Efficient quantum tomography (1508.01907v2)

Abstract: In the quantum state tomography problem, one wishes to estimate an unknown $d$-dimensional mixed quantum state $\rho$, given few copies. We show that $O(d/\epsilon)$ copies suffice to obtain an estimate $\hat{\rho}$ that satisfies $|\hat{\rho} - \rho|_F^2 \leq \epsilon$ (with high probability). An immediate consequence is that $O(\mathrm{rank}(\rho) \cdot d/\epsilon^2) \leq O(d^2/\epsilon^2)$ copies suffice to obtain an $\epsilon$-accurate estimate in the standard trace distance. This improves on the best known prior result of $O(d^3/\epsilon^2)$ copies for full tomography, and even on the best known prior result of $O(d^2\log(d/\epsilon)/\epsilon^2)$ copies for spectrum estimation. Our result is the first to show that nontrivial tomography can be obtained using a number of copies that is just linear in the dimension. Next, we generalize these results to show that one can perform efficient principal component analysis on $\rho$. Our main result is that $O(k d/\epsilon^2)$ copies suffice to output a rank-$k$ approximation $\hat{\rho}$ whose trace distance error is at most $\epsilon$ more than that of the best rank-$k$ approximator to $\rho$. This subsumes our above trace distance tomography result and generalizes it to the case when $\rho$ is not guaranteed to be of low rank. A key part of the proof is the analogous generalization of our spectrum-learning results: we show that the largest $k$ eigenvalues of $\rho$ can be estimated to trace-distance error $\epsilon$ using $O(k^2/\epsilon^2)$ copies. In turn, this result relies on a new coupling theorem concerning the Robinson-Schensted-Knuth algorithm that should be of independent combinatorial interest.

• The paper demonstrates an efficient quantum tomography method that uses O(d/ε) copies to achieve accurate Frobenius norm estimation.
• It extends the approach to perform efficient principal component analysis on quantum states, enabling precise rank-k approximations with fewer copies.
• The study also enhances spectrum estimation by accurately determining the top eigenvalues with a copy complexity of O(k²/ε²), improving verification procedures.

Efficient Quantum Tomography: A Summary

The paper "Efficient Quantum Tomography" by Ryan O'Donnell and John Wright addresses the fundamental problem of estimating an unknown $d$-dimensional mixed quantum state $\rho$ using quantum state tomography. Quantum tomography is an essential process for the verification of quantum technologies and the experimental detection of entanglement. The authors establish that $O(d/\eps)$ copies of the quantum state are sufficient to obtain an estimate $\hat{\rho}$ satisfying the Frobenius norm error $\|\hat{\rho} - \rho\|_F^2 \leq \eps$ with high probability. This result significantly reduces the previously known copy requirement for full tomography from $O(d^3/\eps^2)$ to a linear dependence on the dimension $d$.

Key Contributions

1. Improved Sample Complexity: The paper sets a new standard by showing that $O(d/\eps)$ copies suffice for obtaining an ${\eps}$-accurate estimate in Frobenius error, contrasting the earlier requirement of $O(d^3/\eps^2)$ copies. For the standard trace distance, the requirement is reduced to $O(rank(\rho) \cdot d/\eps^2)$ copies, which translates to $O(d^2/\eps^2)$ copies in the worst-case scenario. This improvement is significant for practical quantum computation scenarios where state preparation and measurement resources are limited.
2. Principal Component Analysis (PCA) Generalization: O'Donnell and Wright extend the proposed tomography methods to enable efficient PCA on quantum states. They provide bounds indicating that $O(k d/\eps^2)$ copies are sufficient to obtain a rank-$k$ approximation where the trace distance error is at most $\eps$ more than that of the best rank-$k$ approximator. This generalization allows the technique to apply even when the quantum state $\rho$ is not assured to be of low rank.
3. Spectrum Estimation Enhancement: One of the innovative results is on spectrum-learning, specifically that the largest $k$ eigenvalues of $\rho$ can be estimated to a trace-distance error $\eps$ using $O(k^2/\eps^2)$ copies. This finding emphasizes the efficiency of the proposed approach in capturing the most significant eigencomponents of quantum states without dependence on the overall dimension $d$.
4. Robust Combinatorial Couplings: The research underpins a new coupling theorem involving the Robinson–Schensted–Knuth (RSK) algorithm. This theorem serves as an essential component of the proposed tomography strategy and holds potential for broader combinatorial and theoretical implications.

Implications and Future Directions

The results presented in the paper have immediate implications for the efficiency of quantum state reconstruction, greatly impacting both theoretical quantum information science and practical quantum computing applications. The reduced copy complexity means that quantum state estimation can be executed with fewer resources, which is critical for scaling up quantum experiments and operations.

Moreover, the methods established for efficient PCA open avenues for further research into structured quantum states, such as those representing quantum many-body systems or arising in quantum machine learning contexts. This suggests future work could explore the integration with quantum algorithms that benefit from PCA, such as variational quantum algorithms and quantum support vector machines.

Finally, the coupling advances in the context of the RSK algorithm encourage further exploration into the intersection of quantum information theory and combinatorics. This could lead to new quantum algorithms that leverage combinatorial structures for improved computational performance.

Overall, this work by O'Donnell and Wright represents a notable advancement in quantum tomography, offering theoretical insights and practical methods that are poised to enhance the efficiency and accuracy of quantum state estimation.