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The fabulous world of GKP codes

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Pith's one-line read The paper shows that GKP codes, viewed as symplectic lattices, admit good families with constant rate and distance scaling $\Delta^2=\Omega(n)$; NTRU lattices are the candidate explicit family.

desk verdict Solid, honest PhD synthesis of previously published GKP lattice results; the NTRU-GKP goodness scaling is a reasonable but unproven extrapolation and should be labeled a conjecture. read the letter →

arxiv 2412.02442 v1 pith:G4WHAVO5 submitted 2024-12-03 quant-ph math-phmath.MP

Jonathan Conrad This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81P7311H0614H52 PACS 03.67.Pp
keywords Gottesman-Kitaev-PreskillcodesbosonicquantumerrorcorrectionsymplecticlatticesNTRUgoodthetafunctionsmodulispaceofellipticcurvesfiberbundlefaulttolerance
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 dissertation argues that Gottesman–Kitaev–Preskill (GKP) codes—stabilizer codes whose logical qubits live in the infinite-dimensional Hilbert space of harmonic oscillators and are fixed by translation symmetries in phase space—are not only an error-correction scheme but a meeting point for lattice theory, number theory, algebraic geometry, and fault tolerance. Its central coding-theoretic result is that good GKP code families exist: lattices $L_n\subset \mathbb{R}^{2n}$ whose encoding rate $\log\det(L_n)$ grows linearly in $n$ while the code distance $\Delta$ grows as $\sqrt{n}$, so $\Delta^2=\Omega(n)$. The existence proof uses a random symmetric-matrix construction that approximates the uniform measure on symplectic lattices. The thesis then proposes an explicit candidate family, NTRU-GKP codes, built from the lattices behind the NTRU cryptosystem, with numerical evidence for the needed $\sqrt{n}$ shortest-vector scaling up to dimension 24. Alongside this, it develops a geometric picture in which single-mode GKP codes are elliptic curves, logical Clifford gates are braids around a trefoil-knot defect in the space of lattices, and fault tolerance is a property of fiber bundles.

What carries the argument

The central object is the symplectically integral lattice $L\subseteq L^\perp$ associated to a GKP stabilizer group $S=\langle D(\xi_1),\dots,D(\xi_{2n})\rangle$, with symplectic form $J$ and Gram matrix $A=MJM^T$. This dictionary converts code distance into the shortest vector of the dual quotient, logical Clifford gates into symplectic automorphisms with integral representation $\mathrm{Sp}^D_{2n}(\mathbb{Z})$, and the space of single-mode codes into the modular curve $\mathrm{SL}_2(\mathbb{Z})\setminus\mathfrak{h}$ with a trefoil-knot defect. The existence proof for good codes uses the Haar measure on symplectic lattices together with the explicit family $M[X]$, which realizes the Gaussian heuristic—the expectation that a random lattice's shortest vector is roughly $\sqrt{n/2\pi e}$ times the $n$-th root of its covolume. NTRU-GKP codes are proposed as a structured, cryptographically motivated instance of the same construction. Theta functions act as generating functions for the lattice distance distribution, and the fiber-bundle framework for fault tolerance turns logical gates into homotopy classes of loops in the moduli space.

What would settle it

Compute exact shortest vectors for randomly sampled NTRU lattices at substantially larger dimension (say $n=100$ or $200$) using exact lattice reduction; if the shortest vector length stops tracking the Gaussian-heuristic $\sqrt{n}$ curve and grows more slowly, the claimed $\Delta=O(\sqrt{n})$ scaling for NTRU-GKP codes fails.

Watch

Extended reading notes

Core claim

The paper establishes that the GKP construction is best understood as a symplectic lattice: stabilizers are displacements by vectors in a lattice $L$, logical Pauli operators are displacements by vectors in the symplectic dual $L^\perp$, and the code distance is $\Delta=\min_{0\neq x\in L^\perp\setminus L}\lVert x\rVert$. With this dictionary, the existence of good codes is proven: for any fixed scaling $d$, the random symmetric matrices $X\in\{-q/2,\dots,q/2\}^{n\times n}$ produce lattices generated by $M[X]=\begin{pmatrix} I & X \\ 0 & qI \end{pmatrix}$, and after rescaling these have shortest vector $\lambda_1\approx\sqrt{n/2\pi e}$ in the large-$q$ limit, giving $\log\det(L_n)=\Omega(n)$ and $\Delta^2=\Omega(n)$. The same dictionary turns NTRU lattices into GKP codes; exact lattice reduction for dimensions up to 24 supports the $\Delta=O(\sqrt{n})$ scaling needed for goodness, though the paper presents this as numerical evidence rather than a theorem.

Load-bearing premise

The load-bearing premise is that random NTRU lattices, which come from a structured polynomial algebra rather than being drawn uniformly at random, have the same shortest-vector statistics as random symplectic lattices; the evidence so far is numerical, for dimensions up to 24.

Editorial extensions

If this is right

  • Good GKP families would make constant-rate bosonic error correction possible in principle: the number of encoded qubits per mode stays bounded away from zero while the minimal logical displacement grows, so concatenation with outer qubit codes is not the only route to scalability.
  • The NTRU-GKP construction provides an explicit family of symplectic lattices whose numerical shortest-vector behavior is consistent with the goodness scaling, and the thesis builds a private quantum channel on this family.
  • The elliptic-curve picture implies that every single-mode GKP code carries a geometric label in the modular curve, with logical Clifford gates corresponding to homotopically non-trivial loops around the zero-distance defect.
  • The distance of a GKP code is fixed by the distance distribution of its lattice, expressible through lattice theta functions; for concatenated codes this reduces distance to the weight enumerator of the qubit code.
  • Tensor products of GKP lattices produce new codes with distance bounded between the individual distances and their product, giving a construction principle beyond concatenation.

Reading between the lines

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

  • Beyond the paper's results, if the NTRU-GKP scaling conjecture holds, approximate shortest-vector solvers would double as practical decoders, linking the hardness of decoding GKP codes to the assumptions behind NTRU-based cryptography.
  • The moduli-space picture for single-mode codes likely extends to multi-mode GKP codes through higher-dimensional abelian varieties; a natural test is whether the topological defect becomes a higher-dimensional submanifold and whether fault-tolerance still corresponds to non-contractible loops around it.
  • One could test the randomness assumption directly by comparing low-order statistics of NTRU lattices (theta series or shortest-vector moments) with those of uniformly random symplectic lattices at dimensions beyond 24, before relying on the $\sqrt{n}$ extrapolation.
  • The fiber-bundle view may give a general method for proving that a gate set is fault tolerant solely from the topology of the code's parameter space, independent of the physical implementation.
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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 / 4 minor

Summary. The manuscript is a doctoral dissertation on Gottesman-Kitaev-Preskill (GKP) codes, presenting them through a lattice-theoretic and algebraic-geometric lens. It develops the stabilizer formalism for GKP codes, proves a Frobenius normal form for symplectically integral lattices, derives distance bounds, proves the existence of good GKP code families by averaging over symmetric matrices over Z_q, analyzes logical Clifford gates via symplectic automorphisms, proposes a moduli-space and fiber-bundle picture of fault tolerance, constructs GKP codes from root lattices and from NTRU lattices, studies the computational complexity of GKP decoding, and discusses experimental implementation. The central mathematical results are largely drawn from the author's published work, and the thesis frames them as a coherent research program.

