Multi-Axial Projective Sphere (MAPS) Overview

• Multi-Axial Projective Sphere (MAPS) is a three-dimensional visualization framework that represents d-valued quantum state-spaces using intersecting axes and explicit phase labels.
• It generalizes the Bloch sphere by mapping each qudit basis level to a unique spatial axis, thereby making relative and global phase information visually explicit.
• The method facilitates intuitive geometric interpretation of complex quantum states, with applications in quantum computing, machine learning, and high-dimensional data visualization.

Searching arXiv for the specified MAPS paper and closely related context. arxiv_search query: (Al-Bayaty, 14 Jun 2026) The Multi-Axial Projective Sphere (MAPS) is a three-dimensional $S^2$-based visualization framework proposed for representing $d$-valued quantum state-spaces of qudits with $d \ge 3$. It is introduced as a generalization of the qubit Bloch sphere that seeks to retain geometric simplicity while making the richer phase structure of higher-dimensional systems explicit. In MAPS, each basis level is assigned its own intersecting spatial axis, and phase values $\pm \omega_d^k$ are displayed on those axes, with the top and bottom hemispheres distinguishing positive and negative phase families. The framework is presented together with a family of diagonal phase-axial operators, denoted $PASS_d$, intended to provide a geometric interpretation of phase swiveling and shifting operations (Al-Bayaty, 14 Jun 2026).

1. Motivation and conceptual setting

The starting point for MAPS is the contrast between the geometric tractability of qubits and the structural complexity of higher-dimensional qudits. For a qubit, the paper emphasizes the standard parametrization

$$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$

and the density-matrix representation

$$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$

with $r=(x,y,z)$, $\|r\|\le 1$, pure states on the boundary $\|r\|=1$, and the maximally mixed state at $d$0. This gives the Bloch sphere its familiar role as a compact visualization of pure and mixed qubit states (Al-Bayaty, 14 Jun 2026).

The paper argues that this geometric picture does not scale naturally to qudits. A qudit state-space has operational dimension $d$1, so a qutrit already occupies an 8-dimensional manifold embedded in $d$2. The physically allowed set becomes the generalized Bloch body $d$3, described as a complicated convex body with nested boundary layers rather than a simple sphere. The paper also states that global and relative phases are not directly visible on the Bloch sphere’s axes, and that for $d$4, visualization commonly requires generalized Bloch vectors, Hopf fibrations, Majorana constellations, phase-space methods, or other high-dimensional constructions (Al-Bayaty, 14 Jun 2026).

Within that context, MAPS is proposed as a way to preserve a single three-dimensional sphere while encoding the additional structure of $d$5-level systems through multiple intersecting axes and explicit phase labels. A plausible implication is that the framework is intended less as a full replacement for high-dimensional state-space formalisms than as a visualization convention that privileges geometric readability.

2. Formal definition of the projective sphere

The paper defines MAPS in “Definition 1” as a generalized $d$6-valued state-space visualization $d$7 framework for $d$8, with $d$9. Its core construction has two parts. First, it is multi-axial: the sphere contains $d \ge 3$0 projectional intersecting spatial axes. Second, it is a projective sphere: these axes are used to project the amplitudes and phases of a $d \ge 3$1-level quantum state into a three-dimensional geometric picture, with the hemispheres separating positive and negative phase families (Al-Bayaty, 14 Jun 2026).

The main components of the definition are given as follows. The $d \ge 3$2 framework consists of $d \ge 3$3 projectional intersecting spatial axes, each axis is mapped to one $d \ge 3$4-valued state-space and labeled the $d \ge 3$5-axis, and each axis has distinct phase values $d \ge 3$6. All axes intersect at the origin and are separated by a “pseudo-orthogonal angle”

$d \ge 3$7

They are also uniformly displaced from the azimuth pole by

$d \ge 3$8

although the source text notes that this notation is typographically malformed. The phase structure is built from the $d \ge 3$9-th roots of unity,

$\pm \omega_d^k$0

The paper further introduces an “orthogonality completeness” condition,

$\pm \omega_d^k$1

which it presents as expressing the idea that the axes must collectively occupy the full sphere boundary without overlap if the visualization is to be complete and distinguishable (Al-Bayaty, 14 Jun 2026).

