Fractal Representation

New Method for Representation of Multi-qubit Systems Using Fractals

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.

Visualizing the Effects of Measurements and Logic Gates On Multi-Qubit Systems Using Fractal Representation

Background and motivation

Galambos and Imre investigate how a two-dimensional representation of a quantum state can expose both its full measurement distribution and the roles of individual qubits. The underlying problem is established: a Bloch sphere gives a useful picture of one qubit, but separate single-qubit pictures do not describe the correlations and phases of a general entangled state. The authors also discuss higher-dimensional geometric approaches, including generalizations involving Hopf fibrations, and argue that their mathematical expressiveness does not automatically make them easy to interpret visually. Their particular concern is that a picture of the whole system should still support reasoning about measurements of only some qubits, changes in subsystem structure, and the action of quantum gates.

This paper develops the fractal representation introduced in the authors' 2011 work, rather than originating the entire approach here. Its emphasis is on rules for reading conditional measurement probabilities, relating the drawing to density matrices, representing mixtures, changing qubit order, and depicting single-qubit and controlled operations. The goal is a visual notation for explaining and manipulating known quantum states. The paper does not present an interactive application, a procedure for experimentally estimating an unknown state, or an empirical assessment of learning and usability.

Encoding amplitudes, phases, and tensor products

For a single qubit,

$$|\psi\rangle=Ae^{i\alpha}|0\rangle+Be^{i\beta}|1\rangle,\qquad A^2+B^2=1,$$

the representation is a horizontal unit-width bar with a black segment of width $A^2$ followed by a white segment of width $B^2$. Black denotes the computational-basis value $0$, and white denotes $1$. A horizontal line within each segment encodes its amplitude's phase through vertical position, with phase interpreted modulo $2\pi$. A zero-phase line coincides with the frame and is therefore not separately visible. Thus segment width encodes a probability, not the amplitude magnitude itself, while the line supplies the phase information that a probability-only display would omit.

Figure 1 of the paper illustrates the single-qubit encoding from which the multi-qubit construction is built.

For an expanded $n$-qubit state $|\psi\rangle=\sum_x c_x|x\rangle$, each computational-basis string $x$ becomes a column containing one black or white bar for each qubit. The column width is $|c_x|^2$, its top-to-bottom colors spell the bit string, and its phase line is placed in the bottom bar. Columns are arranged in ascending binary order, and adjacent compatible bars are merged. Figure 2 shows both the separate columns and the resulting compact drawing. The stacked rows retain a visible correspondence with individual qubits, while the widths and phase lines collectively encode the state-vector coefficients.

For a separated tensor product, the construction instead places scaled copies of each factor's representation beneath the segments of the preceding factor. Figure 3 illustrates this recursive construction with three single-qubit factors. The authors call the resulting patterns fractal because the bar structures repeat at different scales. This recursion also applies when the factors are groups of internally entangled qubits, in which case each repeated object is a multi-qubit drawing. The separated and expanded forms differ in where phase information is stored: moving the component phase lines to the bottom row and adding their heights modulo $2\pi$ produces the combined phases of the expanded state. Conversely, recovering repeated factors requires consistent widths and phases; failure to factor the drawing into such copies indicates inseparability across the proposed division.

The paper also proposes a generalization to subsystems with more than two discrete states. Additional colors identify their basis states, column widths still encode joint probabilities, and complementary-colored phase lines preserve contrast. This is a conceptual extension described with an example of three five-level particles, rather than an evaluated color design or a demonstrated software feature.

Measurement, conditional probability, and qubit order

The central measurement operation is selection and renormalization of the appropriate part of the drawing. When the top qubit is measured in the computational basis, outcome $0$ retains the columns below its black segment, while outcome $1$ retains those below its white segment. The retained width becomes the new unit against which subsequent probabilities are read. For a projector $M$ and an outcome of nonzero probability, this corresponds to

$$|\psi'\rangle=\frac{M|\psi\rangle}{\sqrt{\langle\psi|M|\psi\rangle}}.$$

Figure 4 follows an outcome sequence beginning with $0$ and then $1$. A column representing the remaining outcome is compared with successively smaller reference widths, giving the joint probability and then conditional probabilities such as $P(10\mid 0)$ and $P(0\mid 01)$. The figure therefore explains a probabilistic calculation through width ratios; its successive drawings are schematic states conditioned on specified outcomes, not recorded measurements from hardware. The derivation concerns measurements in the basis encoded by the colors. It does not establish a direct selection rule for every possible measurement basis.

Qubits should be ordered from top to bottom according to the intended measurement sequence. Changing that order requires more than exchanging row labels: the relevant rows are exchanged, the drawing is split into its constituent columns, phase information is retained in the bottom bars, columns are sorted by their new bit strings, and compatible neighbors are merged again. Figure 6 explicitly depicts these stages. This procedure also supports a test of interchangeability: if swapping two qubits leaves the full drawing unchanged, the state is invariant under that exchange. For separated factors, Figure 5 identifies identical qubit states through scaled copies of the same single-qubit bar, including phase information.

