Quantum.Synthesis.Ring

This module provides type classes for rings. It also provides several specific instances of rings, such as the ring ℤ₂ of integers modulo 2, the ring ℚ of rational numbers, the ring ℤ[½] of dyadic fractions, the ring ℤ[i] of Gaussian integers, the ring ℤ[√2] of quadratic integers with radix 2, and the ring ℤ[ω] of cyclotomic integers of degree 8.

Rings

type Ring a = Num a

A type class to denote rings. We make Ring a synonym of Haskell's Num type class, so that we can use the usual notation +, -, * for the ring operations. This is not a perfect fit, because Haskell's Num class also contains two non-ring operations abs and signum. By convention, for rings where these notions don't make sense (or are inconvenient to define), we set abs x = x and signum x = 1.

Rings with particular elements

We define several classes of rings with special elements.

Rings with ½

class Ring a => HalfRing a where

A type class for rings that contain ½.

Minimal complete definition: half. The default definition of fromDyadic uses the expression a*half^n. However, this can give potentially bad round-off errors for fixed-precision types where the expression half^n can underflow. For such rings, one should provide a custom definition, for example by using a/2^n instead.

Minimal complete definition

half

Methods

half :: a

The value ½.

fromDyadic :: Dyadic -> a

The unique ring homomorphism from ℤ[½] to any HalfRing. This exists because ℤ[½] is the free HalfRing.

Instances

HalfRing Double

Defined in Quantum.Synthesis.Ring

Methods

half :: Double

fromDyadic :: Dyadic -> Double

HalfRing Float

Defined in Quantum.Synthesis.Ring

Methods

half :: Float

fromDyadic :: Dyadic -> Float

HalfRing Rational

Defined in Quantum.Synthesis.Ring

Methods

half :: Rational

fromDyadic :: Dyadic -> Rational

HalfRing Rationals

Defined in Quantum.Synthesis.Ring

Methods

half :: Rationals

fromDyadic :: Dyadic -> Rationals

HalfRing Dyadic

Defined in Quantum.Synthesis.Ring

Methods

half :: Dyadic

fromDyadic :: Dyadic -> Dyadic

HalfRing SymReal

Defined in Quantum.Synthesis.Ring.SymReal

Methods

half :: SymReal

fromDyadic :: Dyadic -> SymReal

(HalfRing a, RealFloat a) => HalfRing (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

half :: Complex a

fromDyadic :: Dyadic -> Complex a

Precision e => HalfRing (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

half :: FixedPrec e

fromDyadic :: Dyadic -> FixedPrec e

HalfRing a => HalfRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

half :: Omega a

fromDyadic :: Dyadic -> Omega a

HalfRing a => HalfRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

half :: Cplx a

fromDyadic :: Dyadic -> Cplx a

(Eq a, HalfRing a) => HalfRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

half :: RootTwo a

fromDyadic :: Dyadic -> RootTwo a

(HalfRing a, Nat n) => HalfRing (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

half :: Matrix n n a

fromDyadic :: Dyadic -> Matrix n n a

Rings with √2

class Ring a => RootTwoRing a where

A type class for rings that contain √2.

Minimal complete definition: roottwo. The default definition of fromZRootTwo uses the expression x+roottwo*y. However, this can give potentially bad round-off errors for fixed-precision types, where the expression roottwo*y can be vastly inaccurate if y is large. For such rings, one should provide a custom definition.

Minimal complete definition

roottwo

Methods

roottwo :: a

The square root of 2.

fromZRootTwo :: RootTwoRing a => ZRootTwo -> a

The unique ring homomorphism from ℤ[√2] to any ring containing √2. This exists because ℤ[√2] is the free such ring.

Instances

RootTwoRing Double

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Double

fromZRootTwo :: ZRootTwo -> Double

RootTwoRing Float

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Float

fromZRootTwo :: ZRootTwo -> Float

RootTwoRing SymReal

Defined in Quantum.Synthesis.Ring.SymReal

Methods

roottwo :: SymReal

fromZRootTwo :: ZRootTwo -> SymReal

(RootTwoRing a, RealFloat a) => RootTwoRing (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Complex a

fromZRootTwo :: ZRootTwo -> Complex a

Precision e => RootTwoRing (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

roottwo :: FixedPrec e

fromZRootTwo :: ZRootTwo -> FixedPrec e

Ring a => RootTwoRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Omega a

fromZRootTwo :: ZRootTwo -> Omega a

RootTwoRing a => RootTwoRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Cplx a

fromZRootTwo :: ZRootTwo -> Cplx a

(Eq a, Ring a) => RootTwoRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: RootTwo a

fromZRootTwo :: ZRootTwo -> RootTwo a

(RootTwoRing a, Nat n) => RootTwoRing (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

roottwo :: Matrix n n a

fromZRootTwo :: ZRootTwo -> Matrix n n a

Rings with 1/√2

class (HalfRing a, RootTwoRing a) => RootHalfRing a where

A type class for rings that contain 1/√2.

Minimal complete definition: roothalf. The default definition of fromDRootTwo uses the expression x+roottwo*y. However, this can give potentially bad round-off errors for fixed-precision types, where the expression roottwo*y can be vastly inaccurate if y is large. For such rings, one should provide a custom definition.

Minimal complete definition

roothalf

Methods

roothalf :: a

The square root of ½.

fromDRootTwo :: RootHalfRing a => DRootTwo -> a

The unique ring homomorphism from D[√2] to any ring containing 1/√2. This exists because D[√2] = ℤ[1/√2] is the free such ring.