Significance. If the main claims hold, the paper makes a valuable contribution to the theory of GKP codes. The most significant result is the proof that good GKP code families exist, meaning families with non-vanishing rate and distance scaling Δ^2 = Ω(n), which establishes in principle that GKP codes can go beyond the constant-overhead barrier known for other bosonic codes. The symplectic equivalence and Frobenius normal-form results are clean and useful, and the connection between GKP Clifford gates, symplectic automorphisms, and Riemann-surface/moduli-space structures is a genuinely insightful synthesis. The NTRU-GKP construction is an appealing candidate for an explicit good family and is supported by reproducible numerical evidence for n ≤ 24, but it is not yet a theorem. The manuscript is honest about many open points and makes its numerical code available, which is a strength.

major comments (3)
  1. [§5.3, Fig. 5.3, Corollary 4] The claim that NTRU-GKP codes form an explicit family of good GKP codes is not established. Corollary 4 proves existence by averaging over all symmetric matrices X ∈ U_q, but NTRU lattices are drawn from the much smaller structured family Λ_NTRU = {(u, v) : u ≡ h v mod q} with h ∈ Z_q[x]/(x^n + 1). The equidistribution argument in Theorem 4 does not apply to this subfamily. The numerical results in Fig. 5.3 estimate the average shortest-vector length for n ≤ 24, but the goodness criterion requires a lower-tail statement that, with positive probability, λ_1 ≥ c√n. Matching the Gaussian-heuristic mean does not imply this tail bound, and ring correlations could bias the distribution. Unless a rigorous transfer argument is provided, the NTRU-GKP distance scaling Δ = O(√(n/λ)) should be stated as a conjecture or as numerical evidence, not as a proven property.
  2. [§4.3.3, proof of Corollary 4] There is a dimension-accounting inconsistency in the ball-volume argument. Theorem 4 considers functions f : R^{2n} → R, and L_{q,X} is a lattice in R^{2n}, so the relevant ball volume is V_{2n}(R) = π^n R^{2n}/Γ(n+1). Setting this equal to 1 gives R ≈ √(n/(π e)), not R ≈ √(n/(2π e)) as stated in Corollary 4. The proof appears to use V_n(R) = π^{n/2}R^n/Γ(n/2+1), which is the volume of a ball in R^n. The asymptotic existence claim survives, but the displayed constant and the proof need to be corrected for consistency.
  3. [§4.3.3, Corollary 4 and 6] The passage from the averaged counting statement to an existence statement should be stated more carefully. From the theorem, one obtains that for large q the expected number of short vectors is close to V_{2n}(R). For a fixed R slightly below the Gaussian-heuristic radius, this expectation is less than 1, which indeed implies that some lattice has no nonzero vector shorter than R. The current wording, 'the average property implies the existence of instances,' is acceptable informally, but it is not a statement about typical instances, and the finite-q fluctuation is not quantified. A short remark distinguishing 'there exists' from 'random instances have' would prevent the reader from over-reading Corollary 4.
minor comments (4)
  1. [Fig. 5.1] The table entry for NTRU-GKP codes uses Δ ∼ O(√(n/λ)) without defining λ in the table or its caption; define λ as the NTRU scaling parameter or refer explicitly to the section where it is introduced.
  2. [Chapter 1 and 3] There are several typographical errors in names and terms, e.g., 'Meniccuci' for Menicucci, 'Joza' for Jozsa, and 'Franceosco' for Francesco. These should be corrected in a final revision.
  3. [§5.3.3, Fig. 5.3] The figure caption reports expected shortest-vector lengths from the Gaussian heuristic as λ(n) = √(nq/(π e)), while the text around Corollary 4 uses √(n/(2π e)). Please reconcile the notation and constants so that the reader can compare the numerical data with the theoretical statements.
  4. [§4.4.4] The discussion of the modular discriminant and the bound |Δ(τ)| is interesting but somewhat compressed; in particular, the statement that |Δ(τ)| is lower-bounded away from |τ| = 1 should include the relevant argument or a precise reference, since the constant enters the claimed distance-to-discriminant correspondence.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central existence proof for good GKP families is carried out in the text, and the NTRU-GKP section is a numerically supported candidate family, not a fitted prediction.

full rationale

The derivation chain for the load-bearing claim (Definition 5, Corollaries 4–5, Sec. 4.3.3) is self-contained: Theorem 4 proves an equidistribution statement for the slice U_q of symmetric matrices by an explicit calculation, and Corollary 4 uses the average lattice-point count in balls to infer the existence of lattices with shortest vector at least sqrt(n/2πe). No fitted parameter is renamed as a prediction. The NTRU-GKP construction in Sec. 5.3 is presented as an explicit candidate family with numerical evidence for n ≤ 24; the fact that the random-symplectic equidistribution proof does not formally transfer to the structured NTRU subfamily is a correctness/extrapolation gap, not a circular reduction. Self-citations to the author's prior papers ([57], [58], [55]) appear as chapter provenance, but the load-bearing mathematical ingredients (Frobenius lemma, Haar measures on lattice moduli spaces, theta-function identities, modular-curve classifications) are cited to external sources or proved in the text. No instance was found in which an equation used as input equals the claimed output by construction, nor any parameter fitted to the target quantity and then reported as a prediction.

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

The central proofs are self-contained except for standard mathematical background and the explicitly stated assumptions above. The NTRU construction parameters are design choices from the cryptosystem rather than fitted values, so no free parameters are needed. The thesis introduces no new physical entities such as particles or forces.

assumptions (4)
  • domain assumption Gaussian heuristic for lattice shortest vectors (Sec. 4.3.3)
    Used to conclude that random symplectic lattices have lambda_1 = Omega(sqrt(n)); Theorem 4 gives equidistribution for compactly supported test functions, but the step from average counts to a guaranteed shortest vector still relies on the heuristic.
  • domain assumption Random NTRU lattices behave like uniformly random symplectic lattices (Sec. 5.3.3)
    The NTRU-GKP distance claim assumes the structured NTRU family inherits the shortest-vector statistics of random lattices; only numerical evidence up to n=24 is provided.
  • domain assumption Computational hardness of lattice decoding (Sec. 6.1, 6.3)
    The private quantum channel from NTRU-GKP codes assumes that decoding the relevant lattices is hard for quantum computers, a standard cryptographic assumption not proved in this thesis.
  • domain assumption Gottesman-Zhang fiber-bundle framework as a model of fault tolerance (Sec. 4.4.5)
    Used to interpret the moduli-space construction as fault tolerance; the framework is taken from ref. [96] and is not independently derived.

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Cite this review

Pith. "Pith review of The fabulous world of GKP codes." pith.science (2026). https://pith.science/paper/G4WHAVO5

@misc{pith2026241202442,
  author       = {Pith},
  title        = {Pith review of: The fabulous world of GKP codes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4WHAVO5}},
  note         = {Machine review of arXiv:2412.02442}
}
read the original abstract

Quantum error correction is an essential ingredient in the development of quantum technologies. Its subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with different structure to use. For another, every physical setting also comes with its own imperfections that need to be protected against. Bred by the complexity of a technological ambition, research on quantum error correction has developed into a large field of research that ranges from engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries. While in practice the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group are considered as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover -- through the looking glass of GKP codes -- how quantum error correction fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.

Figures

Figures reproduced from arXiv: 2412.02442 by the authors.

