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Higher-Order Quantum Operations
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abstract
An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
Forward citations
Cited by 13 Pith papers
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Supermaps on generalised theories
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Optimal complex conjugation of unknown isometry channels
The optimal n-use fidelity for complex conjugating an unknown isometry C^d→C^D is derived in closed form, with parallel protocols proven optimal among all general quantum superchannels.
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Causality from the spectrum: Emergence of causal order from process-matrix mereology
A process matrix can be unitarily transformed to a fixed-causal-order process exactly when its eigenvalue multiplicities are divisible by the final output dimension; generic high-dimensional spectra are close to such spectra.
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Sequential quantum processes with group symmetries
A canonical streaming circuit decomposition for (G×H)-invariant quantum combs is derived, and numerical optimization suggests a deterministic 7-query transposition protocol for qutrits that is reported as exact.
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Cloning Quantum Channels
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Storage, Scrambling, and Loss of Information in the Quantum Reservoir Computing Paradigm
The authors show that the process tensor of standard quantum reservoir computing reduces to a classical-quantum state, then use its Holevo quantities to diagnose information scrambling and fading memory in a six-qubit...
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Quantum Circuit Fragments and Link Products in Continuous Variables
A continuous-variable link product for quantum circuit fragments is defined, with an efficient Gaussian covariance-matrix algorithm and demonstrations on non-Markovian and agent-environment processes.
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Enhanced entanglement from quantum ergodicity
Ergodic quantum dynamics, via a non-demolition coupling, generates EPR states with lower purity and higher operator-transfer capacity than infinite-temperature scramblers, with a parametric advantage when initial stat...
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Agency under indefinite causality: operational eternalism in higher-order quantum theory
In indefinite-causal-order quantum theory, an observer is a perspective-dependent grouping of input/output data, with a new 'friendliness' criterion for causally compatible agents.
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Superstate Quantum Mechanics
Superstate Quantum Mechanics generalizes the quantum eigenvalue problem to a matrix-eigenproblem for unitary operators, unifying inverse problems, higher-order maps, and quadratic optimization.
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Language Model for Large-Text Transmission in Noisy Quantum Communications
A BERT-based post-processing module (PQC-BERT) reduces word and sentence error rates when English text is sent over noisy superdense-coding channels, though the protocol only works for natural language and the claimed...
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Tensor network methods for non-perturbative dynamics of open quantum systems
A review that organizes six numerically exact tensor-network methods for open quantum systems into a shared formalism, compares their convergence behavior, and lists software implementations.
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Process Tensor Approaches to Non-Markovian Quantum Dynamics
A review of process tensor methods for non-Markovian open quantum systems, showing how tensor network representations unify several existing approaches.
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