State-o-gram: Quantum Tomography & Visualization

• State-o-gram is a comprehensive framework that integrates quantum tomography, semidefinite programming, and scalable 2D visualization to analyze high-dimensional quantum states.
• It employs Gram matrix formulation and convex relaxation techniques to reconstruct states and measurements from realistic, error-prone experimental data.
• Its novel stacked bar-chart visualization enables clear analysis of interference and phase relations in multi-qubit registers, bridging theoretical insights with experimental applications.

State-o-gram encompasses several related concepts at the intersection of quantum information theory, quantum tomography, visualization, and the theoretical paper of neural and probabilistic models in language processing. Most notably, it refers to methods and tools for the self-consistent reconstruction, analysis, and visualization of complex quantum states—including high-dimensional quantum systems as well as classical vector representations extracted from quantum algorithms. Recently, “state-o-gram” also designates a scalable 2D visualization form for quantum states in multi-qubit registers, addressing classical limitations in quantum state representation.

1. Gram Matrix Formulation in Quantum Tomography

Quantum tomography seeks to reconstruct both the prepared quantum states (density matrices $\rho_w$) and the measurement operators ($E_{vk}$) from experimental data. Traditional approaches often treat either the state or the measurement model as idealized or error-free, assumptions that fail in realistic experimental conditions where state preparation and measurement errors are unavoidable.

The state-o-gram framework re-embeds both $\rho_w$ and $E_{vk}$ as vectors in the real vector space $\operatorname{Herm}(\mathbb{C}^d)$ of Hermitian matrices, with each vector typically of dimension $d^2$ for a $d$-dimensional system. Probabilities from the Born rule, $\operatorname{tr}(\rho_w E_{vk})$, become inner products in this space. All vectorized states and measurements are concatenated into a matrix $P = \left[ P_{st} \mid P_m \right]$; their joint Gram matrix $G = P^T P$ partitions into blocks:

• State–state inner products: $\langle \rho_w, \rho_{w'} \rangle$,
• Measurement–measurement inner products: $\langle E_{vk}, E_{v'k'} \rangle$,
• State–measurement data block: $D = [\operatorname{tr}(\rho_w E_{vk})]$.

Estimating $G$ from the measured data allows simultaneously reconstructing both the states and measurements without designating either as reference.

2. Semidefinite Programming and Convex Relaxation

Estimating $G$ directly from incomplete data is cast as a rank minimization problem:

$$\text{minimize} \;\; \operatorname{rank}(G) \ \text{subject to} \;\; G_\Omega = k,\; G \in \mathcal{G}_{\text{QM}}$$

where $\Omega$ indexes entries fixed by experiment, and $\mathcal{G}_{\text{QM}}$ constrains $G$ to be a quantum Gram matrix. Since rank minimization is NP-hard and nonconvex, convex relaxation replaces the rank by its trace norm surrogate, reducing the problem to an efficiently solvable SDP:

$$\text{minimize} \;\; \operatorname{tr}(G) \ \text{subject to} \;\; G_\Omega = k,\; G \geq 0,\; (W+Vd)I - G \geq 0$$

(W is the number of states, V the number of measurements.) This approach provides the required Gram matrix consistent with all available data and physical constraints.

3. Uniqueness, Rotational Ambiguity, and Global Completeness

The recovered $G$ encodes all mutual inner products, but the explicit representations of $\rho_w$ and $E_{vk}$ are determined only up to a simultaneous rotation in $\operatorname{Herm}(\mathbb{C}^d)$:

$$\vec{\rho}'_w = O \vec{\rho}_w,\;\; \vec{E}'_{vk} = O \vec{E}_{vk}$$

for an orthogonal matrix $O$. This implies multiple realizations of the same physical scenario, differing only by internal rotations that preserve outcome probabilities.

Uniqueness (up to rotation) holds if the preparation and measurement sets are "globally completable"—that is, if their spans cover $\operatorname{Herm}(\mathbb{C}^d)$. The assumption of projectivity and non-degeneracy in measurement further aids overdetermination.

