--- title: New Method for Representation of Multi-qubit Systems Using Fractals authors: - Mate Galambos - Sandor Imre abstract: Visual representation is essential to share ideas, interpret previous achievements or formulate new algorithms quickly and intuitively, however most representations of multy-qubit systems either conceal the properties of individual qubits or fail to visualize entanglement. This study discusses a representation that overcomes these problems through the methodology of fractals. The proposed method visualizes individual qubits as constants of the fractal that corresponds to the whole system. The statistical self similarity allows the total number of qubits to be flexible, making it easy to study subsystems. Generalization of this method through labeled signed binary trees is also presented which makes it possible to create other representations with similar properties. summaryType: survey sourceStatus: null sources: - https://www.thinkmind.org/library/ICQNM/ICQNM_2011/icqnm_2011_3_30_80175.html --- # New Method for Representation of Multi-qubit Systems Using Fractals [Read the original paper](https://www.thinkmind.org/library/ICQNM/ICQNM_2011/icqnm_2011_3_30_80175.html). ## Background and motivation Galambos and Imre propose a recursive two-dimensional representation of pure multi-qubit states, presented at the Fifth International Conference on Quantum, Nano and Micro Technologies in 2011. Their starting point is an existing visualization problem: quantum-state notation captures amplitudes and correlations precisely, but it can be difficult to use when explaining an algorithm, reasoning about part of a system, or communicating a new idea. They cite quantum factorization, cryptography, and communication as application areas motivating more accessible representations, rather than evaluating those applications with their method. The paper contrasts its goals with three approaches to state visualization. A Bloch sphere represents a single qubit, and separate spheres can describe factors of a separable pure state, but those independent views do not capture entanglement between qubits. A sufficiently high-dimensional object can represent the entire state while obscuring its internal organization and the consequences of measuring selected qubits. Generalizations through Hopf fibrations provide a mathematical route beyond the single-qubit sphere, but the authors consider the resulting geometry too complicated for compact visual interpretation. These are the authors' motivating assessments, not findings from a comparative experiment. The desired representation should preserve the structure of a multi-qubit state in a compact two-dimensional image, expose individual qubits and subsystems, accommodate any finite qubit count, and include entangled states. The new contribution is a visual construction that recursively nests probability-and-phase bars, together with a binary-tree framework for designing related constructions. The paper uses fractal self-similarity as an organizing principle: adding a qubit extends a finite hierarchy by another iteration, with the qubit descriptions supplying its building blocks. ## Single-qubit probability and phase encoding For a pure single-qubit state, the paper writes $$ |\varphi\rangle = A e^{i\alpha}|0\rangle + B e^{i\beta}|1\rangle, \qquad A^2+B^2=1. $$ A horizontal bar of total width one is divided into a black segment of width $A^2$ and a white segment of width $B^2$. Black denotes outcome $0$, white denotes outcome $1$, and the black segment always comes first so that the order is unambiguous. The widths encode computational-basis measurement probabilities rather than the unsquared amplitude magnitudes. A gray frame keeps the white region visible against the background and provides boundaries used by the recursive construction. Figure 1 introduces this probability-only bar. Each segment then receives a horizontal line in the opposite color, whose vertical position encodes the phase of that segment's amplitude. The bottom represents zero and the top represents $2\pi$, with phase understood periodically. Figure 2 shows this mapping explicitly, and Figure 3 gives examples for $|0\rangle$, $|1\rangle$, $(|0\rangle-|1\rangle)/\sqrt{2}$, and $(|0\rangle+i|1\rangle)/\sqrt{2}$. The equal-width superpositions have the same measurement probabilities, while their phase-line heights distinguish their different complex amplitudes. Together, width, binary color, and line height supply the information needed for the state description, with physically irrelevant global phase allowed to be omitted. ## Recursive construction and reading a multi-qubit state The construction first orders the qubits according to the sequence in which they will be measured in the intended protocol. For a separable state, the bar for the first qubit spans the full width, and a scaled copy of the second qubit's bar is placed beneath each segment of the first. The same process continues for subsequent qubits. The authors formalize this as a Lindenmayer-style rewriting process: begin with a horizontal gray line of width one, then replace each bottom boundary segment by the next qubit's bar scaled to that boundary's width. Each additional iteration introduces another qubit level and up to twice as many terminal segments. A terminal column identifies a computational-basis string through its black and white regions read from top to bottom. The columns occur in ascending binary order from left to right, and the width of a terminal segment gives the joint probability of its basis string. For separable states, this width is the product of the corresponding single-qubit probabilities along the branch. Thus probability information is already accumulated through the geometry, even when phase information is kept separately at each qubit level. Zero-probability strings have zero width and therefore disappear from the drawing, as the four-qubit example in Figure 6 illustrates for the outcome $1110$. Figure 4 above shows the building blocks for $(|0\rangle+|1\rangle)/\sqrt{2}$, $(|0\rangle-|1\rangle)/\sqrt{2}$, and $-i\sqrt{4/5}|0\rangle+\sqrt{1/5}|1\rangle$. Arrows connect each bar to its repeated, scaled placements in the combined diagram. The first two qubits divide each parent width equally, whereas the third divides it in a four-to-one ratio; the phase lines retain the minus sign of the second qubit and the $-i$ factor of the third. Figure 5 expands the