Concept of 'quantum resource'

Quantum resource theory has recently emerged as a powerful framework for characterizing quantum states. The underlying idea is that a resource is a discriminant for partitioning all possible quantum states into two groups: free states and resource states — i.e., states that are resourceful, or that cost resources to be prepared. The operations that leave the free set invariant are referred to as free operations.  

In quantum physics, the paradigmatic example for a quantum resource is entanglement, but also concepts such as nonlocality, quantum coherence, and quantum correlations have been investigated within the framework of resource theory. Another valuable resource in quantum information is nonstabilizerness (or magic) which represents the hardness to simulate a quantum state classically. Indeed, without magic a quantum computer cannot do anything a classical computer cannot do. With respect to magic, stabilizer states correspond to the free states, and Clifford unitaries to free operations.  

Related reading:

• Chitambar & Gour, Rev. Mod. Phys. 91, 025001 (2019)

Clifford group

In quantum computing, the Clifford group encompasses a set of unitary operations, generated by Hadamard (H), π/2 rotation (S), and controlled NOT (CNOT) gates. The Clifford group has a few interesting properties, some of which are listed below, in no particular order:

• A formal definition of the Clifford group is that it maps the set of N-fold Pauli group (whose elements are Pauli strings for a N-qubit quantum register) into itself. In a group-theoretical language, the Clifford group is the normalizer of the Pauli group. As a consequence, Clifford operations generate periodic orbits in the space of Pauli strings. 

• The Gottesmann-Knill theorem asserts that classical hardware can simulate the Clifford group efficiently (i.e., in polynomial time). The Clifford group and the related stabilizer formalism are relevant in the context of, e.g., quantum error correction since they allow for error mitigation and fault-tolerant quantum computation.  

• Adding a single non-Clifford gate (e.g., a T gate, or π/4 rotation) yields a set suitable for universal quantum computing, i.e., it can approximate any unitary operation with arbitrary precision.  

Related reading:

• Gottesman, Phys. Rev. A 57, 127(1998)

Clifford orbits — a graph theoretical perspective

The problem of finding Clifford orbits can also be formalized within the framework of graph theory. Any Clifford operator can be represented as the adjacency matrix of a directed graph (a.k.a.: digraph) in which unsigned Pauli strings are the vertices (V) and the Clifford operations are directed edges (E). Then, the adjacency matrix A = {aᵢⱼ} is a permutation matrix, where aᵢⱼ connects the pair of vertices (i, j) with a directed edge i → j mapping string i to string j. 

Remark: in order to map the Pauli group onto itself, the elements of the permutation matrix can be ±1 (whereas the phase of ±i can be disregarded since Clifford operators act on strings via conjugation). Assuming a Clifford orbit is closed when string S is mapped onto ±S, corresponds to introducing an orbit parity, effectively shifting all eigenphases of C by π/4 in the case of odd-parity orbits — see Szombathy et al., Phys. Rev. Research 7, 043080 (2025).

For any Clifford operator, orbits do not cross (i.e., two distinct orbits have zero common Pauli strings) and an orbit of length L: {S₁, S₂, ... Sⱼ, ... Sₗ} corresponds to a size |L| component of the graph, i.e., a connected subgraph that is not part of any larger connected subgraph. Hence, finding the orbits of a Clifford operator corresponds to determining the connected component(s) of the associated digraph, which can be done efficiently with e.g., Tarjan's algorithm — complexity O(|E|+|V|).  A graph's number of connected components is an important (topological) invariant, the zero-th Betti number.