Instances

RootHalfRing Double

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Double

fromDRootTwo :: DRootTwo -> Double

RootHalfRing Float

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Float

fromDRootTwo :: DRootTwo -> Float

RootHalfRing SymReal

Defined in Quantum.Synthesis.Ring.SymReal

Methods

roothalf :: SymReal

fromDRootTwo :: DRootTwo -> SymReal

(RootHalfRing a, RealFloat a) => RootHalfRing (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Complex a

fromDRootTwo :: DRootTwo -> Complex a

Precision e => RootHalfRing (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

roothalf :: FixedPrec e

fromDRootTwo :: DRootTwo -> FixedPrec e

HalfRing a => RootHalfRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Omega a

fromDRootTwo :: DRootTwo -> Omega a

RootHalfRing a => RootHalfRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Cplx a

fromDRootTwo :: DRootTwo -> Cplx a

(Eq a, HalfRing a) => RootHalfRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: RootTwo a

fromDRootTwo :: DRootTwo -> RootTwo a

(RootHalfRing a, Nat n) => RootHalfRing (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

roothalf :: Matrix n n a

fromDRootTwo :: DRootTwo -> Matrix n n a

Rings with i

class Ring a => ComplexRing a where

A type class for rings that contain a square root of -1.

Methods

i :: a

The complex unit.

Instances

(Ring a, RealFloat a) => ComplexRing (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

i :: Complex a

Ring a => ComplexRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

i :: Omega a

Ring a => ComplexRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

i :: Cplx a

(Eq a, ComplexRing a) => ComplexRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

i :: RootTwo a

(ComplexRing a, Nat n) => ComplexRing (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

i :: Matrix n n a

Rings with ω

class Ring a => OmegaRing a where

A type class for rings that contain a square root of i, or equivalently, a fourth root of −1.

Methods

omega :: a

The square root of i.

Instances

(RealFloat a, RootHalfRing a) => OmegaRing (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: Complex a

Ring a => OmegaRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: Omega a

(Ring a, RootHalfRing a) => OmegaRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: Cplx a

(Eq a, ComplexRing a, HalfRing a) => OmegaRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: RootTwo a

(OmegaRing a, Nat n) => OmegaRing (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

omega :: Matrix n n a

Rings with particular automorphisms

Rings with complex conjugation

class Adjoint a where

A type class for rings with complex conjugation, i.e., an automorphism mapping i to −i.

When instances of this type class are vectors or matrices, the conjugation also exchanges the roles of rows and columns (in other words, it is the adjoint).

For rings that are not complex, the conjugation can be defined to be the identity function.

Methods

adj :: a -> a

Compute the adjoint (complex conjugate transpose).

Instances

Adjoint Double

Defined in Quantum.Synthesis.Ring

Methods

adj :: Double -> Double

Adjoint Float

Defined in Quantum.Synthesis.Ring

Methods

adj :: Float -> Float

Adjoint Int

Defined in Quantum.Synthesis.Ring

Methods

adj :: Int -> Int

Adjoint Integer

Defined in Quantum.Synthesis.Ring

Methods

adj :: Integer -> Integer

Adjoint Rational

Defined in Quantum.Synthesis.Ring

Methods

adj :: Rational -> Rational

Adjoint Rationals

Defined in Quantum.Synthesis.Ring

Methods

adj :: Rationals -> Rationals

Adjoint Dyadic

Defined in Quantum.Synthesis.Ring

Methods

adj :: Dyadic -> Dyadic

Adjoint Z2

Defined in Quantum.Synthesis.Ring

Methods

adj :: Z2 -> Z2

Adjoint SymReal

Defined in Quantum.Synthesis.Ring.SymReal

Methods

adj :: SymReal -> SymReal

(Adjoint a, Ring a) => Adjoint (Complex a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: Complex a -> Complex a

Precision e => Adjoint (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

adj :: FixedPrec e -> FixedPrec e

(Adjoint a, Ring a) => Adjoint (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: Omega a -> Omega a

(Adjoint a, Ring a) => Adjoint (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: Cplx a -> Cplx a

Adjoint a => Adjoint (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: RootTwo a -> RootTwo a

(Nat n, Adjoint a) => Adjoint (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

adj :: Matrix n n a -> Matrix n n a

Rings with √2-conjugation

class Adjoint2 a where

A type class for rings with a √2-conjugation, i.e., an automorphism mapping √2 to −√2.

When instances of this type class are vectors or matrices, the √2-conjugation does not exchange the roles of rows and columns.

For rings that have no √2, the conjugation can be defined to be the identity function.

Methods

adj2 :: a -> a

Compute the adjoint, mapping a + b√2 to a −b√2.

Instances

Adjoint2 Int

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Int -> Int

Adjoint2 Integer

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Integer -> Integer

Adjoint2 Rational

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Rational -> Rational

Adjoint2 Rationals

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Rationals -> Rationals

Adjoint2 Dyadic

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Dyadic -> Dyadic

Adjoint2 Z2

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Z2 -> Z2

Adjoint2 SymReal

Defined in Quantum.Synthesis.Ring.SymReal

Methods

adj2 :: SymReal -> SymReal

Precision e => Adjoint2 (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

adj2 :: FixedPrec e -> FixedPrec e

(Adjoint2 a, Ring a) => Adjoint2 (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Omega a -> Omega a

(Adjoint2 a, Ring a) => Adjoint2 (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Cplx a -> Cplx a

(Adjoint2 a, Num a) => Adjoint2 (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: RootTwo a -> RootTwo a

(Nat n, Adjoint2 a) => Adjoint2 (Matrix n n a)

Defined in Quantum.Synthesis.Matrix

Methods

adj2 :: Matrix n n a -> Matrix n n a

Normed rings

class Ring r => NormedRing r where

A (number-theoretic) norm on a ring R is a function N : R → ℤ such that N(rs) = N(r)N(s), for all r, s ∈ R. The norm also satisfies N(r) = 0 iff r = 0, and N(r) = ±1 iff r is a unit of the ring.