A central organizational feature is the hemispherical split. The top hemisphere contains all $\pm \omega_d^k$2, the bottom hemisphere contains all $\pm \omega_d^k$3, and the $\pm \omega_d^k$4-axis is treated as special because it always indicates the global phase of the overall qudit state.

3. State representation and geometric mapping

MAPS is built on the standard qudit expansion

$\pm \omega_d^k$5

with $\pm \omega_d^k$6 and normalization

$\pm \omega_d^k$7

The paper also recalls the usual projective equivalence of global phase,

$\pm \omega_d^k$8

and the associated projective state space

$\pm \omega_d^k$9

It additionally references generalized Bloch-vector and Gell-Mann expansions for qudits and qutrits, while noting typographical omissions in the printed expressions reproduced in the source summary (Al-Bayaty, 14 Jun 2026).

The geometric mapping assigns each basis level $PASS_d$0 to a separate axis. The amplitudes and phases of a pure or mixed qudit state are then placed on these axes as points or circles in the top or bottom hemisphere according to whether they belong to the $PASS_d$1 or $PASS_d$2 sign family. When all relative phases are present, the selected points form a polygonal structure: a triangle for qutrits, a quadrilateral for ququadits, a pentagon for quintits, and so forth. The paper explicitly states that the $PASS_d$3-axis always carries the global phase, so the global phase becomes visually readable rather than being removed abstractly through projectivization (Al-Bayaty, 14 Jun 2026).

To describe phase periodicity, the paper gives the generalized relation

$PASS_d$4

where $PASS_d$5 and $PASS_d$6. For $PASS_d$7, it states the specialized form

$PASS_d$8

This construction is presented as preserving a single $PASS_d$9 geometry while increasing representational density through additional axes and additional phase points. This suggests that MAPS prioritizes visible phase organization over exact faithful embedding of the full $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$0 parameter geometry.

4. $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$1 and phase-axial operations

The paper accompanies MAPS with a family of diagonal unitary operators called the $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$2-valued phase-axial swiveling and shifting gate $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$3. They are defined by

$$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$4

with

$$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$5

Their intended interpretation is geometric. A swivel moves a phase from one hemisphere to the other, i.e. between $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$6 and $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$7, while a shift changes the phase within the same hemisphere by moving to another $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$8-th root of unity on the same axis (Al-Bayaty, 14 Jun 2026).

The paper enumerates several cases. If all $$|\psi\rangle=\cos \left( \frac{\theta}{2} \right) |0\rangle + e^{i \phi} \sin \left( \frac{\theta}{2} \right) |1\rangle$$9, then $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$0. If $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$1 while the others are $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$2, the global phase is swiveled. If $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$3 while the others are $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$4, the global phase is shifted. If some $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$5, relative phases are swiveled. If some $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$6, relative phases are shifted. If all $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$7, then all phases are both swiveled and shifted (Al-Bayaty, 14 Jun 2026).

The operators are stated to be unitary. They are Hermitian only if all diagonal entries are $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$8; otherwise the adjoint $$\rho = \frac{1}{2}\left(I_2 + r \cdot \sigma\right),$$9 is obtained by inverting the phases. For $r=(x,y,z)$0, this is expressed as $r=(x,y,z)$1; for $r=(x,y,z)$2, by reversing the phase periodicity of the $r=(x,y,z)$3's. The paper relates $r=(x,y,z)$4 to the standard generalized phase gate

$r=(x,y,z)$5

and states that $r=(x,y,z)$6 generalizes $r=(x,y,z)$7 because each diagonal phase entry can be customized independently. It also claims that the number of permutative $r=(x,y,z)$8 gates grows as

$r=(x,y,z)$9

with explicit examples of 215 permutative $\|r\|\le 1$0 and $\|r\|\le 1$1 gates excluding $\|r\|\le 1$2, and 255 permutative $\|r\|\le 1$3 and $\|r\|\le 1$4 gates excluding $\|r\|\le 1$5 (Al-Bayaty, 14 Jun 2026).

