity, as illustrated by encoding 49 fermionic modes into a 65-qubit heavy-hexagon lattice.

Reading between the lines

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

  • Editorial inference: if path-independence modulo plaquette stabilizers can be established for all system graphs rather than only the square-lattice case shown, the framework becomes a graph-optimization problem: pick the system graph that minimizes a weighted sum of qubits and Pauli weight for the given Hamiltonian.
  • Editorial inference: the same quasi-local trade-off should apply to resource metrics beyond Pauli weight, such as T-gate count or two-qubit gate depth in a Trotter step, because Pauli weight is a dominant contributor to those costs; a direct numerical comparison would test this.
  • Editorial inference: virtual-mode geometries for all-to-all systems suggest that for molecular Hamiltonians with dense active spaces, a similar hub or hierarchical embedding could reduce operator weights before applying Hamiltonian-sparsification methods, though the paper does not perform such a test.
  • Editorial inference: the hyperbolic-geometry examples connect to holographic tensor-network codes; if path equivalence is stable under coarse-graining, the encoded fermionic operators may inherit a notion of geodesic length that could be probed in operator-spreading or entanglement simulations.
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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

4 major / 3 minor

Summary. The paper proposes a general framework for 'custom fermionic codes': given a fermionic Hamiltonian with an interaction graph, one chooses an arbitrary system graph (subject to vertex-count and path-connectivity conditions) and encodes fermionic operators through products of local Majorana operators along paths in that graph. The authors argue that strict locality is not always optimal, and that quasi-local encodings can reduce both qubit counts and Pauli weights. They illustrate this with a square-lattice model with diagonal couplings, with the q=2 SYK model using several virtual geometries (branches, ternary tree, ternary MERA, hyperbolic tiling), with a blocking construction that interpolates between local and linear encodings, and with an encoding adapted to the heavy-hexagon connectivity. The paper also discusses state preparation by truncating a MERA circuit for the toric code.

Significance. If the construction is valid, the paper provides a useful design principle: the system graph is a free resource that can be tuned to reduce qubit count and operator weight, and the examples (especially the SYK reduction from O(N^3) to O(N^2 log N) total Pauli weight) are practically relevant for quantum simulation. The construction is explicit, has no fitted parameters, and subsumes several existing encodings. The connection to topological stabilizer structure and to truncated state preparation is conceptually appealing. However, the central claim rests on path-independence of generalized Jordan-Wigner strings for arbitrary system graphs, and the resource metric used in the SYK comparison is ambiguous; these issues need to be resolved before the claims can be accepted.