Methods

norm :: r -> Integer

Instances

NormedRing Integer

Defined in Quantum.Synthesis.Ring

Methods

norm :: Integer -> Integer

NormedRing a => NormedRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: Omega a -> Integer

NormedRing a => NormedRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: Cplx a -> Integer

(Eq a, NormedRing a) => NormedRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: RootTwo a -> Integer

Floor and ceiling

class Ring r => Floor r where

The floor and ceiling functions provided by the standard Haskell libraries are predicated on many unnecessary assumptions. This type class provides an alternative.

Minimal complete definition: floor_of or ceiling_of.

Minimal complete definition

Nothing

Methods

floor_of :: r -> Integer

Compute the floor of x, i.e., the greatest integer n such that n ≤ x.

ceiling_of :: r -> Integer

Compute the ceiling of x, i.e., the least integer n such that x ≤ n.

Instances

Floor Double

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Double -> Integer

ceiling_of :: Double -> Integer

Floor Float

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Float -> Integer

ceiling_of :: Float -> Integer

Floor Integer

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Integer -> Integer

ceiling_of :: Integer -> Integer

Floor Rational

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Rational -> Integer

ceiling_of :: Rational -> Integer

Floor QRootTwo

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: QRootTwo -> Integer

ceiling_of :: QRootTwo -> Integer

Floor Rationals

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Rationals -> Integer

ceiling_of :: Rationals -> Integer

Precision e => Floor (FixedPrec e)

Defined in Quantum.Synthesis.Ring.FixedPrec

Methods

floor_of :: FixedPrec e -> Integer

ceiling_of :: FixedPrec e -> Integer

Particular rings

The ring ℤ₂ of integers modulo 2

data Z2

The ring ℤ₂ of integers modulo 2.

Constructors

Even

Odd

Instances

Eq Z2

Defined in Quantum.Synthesis.Ring

Methods

(==) :: Z2 -> Z2 -> Bool

(/=) :: Z2 -> Z2 -> Bool

Num Z2

Defined in Quantum.Synthesis.Ring

Methods

(+) :: Z2 -> Z2 -> Z2

(-) :: Z2 -> Z2 -> Z2

(*) :: Z2 -> Z2 -> Z2

negate :: Z2 -> Z2

abs :: Z2 -> Z2

signum :: Z2 -> Z2

fromInteger :: Integer -> Z2

Show Z2

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> Z2 -> ShowS

show :: Z2 -> String

showList :: [Z2] -> ShowS

Adjoint2 Z2

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Z2 -> Z2

Adjoint Z2

Defined in Quantum.Synthesis.Ring

Methods

adj :: Z2 -> Z2

Residue Integer Z2

Defined in Quantum.Synthesis.MultiQubitSynthesis

Methods

residue :: Integer -> Z2

ShowLaTeX (Omega Z2)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Omega Z2 -> String

showlatex_p :: Int -> Omega Z2 -> ShowS

The ring D of dyadic fractions

data Dyadic

A dyadic fraction is a rational number whose denominator is a power of 2. We denote the dyadic fractions by D = ℤ[½].

We internally represent a dyadic fraction a/2ⁿ as a pair (a,n). Note that this representation is not unique. When it is necessary to choose a canonical representative, we choose the least possible n≥0.

Constructors

Dyadic !Integer !Integer

Instances

Eq Dyadic

Defined in Quantum.Synthesis.Ring

Methods

(==) :: Dyadic -> Dyadic -> Bool

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

Num Dyadic

Defined in Quantum.Synthesis.Ring

Methods

(+) :: Dyadic -> Dyadic -> Dyadic

(-) :: Dyadic -> Dyadic -> Dyadic

(*) :: Dyadic -> Dyadic -> Dyadic

negate :: Dyadic -> Dyadic

abs :: Dyadic -> Dyadic

signum :: Dyadic -> Dyadic

fromInteger :: Integer -> Dyadic

Ord Dyadic

Defined in Quantum.Synthesis.Ring

Methods

compare :: Dyadic -> Dyadic -> Ordering

(<) :: Dyadic -> Dyadic -> Bool

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

(>) :: Dyadic -> Dyadic -> Bool

(>=) :: Dyadic -> Dyadic -> Bool

max :: Dyadic -> Dyadic -> Dyadic

min :: Dyadic -> Dyadic -> Dyadic

Real Dyadic

Defined in Quantum.Synthesis.Ring

Methods

toRational :: Dyadic -> Rational

Show DOmega

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> DOmega -> ShowS

show :: DOmega -> String

showList :: [DOmega] -> ShowS

Show DRComplex

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> DRComplex -> ShowS

show :: DRComplex -> String

showList :: [DRComplex] -> ShowS

Show Dyadic

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> Dyadic -> ShowS

show :: Dyadic -> String

showList :: [Dyadic] -> ShowS

ToQOmega Dyadic

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Dyadic -> QOmega

DenomExp DOmega

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DOmega -> Integer

denomexp_factor :: DOmega -> Integer -> DOmega

DenomExp DRootTwo

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DRootTwo -> Integer

denomexp_factor :: DRootTwo -> Integer -> DRootTwo

Adjoint2 Dyadic

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Dyadic -> Dyadic

Adjoint Dyadic

Defined in Quantum.Synthesis.Ring

Methods

adj :: Dyadic -> Dyadic

HalfRing Dyadic

Defined in Quantum.Synthesis.Ring

Methods

half :: Dyadic

fromDyadic :: Dyadic -> Dyadic

ShowLaTeX DOmega

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: DOmega -> String

showlatex_p :: Int -> DOmega -> ShowS

ShowLaTeX Dyadic

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Dyadic -> String

showlatex_p :: Int -> Dyadic -> ShowS

WholePart DOmega ZOmega

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZOmega -> DOmega

to_whole :: DOmega -> ZOmega

WholePart DRootTwo ZRootTwo

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZRootTwo -> DRootTwo

to_whole :: DRootTwo -> ZRootTwo

WholePart Dyadic Integer

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: Integer -> Dyadic

to_whole :: Dyadic -> Integer

ToDyadic Rational Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Rational -> Maybe Dyadic

