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Nine Normal Forms: A Complete Rewrite System for the Single-Phase Three-Qubit Fragment of the Clifford Hierarchy

Ian Richard C Buckley

Published Jun 27, 2026• 0 citations
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Abstract

We prove the first completeness theorem for a multi-qubit fragment of the third Clifford level ($\mathsf{C}3$). Let $\mathcal{F}$ be the single-phase 3-qubit fragment: the 42 states of the form $T_i^{(\pm)} \cdot B_k \cdot |{+}{+}{+}\rangle$, where ${B_k}{k=0}^{6}$ are the seven CZ-type base circuits of the Origami ISA and $T_i^{(\pm)} \in {T_i, T_i^\dagger}$ applies a single T gate to qubit $i$. We define a rewrite system $\mathcal{R} = {R3, R4}$ on $\mathcal{F}$: R3 (T-absorption): $T_i \cdot B_k \cdot |{+}{+}{+}\rangle \to B_k \cdot |{+}{+}{+}\rangle$ when the T gate acts trivially on the reduced state of qubit $i$ in $B_k |{+}{+}{+}\rangle$. R4 (SWAP normalisation): reorder qubits into lexicographically smallest equivalent representative. Theorem 1 (Completeness). Two states $\psi_1, \psi_2 \in \mathcal{F}$ satisfy $\psi_1 = e^{i\theta}\psi_2$ for some $\theta \in \mathbb{R}$ if and only if $\mathcal{R}$ reduces both to the same SWAP-normal form. There are exactly 9 normal forms. The proof follows the standard term-rewriting route: (i) termination via a two-component measure (C$_3$-gate count, lexicographic qubit order); (ii) local confluence by explicit critical-pair analysis; (iii) global confluence by Newman's lemma; (iv) faithfulness by exact state-vector comparison. The 9 normal forms partition into three resource tiers — 1 stabiliser, 6 dark magic ($\mathcal{N} = 0$, $\mathrm{TV} < 1$), and 2 genuine magic ($\mathcal{N} > 0$) — where $\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|$ is the total variation of the discrete Wigner function. A key corollary: the magic class of any state in $\mathcal{F}$ is decidable at compile time by a single rule application of R3, with no state tomography required. This is the first completeness result for any multi-qubit $\mathsf{C}_3$ rewrite system. Prior completeness results (Hadzihasanovic–Ng–Wang 2018; Jeandel–Perdrix–Vilmart 2018) covered single-qubit Clifford+T; this result covers the 3-qubit single-phase case. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 470 (three-tier resource theory) Keywords: quantum resource theory, magic states, Clifford hierarchy, rewrite systems, confluence, normal forms, dark magic, Wigner function, total variation, stabiliser states, T gate, Origami ISA, categorical quantum mechanics, PROP, term rewriting

DOI
10.5281/zenodo.20956674

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Nine Normal Forms: A Complete Rewrite System for the Single-Phase Three-Qubit Fragment of the Clifford Hierarchy

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We prove the first completeness theorem for a multi-qubit fragment of the third Clifford level ($\mathsf{C}3$). Let $\mathcal{F}$ be the single-phase 3-qubit fragment: the 42 states of the form $T_i^{(\pm)} \cdot B_k \cdot |{+}{+}{+}\rangle$, where ${B_k}{k=0}^{6}$ are the seven CZ-type base circuits of the Origami ISA and $T_i^{(\pm)} \in {T_i, T_i^\dagger}$ applies a single T gate to qubit $i$. We define a rewrite system $\mathcal{R} = {R3, R4}$ on $\mathcal{F}$: R3 (T-absorption): $T_i \cdot B_k \cdot |{+}{+}{+}\rangle \to B_k \cdot |{+}{+}{+}\rangle$ when the T gate acts trivially on the reduced state of qubit $i$ in $B_k |{+}{+}{+}\rangle$. R4 (SWAP normalisation): reorder qubits into lexicographically smallest equivalent representative. Theorem 1 (Completeness). Two states $\psi_1, \psi_2 \in \mathcal{F}$ satisfy $\psi_1 = e^{i\theta}\psi_2$ for some $\theta \in \mathbb{R}$ if and only if $\mathcal{R}$ reduces both to the same SWAP-normal form. There are exactly 9 normal forms. The proof follows the standard term-rewriting route: (i) termination via a two-component measure (C$_3$-gate count, lexicographic qubit order); (ii) local confluence by explicit critical-pair analysis; (iii) global confluence by Newman's lemma; (iv) faithfulness by exact state-vector comparison. The 9 normal forms partition into three resource tiers — 1 stabiliser, 6 dark magic ($\mathcal{N} = 0$, $\mathrm{TV} < 1$), and 2 genuine magic ($\mathcal{N} > 0$) — where $\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|$ is the total variation of the discrete Wigner function. A key corollary: the magic class of any state in $\mathcal{F}$ is decidable at compile time by a single rule application of R3, with no state tomography required. This is the first completeness result for any multi-qubit $\mathsf{C}_3$ rewrite system. Prior completeness results (Hadzihasanovic–Ng–Wang 2018; Jeandel–Perdrix–Vilmart 2018) covered single-qubit Clifford+T; this result covers the 3-qubit single-phase case. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 470 (three-tier resource theory) Keywords: quantum resource theory, magic states, Clifford hierarchy, rewrite systems, confluence, normal forms, dark magic, Wigner function, total variation, stabiliser states, T gate, Origami ISA, categorical quantum mechanics, PROP, term rewriting

