Quipper.Algorithms.QLS.QLS

This module contains the Quipper implementation of the Quantum Linear Systems Algorithm.

The algorithm estimates the radar cross section for a FEM scattering problem by using amplitude estimation to calculate probability amplitudes.

The notations are based on the paper

• B. D. Clader, B. C. Jacobs, C. R. Sprouse. Quantum algorithm to calculate electromagnetic scattering cross sections. http://arxiv.org/abs/1301.2340.

Documentation

type OracleARunTime

Arguments

 = Double

Value resolution.

-> Int

Band.

-> Bool

Argflag.

-> ([Qubit], [Qubit], [Qubit])

(x=index,y+node,z+value).

-> Circ ([Qubit], [Qubit], [Qubit])

The type of oracle_A input arguments during runtime.

type OracleBRRunTime

Arguments

 = Double

Magnitude resolution.

-> Double

Phase resolution.

-> ([Qubit], [Qubit], [Qubit])

(x=index,m+magnitude,p+phase).

-> Circ ([Qubit], [Qubit], [Qubit])

The type of oracle_b and oracle_r input arguments during runtime.

data Oracle

A type to encapsulate all three oracles.

Constructors

Oracle

Fields

• oracle_A :: RunTimeParam -> OracleARunTime
• oracle_b :: RunTimeParam -> OracleBRRunTime
• oracle_r :: RunTimeParam -> OracleBRRunTime

dummy_oracle :: Oracle

A set of oracles using only blackboxes.

data RunTimeParam

A type to hold the runtime parameters.

Constructors

RT_param

Fields

• k :: Double

  Wave number.

• theta :: Double

  Incident wave angle.

• phi :: Double

  Direction of desired far field radiation pattern.

• e0 :: Double

  Incident wave amplitude.

• lambda :: Double

  Wavelength.

• xlength :: Double

  x-length of square scattering region.

• ylength :: Double

  y-length of square scattering region.

• scatteringnodes :: [(Int, Int)]

  Metallic region.

• nx :: Int
• ny :: Int
• lx :: Double
• ly :: Double
• kappa :: Double
• epsilon :: Double
• t0 :: Double
• r :: Double
• b_max :: Double
• r_max :: Double
• d :: Int
• nb :: Int
• p2 :: Double
• n0 :: Int
• n1 :: Int
• n2 :: Int
• n4 :: Int
• magnitudeArgflag :: Bool
• phaseArgflag :: Bool

Instances

Show RunTimeParam

Defined in Quipper.Algorithms.QLS.QLS

Methods

showsPrec :: Int -> RunTimeParam -> ShowS

show :: RunTimeParam -> String

showList :: [RunTimeParam] -> ShowS

dummy_RT_param :: RunTimeParam

A convenient set of runtime parameters for testing.

large_RT_param :: RunTimeParam

A set of larger values, for testing scalability.

small_RT_param :: RunTimeParam

A set of smaller values, for manageable yet meaningful output.

expYt :: Timestep -> Qubit -> Circ Qubit

Apply an e^−iYt gate. The timestep t is a parameter.

expYt_at :: Timestep -> Qubit -> Circ ()

Apply an e^−iYt gate. The timestep t is a parameter.

dynamic_lift_double :: Double -> [Bit] -> Circ Double

Read a list of bits and make it into a Double, by multiplying its integer value by the provided factor.

qft_for_show :: [Qubit] -> Circ [Qubit]

A black box gate to stand in as a replacement for QFT.

qlsa_FEM_main :: RunTimeParam -> Oracle -> Circ Double

Main function: for estimating the radar cross section for a FEM scattering problem. The problem can be reduced to the calculation of four angles: φ_b, φ_bx, φ_r1 and φ_r0.

Amplitude Estimation Functions

qlsa_AmpEst_phi_b :: RunTimeParam -> Oracle -> Circ Double

Estimates φ_b, related to the probability of success for the preparation of the known state b, using amplitude amplification.

test_qlsa_AmpEst_phi_b :: Bool -> IO ()

Testing function for qlsa_AmpEst_phi_b.

qlsa_AmpEst_phi_bx :: RunTimeParam -> Oracle -> Circ Double

Estimates φ_bx, related to the probability of success in computing solution value x.

qlsa_AmpEst_phi_bxr :: RunTimeParam -> Oracle -> Bool -> Circ Double

Estimates φ_r0 and φ_r1 (depending on the boolean parameter), related to the overlap of the solution with the arbitrary state r.