ToDyadic Rationals Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Rationals -> Maybe Dyadic

ToDyadic Dyadic Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Dyadic -> Maybe Dyadic

Nat m => Show (Matrix m n DOmega)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DOmega -> ShowS

show :: Matrix m n DOmega -> String

showList :: [Matrix m n DOmega] -> ShowS

Nat m => Show (Matrix m n DRComplex)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DRComplex -> ShowS

show :: Matrix m n DRComplex -> String

showList :: [Matrix m n DRComplex] -> ShowS

Nat m => Show (Matrix m n DRootTwo)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DRootTwo -> ShowS

show :: Matrix m n DRootTwo -> String

showList :: [Matrix m n DRootTwo] -> ShowS

Nat n => ShowLaTeX (Matrix n m DRComplex)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Matrix n m DRComplex -> String

showlatex_p :: Int -> Matrix n m DRComplex -> ShowS

Nat n => ShowLaTeX (Matrix n m DOmega)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Matrix n m DOmega -> String

showlatex_p :: Int -> Matrix n m DOmega -> ShowS

decompose_dyadic :: Dyadic -> (Integer, Integer)

Given a dyadic fraction r, return (a,n) such that r = a/2ⁿ, where n≥0 is chosen as small as possible.

integer_of_dyadic :: Dyadic -> Integer -> Integer

Given a dyadic fraction r and an integer k≥0, such that a = r2^k is an integer, return a. If a is not an integer, the behavior is undefined.

The ring ℚ of rational numbers

newtype Rationals

We define our own variant of the rational numbers, which is an identical copy of the type Rational from the standard Haskell library, except that it has a more sensible Show instance.

Constructors

ToRationals

Fields

• unRationals :: Rational

Instances

Eq Rationals

Defined in Quantum.Synthesis.Ring

Methods

(==) :: Rationals -> Rationals -> Bool

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

Fractional Rationals

Defined in Quantum.Synthesis.Ring

Methods

(/) :: Rationals -> Rationals -> Rationals

recip :: Rationals -> Rationals

fromRational :: Rational -> Rationals

Num Rationals

Defined in Quantum.Synthesis.Ring

Methods

(+) :: Rationals -> Rationals -> Rationals

(-) :: Rationals -> Rationals -> Rationals

(*) :: Rationals -> Rationals -> Rationals

negate :: Rationals -> Rationals

abs :: Rationals -> Rationals

signum :: Rationals -> Rationals

fromInteger :: Integer -> Rationals

Ord Rationals

Defined in Quantum.Synthesis.Ring

Methods

compare :: Rationals -> Rationals -> Ordering

(<) :: Rationals -> Rationals -> Bool

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

(>) :: Rationals -> Rationals -> Bool

(>=) :: Rationals -> Rationals -> Bool

max :: Rationals -> Rationals -> Rationals

min :: Rationals -> Rationals -> Rationals

Real Rationals

Defined in Quantum.Synthesis.Ring

Methods

toRational :: Rationals -> Rational

RealFrac Rationals

Defined in Quantum.Synthesis.Ring

Methods

properFraction :: Integral b => Rationals -> (b, Rationals)

truncate :: Integral b => Rationals -> b

round :: Integral b => Rationals -> b

ceiling :: Integral b => Rationals -> b

floor :: Integral b => Rationals -> b

Show Rationals

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> Rationals -> ShowS

show :: Rationals -> String

showList :: [Rationals] -> ShowS

ToQOmega Rationals

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Rationals -> QOmega

Floor QRootTwo

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: QRootTwo -> Integer

ceiling_of :: QRootTwo -> Integer

Floor Rationals

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: Rationals -> Integer

ceiling_of :: Rationals -> Integer

Adjoint2 Rationals

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Rationals -> Rationals

Adjoint Rationals

Defined in Quantum.Synthesis.Ring

Methods

adj :: Rationals -> Rationals

HalfRing Rationals

Defined in Quantum.Synthesis.Ring

Methods

half :: Rationals

fromDyadic :: Dyadic -> Rationals

ToDyadic Rationals Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Rationals -> Maybe Dyadic

showsPrec_rational :: (Show a, Integral a) => Int -> Ratio a -> ShowS

An auxiliary function for printing rational numbers, using correct precedences, and omitting denominators of 1.

fromRationals :: Fractional a => Rationals -> a

Conversion from Rationals to any Fractional type.

The ring R[√2]

data RootTwo a

The ring R[√2], where R is any ring. The value RootTwo a b represents a + b √2.

Constructors

RootTwo !a !a

Instances

Show DRComplex

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> DRComplex -> ShowS

show :: DRComplex -> String

showList :: [DRComplex] -> ShowS

Parity ZRootTwo

Defined in Quantum.Synthesis.Ring

Methods

parity :: ZRootTwo -> Z2

DenomExp DRootTwo

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DRootTwo -> Integer

denomexp_factor :: DRootTwo -> Integer -> DRootTwo

Floor QRootTwo

Defined in Quantum.Synthesis.Ring

Methods

floor_of :: QRootTwo -> Integer

ceiling_of :: QRootTwo -> Integer

EuclideanDomain ZRootTwo

Defined in Quantum.Synthesis.EuclideanDomain

Methods

rank :: ZRootTwo -> Integer

divmod :: ZRootTwo -> ZRootTwo -> (ZRootTwo, ZRootTwo)

WholePart DRootTwo ZRootTwo

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZRootTwo -> DRootTwo

to_whole :: DRootTwo -> ZRootTwo

Eq a => Eq (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

(==) :: RootTwo a -> RootTwo a -> Bool

(/=) :: RootTwo a -> RootTwo a -> Bool

(Eq a, Fractional a) => Fractional (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

