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

Weizhou Cai, Zi-Jie Chen, Xuanqiang Zhao, Xin Wang, Guang-Can Guo, Luyan Sun, Chang-Ling Zou

TL;DR

Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings, and validate the efficiency and versatility of the proposed framework.

Abstract

Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.

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

TL;DR

Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings, and validate the efficiency and versatility of the proposed framework.

Abstract

Quantum control has been a cornerstone of quantum information science, driving major advances in quantum computing, quantum communication, and quantum sensing. Over the years, it has enabled the implementation of arbitrary completely positive and trace-preserving (CPTP) maps; an important next step is to extend control to Hermitian-preserving and trace-preserving (HPTP) maps, which underpin applications such as entanglement detection, quantum error mitigation, quantum simulation, and quantum machine learning. Here we present an efficient and fully constructive method for implementing arbitrary HPTP maps. Unlike existing methods that decompose an HPTP map into multiple CPTP maps or approximate it using bipartite Hamiltonians with large Hilbert spaces, our approach compiles a target HPTP map into a single executable CPTP map whose Kraus rank is guaranteed to be no larger than the intrinsic rank of the target HPTP map plus one, followed by simple classical post-processing. Numerical results for inverse noise channels used in quantum error mitigation, including bosonic photon loss, confirm substantial reductions in resources and highlight scalability in higher-dimensional settings. Together with our numerical benchmarks, these results validate the efficiency and versatility of the proposed framework, opening a route to broader quantum-information applications enabled by HPTP processing.
Paper Structure (7 equations, 3 figures)

This paper contains 7 equations, 3 figures.

Figures (3)

  • Figure 1: Concept for HPTP maps and their realization. (a) Hierarchy of unitary operations, CPTP maps, and HPTP maps. (b) Binary tree-structured quantum circuit for implementing arbitrary CPTP maps. (c) Quantum circuit for each leaf of the binary tree-structured circuit, consisting a joint unitary operation on a composed quantum system, a projective measurement on the ancilla qubit, and a resetting process for the ancilla qubit. (d) Binary tree-structured quantum circuit for implementing arbitrary HPTP maps. (e) Binary tree-structured quantum circuit for implementing successive HPTP maps.
  • Figure 2: Simulation of QEM through implementing HPTP maps. (a) Mitigation of quantum noise for a single qubit. (b) Mitigation of quantum noise for two qubits.
  • Figure 3: Performance of QEM on a bosonic mode. (a) Variance in mitigating photon loss errors as a function of system dimension. (b) Kraus rank for mitigating photon loss errors as a function of system dimension. The error rate is fixed at $\delta=0.2$ for different system dimensions (see Supplementary Materials SI for the definition of the error rate).