QuantumEyes

QuantumEyes: Towards Better Interpretability of Quantum Circuits

Background and motivation

QuantumEyes addresses the established problem of making quantum circuits understandable to quantum computing developers, researchers, and learners. A circuit diagram identifies gates, qubits, and execution order, but it does not directly show how a gate changes amplitudes, how a basis state arises through earlier operations, or why a particular measurement outcome becomes more probable. The difficulty comes from reasoning about complex-valued amplitudes and matrix transformations, together with the exponential growth from $n$ qubits to $2^n$ computational basis states. For a pure state $|\psi\rangle=\sum_j\alpha_j|j\rangle$, the outcome probability is $p_j=|\alpha_j|^2$, so probabilities alone omit phase information that can affect subsequent interference. The paper uses “measured probability” to mean the probability an outcome would have if a measurement were performed, including at intermediate simulated steps.

The authors situate their work among quantum state representations and visualizations of circuit evolution. State-vector approaches such as the Bloch sphere, decision diagrams, and multi-qubit representations expose aspects of the state but do not always make outcome probabilities easy to compare. Probability-aware approaches, including fractal layouts, geometrical square constructions, and VENUS, make probability more explicit, but the reviewed designs often concentrate on one or two qubits. Circuit visualization tools already display changing probabilities or phases, while algorithm-specific explanations address cases such as Shor's algorithm and the quantum Fourier transform. The remaining opportunity is to coordinate a circuit overview, gate-level explanations, state ancestry, and a geometric account of the amplitude-to-probability relationship within one analysis workflow. QuantumEyes therefore extends an existing interpretability problem with a particular combination of views and interactions, rather than introducing circuit interpretation as a new research problem.

Design process and simulation pipeline

The design was informed by five months of collaboration with six quantum computing experts, including professors, a research scientist, and doctoral students. The authors began with individual interviews, developed a low-fidelity prototype, and refined it through iterative expert feedback. They distilled six requirements into two levels of analysis. Global analysis should summarize the circuit, explain gate effects, and support tracing how states develop. Local analysis should explain probabilities through amplitudes, represent multi-qubit states, and reduce visual clutter as the number of displayed basis states grows.

QuantumEyes obtains its input from Qiskit and AerSimulator: gate sequences and their qubit assignments, intermediate state vectors, and gate transformation matrices. Its processing module calculates basis-state probabilities, separates state-vector components for tracing transformations, and breaks circuit blocks into individual gate steps. Figure 2 presents this architecture as data storage, data processing, and visualization modules. The intermediate state information is available through simulation; the workflow is not a nondestructive observation of every state during a single physical quantum computation. The system also displays the original circuit diagram, providing a familiar reference alongside the new views.

Coordinated views for circuit evolution

Figure 3 explains the three coordinated global views with an illustrative circuit. The Probability Summary View is a stacked area chart whose horizontal axis follows gate steps. At each step, vertical segment lengths encode the probabilities of basis states and sum to one; the connecting areas show how those probabilities change between steps. Colored rectangles group individual steps into circuit blocks, and state labels mark their first appearance. Brushing a range of steps selects the portion to inspect in greater detail.

The State Evolution View aligns steps horizontally with the summary and places groups of basis states vertically according to probability. Each basis state appears as a labeled rounded rectangle, and an enclosing rectangle groups states with the same probability. Pink dotted links connect their evolution across steps, including the splitting and merging associated with Hadamard operations. Gate abbreviations and arrows identify the operation and affected qubit. Two shades of blue distinguish whether an amplitude's real part is positive or nonpositive, while hovering over a state highlights its evolution path in red. This makes it possible to follow a selected output back through earlier contributions, rather than inspecting only independent probability distributions at successive steps. The authors motivate the vertical-position encoding through general perceptual evidence for position, but do not conduct a controlled comparison of this encoding against color or length in this paper.

The Gate Explanation View uses a table-like diagram to unpack an operation in terms of qubit states. Columns correspond to qubits, while rows show the input states, gate operations, and resulting states; a final row combines the displayed results into system basis states. Colored connecting lines illustrate the transformations, and a dotted gray line indicates an unchanged qubit. For example, applying a Hadamard gate to one qubit in $|00\rangle$ produces two basis-state components, while the other qubit remains unchanged. This is an explanation of gate action on basis components, not a claim that an arbitrary entangled system factors into independent single-qubit states.