same construction into successive iterations and identifies the eight final bit strings. These figures are explanatory state diagrams, not screenshots of an interactive application. The authors emphasize that the construction is deterministic: the visual variation comes from the different qubit building blocks. ## Inherited phases, subsystem structure, and entanglement The paper separates the choice of how to display phase from the probability widths, which are always inherited through the hierarchy. In the non-inherited form for separable qubits, phase lines remain at the individual qubit levels, matching a tensor-product description. In the inherited or collapsed form, the phase associated with each entire basis-state amplitude is placed at the lowest segment of its column, matching a sum over basis states, $$ |\psi\rangle = \sum_{x\in\{0,1\}^n} A_x e^{i\alpha_x}|x\rangle. $$ Here the terminal width is $A_x^2$ and its phase-line height represents $\alpha_x$. For a product state, collapsing the phases amounts to adding the phases along each branch modulo $2\pi$. For a general entangled state, the joint amplitudes supply the widths and phases directly; the state cannot in general be constructed by copying independent single-qubit factors. Figure 6 demonstrates the full-state encoding, including missing zero-probability outcomes. Partially inherited representations retain some separate factors while treating an entangled subsystem as a joint building block. Figure 7 gives a concrete example: $$ |\psi\rangle =\frac{|000\rangle-|011\rangle-|100\rangle+|111\rangle}{2} =\frac{|0\rangle-|1\rangle}{\sqrt{2}} \otimes\frac{|00\rangle-|11\rangle}{\sqrt{2}}. $$ The fully inherited drawing puts all four nonzero amplitude phases at the bottom level. The partially inherited drawing places the first qubit's phase separately and repeats the joint representation of the final two qubits beneath it. The first qubit is separable from the pair, whereas the last two qubits remain entangled and cannot be replaced by independent single-qubit bars. This example demonstrates how the chosen factorization can become visible in the hierarchy. It does not establish an automated entanglement detector or an entanglement measure, and using collapsed phases is not itself evidence of entanglement because product states can be drawn in that form too. The paper has two apparent terminology slips: the conversion paragraph on printed page 54 calls phase accumulation “non-inherited,” and the Figure 7 caption later says that the “inherited” representation requires separability. The construction, adjacent explanations, and Figure 7's factored state consistently distinguish local non-inherited phases for separable factors from inherited phases for joint states. That distinction is the one used here. ## Generalization through signed, labeled binary trees The second contribution abstracts the bar construction into a signed, labeled binary tree. Figure 8 draws the tree explicitly: nodes carry complex-number labels and outgoing branches carry the bit values $0$ and $1$. To design another visual representation, the authors propose choosing a building block that encodes a complex number, preferably with an additional visual distinction for the binary value, and a rule connecting two new blocks beneath each previous one. In the bar design, the two colored segments supply the complex-number blocks, color supplies the bit distinction, and the nested layout supplies the connections. The authors mention Bloch spheres as a possible alternative building block but favor two-dimensional, rectangular forms for compact packing. Labels can remain local to individual separable qubits or accumulate ancestor amplitudes along a branch, concentrating full-state information in the leaf labels. Mixed inheritance supports the intermediate subsystem views demonstrated earlier. The bar construction is a particular case in which probability widths are always inherited while phase can remain local, become fully inherited, or use a mixture. The root may carry a complex number of unit magnitude to record global phase, but the authors omit this optional label from the bar representation because global phase is unnecessary for the physical state description. This framework offers a recipe for related encodings rather than an implementation or comparison of multiple alternative designs. ## Evidence, contributions, and limitations The paper's evidence consists of the construction rules, their correspondence to amplitude notation, and worked diagrams. Figures 1 through 3 explain the single-qubit encoding, Figures 4 and 5 explain recursive composition, Figures 6 and 7 show full-state and subsystem phase arrangements, and Figure 8 supplies the structural generalization. There is no reported user study, task-performance comparison, quantitative readability evaluation, runtime benchmark, or interactive software implementation. The figures support the claim that the representation can express the illustrated states, but they do not establish that readers interpret it more accurately or quickly than earlier methods. The main contributions are the combined width/color/phase encoding, a recursive grammar accommodating different qubit counts, selectable levels of phase inheritance that expose a given factorization, and the binary-tree formulation for constructing other representations. The scope is pure-state amplitudes. The paper does not develop mixed-state density-matrix representations, an interaction design, or procedures for animating gates and measurement collapse. Those capabilities should not be inferred from its intended relevance to quantum algorithms and protocols. The formal ability to add arbitrarily many finite levels does not demonstrate perceptual scalability. From the construction, an $n$-qubit state may require $2^n$ nonzero terminal segments, so packing it into fixed width produces increasingly narrow probability regions and phase marks. This is a consequence of the encoding, not a measured limit reported by the authors. The selected qubit order also determines the hierarchy and which subsystem grouping is directly exposed. The paper concludes by emphasizing the recursive connection between fractals and labeled binary trees; it does not specify a separate experimental future-work program. Its proposed route beyond the presented design is the construction of other representations using the same building-block and tree principles.