Figure 1.1
Figure 1.1. High level overview of research areas highlighted in this thesis that [PITH_FULL_IMAGE:figures/full_fig_p015_1_1.png] view at source ↗
Figure 2.1
Figure 2.1. The Wigner function of a vacuum-, squeezed- and approximate GKP [PITH_FULL_IMAGE:figures/full_fig_p026_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. The magic gate injection protocol. The auxiliary magic state is [PITH_FULL_IMAGE:figures/full_fig_p031_3_1.png] view at source ↗
Figures from the paper (31 more)
Figure 4.1
Figure 4.1. Figure 4.1: The symplectic lattices Z 2 (left) and A2 (right) scaled by d = 2 and their respective (dual) unit cells. The logical displacement amplitudes are marked in turquoise and stabilizer displacements are marked in red. 3 − 1 4 compared to λ1 [PITH_FULL_IMAGE:figures/full…
Figure 4.2
Figure 4.2. Figure 4.2: Commutative diagram for the structure of nontrivial Cliffords for [PITH_FULL_IMAGE:figures/full_fig_p061_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: A visualization of the surface Sn, where logical operators for the GKP code are represented as elements of the first homology group indicated by the elements (ei , fi). Logical Clifford transformations are represented by sequences of Dehn-twists of the torus in this …
Figure 4.4
Figure 4.4. Figure 4.4: The trefoil knot corresponding to the one dimensional defect of [PITH_FULL_IMAGE:figures/full_fig_p067_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: The chain of correspondences and maps that associate a GKP code [PITH_FULL_IMAGE:figures/full_fig_p070_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: The Iwasawa- (l.) and Bloch-Messiah (r.) decomposition of sym [PITH_FULL_IMAGE:figures/full_fig_p072_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: The hexagonal GKP code given by Λρ, ρ = e i2π/3 and relative level structure d −1Λρ for d = 3. The points z mod d −1Λρ parametrize the syndrome of the GKP code while the d-torsion points d −1Λρ in in Λρ are interpreted to label logical Pauli operators for the associa…
Figure 4.8
Figure 4.8. Figure 4.8: The fundamental domain F on is marked in grey on the RHS. We illustrate the effect of a squeezing operation τ 7→ (λ ⊕ λ −1 ).τ = λ 2 τ and the corresponding transformation on the lattice Λτ 7→ Λλ2τ / p det(Λλ2τ ). representing basis, the set of symplectic lattices up…
Figure 4.9
Figure 4.9. Figure 4.9: The Seifert fibration describing a decomposition of [PITH_FULL_IMAGE:figures/full_fig_p077_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: A braid on e1 = ℘(ω1/2), e2 = ℘(ω2/2), e3 = ℘((ω1 + ω2)/2) implemented through a rotation (ω1, ω2) 7→ (ω2, −ω1) (l.) corresponding to a GKP Hadamard gate and a sheer (ω1, ω2) 7→ (ω1 + ω2, ω2) (r.) in the case of d = 2. Nontrivial braiding of the three roots ei is in…
Figure 4.11
Figure 4.11. Figure 4.11: The fundamental region F(2) = Γ(2)\h = ∪γ∈Sp2 (Z2)γF is drawn in red and contains the logical Clifford translates of the fundamental region F = SL2Z\h. Understood as GKP codes, this space labels all possible lattices associated with GKP stabilizer groups, i.e. that …
Figure 4.12
Figure 4.12. Figure 4.12: E× → M× forms a universal family of GKP codes, such that every family of single mode GKP codes with non-zero syndrome can be obtained as pullback of this family. The manifolds E× = E×(d), M× = M×(d) implicitly depend on the scaling parameter d. 4.4.5 Towards fiber b…
Figure 4.13
Figure 4.13. Figure 4.13: A fiber bundle π : E → M. The fibers F = π −1 (p) are all equivalent to each other. In the background the additional structure is indicated that defines a connection, which is a choice of parallel transport through the tangent of the total space T E = T H ⊕ T V and …
Figure 4.14
Figure 4.14. Figure 4.14: Illustration of the moduli space of GKP codes [PITH_FULL_IMAGE:figures/full_fig_p083_4_14.png]
Figure 5.1
Figure 5.1. Figure 5.1: Some notable symplectically integral lattices that yield GKP codes. [PITH_FULL_IMAGE:figures/full_fig_p086_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Notable root systems represented as Dynkin diagrams. [PITH_FULL_IMAGE:figures/full_fig_p086_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Shortest vector lengths computed via full [PITH_FULL_IMAGE:figures/full_fig_p099_5_3.png]
Figure 6.1
Figure 6.1. Figure 6.1: Sketch of a Venn diagram illustrating the relationships between rele [PITH_FULL_IMAGE:figures/full_fig_p101_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: Outline of the private quantum channel established using the NTRU [PITH_FULL_IMAGE:figures/full_fig_p111_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: A (top) quantum decoder, a (middle) classical MLD decoder and [PITH_FULL_IMAGE:figures/full_fig_p114_6_3.png]
Figure 7.1
Figure 7.1. Figure 7.1: Illustration of a photonic mode in a cavity of length [PITH_FULL_IMAGE:figures/full_fig_p117_7_1.png]
Figure 7.2
Figure 7.2. Figure 7.2: Generalized stabilizer measurement protocol for a stabilizer given [PITH_FULL_IMAGE:figures/full_fig_p123_7_2.png]
Figure 7.3
Figure 7.3. Figure 7.3: (a) Steane- and (b) Knill stabilizer measurement circuits for a single￾mode GKP code with L = √ 2Z 2 . In (a) individual segments realize measure￾ments of the Sp = e −i2 √ πpˆ and Sq = e i2 √ πqˆ stabilizer sequentially. The Knill circuit (b) can be understood as a l…
Figure 7.4
Figure 7.4. Figure 7.4: CV SWAP gate compiled into a sequence of SUM gates and a π￾rotation. Using this result, one can derive the equivalence between the Knill- and the Steane error correction circuit following the steps displayed in fig. 7.5, which shows a proof based on mostly graphical …
Figure 7.5
Figure 7.5. Figure 7.5: The derivation of the equivalence between the Knill- and the Steane [PITH_FULL_IMAGE:figures/full_fig_p126_7_5.png]
Figure 7.6
Figure 7.6. Figure 7.6: Different applications of the group projector in quantum informa [PITH_FULL_IMAGE:figures/full_fig_p137_7_6.png]
Figure 7.7
Figure 7.7. Figure 7.7: Quantum harmonic oscillator comprising a cavity and a (super- [PITH_FULL_IMAGE:figures/full_fig_p137_7_7.png]
Figure 7.8
Figure 7.8. Figure 7.8: The characteristic function hJJ(x;t) with φ = √ 2π traverses the indicated circle with time. For ωt ≥ 2π the rotating points are smeared out over the circle which represents the first order RWA h (1) JJ (x; t). The delta peaks of the characteristic function hGKP (x) …
Figure 7.9
Figure 7.9. Figure 7.9: (left) probability distribution for n = 1 step of the random walk and (right) the kernel functions νM⊥ for square and hexagonal GKP codes. While the last line, eq. (7.102), shows exactly the approximate projection of the Hamiltonian onto one whose characteristic func…
Figure 7.10
Figure 7.10. Figure 7.10: One possible ordering of the control path as given by a Hamiltonian [PITH_FULL_IMAGE:figures/full_fig_p144_7_10.png]
Figure 7.11
Figure 7.11. Figure 7.11: (a) Wigner functions of the two lowest eigenstates of H (0) av for twirling level N = 1, .., 15 together with their effective squeezing parameter are shown. (b) Finite squeezing parameters for N = 1..30 and (c) the ten lowest eigenenergies of H (0) av are plotted. T…

