Quaternionic and Octonionic Frameworks for Quantum Computation: Mathematical Structures, Models, and Fundamental Limitations

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Report on the Manuscript ID:  quantumrep-3964351
"Quaternionic and Octonionic Frameworks for Quantum Computation: Mathematical Structures, Models, and Fundamental Limitations"

    This manuscript presents a comprehensive and mathematically rigorous investigation of quantum computation formulated over the quaternionic (H) and octonionic (O) normed division algebras. It systematically contrasts the associative quaternionic model, which admits computational equivalence with the standard complex framework, with the non-associative octonionic case, where fundamental obstructions arise. The study integrates algebraic theory, computational modeling, and conceptual implications for physical realizability.
    The paper is highly original, well-structured, and contributes significantly to theoretical quantum information science. It situates quaternionic and octonionic extensions within the context of Hurwitz's theorem and explores both theoretical and potential experimental implications. The paper represents a valuable and innovative contribution. However, there are some suggestions for improvement.

1. In abstract: Add a clearer statement of key findings-e.g., "We prove the polynomial computational equivalence of quaternionic and complex quantum models."
2. The authors should explain more clearly how the proposed quaternionic and octonionic models could be tested or implemented in real experiments.
3. The limits of the octonionic model should be stated more clearly, showing what is feasible and what is not.
4 . The authors must include at least one schematic diagram illustrating the quaternionic-to-complex embedding (Equation 5 or 6) for clarity.
5. Terminology: When using symbols like ι or associator [x,y,z], define them at first appearance in the main text rather than later.
6 . In the section on "Emerging Applications," the authors must mention potential intersections with machine learning or topological quantum computation.
7. Minor grammatical polishing is advised to maintain consistent formal tone (e.g., "quaterbit" vs. "quaternionic qubit").
8. The authors must add some of the recent references.

Author Response

Comments 1:
In abstract: Add a clearer statement of key findings-e.g., "We prove the polynomial computational equivalence of quaternionic and complex quantum models."

Response 1:
Thank you for this suggestion. We have revised the abstract to state our main result explicitly. In particular, we now include a sentence of the form:

“Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: computation over $\mathbb{H}$ is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions.”

This makes the central computational result visible from the very beginning of the paper.
Location in the revised manuscript: Abstract, second sentence.

Comments 2:
The authors should explain more clearly how the proposed quaternionic and octonionic models could be tested or implemented in real experiments.

Response 2:
We agree that this needed to be clearer. In the revised manuscript, we expanded the section on experimental feasibility to include concrete routes for implementation on current or near-term quantum hardware. Specifically, under Experimental Feasibility and Hybrid Architectures we have added a paragraph and itemised discussion (“Concrete routes for quantum-computing experiments”) describing:

• interferometric subcircuits (Peres-type and Procopio-type experiments) reinterpreted as small quaternionic circuits that can be run on photonic platforms,

• embedded quaternionic circuits implemented on existing gate-based processors via the quaternionic–to–complex embedding (superconducting, ion-trap, photonic, etc.), and

• encoded octonionic dynamics realised as digital simulations on few-qubit registers that reproduce octonionic associator effects.

This directly addresses how our models can be tested or instantiated using today’s quantum computing architectures.
Location in the revised manuscript: Section on Experimental Feasibility and Hybrid Architectures, paragraph “Concrete routes for quantum-computing experiments”.

Comments 3:
The limits of the octonionic model should be stated more clearly, showing what is feasible and what is not.

Response 3:
We have added a dedicated paragraph explicitly addressing the scope and limitations of the octonionic model. After presenting our main obstructions and complexity results, we introduce a new paragraph titled “Practical limits of the octonionic model”, where we distinguish between:

• Feasible/realistic uses: small numbers of logical octo-qubits in carefully chosen associative subalgebras, G_2​-covariant encodings, and measurement-only schemes based on equiangular projections, all realised as encoded subspaces of complex Hilbert space; and

• Non-feasible / currently out-of-reach aspects: fully native, large-scale octonionic hardware with generic time-dependent Hamiltonians and unrestricted measurements, where non-associativity makes evolution path-dependent and severely constrains physically reasonable measurement postulates.

We then summarise that, in practice, the octonionic framework is best viewed as a conceptual and small-scale experimental tool, rather than a realistic architecture for scalable fault-tolerant quantum computation.
Location in the revised manuscript: Octonionic results section, paragraph “Practical limits of the octonionic model”.

Comments 4:
The authors must include at least one schematic diagram illustrating the quaternionic-to-complex embedding (Equation 5 or 6) for clarity.

Response 4:
We have added a schematic diagram illustrating the quaternionic–to–complex embedding at the circuit level. In the subsection on circuit-level embeddings, we now include a new figure (Figure 2) showing:

• an upper “quaternionic” circuit on n quaterbits,

• the corresponding lower “complex” circuit on n+1 qubits obtained via the embedding defined in Equations (5)–(6), and

• dashed arrows indicating the embedding of states between H_{\mathbb{H}}^n and \mathcal{H}_{\mathbb{C}}^{n+1}​.

We also added an introductory sentence linking this figure explicitly to Equations (5)–(6), so the reader can immediately see how the algebraic embedding translates into a concrete circuit mapping.
Location in the revised manuscript: Section on quaternionic-to-complex embedding, paragraph “Circuit Diagrams and Practical Implications” and Figure 2.