major comments (4)
  1. [III.A and III.B] The encoded edge operator for a pair of modes that are not directly connected is defined only as a product along a chosen path, but the manuscript does not prove that different paths give equivalent operators in the stabilizer subspace. The square-lattice sentence in Sec. III.A ('the upper and lower path ... are equal up to multiplication by a stabilizer') is a local verification for one plaquette, not a general theorem. For arbitrary system graphs with cycles that are not products of elementary plaquette boundaries, two paths can differ by a loop operator that is a logical operator rather than a stabilizer, making the encoded Hamiltonian ill-defined. This is load-bearing because the construction explicitly allows the system graph to be chosen freely, and the SYK geometries (ternary tree, MERA, hyperbolic tiling) in Sec. III.B rely on paths through cycles. A proof is needed that for every closed loop C, the operator i^{|C|} ∏_{e∈C} ~A_e lies in the stabilizer group generated by Eq. (13), or a characterization of the system graphs for which this holds.
  2. [III.B.1, Eq. (21)] There is a sign error in the mapping of the third term of Eq. (20). A direct calculation from Eqs. (7) and (8) gives B_j A_jk = γ_{2j}γ_{2k-1}, so the term -ι ∑_{j<k} J_{2j,2k-1} γ_{2j}γ_{2k-1} should be encoded as -ι ∑_{j<k} J_{2j,2k-1} B_j ~A_jk, not +ι as written in Eq. (21). As printed, the encoded Hamiltonian differs by a sign on a class of SYK interaction terms, so the resource estimates are attached to a Hamiltonian that does not match Eq. (20).
  3. [III.B and Fig. 4] The quantity plotted as 'worst-case Pauli weight' in Fig. 4 is ambiguous, and the stated O(N^3) scaling for the linear geometry conflates per-operator weight with total Hamiltonian weight. A single Jordan-Wigner string in the linear geometry has Pauli weight O(N), and there are O(N^2) quadratic terms, so the total Pauli weight of the Hamiltonian is O(N^3). The complete-graph and virtual-geometry values O(N^2 log N) are also total weights over the Hamiltonian. If 'worst-case' means the maximum weight of an individual encoded Pauli term, then the linear geometry value should be O(N), not O(N^3). The paper should define the metric precisely and report per-operator and total weights separately, since the headline reduction O(N^3) → O(N^2 log N) is only meaningful as a total-weight statement.
  4. [II, Eq. (13) and Sec. II.A] The identification of the non-contractible loop stabilizer in the 1D periodic chain is not fully justified. The manuscript states that this stabilizer is S = ∏_j Z_j, but a direct computation from the edge operators ~A_{j,j+1}=X_jY_{j+1} (with the sign convention of Eq. (11)) gives a product with a sign factor that depends on the number of modes; for example, for two modes the loop product is -Z_1Z_2, not +Z_1Z_2. This affects whether the stabilized subspace is the even-parity sector, so the relation between loop stabilizers, global parity, and the odd-parity discussion in Sec. II should be checked and stated explicitly.
minor comments (3)
  1. [Fig. 4] The note that the tree, MERA, and hyperbolic data are 'approximate' is not sufficient; the paper should state how the data were obtained for finite N, including which levels are filled and how boundary corrections are treated.
  2. [III.B] The sentence 'all known local fermionic encodings are equivalent to the toric code defined on some lattice up to a constant-depth local Clifford circuit' is an assertion without proof or detailed citation; if it is not central, it should be softened or supported.
  3. [III.D] The heavy-hexagon example in Fig. 7 is only sketched; the text should explain how the 49 encoded fermionic modes are assigned to the grouped vertices and how the couplings between modes that do not share an edge are realized by the generalized Jordan-Wigner strings.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the quasi-local encoding results are constructive computations from the paper's own definitions, with prior-work citations serving as background lemmas rather than as forced conclusions.

full rationale

The paper's central claims are derived by explicit construction rather than by fitting or by importing its own conclusions. The encoded edge and vertex operators (Eqs. 11-12), plaquette stabilizers (Eq. 13), and the generalized Jordan-Wigner strings are defined directly from the chosen system graph, and the reported Pauli-weight reductions for the square-lattice and SYK examples are computed from those definitions without any adjustable parameters. The square-lattice diagonal-coupling example is a direct comparison of qubit counts and operator weights for two explicit system-graph choices, not a prediction from fitted data. The SYK resource estimates likewise follow from counting operator weights in the chosen geometries, with the Fenwick-tree local Majorana basis imported from prior published work. The paper does cite the authors' own earlier work, notably [14], [16], and [17], and [17] provides the underlying local-encoding framework; however, that framework is a peer-reviewed, parameter-free construction whose assumptions do not contain the present results, and the present contribution—relaxing strict locality, choosing arbitrary system graphs, and comparing geometries—is additive to that framework. The manuscript's assertion that different paths around a square plaquette differ by a stabilizer is a specific verification, and the lack of a general proof of path-independence for arbitrary graphs is a correctness or completeness risk, not a circularity: it does not make the output equivalent to the input by definition. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors is invoked to forbid alternatives, and no known result is merely relabeled as novel. Overall, the derivation chain is self-contained at the level of the examples it proves, and the self-citations are not load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The construction rests on well-known stabilizer and Clifford machinery; the main untested premise is the path-consistency of generalized JW strings on arbitrary graphs. No fitted parameters are present; the geometry choices are design decisions rather than fitted values.

