Quantum algorithms for Uhlmann transformation (2509.03619v1)

Abstract: Uhlmann's theorem is a central result in quantum information theory, associating the closeness of two quantum states with that of their purifications. This theorem well characterizes the fundamental task of transforming a quantum state into another state via local operations on its subsystem. The optimal transformation for this task is called the Uhlmann transformation, which has broad applications in various fields; however, its quantum circuit implementation and computational cost have remained unclear. In this work, we fill this gap by proposing quantum query and sample algorithms that realize the Uhlmann transformation in the form of quantum circuits. These algorithms achieve exponential improvements in computational costs, including query and sample complexities, over naive approaches based on state measurements such as quantum state tomography, under certain computational models. We apply our algorithms to the square root fidelity estimation task and particularly show that our approach attains a better query complexity than the prior state-of-the-art. Furthermore, we discuss applications to several information-theoretic tasks, specifically, entanglement transmission, quantum state merging, and algorithmic implementation of the Petz recovery map, providing a comprehensive evaluation of the computational costs.

• The paper introduces novel quantum algorithms for implementing the Uhlmann transformation, leveraging QSVT and DME to optimize resource usage.
• It demonstrates exponential improvements in query and sample complexities across purified and mixed access models compared to naive tomography.
• The algorithms enable practical applications in fidelity estimation, entanglement transmission, and quantum state merging with rigorous error analysis.

Quantum Algorithms for Uhlmann Transformation: Implementation and Implications

Introduction and Motivation

Uhlmann's theorem provides a fundamental operational link between the fidelity of two quantum states and the maximal overlap of their purifications, establishing the existence of an optimal local transformation—termed the Uhlmann transformation—that maps one purification to another. While the theorem is central to quantum information theory, its practical realization as a quantum circuit, and the associated computational costs, have remained largely unexplored. This work addresses this gap by constructing explicit quantum algorithms for the Uhlmann transformation, analyzing their resource requirements, and demonstrating exponential improvements over naive approaches in several computational models.

Algorithmic Framework and Computational Models

The paper develops quantum algorithms for the Uhlmann transformation under three standard computational models:

1. Purified Query Access Model: Access to unitaries that prepare purifications of the input states, with the ability to query these unitaries and their inverses.
2. Purified Sample Access Model: Access to multiple copies of the purified states.
3. Mixed Sample Access Model: Access to multiple copies of the mixed states only.

The algorithms leverage two key primitives: Quantum Singular Value Transformation (QSVT) and Density Matrix Exponentiation (DME). QSVT enables polynomial transformations of block-encoded matrices, while DME allows simulation of unitaries generated by density matrices using multiple copies of the state.

The Uhlmann transformation is implemented by constructing a block-encoding of the operator $\sqrt{\sigma}\sqrt{\rho}$ (or its generalization for purifications), and then applying a QSVT-based sign function to extract the optimal partial isometry. The approach avoids the exponential overhead of full state tomography and classical post-processing.

Main Results: Complexity and Implementation

Upper Bounds

The algorithms achieve the following resource scalings for implementing the Uhlmann transformation to accuracy $\delta$:

Model	Query/Sample Complexity (Best Scaling)
Purified Query Access	$$\mathcal{O}\left(\frac{1}{s_{\min}}\log\frac{1}{\delta}\right)$$
Purified Sample Access	$$\mathcal{O}\left(\frac{1}{\delta s_{\min}^2} \log^2\frac{1}{\delta}\right)$$
Mixed Sample Access	$$\mathcal{O}\left(\frac{d}{\delta^2 s_{\min}^3} \min\left\{\frac{1}{\rho_{\min}^2}+\frac{1}{\sigma_{\min}^2}, \frac{d^2}{\delta^2 s_{\min}^2}\right\}\right)$$

Here, $s_{\min}$ is the minimal nonzero singular value of $\sqrt{\sigma}\sqrt{\rho}$, and $d$ is the Hilbert space dimension. The algorithms require only a logarithmic number of ancilla qubits and polynomially many one- and two-qubit gates in the relevant parameters.

Key technical insight: By maintaining the purifying system throughout the process, the algorithms can apply optimal pure-state fidelity estimation subroutines, yielding significant improvements over prior methods that discard the purifying system.

Lower Bounds

A lower bound of $\Omega(k^{1/3})$ queries is established for the purified query access model, where $k$ is the rank of the reduced states. This is derived via a reduction to the mixedness testing problem and demonstrates a cubic gap between the best known upper and lower bounds in terms of rank scaling.

