Data.Qubit

Qubits and operations on them, using the indexing conventions of <https://arxiv.org/abs/1711.02086/>.

The example below creates a wavefunction for the Bell state and then performs a measurement on its highest qubit.

>>> import Control.Monad.Random (evalRandIO)
>>> import Data.Qubit ((^^*), groundState, measure)
>>> import Data.Qubit.Gate (h, cnot)

-- Construct the Bell state.
>>> let bell = [h 0, cnot 0 1] ^^* groundState 2
>>> bell
0.7071067811865475|00> + 0.7071067811865475|11> @ [1,0]

-- Measure the Bell wavefunction.
>>> bell' <- evalRandIO $ measure [1] bell
>>> bell'
([(1,0)],1.0|00> @ [1,0])

-- Measure it again.
>>> evalRandIO $ measure [1] bell'
([(1,0)],1.0|00> @ [1,0])

-- Measure another Bell wavefunction.
>>> evalRandIO $ measure [1] bell
([(1,0)],1.0|00> @ [1,0])

-- Measure another Bell wavefunction.
>>> evalRandIO $ measure [1] bell
([(1,1)],1.0|11> @ [1,0])

Types

type QIndex = Int

Index for qubits in a wavefunction.

data QState

States of a quibit.

Constructors

QState0

QState1

Instances

Bounded QState

Methods

minBound :: QState

maxBound :: QState

Enum QState

Methods

succ :: QState -> QState

pred :: QState -> QState

toEnum :: Int -> QState

fromEnum :: QState -> Int

enumFrom :: QState -> [QState]

enumFromThen :: QState -> QState -> [QState]

enumFromTo :: QState -> QState -> [QState]

enumFromThenTo :: QState -> QState -> QState -> [QState]

Eq QState

Methods

(==) :: QState -> QState -> Bool

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

Ord QState

Methods

compare :: QState -> QState -> Ordering

(<) :: QState -> QState -> Bool

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

(>) :: QState -> QState -> Bool

(>=) :: QState -> QState -> Bool

max :: QState -> QState -> QState

min :: QState -> QState -> QState

Read QState

Methods

readsPrec :: Int -> ReadS QState

readList :: ReadS [QState]

readPrec :: ReadPrec QState

readListPrec :: ReadPrec [QState]

Show QState

Methods

showsPrec :: Int -> QState -> ShowS

show :: QState -> String

showList :: [QState] -> ShowS

type Amplitude = Complex Double

Amplitude of a state in a wavefunction.

data Wavefunction

A wavefunction for qubits.