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Reference graph

Works this paper leans on

201 extracted references · 80 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aaronson

    S. Aaronson. www.complexityzoo.net

  2. [2]

    Aaronson, A

    S. Aaronson, A. Cojocaru, A. Gheorghiu, and E. Kashefi. Complexity-theoretic limitations on blind delegated quantum computation, 2019

    2019

  3. [3]

    Aaronson and D

    S. Aaronson and D. Gottesman. Improved simulation of stabilizer circuits.Phys. Rev. A, 70:052328, Nov 2004

    2004

  4. [4]

    M. Ajtai. Generating hard instances of lattice problems (extended abstract). In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, page 99–108, New York, NY, USA, 1996. Association for Computing Machinery

    1996

  5. [5]

    V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang. Performance and structure of single-mode bosonic codes.Phys. Rev. A, 97:032346, Mar 2018

    2018

  6. [6]

    Ambainis, M

    A. Ambainis, M. Mosca, A. Tapp, and R. de Wolf. Private quantum channels.IEEE Symp. Found. Comp. Sc., page 547–553, 2000

    2000

  7. [7]

    Y. Aono, T. Espitau, and P. Ngyuen. Random lattices: Theory and practice. https://espitau.github.io/bin/random_lattice.pdf

  8. [8]

    D. Arapura. Notes on low dimensional modular varieties, 2019

    2019

Show all 201 references
  1. [9]

    V. I. Arnol’d.Mathematische Methoden der klassischen Mechanik. Birkhäuser Basel, 1988

    1988

  2. [10]

    Banaszczyk

    W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296(1):625–635, December 1993

    1993

  3. [11]

    B. Q. Baragiola, G. Pantaleoni, R. N. Alexander, A. Karanjai, and N. C. Menicucci. All-gaussian universality and fault tolerance with the gottesman-kitaev-preskill code. Phys. Rev. Lett., 123:200502, Nov 2019

    2019

  4. [12]

    Bargmann

    V. Bargmann. On a hilbert space of analytic functions and an associated integral transform part i.Communications on Pure and Applied Mathematics, 14(3):187–214, August 1961

    1961

  5. [13]

    Beauville

    A. Beauville. Theta functions, old and new. InOpen Problems and Surveys of Contemporary Mathematics, volume 6 ofSurveys of Modern Mathematics, pages 99–131. Higher Education Press et International Press, 2013

    2013

  6. [14]

    P. Benioff. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by turing machines.Journal of Statistical Physics, 22(5):563–591, 1980

    1980

  7. [15]

    C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters. Purification of noisy entanglement and faithful teleportation via noisy channels. Physical Review Letters, 76(5):722–725, jan 1996

    1996

  8. [16]

    A. M. Berge. Symplectic lattices. 1999. 140

    1999

  9. [17]

    Berlekamp, R

    E. Berlekamp, R. McEliece, and H. van Tilborg. On the inherent intractability of certain coding problems (corresp.).IEEE Trans. Inf. Th., 24(3):384–386, 1978

    1978

  10. [18]

    D. J. Bernstein, N. Heninger, and T. Lange. LatticeHacks. https://latticehacks.cr.yp.to/ntru.html

  11. [19]

    Bernstein, J

    D.J. Bernstein, J. Buchmann, and Dahmen E.Post-Quantum Cryptography. Springer Berlin Heidelberg, Berlin, Heidelberg, 2009

    2009

  12. [20]

    Bi and Q

    J. Bi and Q. Chen. Lower bounds of shortest vector lengths in random NTRU lattices. Th. Comp. Sc., 560:121–130, 2014. Networks, Algorithms and complexity: articles from the Turing centenary in Beijing, China

    2014

  13. [21]

    Bi and Q

    J. Bi and Q. Cheng. Lower bounds of shortest vector lengths in random knapsack lattices and random NTRU lattices. Cryptology ePrint Archive, Paper 2011/153,

    2011

  14. [22]

    Birkenhake and H

    C. Birkenhake and H. Lange.Complex Abelian Varieties. Springer Berlin Heidelberg, 2004

    2004

  15. [23]

    Blanes, F

    S. Blanes, F. Casas, J. A. Oteo, and J. Ros. A pedagogical approach to the magnus expansion. European Journal of Physics, 31(4):907–918, jun 2010

    2010

  16. [24]

    M. Blau. Symplectic geometry and geometric quantization. https://ncatlab.org/nlab/files/BlauGeometricQuantization.pdf

  17. [25]

    Bobenko.Introduction to Compact Riemann Surfaces, pages 3–64

    Alexander I. Bobenko.Introduction to Compact Riemann Surfaces, pages 3–64. Number Bd. 2013 in Computational Approach to Riemann Surfaces. Springer Berlin Heidelberg, Berlin, Heidelberg, 2011

    2013

  18. [26]

    J. E. Bourassa, R. N. Alexander, M. Vasmer, A. Patil, I. Tzitrin, T. Matsuura, D. Su, B. Q. Baragiola, S. Guha, G. Dauphinais, and et al. Blueprint for a scalable photonic fault-tolerant quantum computer.Quantum, 5:392, 2021

    2021

  19. [27]

    Bourbaki. Algébre. Springer Berlin Heidelberg, 2007

    2007

  20. [28]

    A. J. Brady, A. Eickbusch, S. Singh, J. Wu, and Q. Zhuang. Advances in bosonic quantum error correction with gottesman–kitaev–preskill codes: Theory, engineering and applications. Progress in Quantum Electronics, 93:100496, January 2024

    2024

  21. [29]

    S. L. Braunstein. Squeezing as an irreducible resource.Phys. Rev. A, 71:055801, 2005

    2005

  22. [30]

    Bravyi and M

    S. Bravyi and M. B. Hastings. Homological product codes. InProceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC ’14, page 273–282, New York, NY, USA, 2014. Association for Computing Machinery

    2014

  23. [31]

    Bravyi and A

    S. Bravyi and A. Kitaev. Universal quantum computation with ideal clifford gates and noisy ancillas. Physical Review A, 71(2), feb 2005

    2005

  24. [32]

    Bravyi, M

    S. Bravyi, M. Suchara, and A. Vargo. Efficient algorithms for maximum likelihood decoding in the surface code.Physical Review A, 90(3), September 2014

    2014

  25. [33]

    Bravyi and B

    S. Bravyi and B. M. Terhal. A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes.New Journal of Physics, 11(4):043029, April 2009

    2009

  26. [34]

    N. P. Breuckmann.Homological quantum codes beyond the toric code. PhD thesis, RWTH Aachen University, 2017

    2017

  27. [35]

    N. P. Breuckmann and J. N. Eberhardt. Balanced product quantum codes.IEEE Transactions on Information Theory, 67(10):6653–6674, oct 2021

    2021

  28. [36]

    N. P. Breuckmann and J. N. Eberhardt. Quantum low-density parity-check codes. PRX Quantum, 2(4), October 2021. 141

    2021

  29. [37]

    N. P. Breuckmann and B. M. Terhal. Constructions and noise threshold of hyperbolic surface codes. IEEE Transactions on Information Theory, 62(6):3731–3744, June 2016

    2016

  30. [38]

    Buser and P

    P. Buser and P. Sarnak. On the period matrix of a riemann surface of large genus (with an appendix by j.h. conway and n.j.a. sloane).Inventiones Mathematicae, 117(1):27–56, dec 1994

    1994

  31. [39]

    F. C. Caramello Jr. Introduction to orbifolds, 2022

    2022

  32. [40]

    K. E. Cahill and R. J. Glauber. Density operators and quasiprobability distributions. Phys. Rev., 177:1882–1902, Jan 1969

    1902

  33. [41]

    K. E. Cahill and R. J. Glauber. Ordered expansions in boson amplitude operators. Phys. Rev., 177:1857–1881, Jan 1969

    1969

  34. [42]

    Campagne-Ibarcq, A

    P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, and M. H. Devoret. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584...