(/) :: RootTwo a -> RootTwo a -> RootTwo a

recip :: RootTwo a -> RootTwo a

fromRational :: Rational -> RootTwo a

(Eq a, Num a) => Num (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

(+) :: RootTwo a -> RootTwo a -> RootTwo a

(-) :: RootTwo a -> RootTwo a -> RootTwo a

(*) :: RootTwo a -> RootTwo a -> RootTwo a

negate :: RootTwo a -> RootTwo a

abs :: RootTwo a -> RootTwo a

signum :: RootTwo a -> RootTwo a

fromInteger :: Integer -> RootTwo a

(Eq a, Ring a) => Ord (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

compare :: RootTwo a -> RootTwo a -> Ordering

(<) :: RootTwo a -> RootTwo a -> Bool

(<=) :: RootTwo a -> RootTwo a -> Bool

(>) :: RootTwo a -> RootTwo a -> Bool

(>=) :: RootTwo a -> RootTwo a -> Bool

max :: RootTwo a -> RootTwo a -> RootTwo a

min :: RootTwo a -> RootTwo a -> RootTwo a

(Show a, Eq a, Ring a) => Show (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> RootTwo a -> ShowS

show :: RootTwo a -> String

showList :: [RootTwo a] -> ShowS

ToQOmega a => ToQOmega (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: RootTwo a -> QOmega

(Eq a, NormedRing a) => NormedRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: RootTwo a -> Integer

(Adjoint2 a, Num a) => Adjoint2 (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: RootTwo a -> RootTwo a

Adjoint a => Adjoint (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: RootTwo a -> RootTwo a

(Eq a, ComplexRing a, HalfRing a) => OmegaRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: RootTwo a

(Eq a, ComplexRing a) => ComplexRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

i :: RootTwo a

(Eq a, HalfRing a) => RootHalfRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: RootTwo a

fromDRootTwo :: DRootTwo -> RootTwo a

(Eq a, Ring a) => RootTwoRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: RootTwo a

fromZRootTwo :: ZRootTwo -> RootTwo a

(Eq a, HalfRing a) => HalfRing (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

half :: RootTwo a

fromDyadic :: Dyadic -> RootTwo a

(ShowLaTeX a, Eq a, Ring a) => ShowLaTeX (RootTwo a)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: RootTwo a -> String

showlatex_p :: Int -> RootTwo a -> ShowS

HalfRing a => RealPart (Omega a) (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

real :: Omega a -> RootTwo a

ToDyadic a b => ToDyadic (RootTwo a) (RootTwo b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: RootTwo a -> Maybe (RootTwo b)

Residue a b => Residue (RootTwo a) (RootTwo b)

Defined in Quantum.Synthesis.MultiQubitSynthesis

Methods

residue :: RootTwo a -> RootTwo b

Nat m => Show (Matrix m n DRComplex)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DRComplex -> ShowS

show :: Matrix m n DRComplex -> String

showList :: [Matrix m n DRComplex] -> ShowS

Nat m => Show (Matrix m n DRootTwo)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DRootTwo -> ShowS

show :: Matrix m n DRootTwo -> String

showList :: [Matrix m n DRootTwo] -> ShowS

Nat n => ShowLaTeX (Matrix n m DRComplex)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Matrix n m DRComplex -> String

showlatex_p :: Int -> Matrix n m DRComplex -> ShowS

The ring ℤ[√2]

type ZRootTwo = RootTwo Integer

The ring ℤ[√2].

zroottwo_root :: ZRootTwo -> Maybe ZRootTwo

Return a square root of an element of ℤ[√2], if such a square root exists, or else Nothing.

The ring D[√2]

type DRootTwo = RootTwo Dyadic

The ring D[√2] = ℤ[1/√2].

The field ℚ[√2]

type QRootTwo = RootTwo Rationals

The field ℚ[√2].

fromQRootTwo :: (RootTwoRing a, Fractional a) => QRootTwo -> a

The unique ring homomorphism from ℚ[√2] to any ring containing the rational numbers and √2. This exists because ℚ[√2] is the free such ring.

The ring R[i]

data Cplx a

The ring R[i], where R is any ring. The reason we do not use the Complex a type from the standard Haskell libraries is that it assumes too much, for example, it assumes a is a member of the RealFloat class. Also, this allows us to define a more sensible Show instance.

Constructors

Cplx !a !a

Instances

Show DRComplex

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> DRComplex -> ShowS

show :: DRComplex -> String

showList :: [DRComplex] -> ShowS

EuclideanDomain ZComplex

Defined in Quantum.Synthesis.EuclideanDomain

Methods

rank :: ZComplex -> Integer

divmod :: ZComplex -> ZComplex -> (ZComplex, ZComplex)

Eq a => Eq (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

(==) :: Cplx a -> Cplx a -> Bool

(/=) :: Cplx a -> Cplx a -> Bool

Fractional a => Fractional (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

(/) :: Cplx a -> Cplx a -> Cplx a

recip :: Cplx a -> Cplx a

fromRational :: Rational -> Cplx a

Num a => Num (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

(+) :: Cplx a -> Cplx a -> Cplx a

(-) :: Cplx a -> Cplx a -> Cplx a

(*) :: Cplx a -> Cplx a -> Cplx a

negate :: Cplx a -> Cplx a

abs :: Cplx a -> Cplx a

signum :: Cplx a -> Cplx a

fromInteger :: Integer -> Cplx a

(Eq a, Show a, Num a) => Show (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> Cplx a -> ShowS

show :: Cplx a -> String

showList :: [Cplx a] -> ShowS

ToQOmega a => ToQOmega (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Cplx a -> QOmega

