Vector Field Visualization of Single-Qubit State Tomography

Vector Field Visualization of Single-Qubit State Tomography

Background and motivation

Quantum characterization, verification, and validation encompasses methods for measuring individual gate errors, reconstructing small quantum states, and benchmarking entire processors. The paper situates quantum state tomography (QST) alongside randomized benchmarking, gate-set tomography, quantum volume, and random-circuit experiments. QST reconstructs a density matrix from repeated measurements and can reveal more state detail than a single performance score, but its data requirements grow rapidly: a general $n$-qubit density matrix has $4^n-1$ independent real parameters. Existing reconstruction approaches include maximum-likelihood estimation (MLE), linear regression (LR), and nuclear-norm-constrained methods. The authors use established tomography and statistical estimation methods to address the practical problem of inspecting how hardware performance changes across the states that an individual qubit can prepare.

The problem of reconstructing and characterizing quantum states is established; the contribution is a coordinated data-collection, reconstruction, and visual-presentation workflow for single-qubit hardware assessment. The authors ask both how accurately a qubit prepares a target state and how consistently it does so across the Bloch sphere. A mean error or purity value can conceal systematic variation with the target state. Plotting reconstructed states as points inside a three-dimensional Bloch sphere also creates ambiguities in a static image because front and back points overlap, and radial depth is difficult to read. The proposed Vector Field Visualisation (VFV) separates the direction and purity of a reconstructed state into complementary visual channels so that these patterns can be compared in a two-dimensional figure.

Tomography protocol and statistical reconstruction

The preparation circuit applies $R_y(\theta)$ followed by $R_z(\phi)$ to $|0\rangle$, producing the ideal Bloch vector

$$\mathbf{a}_{\mathrm{in}}(\theta,\phi)=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).$$

Figures 1 and 2 are circuit schematics explaining the preparation and measurement procedure. Because the hardware measures in the computational basis, additional inverse rotations change the measurement basis before readout. A single-qubit state can be reconstructed from at least three linearly independent projection-valued measures (PVMs), such as the Pauli bases. The experiments instead use four PVMs whose directions form a tetrahedron, with each PVM providing two possible outcomes. Figure 3 contrasts these four directions with the three Pauli directions; the tetrahedral arrangement provides redundant information while keeping the measurement cost manageable. For each target state, the protocol runs four tomography circuits, one for each tetrahedral basis, with $20{,}000$ shots per circuit.

A density matrix is represented as $\rho=\tfrac12(I+\mathbf{a}\cdot\boldsymbol{\sigma})$, where $\boldsymbol{\sigma}$ contains the Pauli matrices and physical states satisfy $\|\mathbf{a}\|\leq1$. Writing $p_{\mathbf{u}}$ for the observed probability of one outcome in measurement direction $\mathbf{u}$, the MLE reconstruction solves

$$\mathbf{a}_{\mathrm{out}}=\underset{\|\mathbf{a}\|\leq1}{\operatorname{argmax}}\sum_{\mathbf{u}}\left[p_{\mathbf{u}}\ln(1+\mathbf{a}\cdot\mathbf{u})+(1-p_{\mathbf{u}})\ln(1-\mathbf{a}\cdot\mathbf{u})\right].$$

This specialization replaces a density-matrix positive-semidefiniteness constraint with a unit-ball constraint in three real dimensions, yielding a concave optimization problem suitable for standard optimization software. MLE is an existing estimator selected for its statistical grounding and practical implementation, rather than a new estimation algorithm introduced by the paper.

The authors assess finite-shot uncertainty through simulation, repeating the sampling and reconstruction process $10^4$ times for each state examined. With $20{,}000$ observations per tetrahedral PVM, they report an empirical 99th-percentile Euclidean reconstruction error no greater than $0.02$ across the angles considered. This motivates the measurement budget used in the hardware experiments. It is a simulated assessment of sampling variability under the assumed measurement model, rather than a guarantee that hardware tomography is free from systematic errors.

Visual encoding and interpretation

VFV combines a Robinson projection of the Bloch sphere, a scalar heatmap, and arrows linking intended and reconstructed state directions. The map’s angular grid locates target states by their spherical coordinates. Each arrow starts at the intended state and ends at the reconstructed state after its direction has been projected onto the sphere’s surface, showing the direction and extent of angular displacement. The arrows therefore expose rotational error, while the heatmap communicates a separate state-quality quantity. All reported heatmaps encode reconstructed purity, $\operatorname{Tr}(\rho_{\mathrm{out}}^2)$; fidelity is mentioned as another possible scalar encoding but is not the quantity displayed in these experiments. A red horizontal marker on each color bar identifies the mean purity across sampled states.