Comparison to Naive and Prior Approaches

Naive tomography-based methods require exponential resources in $d$ due to the need for full state reconstruction and classical post-processing. The prior explicit circuit construction [metger2023stateqipspace] achieves polynomial space but still requires exponential time and sample complexity. The present algorithms remove the exponential scaling in $d$ for the purified access models and reduce the dependence on accuracy $\delta$ to logarithmic or polynomial, depending on the model.

Applications

Fidelity Estimation

The algorithms enable efficient estimation of the square root fidelity between mixed states, with query/sample complexity matching or improving upon the best known results in the purified access models. For example, in the purified query model, the complexity is reduced to $\mathcal{O}(1/\delta)$, matching the lower bound for pure-state fidelity estimation.

Information-Theoretic Protocols

The Uhlmann transformation is a key subroutine in several quantum Shannon-theoretic tasks:

• Entanglement Transmission: The algorithm provides an explicit decoder achieving the optimal fidelity guaranteed by the decoupling approach, with resource requirements that can outperform generalized Yoshida-Kitaev and Petz-like decoders in certain parameter regimes.
• Quantum State Merging: The construction enables explicit realization of the optimal merging protocol, with sample complexity scaling favorably in the relevant parameters.

  Figure 1: A diagram of entanglement transmission. Alice applies a Haar random unitary $U$ to encode her system, which is then sent through a noisy channel. Bob applies the Uhlmann decoder to recover the entangled state.

  

  Figure 2: A diagram of quantum state merging. The Uhlmann transformation enables the transfer of Alice's subsystem to Bob, achieving the optimal merging fidelity.

Petz Recovery Map

The algorithm enables implementation of the Petz recovery map using only access to the quantum channel and a reference state, without requiring a Stinespring dilation unitary. This is achieved by using the Uhlmann transformation to simulate the required isometry, and then applying QSVT and DME to realize the full recovery map.

Implementation Considerations

• Block-Encoding Construction: In the purified query model, block-encodings are constructed via controlled unitaries. In the sample models, DME is used to simulate exponentials of the density matrices, and QSVT is applied to obtain polynomial approximations of the required functions (e.g., sign, square root).
• Error Accumulation: The algorithms carefully track error propagation through each subroutine, ensuring that the final output meets the desired accuracy.
• Resource Overheads: The main computational bottleneck in the mixed sample model is the canonical purification step, which introduces a $d$-dependent overhead. If more efficient purification algorithms are developed, this gap could be closed.

  Figure 3: A quantum circuit preparing the state $\Upsilon$ used in the DME-based block-encoding construction for the Uhlmann transformation.

  

  Figure 4: A diagram of the Stinespring dilation of a quantum channel, illustrating the relationship between the Uhlmann transformation and channel isometries.

Theoretical and Practical Implications

The results provide the first explicit, resource-efficient quantum algorithms for the Uhlmann transformation, enabling its use as a practical subroutine in quantum information processing. The exponential improvements in purified access models make these algorithms suitable for near-term and future quantum devices where full state tomography is infeasible.

The work also clarifies the computational separation between purified and mixed sample access models, suggesting that the hardness of purification is a key barrier to efficient mixed-state processing. This has implications for quantum complexity theory and cryptography, as efficient purification would undermine certain cryptographic assumptions.

Future Directions

• Closing the Upper-Lower Bound Gap: Tightening the cubic gap in rank scaling between upper and lower bounds remains an open problem.
• Efficient Purification Algorithms: Developing more efficient algorithms for canonical purification could bridge the gap between purified and mixed sample models.
• Complexity-Theoretic Connections: The results motivate further paper of the relationship between the Uhlmann transformation, quantum interactive proofs, and cryptographic primitives.
• Extensions to Other Transformations: The techniques may be adapted to other optimal local transformations and recovery maps in quantum information theory.

Conclusion

This work establishes a comprehensive framework for the efficient implementation of the Uhlmann transformation as a quantum circuit, with rigorous resource analysis and broad applicability to quantum information tasks. The algorithms leverage modern quantum algorithmic primitives to achieve exponential improvements over naive methods, and their integration into fidelity estimation, decoupling-based protocols, and recovery maps demonstrates their practical utility. The results also highlight fundamental open questions in quantum algorithmic complexity and the structure of quantum information processing.