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IMPLEMENTING STABILIZER CODES FOR CIRCUIT SIMULATION IN JULIA

Varun Seshadri

Supervised by Prof. Dr. Christian Mendl

09 March 2021

SCCS Kolloqium

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STABILISER FORMALISM

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Pauli And Stabilizer Groups

Pauli Group

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Stabilizer Group

Properties of the Pauli Group:

  1. It is a nonabelian group.
  2. The elements square to
  3. The elements either commute or anti-commute.
  4. The elements are unitary

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Properties of the Stabilizer Group:

  1. It is an abelian group.
  2. The elements square to
  3. The elements either commute or anti-commute.
  4. Since they square to , they have eigenvalues to +1 or -1.
  5. Usually specified by a minimal set of Generators.

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Example:

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Stabilizer Subspace

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Logical Operators for Stabilizer Codes

  • The set of elements are the logical Pauli operators of the k encoded qubits of the subsystem code.
  • These operators preserve code subspace.

Example

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Stabilizer Code Examples

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3 qubit Bit Flip code

3 qubit Phase Flip code

Shor Code

Smallest Code for 1 qubit error

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Gates on Stabilizer Codes

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The Clifford Group:

The Clifford group is the normalizer of the Pauli group.

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Gottesmann Knill Theorem

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GRAPH STATES

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Definition

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Example

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  • The application of Controlled-Z(CZ) gates to an input state initialized as completes the construction.
  • The arrangement of CZs is associated with the adjacency matrix Γ(G) of a given graph G�

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Local Complementation

 

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IMPLEMENTATION

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State representation in simulator.

 

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Gates

 

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Local Complementation Revisited.

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  • The Clifford group is also generated by
  • Every local Clifford operator can be expressed as product of at most 5 of these operators.

A Local Complementation on a graph G, can be represented b a local Clifford operator.

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Two Qubit Gates

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Two Qubit Gates

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Sub Case 2.2: At least one of the operand vertices isisolated or only connected to the other operand vertex

 

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Two Qubit Gates

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Sub Case 2.2: At least one of the operand vertices is isolated or only connected to the other operand vertex

 

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THANK YOU

In particular to Prof. Christian Mendl, Prof. Stefan Filipp and Keefe Huang