assumptions (4)
  • domain assumption All known local fermionic encodings are equivalent to the toric code up to a constant-depth local Clifford circuit.
    Stated in Sec. II (page 2) without proof; it justifies the toric-code interpretation of local encodings and the truncated-state-preparation picture, but is not needed for the core quasi-local construction.
  • domain assumption For any closed loop in the system graph, the product of edge operators around the loop is a stabilizer, and the logical subspace is the charge-flux attached sector.
    Invoked in Sec. II (Eq. 13) and Sec. III.A; it underlies the claim that path choice does not matter modulo stabilizers. The paper does not provide a general proof for arbitrary graphs.
  • standard math A choice of local Majorana operators on a vertex can be changed by a local Clifford circuit without affecting the encoded algebra.
    Used when the paper switches between Jordan-Wigner and Fenwick-tree local bases (Sec. II, III.B). This is a standard property of Clifford unitaries.
  • standard math The MERA circuit of Aguado and Vidal prepares the toric code state level by level.
    Cited [41] and used in Sec. III.C.1 to relate blocking to truncation; treated as a known result.

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Pith. "Pith review of Custom fermionic codes for quantum simulation." pith.science (2026). https://pith.science/paper/OXLNUWHZ

@misc{pith2026200911860,
  author       = {Pith},
  title        = {Pith review of: Custom fermionic codes for quantum simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXLNUWHZ}},
  note         = {Machine review of arXiv:2009.11860}
}
read the original abstract

Simulating a fermionic system on a quantum computer requires encoding the anti-commuting fermionic variables into the operators acting on the qubit Hilbert space. The most familiar of which, the Jordan-Wigner transformation, encodes fermionic operators into non-local qubit operators. As non-local operators lead to a slower quantum simulation, recent works have proposed ways of encoding fermionic systems locally. In this work, we show that locality may in fact be too strict of a condition and the size of operators can be reduced by encoding the system quasi-locally. We give examples relevant to lattice models of condensed matter and systems relevant to quantum gravity such as SYK models. Further, we provide a general construction for designing codes to suit the problem and resources at hand and show how one particular class of quasi-local encodings can be thought of as arising from truncating the state preparation circuit of a local encoding. We end with a discussion of designing codes in the presence of device connectivity constraints.

Figures

Figures reproduced from arXiv: 2009.11860 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Local Majorana modes for a degree [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (a) Two nearest-diagonal-neighbor couplings [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (left) Schematic of an operator coupling two [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4: (Top) Qubit requirements to encode the system [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a) A section of a square lattice of fermionic modes (white circles) with nearest neighbor interaction graph [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Here we show the trade-off between qubit [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: (top left) The heavy-hexagon geometry as [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

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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. A Hybrid Qubit Encoding: Splitting Fock Space into Fermionic and Bosonic Subspaces

    quant-ph 2024-11 conditional novelty 6.0 of 10

    A hybrid fermionic/bosonic qubit encoding splits molecular orbitals into fully resolved spin-orbitals and spin-paired hard-core bosons, reducing quantum resources with a tunable energy error.

Reference graph

Works this paper leans on

58 extracted references · 39 canonical work pages · cited by 1 Pith paper

  1. [1]

    The SYK model consists of 2N Majorana fermions with random strengthq-body interactions cou- pling all Majoranas

    All-to-all coupled fermions We now give an example of a system for which the quantum simulation cost is decreased by using virtual modes. The SYK model consists of 2N Majorana fermions with random strengthq-body interactions cou- pling all Majoranas. Proposals regarding quantum simu- lation of theq = 4 SYK model have previously been put forward [33–35]. W...

    work page

  2. [2]

    A 1D chain with periodic boundary conditions (Linear)

    work page

  3. [3]

    A complete graph with N vertices (Complete Graph)

    work page

  4. [4]

    Ternary tree with physical modes at the leaves and virtual vertices of degree4 (Ternary Tree)

    work page

  5. [5]

    A geometry with a single virtual mode of degreeN connected to all physical modes which only connect to the central virtual mode (N Branches)

    work page

  6. [6]

    Physical modes are identi- fied with legs at the boundary of the disk

    Hyperbolic geometry where each vertex is degree6 and faces are 4-sided. Physical modes are identi- fied with legs at the boundary of the disk. (d = 6 Hyperbolic Tiling) Some of these geometries have taken their inspiration from tensor networks such as MERA [36, 37] as well as existing literature regarding hyperbolic codes [38]. For the geometries containin...