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We prove the first completeness theorem for a multi-qubit fragment of the third Clifford level (C3\mathsf{C}3C3). Let F\mathcal{F}F be the single-phase 3-qubit fragment: the 42 states of the form Ti(±)⋅Bk⋅∣+++⟩T_i^{(\pm)} \cdot B_k \cdot |{+}{+}{+}\rangleTi(±)​⋅Bk​⋅∣+++⟩, where Bkk=06{B_k}{k=0}^{6}Bk​k=06 are the seven CZ-type base circuits of the Origami ISA and Ti(±)∈Ti,Ti†T_i^{(\pm)} \in {T_i, T_i^\dagger}Ti(±)​∈Ti​,Ti†​ applies a single T gate to qubit iii. We define a rewrite system R=R3,R4\mathcal{R} = {R3, R4}R=R3,R4 on F\mathcal{F}F: R3 (T-absorption): Ti⋅Bk⋅∣+++⟩→Bk⋅∣+++⟩T_i \cdot B_k \cdot |{+}{+}{+}\rangle \to B_k \cdot |{+}{+}{+}\rangleTi​⋅Bk​⋅∣+++⟩→Bk​⋅∣+++⟩ when the T gate acts trivially on the reduced state of qubit iii in Bk∣+++⟩B_k |{+}{+}{+}\rangleBk​∣+++⟩. R4 (SWAP normalisation): reorder qubits into lexicographically smallest equivalent representative. Theorem 1 (Completeness). Two states ψ1,ψ2∈F\psi_1, \psi_2 \in \mathcal{F}ψ1​,ψ2​∈F satisfy ψ1=eiθψ2\psi_1 = e^{i\theta}\psi_2ψ1​=eiθψ2​ for some θ∈R\theta \in \mathbb{R}θ∈R if and only if R\mathcal{R}R reduces both to the same SWAP-normal form. There are exactly 9 normal forms. The proof follows the standard term-rewriting route: (i) termination via a two-component measure (C3_33​-gate count, lexicographic qubit order); (ii) local confluence by explicit critical-pair analysis; (iii) global confluence by Newman's lemma; (iv) faithfulness by exact state-vector comparison. The 9 normal forms partition into three resource tiers — 1 stabiliser, 6 dark magic (N=0\mathcal{N} = 0N=0, TV<1\mathrm{TV} < 1TV<1), and 2 genuine magic (N>0\mathcal{N} > 0N>0) — where TV(ψ)=∑u∣Wψ(u)∣\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|TV(ψ)=∑u​∣Wψ​(u)∣ is the total variation of the discrete Wigner function. A key corollary: the magic class of any state in F\mathcal{F}F is decidable at compile time by a single rule application of R3, with no state tomography required. This is the first completeness result for any multi-qubit C3\mathsf{C}_3C3​ rewrite system. Prior completeness results (Hadzihasanovic–Ng–Wang 2018; Jeandel–Perdrix–Vilmart 2018) covered single-qubit Clifford+T; this result covers the 3-qubit single-phase case. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 470 (three-tier resource theory) Keywords: quantum resource theory, magic states, Clifford hierarchy, rewrite systems, confluence, normal forms, dark magic, Wigner function, total variation, stabiliser states, T gate, Origami ISA, categorical quantum mechanics, PROP, term rewriting
We prove the first completeness theorem for a multi-qubit fragment of the third Clifford level (C3\mathsf{C}3C3). Let F\mathcal{F}F be the single-phase 3-qubit fragment: the 42 states of the form Ti(±)⋅Bk⋅∣+++⟩T_i^{(\pm)} \cdot B_k \cdot |{+}{+}{+}\rangleTi(±)​⋅Bk​⋅∣+++⟩, where Bkk=06{B_k}{k=0}^{6}Bk​k=06 are the seven CZ-type base circuits of the Origami ISA and Ti(±)∈Ti,Ti†T_i^{(\pm)} \in {T_i, T_i^\dagger}Ti(±)​∈Ti​,Ti†​ applies a single T gate to qubit iii. We define a rewrite system R=R3,R4\mathcal{R} = {R3, R4}R=R3,R4 on F\mathcal{F}F: R3 (T-absorption): Ti⋅Bk⋅∣+++⟩→Bk⋅∣+++⟩T_i \cdot B_k \cdot |{+}{+}{+}\rangle \to B_k \cdot |{+}{+}{+}\rangleTi​⋅Bk​⋅∣+++⟩→Bk​⋅∣+++⟩ when the T gate acts trivially on the reduced state of qubit iii in Bk∣+++⟩B_k |{+}{+}{+}\rangleBk​∣+++⟩. R4 (SWAP normalisation): reorder qubits into lexicographically smallest equivalent representative. Theorem 1 (Completeness). Two states ψ1,ψ2∈F\psi_1, \psi_2 \in \mathcal{F}ψ1​,ψ2​∈F satisfy ψ1=eiθψ2\psi_1 = e^{i\theta}\psi_2ψ1​=eiθψ2​ for some θ∈R\theta \in \mathbb{R}θ∈R if and only if R\mathcal{R}R reduces both to the same SWAP-normal form. There are exactly 9 normal forms. The proof follows the standard term-rewriting route: (i) termination via a two-component measure (C3_33​-gate count, lexicographic qubit order); (ii) local confluence by explicit critical-pair analysis; (iii) global confluence by Newman's lemma; (iv) faithfulness by exact state-vector comparison. The 9 normal forms partition into three resource tiers — 1 stabiliser, 6 dark magic (N=0\mathcal{N} = 0N=0, TV<1\mathrm{TV} < 1TV<1), and 2 genuine magic (N>0\mathcal{N} > 0N>0) — where TV(ψ)=∑u∣Wψ(u)∣\mathrm{TV}(\psi) = \sum_u |W_\psi(u)|TV(ψ)=∑u​∣Wψ​(u)∣ is the total variation of the discrete Wigner function. A key corollary: the magic class of any state in F\mathcal{F}F is decidable at compile time by a single