    2020

  35. [43]

    Chabaud and S

    U. Chabaud and S. Mehraban. Holomorphic representation of quantum computations. Quantum, 6:831, October 2022

    2022

  36. [44]

    W. L. Chow. On compact complex analytic varieties.American Journal of Mathematics, 71(4):893–914, 1949

    1949

  37. [45]

    I. L. Chuang and M. A. Nielsen. Prescription for experimental determination of the dynamics of a quantum black box.Journal of Modern Optics, 44(11–12):2455–2467, November 1997

    1997

  38. [46]

    C. T. Chubb. General tensor network decoding of 2d pauli codes, 2021

    2021

  39. [47]

    C. T. Chubb and S. T. Flammia. Statistical mechanical models for quantum codes with correlated noise.Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions, 8(2):269–321, May 2021

    2021

  40. [48]

    A. Ciani. Engineering the coupling of superconducting qubits. Dissertation, RWTH Aachen University, Aachen, 2019. Veröffentlicht auf dem Publikationsserver der RWTH Aachen University; Dissertation, RWTH Aachen University, 2019

    2019

  41. [49]

    Ciani, D

    A. Ciani, D. P. DiVincenzo, and B. M. Terhal.Lecture Notes on Quantum Electrical Circuits. TU Delft OPEN Publishing, January 2024

    2024

  42. [50]

    J. I. Cirac and P. Zoller. Quantum computations with cold trapped ions.Phys. Rev. Lett., 74:4091–4094, May 1995

    1995

  43. [51]

    J. Conrad. https://github.com/JonCYeh/NTRUGKP.git

  44. [52]

    J. Conrad. https://github.com/JonCYeh/GKP_DD

  45. [53]

    J. Conrad. Twirling and Hamiltonian engineering via dynamical decoupling for Gottesman-Kitaev-Preskill quantum computing.Phys. Rev. A, 103, 2021

    2021

  46. [54]

    Conrad, A

    J. Conrad, A. Burchards, J. Eisert, and S.T. Flammia. in preparation: Chasing shadows with the gottesman-kitaev-preskill code. 2024

    2024

  47. [55]

    Conrad, A

    J. Conrad, A. Burchards, and S.T. Flammia. Gottesman-kitaev-preskill codes: A rosetta stone. 2024

    2024

  48. [56]

    Conrad, C

    J. Conrad, C. Chamberland, N. P. Breuckmann, and B. M. Terhal. The small stellated dodecahedron code and friends.Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 376(2123):20170323, May 2018

    2018

  49. [57]

    Conrad, J

    J. Conrad, J. Eisert, and F. Arzani. Gottesman-Kitaev-Preskill codes: A lattice perspective. Quantum, 6:648, February 2022. 142

    2022

  50. [58]

    Conrad, J

    J. Conrad, J. Eisert, and J. P. Seifert. Good Gottesman-Kitaev-Preskill codes from the NTRU cryptosystem.Quantum, 8:1398, July 2024

    2024

  51. [59]

    K. Conrad. Lecture notes. https://kconrad.math.uconn.edu/blurbs/, 2024. [Online; accessed 06-May-2024]

    2024

  52. [60]

    Conway and N

    J. Conway and N. Sloane. On the Voronoi regions of certain lattices.SIAM J. Alg. Dis. Meth., 5, 09 1984

    1984

  53. [61]

    Conway and N

    J. Conway and N. Sloane.Sphere packings, lattices and groups, volume 290. Springer, New York, NY, 1988

    1988

  54. [62]

    Coppersmith and A

    D. Coppersmith and A. Shamir. Lattice attacks on ntru. InAdvances in Cryptology - EUROCRYPT ’97, International Conference on the Theory and Application of Cryptographic Techniques, Konstanz, Germany, May 11-15, 1997, Proceeding, volume 1233 ofLecture Notes in Computer Science,...

    1997

  55. [63]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill. Topological quantum memory. Journal of Mathematical Physics, 43(9):4452–4505, September 2002

    2002

  56. [64]

    Deutsch and R

    D. Deutsch and R. Josza. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 439(1907):553–558, December 1992

    1907

  57. [65]

    Stein, D

    The Sage Developers, W. Stein, D. Joyner, D. Kohel, J. Cremona, and B. Eröcal. Sagemath, version 9.6.http://www.sagemath.org, 2022

    2022

  58. [66]

    M. H. Devoret. Quantum Fluctuations in Electrical Circuits. In S. Reynaud, E. Giacobino, and J. Zinn-Justin, editors,Fluctuations Quantiques/Quantum Fluctuations, page 351, January 1997

    1997

  59. [67]

    Dinur, G

    I. Dinur, G. Kindler, and S. Safra. Approximating-cvp to within almost-polynomial factors is np-hard. InProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280), pages 99–109, 1998

    1998

  60. [68]

    P. A. M. Dirac.The Principles of Quantum Mechanics. Clarendon, Oxford, 1958

    1958

  61. [69]

    F. M. Dopico and C. R. Johnson. Parametrization of the matrix symplectic group and applications. SIAM Journal on Matrix Analysis and Applications, 31(2):650–673, 2009

    2009

  62. [70]

    Duivenvoorden, B

    K. Duivenvoorden, B. M. Terhal, and D. Weigand. Single-mode displacement sensor. Phys. Rev. A, 95:012305, 2017

    2017

  63. [71]

    Duivenvoorden, B.M

    K. Duivenvoorden, B.M. Terhal, and D. Weigand. Single-mode displacement sensor. Phys. Rev. A, 95:012305, 2017

    2017

  64. [72]

    Duke, Ö Imamo˜ glu, and Á

    W. Duke, Ö Imamo˜ glu, and Á. Tóth. Modular cocycles and linking numbers.Duke Mathematical Journal, 166(6):1179 – 1210, 2017

    2017

  65. [73]

    Einstein, B

    A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete?Phys. Rev., 47:777–780, May 1935

    1935

  66. [74]

    A. K. Ekert. Quantum cryptography based on bell’s theorem.Phys. Rev. Lett., 67:661–663, Aug 1991

    1991

  67. [75]

    N. D. Elkies. Rational lattices and their theta functions, 2019

    2019

  68. [76]

    Farb and D

    B. Farb and D. Margalit.A Primer on Mapping Class Groups (PMS-49). Princeton University Press, Princeton, 2012

    2012

  69. [77]

    P Feynman

    R. P Feynman. Simulating physics with computers.International Journal of Theoretical Physics, 21:467–488, 1981

    1981

  70. [78]

    Flühmann, T

    C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home. Encoding a qubit in a trapped-ion mechanical oscillator.Nature, 566:513–517, 2019. 143

    2019

  71. [79]

    E. Freitag. Siegel modular forms, pages 8–37. Springer Berlin Heidelberg, Berlin, Heidelberg, 1991

    1991

  72. [80]

    Fremling

    M. Fremling. Quantum hall wave functions on the torus, 2015

    2015

  73. [81]