    work page

  7. [7]

    Ternary MERA-like geometry with virtual vertices of degree4 (Ternary MERA) (a cutout is shown in Fig. 3)

    work page

  8. [8]

    Motta, C

    M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A.J.Minnich, F.G.Brandão, andG.K.-L.Chan, Nature Physics 16, 205 (2020)

    work page 2020

Show all 58 references
  1. [9]

    Bravyi and A

    S. Bravyi and A. Kitaev, Physical Review A71, 022316 (2005)

    2005

  2. [10]

    This will also be a convenient time to address state preparation

    As truncated state preparation We now hope to provide an intuitive picture for the blocking construction. This will also be a convenient time to address state preparation. For simplicity, we will again consider a square lattice of dimensionL×L. As previously mentioned, this en...

  3. [11]

    N branches

    Further generalizations We now discuss a number of ways this above construc- tion can be generalized. Going beyond a square lattice of blocks, one could perform the partitioning of the system of N modes into a set of general sitesS ={s1,...,s |S|} each containing a number of m...

  4. [12]

    E. P. Wigner and P. Jordan, Z. Phys47, 631 (1928)

    1928

  5. [13]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics16, 407 (1961)

    1961

  6. [14]

    where segmented versions of the Bravyi-Kitaev and Fenwicktreeencodingswereexploredforthe2DHubbard model

  7. [15]

    R. P. Feynman, Int. J. Theor. Phys21 (1999)

    1999

  8. [16]

    Lloyd, Science pp

    S. Lloyd, Science pp. 1073–1078 (1996)

    1996

  9. [17]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Reviews of Modern Physics 92, 015003 (2020)

    2020

  10. [18]

    Reiher, N

    M. Reiher, N. Wiebe, K. M. Svore, D. Wecker, and M. Troyer, Proceedings of the National Academy of Sci- ences 114, 7555 (2017)

    2017

  11. [19]

    D. S. Abrams and S. Lloyd, Physical Review Letters79, 2586 (1997)

    1997

  12. [20]

    Derby and J

    C. Derby and J. Klassen, arXiv preprint arXiv:2003.06939 (2020)

    2020 arXiv

  13. [21]

    S. B. Bravyi and A. Y. Kitaev, Annals of Physics298, 210 (2002)

    2002

  14. [22]

    Jiang, A

    Z. Jiang, A. Kalev, W. Mruczkiewicz, and H. Neven, Quantum 4, 276 (2020)

    2020

  15. [23]

    Ball, Physical review letters95, 176407 (2005)

    R. Ball, Physical review letters95, 176407 (2005)

    2005

  16. [24]

    arXiv:2009.11860v1 [quant-ph] 24 Sep 2020 2 Understanding these local encodings is best done through the toric code model

    with focus towards topological and fracton models. arXiv:2009.11860v1 [quant-ph] 24 Sep 2020 2 Understanding these local encodings is best done through the toric code model. This famous model, pro- posed by Kitaev in [25], is a model for a topological quan- tum memory. For our...

    2009 arXiv

  17. [25]

    Verstraete and J

    F. Verstraete and J. I. Cirac, Journal of Statistical Me- chanics: Theory and Experiment2005, P09012 (2005)

    2005

  18. [26]

    J. D. Whitfield, V. Havlíček, and M. Troyer, Physical Review A94, 030301 (2016)

    2016

  19. [27]

    Setia and J

    K. Setia and J. D. Whitfield, The Journal of Chemical Physics 148, 164104 (2018)

    2018

  20. [28]

    R. W. Chien, S. Xue, T. S. Hardikar, K. Setia, and J. D. Whitfield, Physical Review A100, 032337 (2019)

    2019

  21. [29]

    Setia, S

    K. Setia, S. Bravyi, A. Mezzacapo, and J. D. Whitfield, Physical Review Research1, 033033 (2019)

    2019

  22. [30]

    Jiang, J

    Z. Jiang, J. McClean, R. Babbush, and H. Neven, Phys- ical Review Applied12, 064041 (2019)

    2019

  23. [31]

    Steudtner and S

    M. Steudtner and S. Wehner, Physical Review A 99, 022308 (2019)

    2019

  24. [32]