Comments 5:
Terminology: When using symbols like ι or associator [x,y,z], define them at first appearance in the main text rather than later.

Response 5:
We have checked the first appearance of these symbols and made sure they are now defined immediately:

• For the associator [x,y,z][x,y,z][x,y,z], the first occurrence already included the explicit definition [x,y,z]:=(xy)z−x(yz)[x,y,z] := (xy)z - x(yz)[x,y,z]:=(xy)z−x(yz); we have verified that this remains the case.

• For the embedding symbol ι\iotaι, we have revised its first appearance in the main text (in the brief summary of the quaternionic model) to include the explicit formula

  ι(a+bi+cj+dk)=(a+bic+di−c+dia−bi),\iota(a+bi+cj+dk) = \begin{pmatrix} a+bi & c+di\\ -c+di & a-bi \end{pmatrix},ι(a+bi+cj+dk)=(a+bi−c+di​c+dia−bi​),

  matching the detailed definition later in the technical section.

This ensures that both ι\iotaι and the associator are defined where they first appear, as requested.
Location in the revised manuscript: Quaternionic formalism section (first mention of ι\iotaι); algebraic preliminaries section (first mention of [x,y,z][x,y,z][x,y,z]).

Comments 6:
In the section on "Emerging Applications," the authors must mention potential intersections with machine learning or topological quantum computation.

Response 6:
We have significantly expanded Section 3.6 (Emerging Applications and Future Directions — elaboration) to address both of these intersections:

• Under the subsubsection on G2G_2G2​ symmetries and octonionic simulation, we added a paragraph “Connections to topological quantum computation” where we discuss how octonionic and G2G_2G2​-symmetric structures relate to topological quantum computation (TQC), measurement-only schemes, and anyon-based models, citing relevant work (e.g. Nayak et al., Lahtinen–Pachos, Freedman–Shokrian-Zini–Wang).

• In the subsubsection “Hypercomplex Quantum Machine Learning” we now make explicit that we are discussing quantum machine learning, and we connect our quaternionic and octonionic circuit models to recent proposals for quaternionic quantum neural networks and to the classical hypercomplex neural network literature, outlining how hybrid QML architectures can exploit hypercomplex structure on both quantum and classical sides.

These additions directly address the requested connections to machine learning and topological quantum computation.
Location in the revised manuscript: Section 3.6, paragraphs “Connections to topological quantum computation” and “Hypercomplex Quantum Machine Learning”.

Comments 7:
Minor grammatical polishing is advised to maintain consistent formal tone (e.g., "quaterbit" vs. "quaternionic qubit").

Response 7:
Thank you for this remark. In the revised manuscript we have carefully proof-read the text and standardised the terminology throughout. In particular, we now consistently use the expression “quaternionic qubit” instead of the shorter and less standard “quaterbit”, both in the main text and in figure captions. During this pass we also corrected a number of minor typographical and grammatical issues (article use, pluralisation, punctuation) to keep a uniform formal tone.
Location in the revised manuscript: Global change across Sections 1–4 and Appendices (search-and-replace of “quaterbit” by “quaternionic qubit” and minor grammatical edits).

Comments 8:
The authors must add some of the recent references.

Response 8:
We thank the referee for this suggestion. In the revised manuscript we have updated the bibliography and incorporated several more recent and directly relevant works, which are now cited at the appropriate points in the text (foundations, experimental tests, emerging applications, and quantum machine learning). In particular, we have added recent references from 2022 onwards

Reviewer 2 Report

Comments and Suggestions for Authors

The paper explores the possibility of quantum computation based on quaternionic and octonionic numbers, discussing both the underlying formalism and potential material realizations.

However, improvement is needed before the paper can be considered for publication.

For example, Section 2.7 appears to be a central part of the proposed framework, yet it is presented rather briefly. The authors should clarify whether the use of quaternionic (Q) or octonionic (O) numbers requires a reformulation of quantum mechanics itself. Quaternionic quantum mechanics has been studied previously, as I remembered, but its analytical properties—and many of its elegant mathematical features—are not fully reflected in the physical formulation, which already faces considerable challenges. As for the octonionic case, it is unclear whether a comparable study exists, though it is reasonable to expect even greater conceptual and technical difficulties.

The authors should specify clearly at what level their approach affects the standard formulation of quantum mechanics.

Furthermore, the motivation and potential advantages of introducing quaternionic or octonionic structures into quantum computation remain unclear. The authors should provide a stronger connection between these algebraic extensions and the existing qubit framework, emphasizing what improvements or complications may arise from such a generalization.

Author Response

Comments 1:
However, improvement is needed before the paper can be considered for publication.

For example, Section 2.7 appears to be a central part of the proposed framework, yet it is presented rather briefly. The authors should clarify whether the use of quaternionic (Q) or octonionic (O) numbers requires a reformulation of quantum mechanics itself. Quaternionic quantum mechanics has been studied previously, as I remembered, but its analytical properties—and many of its elegant mathematical features—are not fully reflected in the physical formulation, which already faces considerable challenges. As for the octonionic case, it is unclear whether a comparable study exists, though it is reasonable to expect even greater conceptual and technical difficulties.

The authors should specify clearly at what level their approach affects the standard formulation of quantum mechanics.

Response 1:
Thank you very much for this meaningful and insightful comment. We agree that the foundational scope of our approach and its relation to standard quantum mechanics need to be made more explicit. Therefore, we have substantially expanded Section 2.7 (“Octonionic formalism”) to clarify exactly at what level quaternionic and octonionic structures enter our framework and how they relate to the complex formulation of quantum mechanics.