4. Algorithms for Explicit Quantum Realization

While factorizing $G$ (e.g. via Cholesky) yields vector sets reproducing inner products, these do not necessarily map to legitimate density matrices or POVM elements (i.e., positive semidefinite and trace-normalized). Achieving physically valid operators, (Stark, 2012) introduces a heuristic iterative algorithm:

• REGULAR routine: Minimizes $\|P_0^T v - G(:,j)\|^2_2$ for column $j$ with positive semidefinite constraint on $\operatorname{mat}(v)$.
• PARTIAL update: Locally refines columns via neighborhoods defined by Gram matrix distances.
• SELECTION_OF_FASTEST: Prioritizes rapidly moving columns to escape local minima.

Each step enforces quantum constraints and converges to $P$ such that $P^T P\approx G$, guaranteeing all realized operators are quantum mechanically valid. Dynamic switching monitors progress and alternates subroutines to maximize convergence.

5. Applications in Quantum Algorithms and Quantum Information

State-o-gram techniques have implications for quantum algorithm analysis and classical extraction from quantum computations:

• Efficient Read-Out Protocols (Zhang et al., 2020): Protocols (sometimes labeled "state-o-gram") decode quantum states output from algorithms (e.g., HHL) into classical vectors with $\ell^2$ error guarantees, requiring resources polynomial in the effective rank $r$ and error $\epsilon$: $\text{copies} = \mathrm{poly}(r, 1/\epsilon)$, $$\text{oracle queries} = \mathrm{poly}(r, \kappa^r, 1/\epsilon)$$.
• Quantum Gram-Schmidt Algorithms: Enable adaptive, basis-efficient reconstruction and are applicable beyond state read-out to tomography and subspace identification.

These methods enhance self-consistency in tomography, facilitate calibration without reference states, and support robust quantum modeling in device- and semi-device-independent regimes.

6. State-o-gram as a 2D Visualization Tool for Quantum States

The recent work (Schinkel, 25 Aug 2025) introduces state-o-gram as a scalable 2D visualization for arbitrary multi-qubit quantum states, overcoming Bloch Sphere limitations. Its design features:

• Stacked Bar-chart Representation: Each computational basis state is depicted as a bar whose height is proportional to the square of its amplitude (probability), and whose horizontal position reflects its phase angle (from $-\pi$ to $\pi$).
• Color Coding: Unique colors follow basis ordering, offering tractable differentiation as the register scales.
• Scalability: Addresses exponential growth by stacking, not widening, bars—total diagram height remains fixed regardless of qubit count.
• Comprehensiveness: Integrates visualization of all interference patterns and phase relationships, enabling analysis of evolution under gate operations.

A key improvement over prior art (e.g., VENUS) is the support for arbitrarily many qubits without diminishing bar visibility.

7. Empirical Evaluation and Broader Implications

Application to algorithms such as Deutsch-Josza demonstrates state-o-gram's capacity to visualize dynamic phase and interference through quantum circuit steps. For constant or balanced oracles, stacking and phase positions directly identify constructive or destructive interference.

The tool is in the process of integration with simulators like Quirk-E and quirk-s, suggesting future adoption within open-source quantum frameworks. Educational benefits are explicit: state-o-gram aids comprehension of entanglement and interference, with potential utility in real-time circuit debugging and interactive state analysis.

A plausible implication is that the stacked-bar approach may generalize to yet more complex systems as quantum hardware scales, potentially serving as the standard for multidimensional state representation and exploration.

---

State-o-gram thus refers to a set of interrelated frameworks, optimization strategies, and visualization tools for state–measurement tomography, quantum state extraction, neural memorization of probabilistic rules, and scalable graphical representation of high-dimensional quantum states. Its components jointly advance the fidelity, transparency, and tractability of both experimental quantum analysis and theoretical model construction.