State Preparation.

qlsa_StatePrep :: RunTimeParam -> ([Qubit], Qubit) -> OracleBRRunTime -> Double -> Circ ()

Prepares a quantum state x, as specified by an oracle function, entangled with a single qubit flag q marking the desired state.

test_qlsa_StatePrep :: Bool -> IO ()

Testing function for qlsa_StatePrep.

Linear System Solver Functions

qlsa_Solve_x :: RunTimeParam -> ([Qubit], Qubit) -> Oracle -> Circ ()

Implements the QLSA procedure to generate the solution state |x〉.

integer_inverse :: ([Qubit], [Qubit]) -> Circ ()

Implementation of the integer division. The two registers are supposed to be of the same size and represent little-headian unsigned integers, i.e., the head of the list holds the least significant bit.

qlsa_Solve_xr :: RunTimeParam -> ([Qubit], [Qubit], Qubit, Qubit, Qubit, Qubit) -> Oracle -> Circ ()

Implements the complete QLSA procedure to find the solution state |x〉 and then implements the swap protocol required for estimation of 〈x|r〉.

Hamiltonian Simulation Functions.

qlsa_HamiltonianSimulation :: RunTimeParam -> ([Qubit], [Qubit]) -> OracleARunTime -> Circ ()

Uses a quantum register |t〉 to control the implementation of the Suzuki method for simulating a Hamiltonian specified by an oracle function.

test_qlsa_HamiltonianSimulation :: Bool -> IO ()

Testing function for qlsa_HamiltonianSimulation.

qlsa_HsimKernel :: RunTimeParam -> ([Qubit], [Qubit]) -> Int -> Double -> OracleARunTime -> Circ ()

Uses an oracle function and timestep control register (t) to apply 1-sparse Hamiltonian to the input state |t, x〉.

test_qlsa_HsimKernel :: Bool -> IO ()

Testing function for qlsa_HsimKernel.

qlsa_ApplyHmag :: RunTimeParam -> ([Qubit], [Qubit], [Qubit]) -> Double -> Circ ()

Applies the magnitude component of coupling elements in a 1-sparse Hamiltonian.

test_qlsa_ApplyHmag :: Bool -> IO ()

Testing function for qlsa_ApplyHmag.

w :: (Qubit, Qubit) -> Circ ()

Auxiliary function: the W-gate.

test_w :: IO ()

Testing function for w.

Controlled Logic Operations

qlsa_ControlledPhase :: [Qubit] -> Double -> Bool -> Circ ()

Applies a phase shift of φ/2 to the signed input register |φ〉.

qlsa_ControlledRotation :: ([Qubit], Qubit) -> Double -> Bool -> Circ ()

Applies a rotation of φ/2 to the signed input register |φ〉.

Oracles

make_factor_rep :: Double -> Int -> QDouble -> Circ [Qubit]

Map a QDouble into an integer, understood as being scaled by the given factor. Take the factor and the size of the output register as parameter.

inline_oracle_r :: RunTimeParam -> Double -> Double -> ([Qubit], [Qubit], [Qubit]) -> Circ ([Qubit], [Qubit], [Qubit])

Implements the oracle for the arbitrary state |r〉, using the Template Haskell implementation of calcRweights.

inline_oracle_b :: RunTimeParam -> Double -> Double -> ([Qubit], [Qubit], [Qubit]) -> Circ ([Qubit], [Qubit], [Qubit])

Implements the oracle for the known state |b〉, using the Template Haskell implementation of getKnownWeights.

inline_oracle_A :: RunTimeParam -> Double -> Int -> Bool -> ([Qubit], [Qubit], [Qubit]) -> Circ ([Qubit], [Qubit], [Qubit])

Implementation of the oracle calculating the matrix A corresponding to the discretization of the scattering problem, using the Template Haskell implementation of getNodeValuesMoreOutputs.

inline_oracle :: Oracle

Encapsulate the inline oracles in Template Haskell into an object of type Oracle.