⚛️ quantum physics

Three-qubit W state tomography via full and marginal state reconstructions on ibm_osaka

This paper demonstrates a proof-of-principle experiment on IBM's ibm_osaka processor showing that a reduced measurement scheme reconstructing two-qubit marginals not only significantly lowers the overhead of three-qubit W state tomography but also yields higher fidelity than full state reconstruction, thereby validating the theoretical result that two-qubit subsystems can uniquely determine the global pure state.

Original authors: H. Talath, B. P. Govindaraja, B. G. Divyamani, Akshata Shenoy H., A. R. Usha Devi, Sudha

Published 2026-02-03

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CC BY 4.0

Original authors: H. Talath, B. P. Govindaraja, B. G. Divyamani, Akshata Shenoy H., A. R. Usha Devi, Sudha

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a complex, three-dimensional sculpture made of invisible light. You want to know exactly what it looks like from every angle. In the world of quantum computing, this sculpture is called a quantum state, and figuring out its exact shape is called Quantum State Tomography.

Usually, to "see" this invisible sculpture, scientists have to take a massive number of photos from every possible angle. For a three-part quantum object (three qubits), the old way required taking 63 different photos (measurements). This is like trying to reconstruct a statue by taking 63 separate snapshots, which is slow, expensive, and prone to errors because the "camera" (the quantum computer) is a bit shaky and noisy.

This paper presents a smarter, faster way to do it, using IBM's ibm_osaka quantum computer. Here is how they did it, broken down into simple concepts:

1. The "Whole from Parts" Trick

The researchers used a clever shortcut based on a famous mathematical idea: You can often figure out the whole picture just by looking at two of its pieces.

The Analogy: Imagine you have a three-piece puzzle. Usually, to solve the whole puzzle, you need to look at all three pieces. But the researchers found that for this specific type of puzzle (called a W state), if you look closely at just two of the pieces, you can mathematically reconstruct the entire third piece and the whole picture without ever looking at it directly.
The Result: Instead of taking 63 photos of the whole object, they took photos of just two smaller parts. This required only 7 photos per part (14 total), plus a few extra steps to combine them. This is a huge reduction in effort.

2. The Two Experiments

The team ran two different experiments on the IBM quantum computer to prove this works:

Experiment A (The Hard Way): They tried to reconstruct the whole three-qubit state using the new, efficient method of taking 17 specific photos. This is still much better than the old 63-photo method, but it's still a lot of work for a noisy machine.
Experiment B (The Smart Way): They took photos of just two of the three-qubit pairs (the "marginals"). They used 7 photos for each pair. Then, they used a mathematical recipe (developed by a scientist named Diósi) to "stitch" these two partial views together to create the full 3-qubit state.

3. The Surprising Result

When they compared the results, something interesting happened. The version of the state reconstructed from the two smaller parts (Experiment B) was actually more accurate (had higher "fidelity") than the version reconstructed from the full set of measurements (Experiment A).

Why? Think of the quantum computer as a shaky hand trying to draw a picture.
- Experiment A required a long, complex drawing process with many steps (gates). The more steps you take, the more likely your hand is to shake, introducing errors.
- Experiment B required fewer steps. Because the process was shorter and simpler, there were fewer chances for the "hand" to shake.
- The Lesson: Sometimes, doing less work (measuring fewer things) leads to a better result because you avoid the noise and errors that come with doing too much.

4. Cleaning Up the Mess

Quantum computers are "Noisy" (NISQ era). The data they get is often blurry or contains mistakes (like a photo taken in the dark).

The researchers used a "cleaning" technique (Error Mitigation) to fix the blurry photos before trying to reconstruct the state.
They also used a "spectral correction" to make sure the final mathematical description of the state made physical sense (like ensuring a sculpture doesn't have negative weight).

Summary

The paper demonstrates that for certain quantum states (specifically the W state), you don't need to measure everything to know the whole picture. By measuring just two parts and using a smart mathematical trick, you can rebuild the whole state.

The key takeaway: On current, imperfect quantum computers, measuring less can actually be better. It reduces the time the computer is exposed to noise, resulting in a clearer, more accurate picture of the quantum state than trying to measure everything at once. This is a "proof-of-principle" that we can infer the whole from the parts efficiently on real hardware.

Technical Summary: Three-qubit W State Tomography via Full and Marginal State Reconstructions on ibm_osaka

Problem Statement
Quantum State Tomography (QST) is essential for benchmarking quantum hardware but faces a scalability challenge. Reconstructing a generic $N$ -qubit state typically requires $(2^N)^2 - 1$ independent parameters, necessitating an exponential number of Pauli measurement settings (e.g., 63 settings for three qubits). This imposes significant experimental overhead on Noisy Intermediate-Scale Quantum (NISQ) devices, where gate errors and decoherence degrade performance as circuit depth increases. Furthermore, the "quantum marginal problem" investigates whether a global quantum state can be uniquely determined by its reduced density matrices (subsystems). While theoretical results suggest that almost all pure three-qubit states (excluding GHZ states) are uniquely determined by any two of their two-qubit reduced density matrices, experimental validation of this "whole-from-parts" reconstruction on current superconducting hardware remains a critical test of efficiency and fidelity.