1. We now state explicitly that we do not reformulate the postulates of quantum mechanics.
   At the end of Section 2.7 we added a new paragraph titled “Foundational scope of the hypercomplex models”. There, we clarify that throughout the paper, we take the usual Dirac–von Neumann framework—complex Hilbert spaces, Born-rule probabilities, and unitary dynamics—as the underlying physical theory. The quaternionic and octonionic models are introduced as alternative algebraic representations and computational encodings of finite-dimensional systems, which are always equipped with explicit embeddings into ordinary complex Hilbert space. In particular, any protocol we describe can be simulated within the standard qubit model with polynomial overhead, so our work does not propose a new physical theory of quantum mechanics.
   Location in the revised manuscript: Section 2.7, paragraph beginning “Foundational scope of the hypercomplex models.”

2. We relate our framework to previous work on quaternionic and octonionic quantum mechanics.
   Immediately after that, we added a paragraph titled “Relation to quaternionic and octonionic quantum mechanics.” There we briefly review the existing literature on quaternionic quantum mechanics (e.g. Adler’s monograph and related work) and explain that these theories can be embedded into complex quantum mechanics so that observable probabilities are reproduced by complex POVMs and channels, following the perspective of Baez and others. For the octonionic case, we emphasise that non-associativity makes it much harder to construct a full Hilbert-space theory and that existing formulations (e.g. De Leo and collaborators) only work in restricted settings and face substantial technical obstacles. We then state that our approach deliberately avoids making new foundational claims in this direction: the octonionic structures we use are finite-dimensional models that are realised as encoded subspaces of complex Hilbert space, with all probabilities computed using the standard Born rule.
   Location in the revised manuscript: Section 2.7, paragraph beginning “Relation to quaternionic and octonionic quantum mechanics.”

3. We clarify the motivation, advantages, and complications of quantum computation.
   To address the question of why one would introduce quaternionic or octonionic structures at all, we added a third paragraph titled “Computational motivation, advantages, and complications.” There, we explain that the motivation is structural, not to go beyond the computational power of BQP. Quaternionic amplitudes provide a compact parametrisation of SU(2) rotations and arise naturally in information-theoretic comparisons of real, complex, and quaternionic theories, while octonionic amplitudes and G_2​ symmetries underpin continuous families of equiangular projections relevant for measurement-only and topological models of quantum computation (e.g. Freedman–Shokrian-Zini–Wang). We also discuss the complications introduced by non-commutativity and non-associativity—such as subtleties in tensor products, time ordering, and the definition of observables—and emphasise that, once these constraints are taken into account, our later complexity-theoretic results show that the resulting models are polynomially equivalent to standard qubit-based quantum computation.
   Location in the revised manuscript: Section 2.7, paragraph beginning “Computational motivation, advantages, and complications.”

These additions are intended to make it clear that our work does not attempt to overhaul the foundations of quantum mechanics, but rather to study quaternionic and octonionic computational models that are explicitly embedded in the standard complex formalism, while also explaining the structural insights and difficulties that arise from such generalizations.

 

Comments 2:
Furthermore, the motivation and potential advantages of introducing quaternionic or octonionic structures into quantum computation remain unclear. The authors should provide a stronger connection between these algebraic extensions and the existing qubit framework, emphasizing what improvements or complications may arise from such a generalization.

Response 2:
Thank you for this very helpful comment. We agree that the original version did not explain clearly enough why one would introduce quaternionic or octonionic structures in quantum computation, nor how they interface with the standard qubit model. In the revised manuscript we have clarified this at several points:

1. Stronger connection to the qubit framework.
   We now stress explicitly that every quaternionic or octonionic circuit in our model is equipped with an embedding into a standard complex qubit circuit with polynomial overhead. This is explained in more detail at the end of Section 2.7 (new paragraphs “Foundational scope of the hypercomplex models” and “Computational motivation, advantages, and complications”) and cross–referenced from the discussion of the quaternionic–to–complex embedding in Section 3 (subsection “Circuit Diagrams and Practical Implications”). These additions spell out at which level the hypercomplex models modify only the representation of states and gates, while remaining simulable within the usual qubit framework and computing the same complexity class BQP.

2. Motivation and potential advantages.
   In the new paragraph “Computational motivation, advantages, and complications” (Section 2.7), we explicitly discuss the structural reasons for considering quaternionic and octonionic models: quaternionic amplitudes provide a compact parametrisation of SU(2) rotations and arise naturally in information–theoretic reconstructions of finite–dimensional quantum theory, while octonionic amplitudes and G_2​ symmetries underlie continuous families of equiangular projections that are relevant for measurement–only and topological models of quantum computation. In Section 3.6 (“Emerging Applications and Future Directions — elaboration”) we further elaborate on these advantages by adding dedicated paragraphs on “Connections to topological quantum computation” and “Hypercomplex quantum machine learning,” where we describe concrete scenarios in which hypercomplex structure can make certain symmetries and encodings more natural or economical than in a purely complex parametrisation.

   Location in the revised manuscript: Section 2.7, paragraphs starting “Computational motivation, advantages, and complications”; Section 3.6, paragraphs “Connections to topological quantum computation” and “Hypercomplex Quantum Machine Learning.”