DenomExp a => DenomExp (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: Cplx a -> Integer

denomexp_factor :: Cplx a -> Integer -> Cplx a

NormedRing a => NormedRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: Cplx a -> Integer

(Adjoint2 a, Ring a) => Adjoint2 (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Cplx a -> Cplx a

(Adjoint a, Ring a) => Adjoint (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: Cplx a -> Cplx a

(Ring a, RootHalfRing a) => OmegaRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: Cplx a

Ring a => ComplexRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

i :: Cplx a

RootHalfRing a => RootHalfRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Cplx a

fromDRootTwo :: DRootTwo -> Cplx a

RootTwoRing a => RootTwoRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Cplx a

fromZRootTwo :: ZRootTwo -> Cplx a

HalfRing a => HalfRing (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

half :: Cplx a

fromDyadic :: Dyadic -> Cplx a

(ShowLaTeX a, Ring a, Eq a) => ShowLaTeX (Cplx a)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Cplx a -> String

showlatex_p :: Int -> Cplx a -> ShowS

RealPart (Cplx a) a

Defined in Quantum.Synthesis.Ring

Methods

real :: Cplx a -> a

WholePart a b => WholePart (Cplx a) (Cplx b)

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: Cplx b -> Cplx a

to_whole :: Cplx a -> Cplx b

ToDyadic a b => ToDyadic (Cplx a) (Cplx b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Cplx a -> Maybe (Cplx b)

Residue a b => Residue (Cplx a) (Cplx b)

Defined in Quantum.Synthesis.MultiQubitSynthesis

Methods

residue :: Cplx a -> Cplx b

Nat m => Show (Matrix m n DRComplex)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DRComplex -> ShowS

show :: Matrix m n DRComplex -> String

showList :: [Matrix m n DRComplex] -> ShowS

Nat n => ShowLaTeX (Matrix n m DRComplex)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Matrix n m DRComplex -> String

showlatex_p :: Int -> Matrix n m DRComplex -> ShowS

The ring ℤ[i] of Gaussian integers

type ZComplex = Cplx Integer

The ring ℤ[i] of Gaussian integers.

fromZComplex :: ComplexRing a => ZComplex -> a

The unique ring homomorphism from ℤ[i] to any ring containing i. This exists because ℤ[i] is the free such ring.

The ring D[i]

type DComplex = Cplx Dyadic

The ring D[i] = ℤ[½, i] of Gaussian dyadic fractions.

fromDComplex :: (ComplexRing a, HalfRing a) => DComplex -> a

The unique ring homomorphism from D[i] to any ring containing ½ and i. This exists because D[i] is the free such ring.

The ring ℚ[i] of Gaussian rationals

type QComplex = Cplx Rationals

The ring ℚ[i] of Gaussian rationals.

fromQComplex :: (ComplexRing a, Fractional a) => QComplex -> a

The unique ring homomorphism from ℚ[i] to any ring containing the rational numbers and i. This exists because ℚ[i] is the free such ring.

The ring D[√2, i]

type DRComplex = Cplx DRootTwo

The ring D[√2, i] = ℤ[1/√2, i].

fromDRComplex :: (RootHalfRing a, ComplexRing a) => DRComplex -> a

The unique ring homomorphism from D[√2, i] to any ring containing 1/√2 and i. This exists because D[√2, i] = ℤ[1/√2, i] is the free such ring.

The ring ℚ[√2, i]

type QRComplex = Cplx QRootTwo

The field ℚ[√2, i].

fromQRComplex :: (RootTwoRing a, ComplexRing a, Fractional a) => QRComplex -> a

The unique ring homomorphism from ℚ[√2, i] to any ring containing the rational numbers, √2, and i. This exists because ℚ[√2, i] is the free such ring.

The ring ℂ of complex numbers

We provide two versions of the complex numbers using floating point arithmetic.

type CDouble = Cplx Double

Double precision complex floating point numbers.

type CFloat = Cplx Float

Single precision complex floating point numbers.

The ring R[ω]

data Omega a

The ring R[ω], where R is any ring, and ω = e^iπ/4 is an 8th root of unity. The value Omega a b c d represents aω³+bω²+cω+d.

Constructors

Omega !a !a !a !a

Instances

Show DOmega

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> DOmega -> ShowS

show :: DOmega -> String

showList :: [DOmega] -> ShowS

DenomExp DOmega

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DOmega -> Integer

denomexp_factor :: DOmega -> Integer -> DOmega

EuclideanDomain ZOmega

Defined in Quantum.Synthesis.EuclideanDomain

Methods

rank :: ZOmega -> Integer

divmod :: ZOmega -> ZOmega -> (ZOmega, ZOmega)

ShowLaTeX DOmega

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: DOmega -> String

showlatex_p :: Int -> DOmega -> ShowS

ShowLaTeX ZOmega

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: ZOmega -> String

showlatex_p :: Int -> ZOmega -> ShowS

WholePart DOmega ZOmega

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZOmega -> DOmega

to_whole :: DOmega -> ZOmega

Eq a => Eq (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

(==) :: Omega a -> Omega a -> Bool

(/=) :: Omega a -> Omega a -> Bool

Fractional a => Fractional (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

(/) :: Omega a -> Omega a -> Omega a

recip :: Omega a -> Omega a

fromRational :: Rational -> Omega a

Num a => Num (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

(+) :: Omega a -> Omega a -> Omega a

(-) :: Omega a -> Omega a -> Omega a

(*) :: Omega a -> Omega a -> Omega a

negate :: Omega a -> Omega a

abs :: Omega a -> Omega a

signum :: Omega a -> Omega a

fromInteger :: Integer -> Omega a

(Show a, Ring a) => Show (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

showsPrec :: Int -> Omega a -> ShowS

show :: Omega a -> String

showList :: [Omega a] -> ShowS

ToQOmega a => ToQOmega (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Omega a -> QOmega