The projection makes the whole sphere available in one static view, and the separate color encoding retains information about mixedness that would otherwise be difficult to read from radial position. It does not eliminate geometric distortion: the Robinson projection is neither angle-preserving nor equal-area. The figures demonstrate plotted output, not an evaluated interactive interface, and the paper does not describe a specific interaction workflow for inspecting individual states.

Figure 4 presents the same visualization for three data sources associated with qubit 1 of ibm_lagos. The ideal simulation is predominantly yellow, the noisy simulation is more uniformly green, and the hardware result has visible arrows and stronger regional changes in purity. These panels show computed tomography results, whereas Figures 1–3 explain the protocol and measurement geometry.

Hardware and simulation findings

The main comparison samples 200 target states spaced approximately equidistantly around the Bloch sphere. The ideal simulator produces an average reconstructed purity of $0.999$, with only small finite-sampling variation and no visible systematic rotational error. A noisy simulator configured using ibm_lagos calibration data from the time of the experiment produces a lower mean purity of $0.983$ and a standard deviation of $0.004$, but still little visible rotational displacement. The hardware measurements produce a mean purity of $0.972$ and a larger standard deviation of $0.013$. Figure 4c reveals both coherent directional patterns in the arrows and heterogeneous purity across the sphere, including lower-purity regions near the equator. The comparison shows that this particular calibrated simulator misses structure visible in the hardware data; it does not establish that all noisy simulators have this limitation.

The authors then insert an idle delay between state preparation and tomography, as illustrated by the circuit in Figure 5. Figure 6 compares delays of $0$, $800\,dt$, and $1600\,dt$ on the same hardware qubit. Mean purity decreases from $0.972$ to $0.959$ and then $0.945$, while the arrows become longer and show a state-dependent left-to-right displacement that varies across the sphere. The maps make this directional change visible alongside increasing mixedness. The authors describe the experiment as a proof of principle for investigating open-system dynamics and informing better simulation models, rather than a fitted physical model of the observed decoherence.

Comparing estimators to reveal inconsistent measurements

A further experiment reconstructs the same raw data from qubit 3 of ibmq_belem using both MLE and constrained LR. LR minimizes the squared discrepancies between observed and predicted measurement probabilities over the Bloch ball. The authors’ simulated sensitivity analysis suggests that the two reconstructions should agree closely under their measurement assumptions and sampling budget. In Figure 7, however, MLE and LR produce visibly different purity patterns, and a third map highlights regions where their purity difference exceeds $0.02$. This extends the visualization from comparing hardware with simulation to comparing the interpretations that different estimators assign to identical measurements.

The authors regard these discrepancies as signatures of data corruption after state preparation and suggest that the rotations used to change measurement basis may introduce state-dependent errors incompatible with the assumed tomography model. That explanation is a proposed cause, not an independently isolated hardware fault. The comparison also shows why reconstruction differences should be interpreted alongside sampling uncertainty and model assumptions: an estimator can return a physical density matrix even when the measurement statistics do not fit the intended experimental model well.

Contributions, limitations, and future directions

The principal contribution is the integration of a practical single-qubit tomography protocol with a visual representation that separates angular displacement from purity, supported by simulation checks and IBM hardware examples. The paper also provides two open-source packages: sqt supplies tomography bases, reconstruction methods, and data collection, including parallel execution of single-qubit experiments across a chip; sqmap produces visualizations of those results. The implementation described in the paper uses Qiskit as its supported framework, while leaving extension to other frameworks as future work. Parallel single-qubit assessment does not amount to reconstructing the joint state or correlations of an entire processor.

The evaluation establishes that these maps can reveal patterns in the reported datasets, but it includes no controlled user study measuring interpretation accuracy or comparing VFV with alternative visualizations. The single-qubit scope limits data costs but does not solve the exponential measurement requirements of general multiqubit QST. Finite-shot errors, imperfect basis-change operations, and projection distortion also constrain interpretation. The authors propose using the observed patterns to inspire new performance measures, more accurate state-dependent and open-system noise models, and noise-mitigation schemes. They additionally call for more general tomography models that can accommodate effects exposed by estimator disagreement. Those are directions motivated by the experiments; the paper does not implement or evaluate a new mitigation algorithm.