Quantum.Synthesis.Clifford

This module provides an efficient symbolic representation of the Clifford group on one qubit. This group is generated by S, H, and the scalar ω = e^iπ/4. It has 192 elements.

The Clifford group

data Clifford

A type representing single-qubit Clifford operators.

Instances

Eq Clifford

Methods

(==) :: Clifford -> Clifford -> Bool

(/=) :: Clifford -> Clifford -> Bool

Ord Clifford

Methods

compare :: Clifford -> Clifford -> Ordering

(<) :: Clifford -> Clifford -> Bool

(<=) :: Clifford -> Clifford -> Bool

(>) :: Clifford -> Clifford -> Bool

(>=) :: Clifford -> Clifford -> Bool

max :: Clifford -> Clifford -> Clifford

min :: Clifford -> Clifford -> Clifford

Show Clifford

Methods

showsPrec :: Int -> Clifford -> ShowS

show :: Clifford -> String

showList :: [Clifford] -> ShowS

ToClifford Clifford

Methods

to_clifford :: Clifford -> Clifford

ToGates Clifford

Methods

to_gates :: Clifford -> [Gate]

Constructors

clifford_X :: Clifford

The Pauli X-gate as a Clifford operator.

clifford_Y :: Clifford

The Pauli Y-gate as a Clifford operator.

clifford_Z :: Clifford

The Pauli Z-gate as a Clifford operator.

clifford_H :: Clifford

The Hadamard gate as a Clifford operator.

clifford_S :: Clifford

The Clifford operator S.

clifford_SH :: Clifford

The Clifford operator SH.

clifford_E :: Clifford

The Clifford operator E = HS³ω³. This operator is uniquely determined by the properties E³ = I, EXE⁻¹ = Y, EYE⁻¹ = Z, and EZE⁻¹ = X.

clifford_W :: Clifford

The Clifford operator ω = e^iπ/4.

class ToClifford a where

A type class for things that can be exactly converted to a Clifford operator. One particular instance of this is String, so that Clifford operators can be denoted, e.g.,

to_clifford "-iX"

The valid characters for such string conversions are "XYZHSEIWi-".

Minimal complete definition

to_clifford

Methods

to_clifford :: a -> Clifford

Convert any suitable thing to a Clifford operator.

Instances

ToClifford Char

Methods

to_clifford :: Char -> Clifford

ToClifford Axis

Methods

to_clifford :: Axis -> Clifford

ToClifford Clifford

Methods

to_clifford :: Clifford -> Clifford

ToClifford a => ToClifford [a]

Methods

to_clifford :: [a] -> Clifford

Deconstructors

clifford_decompose :: ToClifford a => a -> (Int, Int, Int, Int)

Given a Clifford operator U, return (a, b, c, d) such that

• U = E^aX^bS^cω^d,
• a ∈ {0, 1, 2}, b ∈ {0, 1}, c ∈ {0, …, 3}, and d ∈ {0, …, 7}.

Here, E = HS³ω³. Note that E, X, S, and ω have order 3, 2, 4, and 8, respectively. Moreover, each Clifford operator can be uniquely represented as above.

data Axis

A axis is either I, H, or SH.

Constructors

Axis_I

Axis_H

Axis_SH

Instances

Eq Axis

Methods

(==) :: Axis -> Axis -> Bool

(/=) :: Axis -> Axis -> Bool

Show Axis

Methods

showsPrec :: Int -> Axis -> ShowS

show :: Axis -> String

showList :: [Axis] -> ShowS

ToClifford Axis

Methods

to_clifford :: Axis -> Clifford

ToGates Axis

Methods

to_gates :: Axis -> [Gate]

clifford_decompose_coset :: ToClifford a => a -> (Axis, Int, Int, Int)

Given a Clifford operator U, return (K, b, c, d) such that

• U = KX^bS^cω^d,
• K ∈ {I, H, SH}, b ∈ {0, 1}, c ∈ {0, …, 3}, and d ∈ {0, …, 7}.

Group operations

clifford_id :: Clifford

The identity Clifford operator.

clifford_mult :: Clifford -> Clifford -> Clifford

Clifford multiplication.

clifford_inv :: ToClifford a => a -> Clifford

Clifford inverse.

Conjugation by T

clifford_tconj :: Clifford -> (Axis, Clifford)

Given a Clifford gate C, return an axis K ∈ {I, H, SH} and a Clifford gate C' such that

• CT = KTC'.