Dandelion chart: linking amplitudes and probabilities

The dandelion chart supports local analysis by displaying each complex amplitude $\alpha_j=a_j+i b_j$ as a point at $(a_j,b_j)$. Horizontal and vertical guides expose the real and imaginary components, and a line to the origin makes the point's radial distance and direction visible. A circle links those coordinates to the probability through geometry: its initial radius is the amplitude magnitude $\sqrt{a_j^2+b_j^2}$, so its area satisfies

$$S_j=\pi(a_j^2+b_j^2)=\pi|\alpha_j|^2=\pi p_j.$$

The amplitude point lies on the circle's boundary. In the unscaled configuration, the circles share the origin as their center and can heavily overlap. A slider reduces all radii by the same factor $k$, with $0\leq k\leq1$, while keeping each amplitude point on its circle's boundary. The resulting circles move along their radial directions toward those points as they shrink, producing the dandelion-like arrangement in Figure 4. The area becomes

$$S'_j=\pi k^2(a_j^2+b_j^2)=\pi k^2p_j.$$

For a common nonzero factor, the relative areas still represent relative probabilities while the point coordinates continue to represent the original amplitudes. The geometric argument establishes that scaling preserves this proportionality; it does not establish that all large quantum states can be displayed without overlap.

Paired dandelion charts compare states before and after a selected gate. They expose changes that a probability-only overview can hide: a phase operation changes an amplitude's direction while leaving its magnitude and probability unchanged. Conversely, amplitude redistribution changes circle sizes. Figure 4 is an encoding and interaction demonstration, while Figures 5 and 6 show the charts embedded in the actual Grover and quantum Fourier transform analysis workflows.

Case studies and expert evaluation

The first case study follows an expert examining a two-qubit Grover circuit. In Figure 5, the probability overview supports identifying initialization, the oracle, and amplitude amplification. The initialization produces four equal probabilities of $0.25$, while the oracle changes the sign of the target amplitude for $|11\rangle$ without changing its probability. The dandelion comparison makes that sign change visible, and state tracing and gate explanations support investigating how the target component was generated. The final amplification produces probability one for the target state. The account includes researcher prompts directing the expert to relevant views, so it demonstrates a supported analysis session rather than unaided task performance.

The second case study follows another expert using a three-qubit quantum Fourier transform circuit initialized to $|101\rangle$. Figure 6 combines the probability overview, state evolution, original circuit, and local before-and-after comparisons. The expert distinguishes controlled-phase operations, which rotate amplitudes without changing the displayed probabilities, from Hadamard operations, which redistribute amplitude across more basis states. One comparison shows a roughly $45^\circ$ rotation, and later comparisons illustrate the transition to eight basis states with smaller circles. Adjusting the circle radius helps separate the local marks in this example. Together, the two cases demonstrate the intended global-to-local workflow on small simulated circuits, rather than a scalability benchmark across large algorithms.

The paper also reports interviews with 12 male domain experts from six U.S. educational institutions, averaging 5.9 years of quantum computing experience. The two case-study experts were members of this group, so the case studies do not constitute an additional independent participant sample. Following an introduction to the interface, participants completed six predefined tasks: four concerning global analysis and two concerning the dandelion chart. The task session and verbal explanations took approximately 40 minutes, followed by about 20 minutes of feedback and a 12-item, seven-point questionnaire. Figure 7 presents the response distributions across effectiveness, usability, interaction, and visual design.

The reported mean ratings were 6.02 for effectiveness, 5.88 for usability, 5.54 for interaction, and 6.05 for visual design, with standard deviations of 1.18, 1.65, 1.66, and 0.99 respectively. Experts valued state tracing, familiar circuit references, and the geometric explanation of probability. They also requested folding and unfolding, support for variational circuits, and animated transitions between dandelion charts. These results provide qualitative and self-reported evidence of usefulness for experienced users. The paper does not report a controlled baseline comparison, a quantitative learning-gain assessment, or task-time and accuracy improvements, so the findings do not establish superiority over existing tools or effectiveness for novices.

Contributions, limitations, and future work

The principal contributions are the expert-derived requirements, the coordinated global and local analysis workflow, and the dandelion chart's explicit geometric connection between complex amplitudes and outcome probabilities. The authors report releasing QuantumEyes online and the dandelion chart as an independent NPM package. The contribution concerns visual explanation and interaction with simulated quantum states; it does not introduce a new quantum algorithm or a method for recovering complete intermediate states from hardware.

The implemented system targets static circuits. Although the authors consider the dandelion chart applicable to states from variational quantum circuits, they identify extending the complete workflow to those circuits as future work. They also propose generalizing state evolution paths to Bloch-coefficient or Pauli-probability evolution. Visual scalability remains unresolved beyond the demonstrated two- and three-qubit cases: the three global views require more space as the number of states grows, and shrinking circles cannot guarantee a readable local view for very large state spaces.

The authors further acknowledge that the relationship between state evolution and the original circuit can be difficult to follow, motivating folding and unfolding of basis states and gates. They propose animated transitions for before-and-after comparisons and additional marks to make entanglement more distinguishable in dandelion charts, mentioning sector-length distributions as a possible encoding. These are proposed extensions, not evaluated features of the presented system. Simulator customization and circuit-data loading are also future improvements intended to make the workflow more flexible.