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

- Complete
- "The" ZX-calculus

Z

X

Hadamard

Z

X

Hadamard

Z

X

Hadamard

- Complete
- Spekkens' Toy Theory

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

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