NormedRing a => NormedRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

norm :: Omega a -> Integer

(Adjoint2 a, Ring a) => Adjoint2 (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

adj2 :: Omega a -> Omega a

(Adjoint a, Ring a) => Adjoint (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

adj :: Omega a -> Omega a

Ring a => OmegaRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

omega :: Omega a

Ring a => ComplexRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

i :: Omega a

HalfRing a => RootHalfRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

roothalf :: Omega a

fromDRootTwo :: DRootTwo -> Omega a

Ring a => RootTwoRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

roottwo :: Omega a

fromZRootTwo :: ZRootTwo -> Omega a

HalfRing a => HalfRing (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

half :: Omega a

fromDyadic :: Dyadic -> Omega a

ShowLaTeX (Omega Z2)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Omega Z2 -> String

showlatex_p :: Int -> Omega Z2 -> ShowS

HalfRing a => RealPart (Omega a) (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

real :: Omega a -> RootTwo a

ToDyadic a b => ToDyadic (Omega a) (Omega b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Omega a -> Maybe (Omega b)

Residue a b => Residue (Omega a) (Omega b)

Defined in Quantum.Synthesis.MultiQubitSynthesis

Methods

residue :: Omega a -> Omega b

Nat m => Show (Matrix m n DOmega)

Defined in Quantum.Synthesis.Matrix

Methods

showsPrec :: Int -> Matrix m n DOmega -> ShowS

show :: Matrix m n DOmega -> String

showList :: [Matrix m n DOmega] -> ShowS

Nat n => ShowLaTeX (Matrix n m DOmega)

Defined in Quantum.Synthesis.LaTeX

Methods

showlatex :: Matrix n m DOmega -> String

showlatex_p :: Int -> Matrix n m DOmega -> ShowS

omega_real :: Omega a -> a

An inverse to the embedding R ↦ R[ω]: return the "real rational" part. In other words, map aω³+bω²+cω+d to d.

The ring ℤ[ω]

type ZOmega = Omega Integer

The ring ℤ[ω] of cyclotomic integers of degree 8. Such rings were first studied by Kummer around 1840, and used in his proof of special cases of Fermat's Last Theorem. See also:

• http://fermatslasttheorem.blogspot.com/2006/05/basic-properties-of-cyclotomic.html
• http://fermatslasttheorem.blogspot.com/2006/02/cyclotomic-integers.html
• Harold M. Edwards, "Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory".

fromZOmega :: OmegaRing a => ZOmega -> a

The unique ring homomorphism from ℤ[ω] to any ring containing ω. This exists because ℤ[ω] is the free such ring.

zroottwo_of_zomega :: ZOmega -> ZRootTwo

Inverse of the embedding ℤ[√2] → ℤ[ω]. Note that ℤ[√2] = ℤ[ω] ∩ ℝ. This function takes an element of ℤ[ω] that is real, and converts it to an element of ℤ[√2]. It throws an error if the input is not real.

The ring D[ω]

type DOmega = Omega Dyadic

The ring D[ω]. Here D=ℤ[½] is the ring of dyadic fractions. In fact, D[ω] is isomorphic to the ring D[√2, i], but they have different Show instances.

fromDOmega :: (OmegaRing a, HalfRing a) => DOmega -> a

The unique ring homomorphism from D[ω] to any ring containing ω and 1/2. This exists because D[ω] is the free such ring.

The field ℚ[ω]

type QOmega = Omega Rationals

The field ℚ[ω] of cyclotomic rationals of degree 8.

fromQOmega :: (OmegaRing a, Fractional a) => QOmega -> a

The unique ring homomorphism from ℚ[ω] to any ring containing the rational numbers and ω. This exists because ℚ[ω] is the free such ring.

Conversion to dyadic

class ToDyadic a b | a -> b where

A type class relating "rational" types to their dyadic counterparts.

Methods

maybe_dyadic :: a -> Maybe b

Convert a "rational" value to a "dyadic" value, if the denominator is a power of 2. Otherwise, return Nothing.

Instances

ToDyadic Rational Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Rational -> Maybe Dyadic

ToDyadic Rationals Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Rationals -> Maybe Dyadic

ToDyadic Dyadic Dyadic

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Dyadic -> Maybe Dyadic

ToDyadic a b => ToDyadic (Omega a) (Omega b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Omega a -> Maybe (Omega b)

ToDyadic a b => ToDyadic (Cplx a) (Cplx b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: Cplx a -> Maybe (Cplx b)

ToDyadic a b => ToDyadic (RootTwo a) (RootTwo b)

Defined in Quantum.Synthesis.Ring

Methods

maybe_dyadic :: RootTwo a -> Maybe (RootTwo b)

ToDyadic a b => ToDyadic (Vector n a) (Vector n b)

Defined in Quantum.Synthesis.Matrix

Methods

maybe_dyadic :: Vector n a -> Maybe (Vector n b)

ToDyadic a b => ToDyadic (Matrix m n a) (Matrix m n b)

Defined in Quantum.Synthesis.Matrix

Methods

maybe_dyadic :: Matrix m n a -> Maybe (Matrix m n b)

to_dyadic :: ToDyadic a b => a -> b

Convert a "rational" value to a "dyadic" value, if the denominator is a power of 2. Otherwise, throw an error.

Real part

class RealPart a b | a -> b where

A type class for rings that have a "real" component. A typical instance is a = DRComplex with b = DRootTwo.

Methods

real :: a -> b

Take the real part.

Instances

RealPart (Cplx a) a

Defined in Quantum.Synthesis.Ring

Methods

real :: Cplx a -> a

HalfRing a => RealPart (Omega a) (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

real :: Omega a -> RootTwo a

Rings of integers

class WholePart a b | a -> b where

A type class for rings that have a distinguished subring "of integers". A typical instance is a = DRootTwo, which has b = ZRootTwo as its ring of integers.

