Qubit Graphical Calculi

Diagrams are read bottom-to-top,

and here are some sample projects for Quantomatic

General Form
Universal Quantum Computation
Approximately Universal
Stabilizer
Other
ZW_R
W
X
  • a commutative ring
  • Complete for any commutative ring
  • Two types of wire-crossing
  • "The" ZW-calculus refers to either or to phase-free , which are equivalent
W
X
  • Used for completeness
W
X
  • Used for completeness
Z
X
Hadamard
Lambda box
  • the phase group
  • (lambda boxes are only used in the universal segment)
  • Empty boxes are assumed to be Hadamard nodes
Z
X
Hadamard
Z
X
Hadamard
Z
X
Hadamard
Z
X
Hadamard
Circuits
CNOT
Z rotations
Hadamard
  • Various sets of generators exist
  • It can be convenient to distinguish between up-down rails and side-side bridges
Exact Universal Circuits
CNOT
Z rotations
Hadamard
Clifford+T Circuits
CNOT
S
Hadamard
T
  • and are names for those particular Z-rotations.
Stabilizer Circuits
CNOT
S
Hadamard
The Clifford Group
S
Hadamard
ZH
Z
Generalised Hadamard
  • can be in any commutative ring, but we assume
  • If then the phase can be omitted.
  • The generalised Hadamard is not to be confused with the lambda-boxes of ZX
ZH
Z
Generalised Hadamard