Density matrices and mixed states

The representation connects to a pure-state density matrix through the same amplitudes used to construct its columns. If $c_x=\sqrt{p_x}e^{i\gamma_x}$, then

$$\rho_{xy}=c_xc_y^*=\sqrt{p_xp_y}\,e^{i(\gamma_x-\gamma_y)}.$$

Diagonal elements are therefore the column widths, while off-diagonal elements require a geometric mean of widths and a difference of phase-line heights. Figure 7 supplies a geometric construction using the parabola $y=x^2$ to obtain square roots and products, then shows the corresponding complex vectors. This connection explains why the width alone cannot provide the quantities needed for interference or density-matrix coherences. The authors also discuss constructing a pure-state drawing from a density matrix, while noting that the drawing's correspondence with a state vector is not a one-to-one correspondence with a density matrix. In particular, the density matrix does not retain the vector's overall phase.

A mixed state is drawn from an explicitly supplied ensemble,

$$\rho=\sum_k w_k|\psi_k\rangle\langle\psi_k|,\qquad \sum_k w_k=1.$$

Each pure-state drawing is scaled horizontally by $w_k$ and placed beside the others. Extended gray separators and triangular markers identify boundaries between ensemble components so that they are not mistaken for ordinary neighboring columns of a single pure state. Figure 8 illustrates a mixture of $|00\rangle$ and $|11\rangle$. A joint outcome probability is obtained by adding the widths of every matching column across the components. Figure 9 extends the selection-and-renormalization argument to sequential measurements, retaining matching regions across the ensemble and using their combined width as the reference for conditional probabilities. Density matrices are recovered by calculating each component's density matrix and adding them with the ensemble weights.

The paper further suggests a graphical route to reduced subsystem states: reorder the desired subsystem below the other qubits and treat its pieces under different upper-bit configurations as ensemble components. Figure 10 illustrates the proposed construction for two subsystems. The authors explicitly state that they lack a general proof of this procedure. It should therefore be understood as a proposed connection with reduced density matrices, rather than a fully established replacement for computing partial traces.

Visual rules for quantum gates

The gate examples describe transformations of this notation, with color changes, phase-line movement, regrouping, and column reordering reflecting the underlying linear operation. Figure 11 shows Pauli $X$: invert the colors in the affected row, then restore ascending column order and merge compatible bars. Figure 12 adds the phase shifts required for Pauli $Y$ along with its bit flip. For Pauli $Z$, the colors and probabilities remain unchanged, while phases of columns whose target bit is $1$ shift by $\pi$. Figure 13 depicts this $Z$ transformation through half-height shifts of the relevant bottom-row phase lines, despite its caption mistakenly naming Pauli $Y$. Changes in phase can require splitting or merging bars even when the binary ordering stays unchanged.

The Hadamard example makes the need for complex amplitudes especially clear. The affected qubit is first moved to the bottom row. Columns are grouped by the values of all the other qubits, so that the two amplitudes in each group differ only in the target bit. The Hadamard acts on each pair by its normalized sum and difference, and the updated amplitudes determine the new widths and phases. Figure 14 shows the group boundaries and the altered bottom-row bars. These groups behave algebraically like two-component vectors; they do not imply that an entangled target qubit has become an independent pure qubit.

The paper generalizes this grouping rule to any single-qubit matrix $U$ using the block structure of $I\otimes\cdots\otimes I\otimes U$. A gate on another row can be handled by reordering the qubits before and after the operation. Controlled gates apply the corresponding transformation only within columns where the control bit is $1$, identified by a white bar in the control row. Figure 15 illustrates CNOT by flipping the target colors only in those columns, followed by reordering. This conditional change provides a way to discuss how CNOT can create entanglement, without treating every CNOT application as necessarily entangling. For a general controlled single-qubit gate, the same restriction is applied to the bottom-row amplitude groups.

Contributions, evidence, and limitations

The main contribution is a set of connected graphical rules that extends the earlier fractal state notation to measurement, mixed states, subsystem rearrangement, and gate action. The representation ties outcome probabilities to widths, basis values to colors, and phases to line heights, while its recursive structure connects tensor products with repeated graphical components. Its mathematical explanations and fifteen schematic figures demonstrate how to carry out these operations on illustrative states. The paper contains no controlled user study, quantitative comparison with other visualizations, runtime benchmark, or reported implementation of an interactive interface. Claims that the notation improves intuition or communication are motivations and intended benefits, rather than measured outcomes.

The most explicit unresolved issue is the proposed construction of subsystem density matrices without directly calculating partial traces, for which the authors report no general proof. The mixed-state discussion is framed around ensembles with a small number of distinct pure components, and its drawing depends on the chosen ensemble decomposition. Although the construction can describe finite multi-qubit systems, that expressiveness does not establish visual scalability: a generic expanded state still contains up to $2^n$ basis columns, and small probabilities produce correspondingly narrow features. This is a consequence of the encoding, not a scalability experiment reported by the authors. The paper does not present a separate future-work agenda; establishing the proposed subsystem rule and empirically evaluating readability remain open issues evident from the work's scope.