---
title: state-o-gram – A Novel 2D Visualization for Quantum States
authors:
  - Fritz Schinkel
abstract: Quantum computing is rapidly gaining popularity, necessitating intuitive visualization tools for complex quantum states. While the Bloch Sphere effectively visualizes single-qubit states, it fundamentally lacks scalability for multi-qubit systems. Existing multi-qubit visualization attempts, such as VENUS, have shown promise but often face limitations in scalability beyond a few qubits. This paper introduces state-o-gram, a novel 2D visualization approach designed to intuitively represent quantum states for an arbitrary number of qubits. state-o-gram effectively visualizes probability amplitudes and phase angles in a unified 2D framework, addressing the limitations of prior art. We detail its design principles, visual elements, and application to multi-qubit systems, aiming to provide a scalable and intuitive tool for quantum state analysis. We evaluate the applicability by visualizing the states throughout the Deutsch-Josza algorithm.
summaryType: survey
sourceStatus: null
sources:
  - https://doi.org/10.48550/arXiv.2508.18390
  - http://arxiv.org/abs/2508.18390
---

# state-o-gram: A Novel 2D Visualization for Quantum States

[Read the original paper](https://doi.org/10.48550/arXiv.2508.18390).

## Background and motivation

Understanding a quantum algorithm requires following both the magnitudes and phases of its state amplitudes.
A computational-basis probability histogram makes measurement outcomes explicit, but it loses the phase information that determines how amplitudes combine under later gates.
For an $n$-qubit register, it also needs up to $2^n$ bars.
Schinkel addresses the established problem of representing this growing state space in a form that helps explain, inspect, and verify quantum algorithms.
The proposed contribution is a two-dimensional encoding called state-o-gram, together with small worked examples and a simulator integration.

The paper places this design among several earlier approaches.
The Bloch sphere is useful for individual qubits, but a collection of single-qubit views does not describe an arbitrary joint state and its correlations.
The related-work discussion also covers sphere-based extensions, Q-sphere, fractal representations, density-matrix mappings, and circuit-oriented tools.
It identifies scalability, difficulty interpreting geometric or mathematical encodings, and the need to follow intermediate states as recurring concerns.
VENUS is a particularly close point of comparison because it uses two-dimensional geometry to relate amplitudes and probabilities for one- and two-qubit states.
Schinkel seeks an encoding whose construction extends beyond two qubits while retaining explicit probability and phase information.
These comparisons motivate the design; the paper does not experimentally compare state-o-gram with those systems.

## A stacked representation of probability and phase

For a normalized pure state

$$
|\psi\rangle=\sum_{k=0}^{2^n-1}\alpha_k|k\rangle,
\qquad
\alpha_k=\sqrt{p_k}\,e^{i\phi_k},
\qquad
\sum_k p_k=1,
$$

state-o-gram represents each nonzero component with a narrow vertical bar.
The bar's horizontal position is the phase $\phi_k$ on an axis from $-\pi$ to $\pi$.
Its height is the measurement probability $p_k=|\alpha_k|^2$, rather than the amplitude magnitude $|\alpha_k|$.
Bars are stacked in computational-basis order: the bottom of bar $k$ is at $\sum_{j<k}p_j$, and its top is at $\sum_{j\leq k}p_j$.
Consequently, a bar's own vertical extent expresses its probability, while its absolute vertical position depends on the preceding components.
The total stack covers 100% even when the bars occupy different phase positions.

Basis-state labels identify the components, and the examples use colors progressing from blue through purple to red according to computational-basis index.
A label is placed near the top of its bar.
Components with zero amplitude receive no bar and are listed separately in a gray box in the relevant examples.
The fixed phase range and normalized total height allow states with different distributions to occupy the same chart dimensions.
This conserves display area but does not remove the exponential number of possible components.

Figure 1 explains the encoding with

$$
|\psi\rangle=\frac{i}{\sqrt{2}}|0\rangle-\frac{i}{\sqrt{2}}|1\rangle.
$$

The blue $|0\rangle$ bar occupies the lower half of the chart at $\pi/2$, while the red $|1\rangle$ bar occupies the upper half at $-\pi/2$.
Both have height 50%, although their upper endpoints differ.
The example directly separates the probability channel from the phase channel.



Figure 1 from the paper: equal probabilities are encoded by equal bar heights, and the coefficients $i/\sqrt{2}$ and $-i/\sqrt{2}$ appear at phases $\pi/2$ and $-\pi/2$.

Figure 2 extends the construction to three qubits with eight equally probable components and phases at $\pm\pi/2$.
Figure 3 varies both probabilities and phases, showing that the construction is not restricted to uniform distributions or to those two angles.
Figure 4 displays the four Bell states.
Each Bell-state panel has two nonzero components with probability $1/2$, while the other two basis states are listed below the plot.
The positive and negative Bell-state variants differ in the phase placement of one component.
These panels show that the representation can encode these entangled states; they do not introduce a separate visual measure or detector of entanglement.



