VENUS

VENUS: A Geometrical Representation for Quantum State Visualization

Background and motivation

VENUS is a design study of how to make the complex amplitudes, measurement probabilities, and normalization of small quantum states visible in one coordinated two-dimensional representation. Its intended users are quantum computing researchers and developers inspecting simulated states while developing algorithms or quantum machine learning models. The underlying visualization problem is established: circuit diagrams describe operations, but users also need to understand the states produced between those operations. A single-qubit state has complex amplitudes, and a joint two-qubit state requires four amplitudes whose relationships cannot be recovered from independent views of the two qubits alone.

The paper positions VENUS against the conventional Bloch sphere, extensions of sphere representations, IBM's Q-sphere, fractal representations, triangle-based state representations, and graphical circuit tools such as Quirk and ShorVis. Its criticism of the conventional Bloch sphere concerns its single-qubit scope, the difficulty of reading angles in three dimensions, and the indirect relationship between those angles and computational-basis measurement probabilities. The abstract makes a broader claim about inability to visualize superposition, but the detailed design motivation is more specific: the authors want probabilities and amplitude components to be directly visible and geometrically linked. The study does not establish that previous visualizations cannot represent superposition, nor that all two-dimensional representations outperform all three-dimensional ones. VENUS contributes a different encoding for this existing problem, concentrating on one- and two-qubit states rather than a general solution for arbitrarily large systems.

Design process and requirements

The authors worked with five quantum computing experts over more than five months. Three experts participated in preliminary interviews about their existing practices and problems with Bloch-sphere views, while two other experts tried an online prototype in their routine work over the following four months. Meetings approximately every two weeks informed revisions of the visual design. This formative group was separate from the fourteen experts involved in the subsequent evaluation.

The resulting requirements combine functionality and usability. The representation should support both single-qubit and joint two-qubit states, show computational-basis probabilities without requiring users to calculate them from amplitudes, and expose the real and imaginary components of those amplitudes. It should also connect these quantities through their mathematical relationships, use two-dimensional shapes, and be accessible through a web interface. The central design decision is therefore to make amplitude and probability encodings share geometry, rather than place unrelated numerical and graphical displays beside one another.

Geometric encoding of a single-qubit state

For a state

$$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\qquad \alpha=a+bi,\quad \beta=c+di,$$

the real and imaginary components are real numbers satisfying

$$a^2+b^2+c^2+d^2=1.$$

VENUS constructs a colored right triangle for each amplitude. The two perpendicular sides have lengths equal to the absolute values of its real and imaginary components. Black segments encode real components, gray segments encode imaginary components, and a doubled segment marks a negative value. The hypotenuse consequently has length equal to the complex amplitude's magnitude. Cyan and red distinguish the basis states $|0\rangle$ and $|1\rangle$. If an imaginary component is zero, its triangle degenerates and the real-component segment lies along the corresponding semicircle's diameter.

A semicircle is drawn with each amplitude triangle's hypotenuse as its diameter. For the amplitude $\alpha$, its area is

$$S_\alpha=\frac{\pi}{8}(a^2+b^2)=\frac{\pi}{8}|\alpha|^2.$$

Thus semicircle area is proportional to the probability of measuring the associated basis state, rather than to amplitude magnitude itself. Changes in the real and imaginary components reshape the triangle, while changes in their squared sum change the semicircle area. This distinction allows a phase-related change to be visible even when the measurement probability stays fixed. A white auxiliary right triangle joins the two amplitude magnitudes as its perpendicular sides. Its hypotenuse forms the unit-length base of the complete construction because $|\alpha|^2+|\beta|^2=1$. The normalization constraint therefore controls the geometry connecting the two colored regions.

Figure 1 of the paper is a schematic explanation of the encoding, with a single-qubit construction on the left and a joint two-qubit construction on the right. The annotation groups identify the amplitude components, area-probability relationship, and normalization constraint.

Joint two-qubit representation and interaction

For a joint state

$$|\psi\rangle=\alpha|00\rangle+\beta|01\rangle+\gamma|10\rangle+\delta|11\rangle,$$

VENUS repeats the amplitude triangle and semicircle construction four times. Four colors distinguish the computational-basis outcomes, and three white auxiliary triangles combine the amplitude magnitudes into a hierarchy whose bottom hypotenuse remains one unit long. This construction encodes the normalization condition $|\alpha|^2+|\beta|^2+|\gamma|^2+|\delta|^2=1$. The individual auxiliary triangles need not all have unit-length hypotenuses; the unit-length constraint belongs to the complete normalized construction.

Because it retains the joint amplitudes, this view can display entangled two-qubit pure states as well as separable ones. Its representation of entanglement is through the joint state, not through a separately defined entanglement measure. The semicircle areas alone show computational-basis probabilities; the signed real and imaginary components carry additional state information. This distinction matters when interpreting the paper's entanglement claims, since showing several nonzero outcome probabilities is not by itself a test for entanglement.

The implemented web interface updates its geometry from manually entered real and imaginary amplitude components. The reported interactions include switching qubit display order and revealing exact values through tooltips. In the Grover example, hovering over an auxiliary triangle exposes a probability for a shared single-qubit value in the displayed joint state. The paper's case-study figures juxtapose circuit stages and exported state views; they demonstrate an analysis workflow, while automatic extraction of states from circuits remains proposed future work.