    Y.-A. Chen, A. Kapustin, and Ð. Radičević, Annals of Physics 393, 234 (2018)

    2018

  25. [33]

    Chen, arXiv preprint arXiv:1911.00017 (2019)

    Y.-A. Chen, arXiv preprint arXiv:1911.00017 (2019)

    2019 arXiv

  26. [34]

    S. K. Shukla, T. D. Ellison, and L. Fidkowski, Physical Review B101, 155105 (2020)

    2020

  27. [35]

    Tantivasadakarn, Physical Review Research2, 023353 (2020)

    N. Tantivasadakarn, Physical Review Research2, 023353 (2020)

    2020

  28. [36]

    A. Y. Kitaev, Annals of Physics303, 2 (2003)

    2003

  29. [37]

    Levin and X.-G

    M. Levin and X.-G. Wen, Physical Review B67, 245316 (2003)

    2003

  30. [38]

    M. A. Levin and X.-G. Wen, Physical Review B 71, 045110 (2005)

    2005

  31. [39]

    Wen, Physical review letters90, 016803 (2003)

    X.-G. Wen, Physical review letters90, 016803 (2003)

    2003

  32. [40]

    Nussinov, G

    Z. Nussinov, G. Ortiz, and E. Cobanera, Physical Review B 86, 085415 (2012)

    2012

  33. [41]

    Zohar and J

    E. Zohar and J. I. Cirac, Physical Review B98, 075119 (2018)

    2018

  34. [42]

    Sachdev and J

    S. Sachdev and J. Ye, Physical Review Letters70, 3339 (1993)

    1993

  35. [43]

    Sachdev, Physical Review X5, 041025 (2015)

    S. Sachdev, Physical Review X5, 041025 (2015)

    2015

  36. [44]

    García-Álvarez, I

    L. García-Álvarez, I. Egusquiza, L. Lamata, A. Del Campo, J. Sonner, and E. Solano, Physical Review Letters119, 040501 (2017)

    2017

  37. [45]

    R.Babbush, D.W.Berry, andH.Neven, PhysicalReview A 99, 040301 (2019)

    2019

  38. [46]

    Cao, Y.-N

    Y. Cao, Y.-N. Zhou, T.-T. Shi, and W. Zhang, arXiv preprint arXiv:2003.01514 (2020)

    2020 arXiv

  39. [47]

    Vidal, Physical Review Letters93, 040502 (2004)

    G. Vidal, Physical Review Letters93, 040502 (2004)

    2004

  40. [48]

    Swingle, Physical Review D86, 065007 (2012)

    B. Swingle, Physical Review D86, 065007 (2012)

    2012

  41. [49]

    Pastawski, B

    F. Pastawski, B. Yoshida, D. Harlow, and J. Preskill, Journal of High Energy Physics2015, 149 (2015)

    2015

  42. [50]

    S. Xu, L. Susskind, Y. Su, and B. Swingle, arXiv preprint arXiv:2008.02303 (2020)

    2020 arXiv

  43. [51]

    Chen, Z.-C

    X. Chen, Z.-C. Gu, and X.-G. Wen, Physical Review B 82, 155138 (2010)

    2010

  44. [52]

    Aguado and G

    M. Aguado and G. Vidal, Physical Review Letters100, 070404 (2008)

    2008

  45. [53]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Physical Review Letters97, 050401 (2006)

    2006

  46. [54]

    Y.Gu, X.-L.Qi, andD.Stanford, JournalofHighEnergy Physics 2017, 125 (2017)

    2017

  47. [55]

    Song, C.-M

    X.-Y. Song, C.-M. Jian, and L. Balents, Physical Review Letters 119, 216601 (2017)

    2017

  48. [56]

    Chamberland, G

    C. Chamberland, G. Zhu, T. J. Yoder, J. B. Hertzberg, and A. W. Cross, Physical Review X10, 011022 (2020)

    2020

  49. [57]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. Brandao, D. A. Buell, et al., Nature574, 505 (2019)

    2019

  50. [58]

    A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, and J. M. Gambetta, Physical Review A100, 032328 (2019)

    2019

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