3. Complications introduced by the generalisation.
   In the same new paragraph in Section 2.7 we also emphasise the drawbacks of moving to H and O: on the quaternionic side, non-commutativity makes gate ordering and tensor products more subtle and can impose superselection–like constraints; on the octonionic side, non-associativity severely restricts admissible dynamics and measurement maps and requires working in carefully chosen associative subalgebras or encoded subspaces. We explicitly state that, once these constraints are properly taken into account and the embeddings into complex Hilbert spaces are made explicit, our later complexity–theoretic results show that the overall computational power remains polynomially equivalent to the standard qubit model.

   Location in the revised manuscript: Section 2.7, paragraph starting “Computational motivation, advantages, and complications.”

We hope these additions now make clear both the motivation and the structural benefits of introducing quaternionic and octonionic models in quantum computation, as well as the complications and limitations that accompany such generalisations, and how they are systematically related back to the familiar qubit framework.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The authors study models of quantum computing based on quaternions and octonions. This allows them to propose a quaternion-based model and study obstructions for an octonionic model.

The manuscript is well-written with a good introduction ot quaternions and octonions for a wider readership. However, when it comes to contributions, it is very condensed with a lot of content pushed into the appendix which makes it difficult to follow or know what is actually contributed. Some examples of those are:
- l161 notation "H\oplus He_6" not explained and only becomes clear from later context
- unclear how Eq (7) leads to Eq (8)
- sector change sentence in sec 2.11 is not clear. I can see how this is ok with octonionic states, but not sure how this extends to operations
- some major "what we did" is in the appendix while the conclusion is just three bullet points

More generally, there is confusion in the manuscript about operations and states. The authors jump between quaternionic states and quaternionic operations without much notice making it hard to follow; e.g. in line 248ff where what seems to be state amplitudes are talked about in the context of operations.

This is particularly difficult to follow where context changes a lot and int implicit context is not always clear. E.g. SU(2) seems to be used with complex and quaternionic entries but with the frequent changes in setting it is hard to follow which SU(2) is referred to, especially when the authors are talking about equivalence between quaternionic and complex gates.

As such, the first point that I think needs addressing is:

Point 1. Clarification of notation and improvements to contextualization of what is said. 

In line with the contextualization, it should be noted that I like the side by side contrasting. However, quite a lot about the quaternions follows from the embetting \iota: H \hookrightarrow M_2(C). Bounded operators L(H) on a Hilbert space H are isomorphic to H\otimes H, so quaternions as SU(2) operations are naturally embedded into H\otimes H, i.e. there is a natural embedding of any quaternionic state into a qubit state on a space double the size. This is implicitly acknowledged in line 247, but not explicitly mentioned. A lot of what is said can be easily derived from this embedding and isomorphism.

As such, a reader familiar with this background knowledge is likely to see this as a good expose to show what is true but it feels like a review of known or easily concluded facts. I strongly recommend more of an emphasis of the original contribution here, or even a clear separation of known content vs original contribution.

A reader unfamiliar with this background likely is going to see the argument a bit like "because I say so" which is very unsatisfying when there is such a fundamental reason behind it.

This also means that I am somewhat missing the point as the model is clearly contained in a quantum computing model and also clearly contains a quantum computing model which makes the equivalence relatively easy to see. The authors even cite a reference that shows this can be done with a single ancilla instead of requiring a doubled system. Thus, it is not clear what the main contribution of "A complete H-based model" means given that the embeddings into the standard computational model are known and even more efficient representations than the naive embedding are cited.

Meanwhile, the octonion discussions seem to be a variation of the same issues repeated. Again, it is difficult to dissect what is a genuine contribution of a mitigation vs an fundamental obstruction and what exactly on each of these points is novel within the here presented work.

Hence, the second main point is:
Point 2. Stronger separation of novel contributions from existing knowledge

Finally, I am missing the motivation for the work. Of course it is possible to embed quaternionic and octonionic circuits. But why? The are examples of why I would care about such a computing hardware are very vague. Since there is equivalence between the quaternionic and standard computational framework, there need to be some stronger arguments to do this than statements such as l640ff "This mathematical elegance has a profound physical implication. If a quantum hardware platform could be controlled with pulses that directly map to the four real components of a quaternion (for instance, using the amplitude and phase of two orthogonal control fields), then an arbitrary rotation could, in principle, be implemented in a single operational step. This contrasts sharply with the three sequential operations required by the Euler decomposition." This (and the other generic examples given) is a bit of a false equivalence as it is not clear whether such an operation for the quaternions is even possible. Otherwise I can do the same argument for a generic unitary operation and conclude that all quantum computations could be done in a single compute step. Now that would be even better and have profound implications for complexity theory, but it is obviously a way over-simplified argument to make. Yet, for the special case of quaternionic/octonionic operations, this is precisely the argument that is put forward in sentences like l622ff "This section argues that this representational efficiency can translate into tangible advantages in circuit simplification and algorithmic design, particularly for quantum simulation and variational algorithms." Additionally, this argument needs to also be contrasted with qudits. Are we simply looking at a special d=4 qudit representation with the additional assumption that single qudit operations can be performed in a single compute step?

Thus, if this is the line of argumentation for pursuing this idea, then I think this needs to be justified more. If the motivation lies in a different direction, the motivation for the presented work should be made clearer.

Point 3. Clearer motivation of the presented work.