    N. Gama, N. Howgrave-Graham, and P. Q. Nguyen. Symplectic lattice reduction and ntru. In Serge Vaudenay, editor,Advances in Cryptology - EUROCRYPT 2006, pages 233–253, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg

    2006

  74. [82]

    Ganeshan and M

    S. Ganeshan and M. Levin. Formalism for the solution of quadratic hamiltonians with large cosine terms.Physical Review B, 93(7), February 2016

    2016

  75. [83]

    Ganeshan and M

    S. Ganeshan and M. Levin. Formalism for the solution of quadratic hamiltonians with large cosine terms.Phys. Rev. B, 93:075118, Feb 2016

    2016

  76. [84]

    Ganeshan and M

    S. Ganeshan and M. Levin. Ungappable edge theories with finite-dimensional hilbert spaces. Phys. Rev. B, 105:155137, Apr 2022

    2022

  77. [85]

    T. Gannon. Lattices and theta functions. PhD thesis, McGill University (Canada), 1991

    1991

  78. [86]

    T. Gannon. Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics. Cambridge University Press, July 2023

    2023

  79. [87]

    Garibaldi

    S. Garibaldi. e8, the most exceptional group.Bulletin of the American Mathematical Society, 53(4):643–671, June 2016

    2016

  80. [88]

    Cambridge University Press, 2004

    Christopher Gerry and Peter Knight.Introductory Quantum Optics. Cambridge University Press, 2004

    2004

  81. [89]

    Knots and dynamics.Proceedings oh the International Congress of Mathematicians, Vol

    É Ghys. Knots and dynamics.Proceedings oh the International Congress of Mathematicians, Vol. 1, 2006-01-01, ISBN 978-3-03719-022-7, pags. 247-277, 1, 01 2006

    2006

  82. [90]

    E. Ghys. Lorenz and modular flows: A visual introduction, 2006

    2006

  83. [91]

    S. M. Girvin.Circuit QED: superconducting qubits coupled to microwave photons, page 113–256. Oxford University PressOxford, jun 2014

    2014

  84. [92]

    S. M. Girvin. Circuit qed: superconducting qubits coupled to microwave photons. In Quantum Machines: Measurement and Control of Engineered Quantum Systems: Lecture Notes of the Les Houches Summer School. Oxford University Press, Oxford, 2014

    2014

  85. [93]

    Glancy and E

    S. Glancy and E. Knill. Error analysis for encoding a qubit in an oscillator.Physical Review A, 73(1), January 2006

    2006

  86. [94]

    Gottesman

    D. Gottesman. Stabilizer codes and quantum error correction, 1997

    1997

  87. [95]

    Gottesman, A

    D. Gottesman, A. Kitaev, and J. Preskill. Encoding a qubit in an oscillator.Phys. Rev. A, 64:012310, 2001

    2001

  88. [96]

    Gottesman and L

    D. Gottesman and L. L. Zhang. Fibre bundle framework for unitary quantum fault tolerance. 2017

    2017

  89. [97]

    D. E. Gottesman.Stabilizer codes and quantum error correction. PhD thesis, California Institute of Technology, 1997

    1997

  90. [98]

    Griffiths

    P. Griffiths. Introduction to Algebraic Curves. American Mathematical Society, December 1989

    1989

  91. [99]

    A. L. Grimsmo, J. Combes, and B. Q. Baragiola. Quantum computing with rotation-symmetric bosonic codes.Phys. Rev. X, 10:011058, Mar 2020. 144

    2020

  92. [100]

    L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, page 212–219, New York, NY, USA, 1996. Association for Computing Machinery

    1996

  93. [101]

    Grushevsky

    S. Grushevsky. The schottky problem, 2010

    2010

  94. [102]

    R. Hain. Lectures on moduli spaces of elliptic curves, 2014

    2014

  95. [103]

    hall viscosity

    F. D. M. Haldane. "hall viscosity" and intrinsic metric of incompressible fractional hall fluids, 2009

    2009

  96. [104]

    Halevi and T

    S. Halevi and T. Malkin. Lecture Notes: Lattices and homomorphic encryption, Spring 2013. https://www.cs.columbia.edu/~tal/6261/SP13/. Online; accessed 04 December 2022

    2013

  97. [105]

    Hänggli, M

    L. Hänggli, M. Heinze, and R. König. Enhanced noise resilience of the surface–gottesman-kitaev-preskill code via designed bias.Phys. Rev. A, 102, 2020

    2020

  98. [106]

    Harrington and J

    J. Harrington and J. Preskill. Achievable rates for the Gaussian quantum channel. Phys. Rev. A, 64:062301, 2001

    2001

  99. [107]

    J. W. Harrington.Analysis of quantum error-correcting codes: Symplectic lattice codes and toric codes. PhD thesis, California Institute of Technology, 2004

    2004

  100. [108]

    Hermanns, J

    M. Hermanns, J. Suorsa, E. J. Bergholtz, T. H. Hansson, and A. Karlhede. Quantum hall wave functions on the torus.Physical Review B, 77(12), March 2008

    2008

  101. [109]

    Hoffstein, J

    J. Hoffstein, J. Pipher, and J. H. Silverman. Ntru: A ring-based public key cryptosystem. In Joe P. Buhler, editor,Algorithmic Number Theory, Lecture Notes in Computer Science, page 267–288, Berlin, Heidelberg, 1998. Springer

    1998

  102. [110]

    Hsieh and F

    M.-H. Hsieh and F. Le Gall. NP-hardness of decoding quantum error-correction codes. Phys. Rev. A, 83(5):052331, 2011

    2011

  103. [111]

    Hänggli and R

    L. Hänggli and R. König. Oscillator-to-oscillator codes do not have a threshold.IEEE Transactions on Information Theory, 68(2):1068–1084, 2022

    2022

  104. [112]

    Iyer and D

    P. Iyer and D. Poulin. Hardness of decoding quantum stabilizer codes.IEEE Trans. Inf. Theor., 61(9):5209–5223, sep 2015

    2015

  105. [113]

    Kliesch, R

    M. Kliesch, R. Kueng, J. Eisert, and D. Gross. Guaranteed recovery of quantum processes from few measurements.Quantum, 3:171, August 2019

    2019

  106. [114]

    E. Knill. Fault-tolerant postselected quantum computation: Schemes, 2004

    2004

  107. [115]

    E. Knill. Quantum computing with realistically noisy devices.Nature, 434(7029):39–44, March 2005

    2005

  108. [116]

    Knill, R

    E. Knill, R. Laflamme, and L. Viola. Theory of quantum error correction for general noise. Phys. Rev. Lett., 84:2525–2528, 2000

    2000

  109. [117]

    J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf. Charge-insensitive qubit design derived from the cooper pair box.Phys. Rev. A, 76:042319, Oct 2007

    2007

  110. [118]

    X. C. Kolesnikow, R. W. Bomantara, A. C. Doherty, and A. L. Grimsmo. Gottesman-kitaev-preskill state preparation using periodic driving.Physical Review Letters, 132(13), March 2024

    2024

  111. [119]

    Koliofoti and R

    C. Koliofoti and R. Riwar. Compact description of quantum phase slip junctions.npj Quantum Information, 9(1), December 2023

    2023

  112. [120]

    Konno, W

    S. Konno, W. Asavanant, F. Hanamura, H. Nagayoshi, K. Fukui, A. Sakaguchi, R. Ide, F. China, M. Yabuno, S. Miki, H. Terai, K. Takase, M. Endo, P. Marek, R. Filip, P. van Loock, and A. Furusawa. Logical states for fault-tolerant quantum computation with propagating light.Scienc...

