Fault-Tolerant Quantum Computation with Higher-Dimensional Systems

Daniel Gottesman

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Fault-Tolerant Quantum Computation with Higher-Dimensional Systems

Daniel Gottesman

Quantum Physics · Submitted 2 Feb 1998

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.

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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,924 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Instead of a quantum computer where the fundamental units are 2-dimensional qubits, we can consider a quantum computer made up of d-dimensional systems. There is a straightforward generalization of the class of stabilizer codes to d-dimensional systems, and I will discuss the theory of fault-tolerant computation using such codes. I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Instead of a quantum computer where the fundamental units are 2-dimensional qubits, we can consider a quantum computer made up of d-dimensional systems.	background 0.99	0.09	0.03	hedged possibility 0.66
s1	There is a straightforward generalization of the class of stabilizer codes to d-dimensional systems, and I will discuss the theory of fault-tolerant computation using such codes.	contribution 0.59	0.32	0.07	asserted 0.98
s2	I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.	result 0.96	0.93	0.88	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
relations to other papers

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P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

Instead of a quantum computer where the fundamental units are 2-dimensional qubits, we can consider a quantum computer made up of d-dimensional systems. There is a straightforward generalization of the class of stabilizer codes to d-dimensional systems, and I will discuss the theory of fault-tolerant computation using such codes. I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.

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References

Forward citations

Cited by 3 Pith papers

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

Local Universality and Structural Certificates for Minimal Fixed-Depth Two-Qutrit Gate Decomposition
quant-ph 2026-07 conditional novelty 7.0 of 10

A four-copy fixed-core architecture with five local SU(3)⊗SU(3) layers is shown to be locally universal for two-qutrit gates, via an explicit Clifford core that makes the differential an exact isometry.
Non-Abelian dynamics on a cube: improving quantum compilation through qudit-based simulations
quant-ph 2025-06 conditional novelty 7.0 of 10

A qudit-based circuit for SU(2) lattice gauge theory on a cube, with improved decompositions for uniformly-controlled rotations and new elementary-gate resource estimates.
Fault-Tolerant Resource Comparison of Qudit and Qubit Encodings for Diagonal Quadratic Operators
quant-ph 2026-04 unverdicted novelty 6.0 of 10

The paper derives explicit finite-d break-even synthesis costs for qudit vs. qubit encodings of diagonal quadratic operators in product-formula and LCU simulations, identifying low-d regions where qudits yield savings.

How later papers used it

73 later papers cite it

Evidence: substantial. 73 of the citing papers share no author with it, and 9 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 183 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 built on it and 1 used its method or tool.

Builds on it 2 of 8 papers with this label

Generalized the well-known stabilizer formalism in prime dimension as in this paper to cover non-commuting Z_r measurements.

A linearized stabilizer formalism for systems of finite dimension (2011) · 0.96 · paraphrased: the licence does not allow quoting
We can naturally extend [ref] the action of $Π_n$ to Hilbert spaces of dimension $d>2$ by constructing the following $d$-dimensional generalizations of $σ_X$ and $σ_Z$

Topological Preparation of Non-Stabilizer States and Clifford Evolution in $SU(2)_1$ Chern-Simons Theory (2025) · The Qudit Pauli Group · 0.80

Uses its method or tool 1 paper

Used the nonbinary stabilizer formalism of this paper to efficiently represent tensor networks with prime bond dimensions.

Entanglement phase transitions in random stabilizer tensor networks (2021) · 0.80 · paraphrased: the licence does not allow quoting

How the literature uses it

47 arXiv papers cite this in 132 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 83 sentences without a confident label

Builds on it 7 papers

[ref] , where we give a parallel construction of the generalized Pauli group, which can be prepared by topological operations for $SU(N)_1$, $N$ odd prime, as a higher dimensional generalization of the Pauli group, in analogy with the general discussion of Gottesman [ref] for qudits, with $d$ an odd prime.

SU(N)1 Chern-Simons theory, the Clifford group, and Entropy Cone · Stabilizer cone for $SU(N)_1$ · 0.89
Our circuit model extends the so-called model of normalizer circuits over Abelian groups [ref] , which, in turn, generalizes the better-known model of Clifford circuits [ref] .

The computational power of normalizer circuits over black-box groups · Introduction · 0.99
We extend the PSF [ref] , a powerful tool for describing quantum many-body states as eigenstates of commuting groups of Pauli operators, and the GSF, the abelian group extension of [ref] , to a stabilizer formalism using commuting hypergroups of generalized Pauli operators.

Abelian Hypergroups and Quantum Computation · Main results · 1.00

Uses its method or tool 1 paper

This allows for an efficient representation of the tensor network with prime bond dimensions using the nonbinary stabilizer formalism [ref] , which we now briefly review.

Entanglement phase transitions in random stabilizer tensor networks · Random stabilizer tensor networks · 0.80

Cites it as background 23 papers

These circuits were introduced in [ref] by one of us, as a class of quantum computations generalizing the Clifford circuits, as well as standard extensions of the latter to qudits [ref] .

Classical simulations of Abelian-group normalizer circuits with intermediate measurements · Introduction · 0.93
Normalizer circuits [ref] are a family of quantum circuits that generalize the so-called Clifford circuits [ref] to Hilbert spaces associated with finite Abelian groups $G$.

Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems · Introduction · 0.90
Aside from generalizations of Clifford circuits [ref] (which includes normalizer circuits), many other classes of restricted quantum circuits have been studied in the literature.

The computational power of normalizer circuits over black-box groups · Relationship to previous work · 0.90

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