Otherwise, as far as I can tell, the work is all correct. It is interesting and fits well within the scope of Quantum Reports. Thus, if the above can be addressed, I would be more than happy to recommend the work for publication.

Author Response

Comments 1:
Point 1. Clarification of notation and improvements to contextualization of what is said. The manuscript is well-written with a good introduction to quaternions and octonions for a wider readership. However, when it comes to contributions, it is very condensed with a lot of content pushed into the appendix which makes it difficult to follow or know what is actually contributed. Some examples of those are:
– l161 notation “\mathbb H \oplus \mathbb H e_6​” not explained and only becomes clear from later context
– unclear how Eq. (7) leads to Eq. (8)
– sector change sentence in Sec. 2.11 is not clear. I can see how this is ok with octonionic states, but not sure how this extends to operations
– some major “what we did” is in the appendix while the conclusion is just three bullet points

More generally, there is confusion in the manuscript about operations and states. The authors jump between quaternionic states and quaternionic operations without much notice making it hard to follow; e.g. in line 248ff where what seems to be state amplitudes are talked about in the context of operations.

This is particularly difficult to follow where context changes a lot and implicit context is not always clear. E.g. SU(2) seems to be used with complex and quaternionic entries but with the frequent changes in setting it is hard to follow which SU(2) is referred to, especially when the authors are talking about equivalence between quaternionic and complex gates.

Response 1:
Thank you very much for this detailed and constructive comment. We agree that the original version did not always make the notation and context explicit enough, especially around the transition from quaternionic to complex representations and in the octonionic discussion. Therefore, we have revised the manuscript to clarify the notation and to separate more clearly the different levels (states vs. operations, complex vs. quaternionic vs. octonionic). The main changes are:

1. Explicit explanation of \mathbb{H}\oplus \mathbb{H}e_6​ (Cayley–Dickson notation).
   In the subsection on octonionic preliminaries we now explain the notation
   \mathbb{O} = \mathbb{H} \oplus \mathbb{H}e_6​ before using it. We state that every octonion can be written uniquely as x = a + b e_6​ with a,b∈H, identify x with the pair (a,b), and then write the Cayley–Dickson product explicitly for pairs (a,b),(c,d)∈H⊕He6​. We also add a sentence explaining that this notation will be used repeatedly later when discussing associative triples and confinement strategies.
   Location in the revised manuscript: Section 2 (hypercomplex preliminaries), paragraph “Cayley–Dickson product”.

2. Clarifying which SU(2) is used.
   To avoid confusion between complex and quaternionic SU(2), we added a short notation paragraph stating that, unless explicitly stated otherwise, SU(2) denotes the standard group of 2×2 complex special unitaries acting on a single complex qubit, and that “quaternionic gates realised as SU(2) operations” always refer to their image under the embedding ι:H↪M2(C).
   Location in the revised manuscript: Section 2 (Quaternionic and octonionic preliminaries), paragraph “Notation on SU(2) and context”.

3. Separating “states” and “operations” in the quaternionic formalism.
   In the subsection “Quaternionic formalism” we have inserted a new paragraph explicitly distinguishing quaternionic states (vectors in a right H-module, with components in H) from quaternionic operations (right-H-linear maps/gates acting on those states). We also state that whenever we move from statements about states (amplitudes, inner products) to statements about operations (unitaries, channels), we will indicate this explicitly. This addresses the confusion you pointed out around lines 247–248.
   Location in the revised manuscript: Section 2 (Quaternionic formalism), paragraph “States versus operations.”

4. Detailed derivation from the quaternionic Hadamard H_H​ to its complex matrix H_C​.
   In the subsection “Circuit Diagrams and Practical Implications” we have expanded the discussion around the quaternionic Hadamard example. We now write H_H explicitly as
   H_H=A+Bj with

   
   

   and show how the embedding from Eq. (5) produces the block matrix
   
   yielding the explicit 4×4 complex matrix in Eq. (7). This makes the previously implicit step “Eq. (7) → Eq. (8)” fully explicit.
   Location in the revised manuscript: Section 3, subsection “Circuit Diagrams and Practical Implications,” paragraph following Eq. (7).

5. Clarifying “switching associative sectors” in the octonionic remark.
   In the remark on octonionic costs and limits we have rewritten the text to explain what we mean by “switching associative sectors.” We now define this as moving states and gates between different chosen quaternionic subalgebras Hα⊂O (or equivalently, changing the parenthesisation class of multi-octonion products), and we explain how such sector changes affect both states and operations in our encoded complex representation. We also clarify that practical octonionic architectures must confine dynamics to a fixed associative subalgebra or compensate for sector changes using our ASD/synthesis strategies.
   Location in the revised manuscript: Section 3 (Octonionic obstructions and costs), Remark “Octonionic costs and limits”.

Taken together, these changes are intended to address your Point 1 by making the notation explicit, clearly separating states from operations, specifying the meaning of SU(2) throughout, and spelling out the key embedding step from the quaternionic Hadamard gate to its complex realisation, as well as the meaning of associative-sector changes in the octonionic model.

 

Comments 2:
Point 2. Stronger separation of novel contributions from existing knowledge. The referee notes that much of the quaternionic material follows from the embedding ι:H↪M_2(C) and from the isomorphism L(H)≅H⊗H, and that this is only implicitly acknowledged in the original version. As a result, a reader familiar with the background might see large parts of the text as a review of known or easily derived facts, while a reader unfamiliar with the background might feel that some arguments are “because I say so”. The referee therefore asks for a clearer separation between background/review content and the genuinely new contributions, both for the quaternionic and for the octonionic parts, and in particular for what we called “a complete H-based model”.