rule application of R3, with no state tomography required. This is the first completeness result for any multi-qubit C3\mathsf{C}_3C3​ rewrite system. Prior completeness results (Hadzihasanovic–Ng–Wang 2018; Jeandel–Perdrix–Vilmart 2018) covered single-qubit Clifford+T; this result covers the 3-qubit single-phase case. Part of the Origami ISA series. Companion papers: 466 (CZ-type bases) · 467 (magic orbits) · 468 (ISA as coloured PROP) · 470 (three-tier resource theory) Keywords: quantum resource theory, magic states, Clifford hierarchy, rewrite systems, confluence, normal forms, dark magic, Wigner function, total variation, stabiliser states, T gate, Origami ISA, categorical quantum mechanics, PROP, term rewriting
We present a complete set of 161616 rewrite rules for multi-qudit Clifford circuits in all odd prime dimensions. Completeness means that any two Clifford circuits representing the same linear map can be rewritten into each other using these rules. Each rule involves at most three qudits and admits an intuitive interpretation. To establish completeness, we first work at the symplectic level, using the isomorphism between the symplectic group Sp(2n,Zp)\mathrm{Sp}(2n,\mathbb{Z}_p)Sp(2n,Zp​) and the quotient of the Clifford group by the Pauli group. We construct a circuit normal form that captures the stabiliser tableau of a Clifford operator and is unique up to Pauli correction. Using this normal form, we derive a complete set of symplectic relations, which we then lift to Clifford relations by incorporating Pauli corrections. We also formally verify completeness in the Agda proof assistant.
We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer.This is the first completeness result for any fragment of quantum circuits in odd prime dimensions.We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting.Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form.We then simplify the rewrite rules in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.
The Clifford hierarchy of unitary operators is a foundational concept for universal quantum computation. It was introduced to show that universal quantum computation can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. [S. X. Cui et al., Phys. Rev. A 95, 012329 (2017)] recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal unitaries whose action on a computational basis qudit state is described by a 2kth{2}^{k}\mathrm{th}2kth root of unity raised to some polynomial function of the state, and they established the level of such unitaries in the hierarchy as a function of kkk and the degree of the polynomial. For qubit systems, we consider kthk\mathrm{th}kth-level diagonal unitaries that can be described just by quadratic forms of the state over the ring Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ of integers modulo 2k{2}^{k}2k. The quadratic forms involve symmetric matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ that can be used to efficiently describe all two-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal unitaries from lower levels. The result involves symplectic matrices over Z2k{\mathbb{Z}}_{{2}^{k}}Z2k​ and hence our perspective unifies a subgroup of diagonal gates in the Clifford hierarchy with the binary symplectic framework for gates in the Clifford group. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to expensive magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+T\mathrm{Clifford}+TClifford+T set of gates, perhaps leading to shorter depth circuits. Our results suggest that some nondiagonal gates in the hierarchy might also be understood by generalizing other binary symplectic matrices to integer rings.
Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. A finite rule set is most informative when it separates the genuine algebra of a circuit fragment from the structural treatment of wires. This paper gives six near-Clifford fragments a common PROP treatment, where wire permutations are structural: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, and qutrit Clifford. Starting from prior completeness theorems, we transfer completeness into this setting and remove redundant non-structural rules, then check minimality by separating interpretations tailored to individual axioms; the resulting presentations are minimal in all arities for qubit Clifford, real Clifford, and CNOT-dihedral, minimal in bounded ranges for the remaining fragments, and comparable by one transfer-and-separation pattern.