Design alternatives

Figure 2 documents three alternatives considered during co-design. The first translates Bloch-sphere axes and rotation information into two-dimensional circular indicators, but does not provide the direct amplitude and probability view the experts wanted. The second uses line lengths within an equilateral-triangle construction inspired by Viviani's theorem, together with amplitude bars, but was rejected because it did not extend suitably to the intended two-qubit representation. The third combines right triangles with square areas to link amplitude magnitudes and probabilities, yet the experts found the relationships across the whole state insufficiently clear. These are design rationales derived from the formative process, rather than results of a controlled comparison of the alternatives. The final representation uses connected right triangles and semicircles to bring amplitude components, outcome probabilities, and normalization into one geometry.

Case studies and what the figures show

The first case follows one expert examining a simulated single-qubit classifier built with TorchQuantum, using two features from the Iris dataset. Figure 3A shows one validation example at epochs 1, 25, 50, and 100. The displayed probability of $|0\rangle$ changes from $0.69$ to $0.44$, $0.24$, and $0.26$, so the probability of the target label $|1\rangle$ rises from $0.31$ initially to $0.74$ at epoch 100. The expert used the changing semicircle areas and amplitude segments to inspect the learning trajectory and judged this example's output to have largely stabilized around epoch 50. This is an interpretation of one example, not a reported improvement in classifier accuracy caused by VENUS.

Figure 3B follows a separate input through an $R_Y$ and an $R_Z$ data-embedding gate, followed by an $R_Y$ and an $R_Z$ model layer. The plotted $|0\rangle$ probabilities are $0.56$, $0.56$, $0.10$, and $0.10$. The unchanged semicircle areas across each $R_Z$ step accompany changes in the internal amplitude segments, illustrating how the design distinguishes phase-related component changes from changes in computational-basis probabilities. These plotted values provide a clearer account of the example than some contradictory wording in the case narrative about an increasing probability of state zero. The expert's suggestions about parameter initialization and noise resilience are reported impressions; the paper does not experimentally validate those suggestions.

Figure 3 connects the classifier's state evolution to its circuit stages. Its labels report the probability of measuring zero, while the red semicircle represents the probability of measuring one.

The second case examines a two-qubit Grover search targeting $|11\rangle$. Figure 4 shows equal semicircle areas after the initial Hadamard gates, a doubled amplitude segment identifying the oracle's sign change for the target, and a single remaining purple semicircle with probability $1.00$ after one search iteration. The expert then inspected a circuit extended by another iteration, whose final view shows four equal outcome probabilities of $0.25$. Intermediate views and an auxiliary-triangle hover helped the expert compare the two executions. This example demonstrates inspection of sign changes, amplitude amplification, and the consequences of an extra iteration in a four-item search; it does not evaluate the visualization on a larger search space.

Expert evaluation and findings

Fourteen quantum computing experts, different from the co-design participants, took part in an approximately 95-minute procedure: a 20-minute introduction, 45 minutes of tasks, and a 30-minute interview and rating session. Their research covered areas including quantum machine learning, error modeling, chemistry, compilers, systems, and simulation. They used examples related to their own research and completed three kinds of tasks: exploring a joint two-qubit state, identifying and comparing outcome probabilities, and reading real and imaginary amplitude components and their connection to probability. For the probability task, interaction that displayed exact values was allowed after the initial answer.

Table 1 specifies nine seven-point Likert questions, and Figure 5 presents their response distributions as stacked bars. The authors report favorable average ratings for two-qubit analysis, probability observation, state-vector exploration, and overall usability, with probability observation receiving the highest reported task-category mean. Participants valued the direct probability view, the availability of amplitude components, and the ability to switch qubit order. They suggested uses in debugging, circuit analysis, and instruction. The figure also contains neutral and negative responses, so the results should be read as generally positive feedback rather than universal agreement on every question.

The evaluation supports perceived usefulness and illustrates how experts reasoned with the representation. It does not report a controlled head-to-head task-performance experiment against the Bloch sphere, an objective accuracy or completion-time comparison, or measured educational gains among novices. Comments about potential use in quantum error correction or circuit cutting indicate possible applications, not implemented and validated extensions in this paper.

Contributions, limitations, and future work

The principal contribution is a mathematically connected geometric encoding that exposes signed amplitude components and measurement probabilities for one- and two-qubit states. The accompanying design process documents why experts wanted those relationships to be visible, and the web implementation and case studies show how the representation can support inspection of small simulated quantum programs. The work's value lies in this particular connection between geometry and state semantics, rather than a new simulation algorithm or a new method for detecting entanglement.

The authors identify three practical limitations: the focus on one and two qubits, lack of support for quantum-noise analysis in their noise-free simulation workflow, and the inconvenience of entering amplitude components manually. The amplitude-based construction and evaluation concern pure-state examples; a general mixed-state or noisy-state visualization is not established. Adding further levels of triangles is suggested as a direction, but a readable and evaluated extension beyond two qubits is not delivered. The paper also proposes easier input, possible conversion from Bloch-sphere views, and automatic loading of states from quantum circuits. Its expert feedback motivates these extensions, while broader scalability, quantitative comparative benefits, and novice learning outcomes remain outside the reported evidence.