Generators and Relations for Real Stabilizer Operators

  title={Generators and Relations for Real Stabilizer Operators},
  author={Justin Makary and Neil J. Ross and Peter Selinger},
Real stabilizer operators, which are also known as real Clifford operators, are generated, through composition and tensor product, by the Hadamard gate, the Pauli Z gate, and the controlled-Z gate. We introduce a normal form for real stabilizer circuits and show that every real stabilizer operator admits a unique normal form. Moreover, we give a finite set of relations that suffice to rewrite any real stabilizer circuit to its normal form. 


Generators and relations for n-qubit Clifford operators
  • P. Selinger
  • Mathematics, Computer Science
  • Log. Methods Comput. Sci.
  • 2015
A rewrite system by which any Clifford circuit can be reduced to normal form is presented and it is proved that every Clifford operator has a unique normal form. Expand
A finite presentation of CNOT-dihedral operators
It is shown that in the presence of certain structural rules only finitely many circuit identities are required to reduce an arbitrary CNOT-dihedral circuit to its normal form. Expand
Pivoting makes the ZX-calculus complete for real stabilizers
An angle-free version of the ZX-calculus is derived and it is shown that it is complete for real stabilizer quantum mechanics and does not imply local complementation of graph states. Expand
Circuit Relations for Real Stabilizers: Towards TOF+H
This work completes the category CNOT generated by the controlled not gate and the computational ancillary bits, presented by circuit relations, to the real stabilizer fragment of quantum mechanics, and discusses how this could potentially lead to a complete axiomatization, in terms of circuit Relations, for the approximately universal fragment ofquantum mechanicsgenerated by the Toffoli gate, Hadamard gate and computational anCillary bits. Expand
A ZX-Calculus with Triangles for Toffoli-Hadamard, Clifford+T, and Beyond
A ZX-calculus augmented with triangle nodes is considered, and the form of the matrices it represents is precisely shown, and an axiomatisation is provided which makes the language complete for the Toffoli-Hadamard quantum mechanics. Expand
The ZX−calculus is complete for stabilizer quantum mechanics
The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitaryExpand
ZH: A Complete Graphical Calculus for Quantum Computations Involving Classical Non-linearity
We present a new graphical calculus that is sound and complete for a universal family of quantum circuits, which can be seen as the natural string-diagrammatic extension of the approximatelyExpand
The Heisenberg Representation of Quantum Computers
Since Shor`s discovery of an algorithm to factor numbers on a quantum computer in polynomial time, quantum computation has become a subject of immense interest. Unfortunately, one of the key featuresExpand
Improved Simulation of Stabilizer Circuits
The Gottesman-Knill theorem, which says that a stabilizer circuit, a quantum circuit consisting solely of controlled-NOT, Hadamard, and phase gates can be simulated efficiently on a classical computer, is improved in several directions. Expand
The Invariants of the Clifford Groups
A simpler proof of Runge@apos;s 1996 result that the space of invariants for C, the automorphism group of the Barnes-Wall lattice Lm, of degree 2k is spanned by the complete weight enumerators of the codes C, where C ranges over all binary self-dual codes of length 2k. Expand