    2024

  113. [121]

    Lachance-Quirion, M.N

    D. Lachance-Quirion, M.N. Lemonde, Simoneau J.O., L. St-Jean, P. Lemieux, S. Turcotte, W. Wright, A. Lacroix, J. Fréchette-Viens, R. Shillito, F. Hopfmueller, M. Tremblay, N. E. Frattini, J.C. Lemyre, and P. St-Jean. Autonomous quantum error correction of gottesman-kitaev-pres...

    2023

  114. [122]

    D. T. Le, A. Grimsmo, C. Müller, and T. M. Stace. Doubly nonlinear superconducting qubit. Physical Review A, 100(6), December 2019

    2019

  115. [123]

    Lenstra, H

    A. Lenstra, H. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982

    1982

  116. [124]

    Levitt and R

    M.H. Levitt and R. Freeman. Nmr population inversion using a composite pulse. Journal of Magnetic Resonance (1969), 33(2):473–476, 1979

    1969

  117. [125]

    S. Lloyd. A potentially realizable quantum computer.Science, 261(5128):1569–1571, 1993

    1993

  118. [126]

    Loss and D

    D. Loss and D. P. DiVincenzo. Quantum computation with quantum dots.Physical Review A, 57(1):120–126, January 1998

    1998

  119. [127]

    Lyubashevsky and D

    V. Lyubashevsky and D. Micciancio. Generalized compact knapsacks are collision resistant. InProceedings of the 33rd International Conference on Automata, Languages and Programming - Volume Part II, ICALP’06, page 144–155, Berlin, Heidelberg, 2006. Springer-Verlag

    2006

  120. [128]

    A. M. Macbeath and C. A. Rogers. A modified form of Siegel’s mean value theorem. II. Math. Proc. Cambr. Phil. Soc., 54(3):322–326, 1958

    1958

  121. [129]

    Y. Manin. Computable and Uncomputable.Sovetskoye Radio, Moscow, 128, 1980

    1980

  122. [130]

    Martinet

    J. Martinet. Perfect Lattices in Euclidean Spaces. Springer Berlin Heidelberg, 2003

    2003

  123. [131]

    Matsusaka and J

    T. Matsusaka and J. Ueki. Modular knots, automorphic forms, and the rademacher symbols for triangle groups.Research in the Mathematical Sciences, 10(1), December 2022

    2022

  124. [132]

    A. May. Cryptanalysis of ntru. preprint

  125. [133]

    A. May. Auf polynomgleichungen basierende public-key-kryptosysteme, 1999

    1999

  126. [134]

    A public-key cryptosystem based on algebraic.Coding Thv, 4244:114–116, 1978

    Robert J McEliece. A public-key cryptosystem based on algebraic.Coding Thv, 4244:114–116, 1978

    1978

  127. [135]

    N. C. Menicucci, S. T. Flammia, and P. van Loock. Graphical calculus for gaussian pure states. Physical Review A, 83(4), apr 2011

    2011

  128. [136]

    L. J. Mensen, B. Q. Baragiola, and N. C. Menicucci. Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits.Phys. Rev. A, 104:022408, 2021

    2021

  129. [137]

    Micciancio

    D. Micciancio. Cse 206a: Lattice algorithms and applications, 2014

    2014

  130. [138]

    J. Milnor. Introduction to Algebraic K-Theory. (AM-72), Volume 72. Princeton University Press, Princeton, 1972

    1972

  131. [139]

    T. Mori. Floquet prethermalization in periodically driven classical spin systems. Physical Review B, 98(10), Sep 2018

    2018

  132. [140]

    D. W. Morris. Introduction to arithmetic groups, 2015

    2015

  133. [141]

    Mosca, A

    M. Mosca, A. Tapp, and R. de Wolf. Private quantum channels and the cost of randomizing quantum information, 2000

    2000

  134. [142]

    D. Mumford. Tata lectures on Theta I. Birkhäuser Boston, 2007

    2007

  135. [143]

    Nakahara

    M. Nakahara. Geometry, topology and physics. IOP 2003, 2003. Bristol, UK: Hilger (1990) 505 p. (Graduate student series in physics). 146

    1990

  136. [144]

    Nathan, L

    F. Nathan, L. O’Brien, K. Noh, M. H. Matheny, A. L. Grimsmo, L. Jiang, and G. Refael. Self-correcting gkp qubit and gates in a driven-dissipative circuit, 2024

    2024

  137. [145]

    E. Nelson. A proof of liouville’s theorem.Proc. Am. Math. Soc., 12(6):995, 1961

    1961

  138. [146]

    M. A. Nielsen and I. L. Chuang.Quantum Computation and Quantum Information. Cambridge University Press, 2000

    2000

  139. [147]

    K. Noh, V. V. Albert, and L. Jiang. Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman-Kitaev-Preskill codes.IEEE Trans. Inf. Th., 65:2563–2582, 2019

    2019

  140. [148]

    Noh and C

    K. Noh and C. Chamberland. Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code.Phys. Rev. A, 101:012316, 2020

    2020

  141. [149]

    K. Noh, S. M. Girvin, and L. Jiang. Encoding an oscillator into many oscillators. Phys. Rev. Lett., 125:080503, 2020

    2020

  142. [150]

    O.T. O’Meara. Symplectic Groups. Mathematical Surveys and Monographs. American Mathematical Society, 1978

    1978

  143. [151]

    Panteleev and G

    P. Panteleev and G. Kalachev. Asymptotically good quantum and locally testable classical ldpc codes, 2022

    2022

  144. [152]

    Patel, Igor L

    Ketan N. Patel, Igor L. Markov, and John P. Hayes. Optimal synthesis of linear reversible circuits.Quantum Info. Comput., 8(3):282–294, mar 2008

    2008

  145. [153]

    Preskill

    J. Preskill. Quantum computing 40 years later, 2023

    2023

  146. [154]

    E. M. Rains. Quantum weight enumerators.IEEE Trans. Inf. Th., 44:1388–1394, 1998

    1998

  147. [155]

    F. K. C. Rankin and H. P. F. Swinnerton-Dyer. On the zeros of eisenstein series. Bulletin of the London Mathematical Society, 2(2):169–170, July 1970

    1970

  148. [156]

    N. Read. Non-abelian adiabatic statistics and hall viscosity in quantum hall states and px + ipy paired superfluids. Physical Review B, 79(4), January 2009

    2009

  149. [157]

    O. Regev. Lecture Notes: Lattices in Computer Science. https://cims.nyu.edu/~regev/teaching/lattices_fall_2009/. Online; accessed 05 December 2022

    2022

  150. [158]

    O. Regev. On lattices, learning with errors, random linear codes, and cryptography. In Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’05, page 84–93, New York, NY, USA, 2005. Association for Computing Machinery

    2005

  151. [159]

    O. Regev. On the Complexity of Lattice Problems with Polynomial Approximation Factors, pages 475–496. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010

    2010

  152. [160]

    Rojkov, P

    I. Rojkov, P. M. Röggla, M. Wagener, M. Fontboté-Schmidt, S. Welte, J. Home, and F. Reiter. Two-qubit operations for finite-energy gottesman-kitaev-preskill encodings, 2023

    2023

  153. [161]

    Rosenberg

    J. Rosenberg. A selective history of the stone-von neumann theorem.Contemp. Math., 365, 01 2004

    2004

  154. [162]