Methods

from_whole :: b -> a

The embedding of the ring of integers into the larger ring.

to_whole :: a -> b

The inverse of from_whole. Throws an error if the given element is not actually an integer in the ring.

Instances

WholePart () ()

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: () -> ()

to_whole :: () -> ()

WholePart DOmega ZOmega

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZOmega -> DOmega

to_whole :: DOmega -> ZOmega

WholePart DRootTwo ZRootTwo

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: ZRootTwo -> DRootTwo

to_whole :: DRootTwo -> ZRootTwo

WholePart Dyadic Integer

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: Integer -> Dyadic

to_whole :: Dyadic -> Integer

WholePart a b => WholePart [a] [b]

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: [b] -> [a]

to_whole :: [a] -> [b]

WholePart a b => WholePart (Cplx a) (Cplx b)

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: Cplx b -> Cplx a

to_whole :: Cplx a -> Cplx b

(WholePart a a', WholePart b b') => WholePart (a, b) (a', b')

Defined in Quantum.Synthesis.Ring

Methods

from_whole :: (a', b') -> (a, b)

to_whole :: (a, b) -> (a', b')

WholePart a b => WholePart (Vector n a) (Vector n b)

Defined in Quantum.Synthesis.Matrix

Methods

from_whole :: Vector n b -> Vector n a

to_whole :: Vector n a -> Vector n b

WholePart a b => WholePart (Matrix m n a) (Matrix m n b)

Defined in Quantum.Synthesis.Matrix

Methods

from_whole :: Matrix m n b -> Matrix m n a

to_whole :: Matrix m n a -> Matrix m n b

Common denominators

class DenomExp a where

A type class for things from which a common power of 1/√2 (a least denominator exponent) can be factored out. Typical instances are DRootTwo, DRComplex, as well as tuples, lists, vectors, and matrices thereof.

Methods

denomexp :: a -> Integer

Calculate the least denominator exponent k of a. Returns the smallest k≥0 such that a = b/√2^k for some integral b.

denomexp_factor :: a -> Integer -> a

Factor out a kth power of 1/√2 from a. In other words, calculate a√2^k.

Instances

DenomExp ()

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: () -> Integer

denomexp_factor :: () -> Integer -> ()

DenomExp DOmega

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DOmega -> Integer

denomexp_factor :: DOmega -> Integer -> DOmega

DenomExp DRootTwo

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: DRootTwo -> Integer

denomexp_factor :: DRootTwo -> Integer -> DRootTwo

DenomExp a => DenomExp [a]

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: [a] -> Integer

denomexp_factor :: [a] -> Integer -> [a]

DenomExp a => DenomExp (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: Cplx a -> Integer

denomexp_factor :: Cplx a -> Integer -> Cplx a

(DenomExp a, DenomExp b) => DenomExp (a, b)

Defined in Quantum.Synthesis.Ring

Methods

denomexp :: (a, b) -> Integer

denomexp_factor :: (a, b) -> Integer -> (a, b)

DenomExp a => DenomExp (Vector n a)

Defined in Quantum.Synthesis.Matrix

Methods

denomexp :: Vector n a -> Integer

denomexp_factor :: Vector n a -> Integer -> Vector n a

DenomExp a => DenomExp (Matrix m n a)

Defined in Quantum.Synthesis.Matrix

Methods

denomexp :: Matrix m n a -> Integer

denomexp_factor :: Matrix m n a -> Integer -> Matrix m n a

denomexp_decompose :: (WholePart a b, DenomExp a) => a -> (b, Integer)

Calculate and factor out the least denominator exponent k of a. Return (b,k), where a = b/(√2)^k and k≥0.

showsPrec_DenomExp :: (WholePart a b, Show b, DenomExp a) => Int -> a -> ShowS

Generic show-like method that factors out a common denominator exponent.

Conversion to ℚ[ω]

QOmega is the largest one of our "exact" arithmetic types. We define a toQOmega family of functions for converting just about anything to QOmega.

class ToQOmega a where

A type class for things that can be exactly converted to ℚ[ω].

Methods

toQOmega :: a -> QOmega

Conversion to QOmega.

Instances

ToQOmega Integer

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Integer -> QOmega

ToQOmega Rational

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Rational -> QOmega

ToQOmega Rationals

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Rationals -> QOmega

ToQOmega Dyadic

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Dyadic -> QOmega

ToQOmega a => ToQOmega (Omega a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Omega a -> QOmega

ToQOmega a => ToQOmega (Cplx a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: Cplx a -> QOmega

ToQOmega a => ToQOmega (RootTwo a)

Defined in Quantum.Synthesis.Ring

Methods

toQOmega :: RootTwo a -> QOmega

Parity

class Parity a where

A type class for things that have parity.

Methods

parity :: a -> Z2

Return the parity of something.

Instances

Integral a => Parity a

Defined in Quantum.Synthesis.Ring

Methods

parity :: a -> Z2

Parity ZRootTwo

Defined in Quantum.Synthesis.Ring

Methods

parity :: ZRootTwo -> Z2

Auxiliary functions

lobit :: Integer -> Integer

Return the position of the rightmost "1" bit of an Integer, or -1 if none. Do this in time O(n log n), where n is the size of the integer (in digits).

log2 :: Integer -> Maybe Integer

If n is of the form 2^k, return k. Otherwise, return Nothing.

hibit :: Integer -> Int

Return 1 + the position of the leftmost "1" bit of a non-negative Integer. Do this in time O(n log n), where n is the size of the integer (in digits).

intsqrt :: Integral n => n -> n

For n ≥ 0, return the floor of the square root of n. This is done using integer arithmetic, so there are no rounding errors.