Figure 2 from the paper: computational-basis order determines the vertical stacking, while phase determines the horizontal position.

## Worked gate examples

The paper evaluates applicability through mathematical explanations and visual examples.
Figure 5 follows a qubit initialized to $|0\rangle$, then applies a Hadamard gate to obtain $(|0\rangle+|1\rangle)/\sqrt{2}$, and applies Hadamard again to recover $|0\rangle$.
The middle chart contains two equal-height bars at phase zero; the initial and final charts contain only the $|0\rangle$ component.
This sequence illustrates how gate operations change the decomposition represented by the chart.

Figure 6 applies Hadamard gates to all three qubits for each of the eight computational-basis inputs.
Every resulting state has eight equally probable components, but their signs differ with the input.
For input $|000\rangle$, all bars lie at phase zero.
The other inputs produce patterns split between phases zero and $\pi$.
The figure therefore distinguishes states that share an identical probability distribution but differ in phase, which a probability-only histogram would conceal.



Figure 6 from the paper: equal probabilities persist across the eight examples, while the phase patterns change.

## Explaining the Deutsch–Jozsa algorithm

The main algorithm example uses two input qubits and one output ancilla.
The oracle represents a Boolean function promised to be either constant or balanced.
The paper explains why the final input-register measurement is $00$ for a constant function and never $00$ for a balanced function.
Its complexity discussion concerns the known oracle-query distinction: one quantum oracle call suffices for the exact promised decision, whereas a deterministic classical procedure may require $2^{n-1}+1$ queries in the worst case.
This is an explanation of an existing algorithm, rather than a new speedup or a measured runtime result for the visualization.

Figure 7 first uses a constant-zero oracle with the output qubit initialized to zero to display its value table.
It then changes the preparation so that the ancilla becomes $(|0\rangle-|1\rangle)/\sqrt{2}$.
After the oracle, components with ancilla zero appear at phase zero and those with ancilla one at phase $\pi$.
The final Hadamard gates on the input register yield

$$
\frac{|000\rangle-|001\rangle}{\sqrt{2}},
$$

so measurement of the two input qubits always returns $00$.
The constant-one case has the same input measurement outcome.

Figure 8 explains the balanced case using annotated state plots.
Colored boxes identify selected components, and yellow arrows connect them to separate illustrations of their Hadamard-transformed contributions.
For example, the contributions from $|000\rangle$ and $-|100\rangle$ to the final $|000\rangle$ amplitude have opposite signs and cancel.
The argument is repeated across pairs and for the ancilla-one sector, eliminating both $|000\rangle$ and $|001\rangle$ from the final state.
The displayed balanced example ends with the components $|100\rangle$ and $-|101\rangle$, whose input register is $10$.
The general conclusion is the absence of input outcome $00$, rather than the claim that every balanced function produces $10$.

The explanatory mechanism depends on following a gate transformation and comparing contributions to the same resulting basis state.
Opposite-phase bars for different basis states do not cancel merely because they are visible together in a chart.
The intermediate contribution diagrams and arrows in Figure 8 make this reasoning explicit.
They are annotations in the paper's explanation, not evidence of an implemented brushing or selection interaction.

## Implementation and contribution

Schinkel reports integrating state-o-gram into quirk-s, a version of the open-source Quirk or Quirk-E simulator.
The stated workflow allows an algorithm to be tracked step by step and its intermediate states to be displayed as state-o-grams.
The figures place circuit fragments alongside the corresponding state views, providing examples of how the plots support reasoning about gate effects.
The paper does not document a separate interaction design in detail, so specific playback controls, linked highlighting, or interactive filtering should not be inferred from those figures.

The main contribution is the combination of cumulative probability stacking and explicit phase position in a compact two-dimensional display.
The Hadamard and Deutsch–Jozsa examples show how this encoding supports visual explanations that require phase information.
The simulator integration supplies a practical setting for examining intermediate states.
The evidence supports these demonstrated uses; it does not establish improved learning, faster debugging, or better performance than other visualization methods.

## Limitations and future work

The author explicitly acknowledges that state-o-grams can become fragmented as individual bars become very thin.
The proposed response is that collective arrangements may remain informative even when individual components are difficult to inspect.
The encoding is defined for an arbitrary number of qubits, but the paper's examples use one-, two-, and three-qubit registers.
There is no large-register readability study, user experiment, comparative task evaluation, or rendering-performance benchmark.
Practical scalability and the claimed intuitiveness therefore remain open evaluation questions.

The method is presented through pure-state amplitude vectors.
The paper does not develop an extension for mixed states or noisy-state density matrices.
Its Bell-state illustrations also do not establish a general procedure for assessing entanglement.
The stated future implementation work is integration with additional quantum-computing frameworks, with Qulacs named as an example.
Educational and programming benefits are proposed applications supported by the worked explanations, rather than measured outcomes.