Response 2:
We thank the referee for this important observation. We agree that the original version did not sufficiently distinguish between background material and our own contributions, and that the foundational role of the embedding ι:H↪M_2(C) and the operator–state isomorphism could have been made more explicit. In the revised manuscript we have addressed this point in several ways:

1. Explicit split between review and original work in the “Main contributions” paragraph.
   In the introduction, the “Main contributions” paragraph has been rewritten so that each item is now explicitly tagged either as review/synthesis or as original. We first state what is being summarised from the existing literature (e.g., foundational work on quaternionic quantum mechanics by Finkelstein and Adler, quaternionic circuit and channel simulations by Fernández–Schneeberger, Graydon, and Gantner, and octonionic measurement-only schemes and physical motivations from Baez and Freedman et al.). We then clearly separate what is new in this paper:
   – the formulation of a complete quaternionic circuit model (states, channels, measurements, tensor products) together with an explicit, ancilla-efficient embedding into standard complex circuits, and
   – the systematic obstruction and mitigation analysis for octonionic gate-based models, including the structured strategies (M1)–(M4) and their resource trade-offs.
   This is intended to make it immediately clear to both expert and non-expert readers which parts of the paper are expository and which contain new results.

2. Quaternionic formalism: what is standard and what is new.
   At the beginning of the subsection “Quaternionic formalism”, we now add a short contextual paragraph explaining that the basic structures we use (right H-modules, adjoints, and the possibility of simulating quaternionic dynamics on complex Hilbert spaces) are rooted in standard references such as Finkelstein, Adler and others, and in the more recent works of Fernández–Schneeberger, Graydon and Gantner. We then state explicitly that our contribution is to package these ingredients into a self-contained circuit model tailored to quantum information (including channels and measurement patterns), and to connect this model directly to the complexity-theoretic equivalence with BQP proved later in the paper.
   In this way we acknowledge, up front, that many of the structural facts about quaternionic gates follow from the embedding ι\iotaι and the operator–state isomorphism, and we clarify that our original contribution lies in the circuit-level formulation, the explicit ancilla-efficient encoding, and the complexity analysis.

3. Octonionic formalism and obstructions: positioning with respect to prior work.
   Similarly, at the start of the “Octonionic formalism” subsection we now explicitly position our discussion with respect to existing work: we note that there is comparatively little quantum-information level work on octonions, and that our starting point is the general account in Baez and the octonion-based measurement-only scheme of Freedman–Shokrian-Zini–Wang. We then state that the goal of our octonionic analysis is not to propose yet another measurement-only model, but to examine from first principles which parts of the standard gate-based framework survive when one insists on using O and where fundamental obstructions arise.

4. Marking the novelty of the obstruction and mitigation analysis (M1)–(M4).
   In the section “Octonionic model: Formal obstructions and limits”, at the beginning of the subsection we now state explicitly that, in contrast to the existing measurement-only construction of Freedman et al., the results that follow — namely, the identification of structural obstructions (loss of associative tensor products, path-dependent dynamics, nonlocal measurement postulates) and the organisation of mitigation strategies into (M1)–(M4) — are, to the best of our knowledge, new. We also emphasise that this yields a systematic picture of when an octonionic description can be realised as an encoded or effective complex-qubit model, and where irreducible costs appear.

5. Complexity proposition: making the link to existing simulations explicit.
   In the proposition describing the complexity of the quaternionic model, we have added a sentence explicitly connecting our result to earlier work by Fernández–Schneeberger, Graydon, and Gantner. We explain that our proposition can be viewed as a circuit-level reformulation of their simulation results, now phrased in terms of explicit embeddings of general quaternionic channels and measurements and directly tied to the equivalence with BQP. This makes clear that the underlying idea of simulating quaternionic models by complex ones is not new, but that our contribution is to provide a unified treatment at the level of a full circuit model and its complexity class.

We hope that these changes address the Point 2 by (i) clearly separating background/review material from what we claim as original contributions, in both the quaternionic and octonionic parts, and (ii) making the role of the fundamental embedding and operator–state isomorphism explicit so that the logical structure of the arguments is transparent to both expert and non-expert readers.

 

Comments 3:
Point 3. Clearer motivation of the presented work. Finally, I am missing the motivation for the work. Of course it is possible to embed quaternionic and octonionic circuits. But why? The examples of why I would care about such a computing hardware are very vague. Since there is equivalence between the quaternionic and standard computational framework, there need to be some stronger arguments to do this than statements such as l640ff “This mathematical elegance has a profound physical implication. If a quantum hardware platform could be controlled with pulses that directly map to the four real components of a quaternion (for instance, using the amplitude and phase of two orthogonal control fields), then an arbitrary rotation could, in principle, be implemented in a single operational step. This contrasts sharply with the three sequential operations required by the Euler decomposition.” This (and the other generic examples given) is a bit of a false equivalence as it is not clear whether such an operation for the quaternions is even possible. Otherwise I can do the same argument for a generic unitary operation and conclude that all quantum computations could be done in a single compute step. Now that would be even better and have profound implications for complexity theory, but it is obviously a way over-simplified argument to make. Yet, for the special case of quaternionic/octonionic operations, this is precisely the argument that is put forward in sentences like l622ff “This section argues that this representational efficiency can translate into tangible advantages in circuit simplification and algorithmic design, particularly for quantum simulation and variational algorithms.” Additionally, this argument needs to also be contrasted with qudits. Are we simply looking at a special d=4 qudit representation with the additional assumption that single qudit operations can be performed in a single compute step? Thus, if this is the line of argumentation for pursuing this idea, then I think this needs to be justified more. If the motivation lies in a different direction, the motivation for the presented work should be made clearer.

