Higher-order quantum transformations

From quantum circuit design to indefinite causal order

Presented at La Ricotta Summer School: The Quantum in the Haystack (Castro Cadelas, Spain, 2026).

Course dates: 6–10 July 2026.

This course introduces the formalism of higher-order quantum transformations, a framework for describing transformations between quantum operations. We will see how higher-order transformations provide a natural characterization of quantum circuits while also enabling the study of quantum computation beyond the quantum circuit model, including scenarios with indefinite causal order. The course explores applications of this formalism to the design of optimal protocols and the derivation of fundamental no-go theorems for quantum information-processing tasks such as unitary transformations, channel discrimination, and parameter estimation. One lecture and one tutorial are devoted to semidefinite programming, a powerful analytical and numerical optimization tool that plays a central role in the higher-order framework and is widely used throughout quantum information science. We conclude by examining the problem of quantifying the resources required to simulate higher-order quantum operations using standard quantum circuits, and by surveying current challenges and open research directions in the field.

Course materials

View or download the PDFs below, listed in teaching order.

  1. Course outline (PDF)
  2. Lecture I — The higher-order quantum formalism: Channels and superchannels (PDF)
  3. Tutorial I — Case study: Quantum channel discrimination as a higher-order problem (PDF)
  4. Lecture II — General superchannels: Computation with indefinite causal order (PDF)
  5. Lecture III — Semidefinite programming (PDF)
  6. Tutorial II — Case study: Quantum channel discrimination as a semidefinite programming problem (PDF)

Back to Courses