Instances

Eq Wavefunction

Methods

(==) :: Wavefunction -> Wavefunction -> Bool

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

Floating Wavefunction

Methods

pi :: Wavefunction

exp :: Wavefunction -> Wavefunction

log :: Wavefunction -> Wavefunction

sqrt :: Wavefunction -> Wavefunction

(**) :: Wavefunction -> Wavefunction -> Wavefunction

logBase :: Wavefunction -> Wavefunction -> Wavefunction

sin :: Wavefunction -> Wavefunction

cos :: Wavefunction -> Wavefunction

tan :: Wavefunction -> Wavefunction

asin :: Wavefunction -> Wavefunction

acos :: Wavefunction -> Wavefunction

atan :: Wavefunction -> Wavefunction

sinh :: Wavefunction -> Wavefunction

cosh :: Wavefunction -> Wavefunction

tanh :: Wavefunction -> Wavefunction

asinh :: Wavefunction -> Wavefunction

acosh :: Wavefunction -> Wavefunction

atanh :: Wavefunction -> Wavefunction

log1p :: Wavefunction -> Wavefunction

expm1 :: Wavefunction -> Wavefunction

log1pexp :: Wavefunction -> Wavefunction

log1mexp :: Wavefunction -> Wavefunction

Fractional Wavefunction

Methods

(/) :: Wavefunction -> Wavefunction -> Wavefunction

recip :: Wavefunction -> Wavefunction

fromRational :: Rational -> Wavefunction

Num Wavefunction

Methods

(+) :: Wavefunction -> Wavefunction -> Wavefunction

(-) :: Wavefunction -> Wavefunction -> Wavefunction

(*) :: Wavefunction -> Wavefunction -> Wavefunction

negate :: Wavefunction -> Wavefunction

abs :: Wavefunction -> Wavefunction

signum :: Wavefunction -> Wavefunction

fromInteger :: Integer -> Wavefunction

Show Wavefunction

Methods

showsPrec :: Int -> Wavefunction -> ShowS

show :: Wavefunction -> String

showList :: [Wavefunction] -> ShowS

data Operator

An operator on wavefunctions.

Instances

Eq Operator

Methods

(==) :: Operator -> Operator -> Bool

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

Floating Operator

Methods

pi :: Operator

exp :: Operator -> Operator

log :: Operator -> Operator

sqrt :: Operator -> Operator

(**) :: Operator -> Operator -> Operator

logBase :: Operator -> Operator -> Operator

sin :: Operator -> Operator

cos :: Operator -> Operator

tan :: Operator -> Operator

asin :: Operator -> Operator

acos :: Operator -> Operator

atan :: Operator -> Operator

sinh :: Operator -> Operator

cosh :: Operator -> Operator

tanh :: Operator -> Operator

asinh :: Operator -> Operator

acosh :: Operator -> Operator

atanh :: Operator -> Operator

log1p :: Operator -> Operator

expm1 :: Operator -> Operator

log1pexp :: Operator -> Operator

log1mexp :: Operator -> Operator

Fractional Operator

Methods

(/) :: Operator -> Operator -> Operator

recip :: Operator -> Operator

fromRational :: Rational -> Operator

Num Operator

Methods

(+) :: Operator -> Operator -> Operator

(-) :: Operator -> Operator -> Operator

(*) :: Operator -> Operator -> Operator

negate :: Operator -> Operator

abs :: Operator -> Operator

signum :: Operator -> Operator

fromInteger :: Integer -> Operator

Show Operator

Methods

showsPrec :: Int -> Operator -> ShowS

show :: Operator -> String

showList :: [Operator] -> ShowS

Construction

qubit

Arguments

:: QIndex

The index of the qubit in the wavefunction.

-> (Amplitude, Amplitude)

The amplitude of the 0 and 1 states, respectively.

-> Wavefunction

The wavefunction for the qubit.

Construct a qubit from the amplitudes of its states.

The squares of the norms of the amplitudes must sum to one.

pureQubit

Arguments

:: QIndex

Which qubit.

-> QState

The state of the qubit.

-> Wavefunction

The wavefunction.

Construct a qubit with a pure state.

qubits

Arguments

:: [Amplitude]

The amplitudes.

-> Wavefunction

The wavefunction.

Construct a wavefunction for the amplitudes of its qubit states.

Amplitudes ordered so that the 0 state appears before the 1 state and the lower qubit indices cycle faster than then higher qubit indices. For example, a two-qubit state has its amplitudes ordered |00>, |01>, |10>, |11>. This ordering can be generated as follows, where qubits are orderd from higher indices to lower ones:

>>> sequence $ replicate 3 [minBound..maxBound] :: [[QState]]
[[0,0,0],[0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0],[1,1,1]]

The squares of the norms of the amplitudes must sum to one.

groundState

Arguments

:: Int

Number of qubits.

-> Wavefunction

The ground state wavefunction.

Construct the ground state where each qubit is in state 0.

pureState

Arguments

:: [QState]

The state of each qubit, ordered from higher index to lower index.

-> Wavefunction

The wavefunction.

Constructa a pure state.

qubitsOperator

Arguments

:: [QIndex]

The qubit indices for which the operator applies, in descending order according to <https://arxiv.org/pdf/1711.02086/>.

-> [Amplitude]

The amplitudes of the operator matrix, in row-major order with the states ordered from higher indices to lower ones.