Response 3:
We are grateful for this thoughtful critique of the motivation and have substantially revised the relevant parts of the manuscript to address it. In particular, we have removed the potentially misleading “single-step” hardware argument and rephrased the discussion so that our motivation is clearly structural and informational rather than a claim of generic speed-up or depth collapse. Concretely:

1. Removal of the “single operational step” claim and clarification of its status.
   We have deleted the sentence suggesting that, if one could directly control the four real components of a quaternion, then “an arbitrary rotation could, in principle, be implemented in a single operational step,” as well as the associated “profound physical implication” wording. In its place, we now emphasise that unit quaternions provide a compact, redundancy-free parametrisation of single-qubit rotations in SU(2), and we explicitly state that this should not be interpreted as evidence that arbitrary rotations are physically realisable as a single gate in realistic devices. The revised paragraph stresses that any such operation must still be implemented via whatever Hamiltonians and control channels the hardware provides, and remains constrained by bandwidth, noise, calibration, and similar issues. Our quaternionic description is thus framed as a representation tool (for compact parametrisations and symmetry-aware design), not as a claim of collapsing circuit depth or complexity.
   Location in the revised manuscript: Section on hardware and representational considerations (previously containing the “profound physical implication” sentence), where the quaternionic rotation discussion is now rewritten in these more cautious terms.

2. Reframing the “representational efficiency” paragraph.
   The sentence “This section argues that this representational efficiency can translate into tangible advantages in circuit simplification and algorithmic design, particularly for quantum simulation and variational algorithms” has been replaced by a more precise and modest statement. We now write that we explore how quaternionic and octonionic parametrisations can inform circuit design and algorithmic heuristics, for example by suggesting symmetry-adapted parametrisations and ansätze for quantum simulation and variational algorithms. Importantly, we explicitly state that we do not claim a generic asymptotic reduction in worst-case circuit complexity, and we emphasise that any practical advantage would be task- and architecture-dependent.
   Location in the revised manuscript: Same subsection in the “Emerging applications” / design discussion, immediately after the explanation of the representational role of hypercomplex parametrisations.

3. Explicit contrast with qudits and high-dimensional encodings.
   In line with your suggestion, we now explicitly compare our hypercomplex viewpoint with the established use of high-dimensional qudits. We point out that, much like qudit encodings trade local dimension against circuit width or depth, quaternionic and octonionic parametrisations can be viewed as particular structured encodings that may help in designing compact, symmetry-aware circuits, without changing the underlying BQP power of the model. We also cite recent work on qudits and high-dimensional quantum computing to make clear that this perspective is closely aligned with the broader literature on multilevel systems, and that we are not proposing hypercomplex hardware as something fundamentally beyond that paradigm.
   Location in the revised manuscript: The same paragraph that discusses representational efficiency and ansatz design now explicitly mentions the analogy with qudit-based encodings and includes a citation to recent reviews on qudits.

4. Clarifying the overarching motivation.
   Finally, we have revised the introductory and concluding discussion to make our motivation more explicit: our primary goal is to understand what is gained and what is lost when one insists on making quaternionic or octonionic structure manifest at the level of models of quantum computation. For the quaternionic case, the value lies in a unified, circuit-level formulation that is equivalent to BQP but highlights symmetry and parametrisation structures that may be useful in specific algorithmic or hardware contexts. For the octonionic case, the emphasis is on identifying structural obstructions and the cost of possible mitigation strategies, thereby clarifying the limitations of any putative octonionic computing architecture. We state explicitly that the work is not proposing a new route to asymptotic computational speed-ups, but rather aims to provide a clear and systematic picture of hypercomplex models within the familiar landscape of quantum information and complexity.

We hope that these changes make the motivation of the paper more transparent and avoid any suggestion that we are claiming generic “single-step” advantages. Instead, the revised text presents quaternionic and octonionic models as structurally rich, but computationally equivalent, viewpoints whose potential benefits are in representation, symmetry, and problem-structured design rather than in fundamental complexity-theoretic improvements.

   

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

          I have conflicting thoughts regarding this article. It is clear that the authors have put tremendous effort to communicate their vision. They are indeed experts in this particular area, as one can easily ascertain by the depth of their exposition and the numerous references included. I also believe that this work can serve as a starting point towards alternative formal frameworks for doing quantum mechanics in general.

          However, I am not really convinced about the relevance to current quantum computing. I am not a mathematician, so many subtle details are lost on me. I am a computer scientist that actively writes quantum programs, mainly within the Qiskit framework. Accordingly, my one fundamental question is where are the quantum circuits? I see no circuits from any currently available quantum computer. I believe the only reasonable and beyond any doubt way to convince anyone is to provide with a circuit that can be replicated and run. For example, I am stuck at contemplating the matrix in equation (8), lines 344-345. Is this really a unitary matrix? How can it be decomposed by existing quantum gates available today? I want to see the Qiskit (or whatever alternative the authors prefer) quantum circuit, so that I can verify it myself.