Methodology
The authors implemented two distinct QST protocols on the 127-qubit IBM superconducting processor, ibm_osaka, using qubits q97, q98, and q99. The target state was the three-qubit W state, $|W_{ABC}\rangle = \frac{1}{\sqrt{3}}(|100\rangle + |010\rangle + |001\rangle)$ .

Full Three-Qubit Tomography (17 Settings):
Instead of the standard 63 Pauli settings, the authors proposed a resource-efficient scheme using only 17 measurement settings. This scheme utilizes single- and two-qubit unitary gates ( $H$ , $CNOT$, $R_x(\pi/2)$ ) applied prior to computational basis measurements. The settings are designed to determine the real and imaginary parts of the density matrix elements $\rho_{ABC}$ .
Marginal (Reduced-State) Tomography (7 Settings per Marginal):
To test the "whole-from-parts" hypothesis, the authors performed two-qubit tomography on the subsystems $\rho_{AB}$ and $\rho_{BC}$ . This required only 7 measurement settings per marginal (14 total). The resulting experimental reduced density matrices were then fed into the Diósi reconstruction protocol. This theoretical framework allows for the unique reconstruction of a generic three-party pure state from any two of its bipartite reduced density matrices by solving for undetermined phase factors.
Error Mitigation and Analysis:
- SPAM Mitigation: Readout errors were characterized using calibration matrices ( $F$ ) and mitigated via matrix inversion on the probability vectors.
- Physicality Correction: To ensure the reconstructed density matrices were physically valid (positive semidefinite), a spectral correction procedure was applied, setting negative eigenvalues to zero and renormalizing.
- Maximum Likelihood Estimation (MLE): To further validate results and avoid rank-deficiency issues, MLE was performed using a noisy reference state ( $\rho_{noisy} = 0.15 \frac{I}{8} + 0.85 |W\rangle\langle W|$ ) as input. Bootstrap analysis with 100 resamples was used to estimate statistical uncertainties.

Key Contributions

Efficient 17-Setting Scheme: The paper presents a specific 17-setting QST protocol for arbitrary three-qubit states, significantly reducing the measurement overhead compared to the conventional 63-setting Pauli scheme.
Experimental "Whole-from-Parts" Demonstration: The work provides a proof-of-principle experimental demonstration on an IBM quantum processor that a global pure three-qubit state can be reconstructed from its two-qubit marginals, validating the theoretical result that two reduced density matrices suffice to determine the whole for W states.
Comparative Fidelity Analysis: The study directly compares the fidelity of the state reconstructed via full tomography against the state reconstructed via the marginal approach.

Results
The experiments were conducted over five independent trials involving approximately 70,000 measurement shots.

Fidelity Comparison: The fidelity of the W state reconstructed from its two-qubit marginals ( $\rho^{II}_{ABC}$ $ρ_{A B C}^{I I}$ ) was consistently higher than that obtained from the full three-qubit tomography ( $\rho^{I}_{ABC}$ $ρ_{A B C}^{I}$ ).
- Mitigated Linear Inversion: Trial fidelities for the marginal approach ranged from ~0.96 to 0.99, while the full approach ranged from ~0.85 to 0.93.
- MLE Analysis: The mean fidelity for the 17-setting full tomography was $0.9313 \pm 0.0033$ , whereas the marginal tomography yielded $0.9359 \pm 0.0036$ .
Error Sources: The authors attribute the lower fidelity in the full tomography scheme to the increased circuit depth and the higher number of CNOT gates required, which amplifies SPAM and gate errors inherent to NISQ hardware. The marginal approach, utilizing shallower circuits, proved more robust against these noise sources.

Significance and Claims
The paper claims to demonstrate a practical advantage of subsystem-based tomography over full state tomography on current quantum hardware. By showing that reconstructing a global state from partial information (two-qubit marginals) yields higher fidelity than direct full-state reconstruction, the authors highlight a pathway to more efficient and reliable quantum state characterization. The work serves as a proof-of-concept that the theoretical result regarding the unique determination of pure states from marginals is experimentally viable on superconducting processors, offering a resource-efficient alternative for benchmarking and validating quantum states in the NISQ era. The authors remain modest, noting that the improved fidelity is likely a consequence of reduced circuit complexity rather than a fundamental superiority of the reconstruction algorithm itself, and that the results are specific to the noise characteristics of the ibm_osaka processor.

1. The "Whole from Parts" Trick

2. The Two Experiments

3. The Surprising Result

4. Cleaning Up the Mess

Summary

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