Efficient implementation of arbitrary Hermitian-preserving and trace-preserving maps

Technical Summary: Efficient Implementation of Arbitrary Hermitian-Preserving and Trace-Preserving Maps

Problem Statement
Quantum control has successfully enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps, which describe physically realizable quantum processes. However, a critical frontier remains the implementation of Hermitian-preserving and trace-preserving (HPTP) maps. Unlike CPTP maps, HPTP maps do not guarantee positivity, meaning they can represent non-physical processes such as partial transposes (used for entanglement detection), inverse noise channels (central to quantum error mitigation, or QEM), and non-CP intermediate maps in non-Markovian dynamics. Existing methods to realize HPTP maps face significant limitations:

1. Decomposition approaches: Decomposing an HPTP map into two CPTP maps followed by post-processing requires semi-definite programming for optimization and has been experimentally limited to single-qubit systems.
2. Hamiltonian approaches: Constructing bipartite Hamiltonians to approximate HPTP maps requires an ancilla system with the same dimension as the target system, leading to high resource costs.

Methodology
The authors propose a fully constructive, resource-efficient method to implement arbitrary HPTP maps using a single executable CPTP map followed by simple classical post-processing. The core of the method involves three steps:

1. Normalization: Given a target HPTP map $\mathcal{N}$ with Kraus operators $\{K_i\}$ and signs $\alpha(i) \in \{1, -1\}$ (derived from the eigenvalues of the Choi matrix), the authors normalize the sum $\sum K_i^\dagger K_i$ by a factor $\gamma$ (the largest eigenvalue of this sum). This yields rescaled operators $\tilde{K}_i = K_i / \sqrt{\gamma}$.
2. Completion: To ensure the resulting map is a valid CPTP map, an additional Kraus operator is introduced: $\tilde{K}_{r+1} = (I - \sum_{i=1}^r \tilde{K}_i^\dagger \tilde{K}_i)^{1/2}$. This guarantees the completeness condition $\sum_{i=1}^{r+1} \tilde{K}_i^\dagger \tilde{K}_i = I$. The resulting CPTP map, $\mathcal{E}_\mathcal{N}$, has a Kraus rank of at most $r+1$, where $r$ is the intrinsic rank of the target HPTP map.
3. Post-processing: The physical circuit implements $\mathcal{E}_\mathcal{N}$. To recover the expectation value of an observable $O$ under the original HPTP map $\mathcal{N}$, the measurement outcomes are classically post-processed. Specifically, outcomes corresponding to the first $r$ Kraus branches are weighted by $\alpha(i)\gamma$, while outcomes corresponding to the $(r+1)$-th branch are weighted by zero.

Circuit Implementation
The method utilizes a binary-tree-structured quantum circuit, previously developed for CPTP maps, to implement the constructed CPTP map $\mathcal{E}_\mathcal{N}$.

• Resources: The protocol requires only a single two-level ancilla qubit (reused via measurement, reset, and feedback) and a classical register to store measurement outcomes.
• Depth: The circuit depth scales as $\lceil \log_2(r+1) \rceil$, offering logarithmic scaling with the Kraus rank.
• Successive Maps: For sequences of HPTP maps, the authors propose concatenating the corresponding quantum circuits and performing a single post-processing step at the end, avoiding the need to re-prepare quantum states between steps.

Key Results
The paper validates the method through numerical simulations and theoretical analysis:

• Quantum Error Mitigation (QEM): The method was applied to invert noise channels (amplitude damping, depolarizing, and dephasing) for single and two-qubit systems. Numerical results show substantial reductions in sampling overhead (variance) compared to previous decomposition methods.
• Bosonic Systems: Simulations on bosonic photon loss channels demonstrate that the proposed method scales linearly with system dimension $d$ (Kraus rank $\approx d+1$). In contrast, previous decomposition methods result in nearly full-rank CPTP maps ($d^2$), rendering them impractical for high-dimensional systems.
• Variance Analysis: The authors derive an analytical expression for the single-shot variance of the estimator. They show that the sampling overhead is transparent and analytically guaranteed, contrasting with methods relying on numerical optimization where overhead is determined by coefficients of decomposed maps.
• Haar-Averaged Variance: Theoretical predictions for Haar-random input states and observables confirm the efficiency of the method, particularly for unital channels.

Significance and Claims
The authors claim that this work provides a practical route to broader quantum information applications enabled by HPTP processing. Key advantages highlighted include:

• Resource Efficiency: The method requires significantly fewer quantum resources (specifically, a single two-level ancilla) compared to Hamiltonian-based approaches and avoids the high-dimensional ancilla requirements of previous methods.
• Constructive Nature: The protocol is fully constructive and deterministic, eliminating the need for numerical semi-definite programming optimization.
• Scalability: The logarithmic circuit depth and linear scaling of Kraus rank with system dimension make the approach suitable for near-term hardware and high-dimensional settings.
• Versatility: By enabling the efficient realization of non-physical maps, the framework supports applications in entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning.

The paper concludes that these results validate the efficiency and versatility of the proposed framework, offering a scalable solution for implementing arbitrary HPTP maps without the resource bottlenecks of prior techniques.