          Alternatively, the title may be rephrased so that it doesn’t contain quantum computation. I don’t have any other meaningful remarks because I feel they are less important, since the paper is competently organized and the command of the English language is quite good.

Author Response

Comments 1:
However, I am not really convinced about the relevance to current quantum computing. I am not a mathematician, so many subtle details are lost on me. I am a computer scientist that actively writes quantum programs, mainly within the Qiskit framework. Accordingly, my one fundamental question is where are the quantum circuits? I see no circuits from any currently available quantum computer. I believe the only reasonable and beyond any doubt way to convince anyone is to provide with a circuit that can be replicated and run. For example, I am stuck at contemplating the matrix in equation (8), lines 344-345. Is this really a unitary matrix? How can it be decomposed by existing quantum gates available today? I want to see the Qiskit (or whatever alternative the authors prefer) quantum circuit, so that I can verify it myself.

Response 1:
Thank you very much for this detailed and helpful comment. We agree that the manuscript should make the connection to actual quantum circuits as clear as possible, and that the quaternionic–to–complex embedding should be illustrated by a circuit that can in principle be implemented and verified on current hardware.

(i) Correction of the matrix in the text.
In the original version there was a sign typo in the embedded 4×4 matrix associated with the quaternionic Hadamard gate. This has now been corrected in Eq. (7) of the revised manuscript (previously Eq. (8)). The corrected matrix reads

which is consistent with the block embedding

used throughout the paper. Location in the revised manuscript: Section 2.8.1, Eq. (6).

(ii) Unitarity and gate decomposition (technical note).
To address your question “Is this really a unitary matrix? How can it be decomposed by existing quantum gates?”, we have carried out the full calculation showing that the corrected matrix H_C​ is unitary and that it can be decomposed into a product of standard two–qubit gates (Hadamard, CNOT, CZ). These algebraic steps would be rather long to include in the main text, so we provide them in a separate technical note titled unitary.pdf, uploaded together with the revised manuscript for the reviewers and editor. In that note we explicitly compute H†_C H_C=I_4​ and give one concrete decomposition of H_C​ into elementary gates.

To make the example directly reproducible in standard software, we now explain in the text that the same unitary can be implemented in Qiskit by the simple circuit

qc.h(1)        # I ⊗ H
qc.cx(1, 0)    # CNOT with control qubit 1, target qubit 0
qc.cz(1, 0)    # CZ with control qubit 1, target qubit 0

If the editor considers it useful for the broader readership, we are happy to incorporate this worked example (or a shortened version of it) as an appendix or as an extended example at the end of the subsection “Circuit Diagrams and Practical Implications” in a further revision.

(iii) Clarifying the presence and role of circuits.
In the main text we now emphasise more clearly that every quaternionic gate in our model induces an explicit complex unitary on an enlarged register, which can be compiled into standard gate sets on any gate-based quantum computer. We added a brief comment in the subsection “Circuit Diagrams and Practical Implications” explaining that the embedded quaternionic Hadamard H_C​ can be realised as a two–qubit circuit composed of single–qubit and controlled gates, and we refer to the accompanying technical note for the full derivation and Qiskit example.
Location in the revised manuscript: Section 3.X, paragraph following Eq. (7).

Comments 2:
Alternatively, the title may be rephrased so that it doesn’t contain quantum computation. I don’t have any other meaningful remarks because I feel they are less important, since the paper is competently organized and the command of the English language is quite good.

Response 2:
Thank you for this suggestion and for your positive overall assessment of the manuscript. We carefully considered the possibility of changing the title, but we believe that including “quantum computation” remains appropriate, for the following reasons.

First, the central results of the paper are explicitly computational: we introduce quaternionic and octonionic circuit models and prove that they are polynomially equivalent to the standard complex quantum circuit model (i.e., they compute exactly the class BQP up to polynomial overhead). This is now stressed more clearly in the revised introduction and in the concluding discussion of our complexity-theoretic results.
Location in the revised manuscript: Section 1 (Introduction, last paragraphs) and Section 4 (Summary and Outlook), where we explicitly state that our focus is on models of quantum computation rather than on reformulating quantum mechanics.

Second, in response to your earlier comment, we have made the circuit aspects more explicit by:

• emphasising that every quaternionic circuit on nnn quaterbits induces a concrete complex circuit on n+1n{+}1n+1 qubits, and

• providing a fully worked two–qubit example (in an accompanying technical note) showing how the embedded quaternionic Hadamard gate corresponds to a unitary that can be decomposed into standard gates and implemented in Qiskit.
  Reference to the technical note unitary.pdf.

Taken together, these clarifications underline that the manuscript is fundamentally about quantum computational models and their circuit embeddings, rather than about abstract algebra alone. For this reason, we have retained the current title including “quantum computation.” Of course, we are happy to adopt a different wording if the editor ultimately prefers a more conservative title.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors did a good job of responding to most of the comments, which improved the paper.

Reviewer 3 Report

Comments and Suggestions for Authors

I thank the authors for the much improved manuscript. All my points have been addressed and I am more than happy to recommend the manuscript for publication.

Reviewer 4 Report

Comments and Suggestions for Authors

Having read the latest pdf version with track changes, I can confirm that the authors have addressed in a convincing manner, the issues I raised during the first revision of their paper. As a result, I believe that the paper has been improved further and can be published in its present form.