    Royer, S

    B. Royer, S. Singh, and S. M. Girvin. Stabilization of finite-energy gottesman-kitaev-preskill states. Phys. Rev. Lett., 125:260509, Dec 2020

    2020

  155. [163]

    Royer, S

    B. Royer, S. Singh, and S. M. Girvin. Encoding qubits in multimode grid states.PRX Quantum, 3:010335, Mar 2022

    2022

  156. [164]

    Rubio-Abadal, M

    A. Rubio-Abadal, M. Ippoliti, S. Hollerith, D. Wei, J. Rui, S. L. Sondhi, V. Khemani, C. Gross, and I. Bloch. Floquet prethermalization in a bose-hubbard system.Phys. Rev. X, 10:021044, May 2020. 147

    2020

  157. [165]

    Rymarz, S

    M. Rymarz, S. Bosco, A. Ciani, and D. P. DiVincenzo. Hardware-encoding grid states in a nonreciprocal superconducting circuit.Phys. Rev. X, 11:011032, Feb 2021

    2021

  158. [166]

    Sarnak and P

    P. Sarnak and P. Buser. On the period matrix of a Riemann surface of large genus (with an Appendix by J. H. Conway and N. J. A. Sloane).Inventiones mathematicae, 117:27–56, 1994

    1994

  159. [167]

    Schmidt and P

    F. Schmidt and P. van Loock. Quantum error correction with higher Gottesman-Kitaev-Preskill codes: Minimal measurements and linear optics.Phys. Rev. A, 105:042427, Apr 2022

    2022

  160. [168]

    C. P. Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theor. Comput. Sci., 53:201–224, 1987

    1987

  161. [169]

    Schumacher and M

    B. Schumacher and M. D. Westmoreland. Approximate quantum error correction. Quantum Information Processing, 1(1/2):5–12, 2002

    2002

  162. [170]

    I.E. Segal. Mathematical Problems of Relativistic Physics: With an Appendix on Group Representations in Hilbert Space. Lectures in applied mathematics; proceedings of the Summer seminar, Boulder, Colorado, 1960, 2. American Mathematical Society, 1967

    1960

  163. [171]

    H. Seifert. Topologie Dreidimensionaler Gefaserter Räume.Acta Mathematica, 60(none):147 – 238, 1933

    1933

  164. [172]

    Sellem, A

    L. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq. A gkp qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb, 2023

    2023

  165. [173]

    Shor and R

    P. Shor and R. Laflamme. Quantum analog of the macwilliams identities for classical coding theory.Phys. Rev. Lett., 78:1600–1602, 1997

    1997

  166. [174]

    P. W. Shor. Scheme for reducing decoherence in quantum computer memory.Phys. Rev. A, 52:R2493–R2496, Oct 1995

    1995

  167. [175]

    P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.SIAM Journal on Computing, 26(5):1484–1509, 1997

    1997

  168. [176]

    Peter W. Shor. Introduction to quantum algorithms, 2001

    2001

  169. [177]

    Silverman

    J. Silverman. Lecture notes: An introduction to lattices, lattice reduction, and lattice-based cryptography. https://www.ias.edu/sites/default/files/Silverman_ PCMI_Note_DistributionVersion_220705.pdf. Online; accessed 05 December 2022

    2022

  170. [178]

    J. H. Silverman.The Arithmetic of Elliptic Curves. Springer New York, 2009

    2009

  171. [179]

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret. Real-time quantum error correction beyond break-even.Nature, 616(7955):50–55, mar 2023

    2023

  172. [180]

    A. M. Steane. Active stabilization, quantum computation, and quantum state synthesis. Physical Review Letters, 78(11):2252–2255, March 1997

    1997

  173. [181]

    Stehlé and R

    D. Stehlé and R. Steinfeld. Making ntru as secure as worst-case problems over ideal lattices. In K. G. Paterson, editor,Advances in Cryptology – EUROCRYPT 2011, pages 27–47, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg

    2011

  174. [182]

    B. M. Terhal. Quantum error correction for quantum memories.Reviews of Modern Physics, 87(2):307–346, apr 2015

    2015

  175. [183]

    B. M. Terhal, J. Conrad, and C. Vuillot. Towards scalable bosonic quantum error correction. Quantum Science and Technology, 5:043001, 2020. 148

    2020

  176. [184]

    B. M. Terhal and D. J. Weigand. Encoding a qubit into a cavity mode in circuit QED using phase estimation.Physical Review A, 93(1), jan 2016

    2016

  177. [185]

    Tillich and G

    J.-P. Tillich and G. Zemor. Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the block length.IEEE Trans. Inf. Th., 60:1193–1202, 2014

    2014

  178. [186]

    D. Tong. Lectures on the quantum hall effect, 2016

    2016

  179. [187]

    Tzitrin, J

    I. Tzitrin, J. E. Bourassa, N. C. Menicucci, and K. K. Sabapathy. Progress towards practical qubit computation using approximate Gottesman-Kitaev-Preskill codes. Phys. Rev. A, 101:032315, 2020

    2020

  180. [188]

    A. Vardy. The intractability of computing the minimum distance of a code.IEEE Trans. Inf. Th., 43(6):1757–1766, 1997

    1997

  181. [189]

    Viazovska

    M. Viazovska. The sphere packing problem in dimension8. Annals of Mathematics, 185(3), May 2017

    2017

  182. [190]

    von Neumann

    J. von Neumann. Uber einen satz von herrn m. h. stone.Annals of Mathematics, 33(3):567–573, 1932

    1932

  183. [191]

    Vuillot, H

    C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal. Quantum error correction with the toric gottesman-kitaev-preskill code.Physical Review A, 99(3), mar 2019

    2019

  184. [192]

    Vuillot, H

    C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal. Quantum error correction with the toric Gottesman-Kitaev-Preskill code.Phys. Rev. A, 99:032344, 2019

    2019

  185. [193]

    Weedbrook, S

    C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T.C. Ralph, J.H. Shapiro, and S. Lloyd. Gaussian quantum information.Rev. Mod. Phys., 84(2):621–669, May 2012

    2012

  186. [194]

    D. J. Weigand. https://github.com/dweigand/qubit-oscillator

  187. [195]

    D. J. Weigand and B. M. Terhal. Generating grid states from schrödinger-cat states without postselection. Physical Review A, 97(2), feb 2018

    2018

  188. [196]

    D. J. Weigand and B. M. Terhal. Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit qed.Physical Review A, 101(5), May 2020

    2020

  189. [197]

    A. Weil. Oeuvres scientifiques - collected papers I. Springer Collected Works in Mathematics. Springer, Berlin, Germany, November 2014

    2014

  190. [198]

    https://www.ams.org/cgi-bin/notices/nxgnotices.pl?fm=gen&cnt=whatis

    What is... "https://www.ams.org/cgi-bin/notices/nxgnotices.pl?fm=gen&cnt=whatis"

  191. [199]

    Elliptic Modular Forms and Their Applications, pages 1–103

    Don Zagier. Elliptic Modular Forms and Their Applications, pages 1–103. Springer Berlin Heidelberg, Berlin, Heidelberg, 2008

    2008

  192. [200]

    J. Zak. Finite translations in solid-state physics.Phys. Rev. Lett., 19:1385–1387, Dec 1967. Declaration of authorship • Name: Conrad • First name: Jonathan I declare to the Freie Universität Berlin that I have completed the submitted dissertation independently and without the...

    1967

  193. [2011]

    https://eprint.iacr.org/2011/153

    2011

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