5. Representative realizations for $\|r\|\le 1$6

The paper develops concrete MAPS instances for qutrits, ququadits, and quintits. In each case, the increase in $\|r\|\le 1$7 is reflected by an increase in the number of axes, a decrease in the angular spacing $\|r\|\le 1$8, and a corresponding increase in polygonal complexity while retaining a single 3D sphere geometry (Al-Bayaty, 14 Jun 2026).

System	Axes and phase set	Example geometry or state
Qutrit, $\|r\|\le 1$9	Axes $\|r\|=1$0; $\|r\|=1$1; $\|r\|=1$2	Mixed state after a Chrestenson transform is drawn as a triangle
Ququadit, $\|r\|=1$3	Four axes; $\|r\|=1$4; $\|r\|=1$5	Mixed states are shown as a quadrilateral polygon
Quintit, $\|r\|=1$6	Five axes; $\|r\|=1$7; $\|r\|=1$8	Mixed-state geometry becomes a pentagon

For the qutrit case, the paper gives the states

$\|r\|=1$9

and

$d$00

It also uses qutrit examples to display global phases, including $d$01, $d$02, $d$03, and $d$04 (Al-Bayaty, 14 Jun 2026).

For the ququadit case, the example state is

$d$05

For the quintit case, the paper gives

$d$06

These examples support the paper’s visual thesis that higher $d$07 does not require a higher-dimensional display space; instead, the same $d$08 picture is made denser by additional axes and more elaborate phase polygons.

6. Claimed properties, limitations, and proposed applications

The paper repeatedly attributes several advantages to MAPS: ease of illustration, structural simplicity, natural representation of $d$09-valued states, explicit visualization of global and relative phases, direct geometric interpretation of phase operations, and preservation of quantum properties, distinguishability, entanglement structure, and unitary dynamics. It also presents MAPS as avoiding the need for complicated high-dimensional geometry, nested Hopf fibrations, generalized Bloch-body inequalities, and full matrix-based descriptions when reasoning visually about quantum operators (Al-Bayaty, 14 Jun 2026).

At the same time, the paper’s own discussion places practical limits on these claims. It does not provide a formal proof that MAPS is a complete mathematical replacement for standard qudit geometry. Its axes, “pseudo-orthogonal angles,” and “orthogonality completeness” are introduced as visualization conventions rather than established quantum-geometric structures. The discussion also notes that a two-dimensional phase plane is suitable only for $d$10 and becomes inconsistent for $d$11, with MAPS proposed specifically as a response to that inconsistency. This suggests that MAPS is best understood as a geometric visualization framework rather than as a substitute for the full $d$12-parameter formalism.

Beyond state visualization, the paper explicitly proposes MAPS for machine learning, quantum machine learning, quantum chemistry, and high-dimensional data visualization more generally. In that broader framing, each MAPS axis may represent a feature or dimension of data with corresponding distinct values, numerical or textual. It also suggests future use in quantum arithmetic circuits, comparators, counters, and quantum circuit simulators, specifically mentioning QuDiet, GCAMPS, and the author’s own ternary and hybrid circuit simulators (Al-Bayaty, 14 Jun 2026).

The resulting picture is that MAPS is a projective-sphere formalism intended to preserve the visual economy of the Bloch sphere while making higher-dimensional phase structure directly inspectable. Its distinctive feature is not a rederivation of qudit state-space topology, but the assignment of one intersecting axis per basis level together with an explicit top-versus-bottom hemisphere encoding of $d$13.