-> Operator

The wavefunction operator.

Construct an operator on qubit wavefunctions.

Amplitudes in row-major order where amplitudes are ordered so that the 0 state appears before the 1 state and the lower qubit indices cycle faster than then higher qubit indices. For example, a three-qubit operator has its amplitudes ordered <00|00>, <00|01>, <00|10>, <00|11>, <01|00>, <01|01>, <01|10>, <01|11>, <10|00>, <10|01>, <10|10>, <10|11>, <11|00>, <11|01>, <11|10>, <11|11>, where states in the bras and kets are correspond to the order of the first argument to qubitsOperator. This ordering can be generated as follows:

>>> fmap (splitAt 2) . sequence $ replicate (2 * 2) [minBound..maxBound] :: [([QState], [QState])]
[([0,0],[0,0]),([0,0],[0,1]),([0,0],[1,0]),([0,0],[1,1]),([0,1],[0,0]),([0,1],[0,1]),([0,1],[1,0]),([0,1],[1,1]),([1,0],[0,0]),([1,0],[0,1]),([1,0],[1,0]),([1,0],[1,1]),([1,1],[0,0]),([1,1],[0,1]),([1,1],[1,0]),([1,1],[1,1])]

The operator must be unitary.

Properties

wavefunctionOrder :: Wavefunction -> Int

Number of qubits in a wavefunction.

wavefunctionIndices

Arguments

:: Wavefunction

The wavefunction.

-> [QIndex]

List of qubit indices.

Qubit indices in a wavefunction.

wavefunctionAmplitudes

Arguments

:: Wavefunction

The wavefunction.

-> [([QState], Amplitude)]

List of qubit states and their amplitudes, where indices of states are ordered according to wavefunctionIndices.

Amplitudes of states in a qubit wavefunction.

rawWavefunction :: Wavefunction -> Tensor Amplitude

The Tensor encoding the wavefunction.

operatorOrder :: Operator -> Int

Number of qubits for an operator.

operatorIndices

Arguments

:: Operator

The operator.

-> [QIndex]

List of qubit indices.

Qubit indices in an operator.

operatorAmplitudes

Arguments

:: Operator

The wavefunction.

-> [(([QState], [QState]), Amplitude)]

List of qubit state transitions and their amplitudes, in row-major order with the states ordered according to operatorIndices.

Amplitudes of state transitions in a qubit operator.

rawOperator :: Operator -> Tensor Amplitude

The Tensor encoding the operator.

Operations

(^*^) :: Operator -> Operator -> Operator infixr 7

Apply two operators in sequence.

(^*) :: Operator -> Wavefunction -> Wavefunction infixr 6

Apply an operator to a wavefunction.

(*^) :: Wavefunction -> Operator -> Wavefunction infixl 6

Apply an operator to a wavefunction.

(^^*) :: Foldable t => t Operator -> Wavefunction -> Wavefunction infixr 6

Apply a sequence of operators to a wavefunction.

(*^^) :: Foldable t => Wavefunction -> t Operator -> Wavefunction infixl 6

Apply a sequence of operators to a wavefunction.

probabilities

Arguments

:: [QIndex]

Which qubits.

-> Wavefunction

The wavefunciton.

-> [([(QIndex, QState)], Double)]

The probabilities for the combinations of qubit states.

Probabilities of a selection of qubits.

project

Arguments

:: [(QIndex, QState)]

The qubits for the state.

-> Wavefunction

The wavefunction.

-> Wavefunction

The projected wavefunction.

Project a wavefunction onto a particular state.

measure

Arguments

:: RandomGen g

=> [QIndex]

Which qubits to measure.

-> Wavefunction

The wavefunction.

-> Rand g ([(QIndex, QState)], Wavefunction)

Action for the resulting measurement and wavefunction.

Measure qubits in a wavefunction.

wavefunctionProbability :: Wavefunction -> Amplitude

The total probability for the wave function, which should be 1.