Safe Haskell	None

Quipper.Algorithms.GSE.GSE

Contents

Basic time step
Iteration
Main circuit

Description

This module provides the main circuit for the GSE algorithm. This circuit consists of a state preparation, followed by a large number of Hamiltonian simulation terms for small time steps, followed by an inverse Quantum Fourier Transform and a final measurement.

Synopsis

exp_pq :: ((Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ ()
exp_pqrs :: ((Int, Int, Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ ()
exp_pqrs_orthodox :: ((Int, Int, Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ ()
unitary_hat_at :: GSEData -> Int -> Double -> Double -> Bool -> Int -> [Qubit] -> Qubit -> Circ ()
gse :: Int -> Int -> Int -> GSEData -> Double -> Double -> (Int -> Int) -> Bool -> Circ [Bit]

Basic time step

These functions provide one- and two-electron operators for an individual Trotter time step θ. Each operator consists of a large number of individual Hamiltonian terms.

exp_pq :: ((Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ () Source #

Apply the one-electron operator e^-iθH, where H = h_pq a_p^†a_q if p = q and H = h_pq (a_p^†a_q + a_q^†a_p) otherwise, to every pair of qubits p≥q in a register |ψ〉. The inputs are Hamiltonian data h, the angle θ, the register |ψ〉, and a control qubit.

exp_pqrs :: ((Int, Int, Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ () Source #

Apply the two-electron operator e^-iθH, where H = h_pqrs a_p^†a_q^†a_ra_s if (p,q) = (s,r) and H = a_p^†a_q^†a_ra_s + a_s^†a_r^†a_qa_p otherwise, to every quadruple (p, q, r, s) of qubits in a register |ψ〉. To ensure that terms are enumerated exactly once, we only consider indices where (p, q) ≥ (s, r) in the lexicographic order (i.e., p>s or (p=s and q≥r). The inputs are Hamiltonian data h, the angle θ, the register |ψ〉, and a control qubit.

exp_pqrs_orthodox :: ((Int, Int, Int, Int) -> Double) -> Double -> [Qubit] -> Qubit -> Circ () Source #

Like exp_pqrs, but use the "orthodox" circuit template for the Coulomb operator.

Iteration

The following function iterates the basic Trotter timestep N_k times, and also normalizes the maximum energy E_max.

unitary_hat_at Source #

Arguments

:: GSEData	The integral data h_pq and h_pqrs.
-> Int	The Trotter iteration count N_k.
-> Double	The Hamiltonian scaling parameter τ.
-> Double	The maximum energy E_max.
-> Bool	Use the "orthodox" Coulomb operator?
-> Int	The control qubit index k.
-> [Qubit]	The state \|ψ〉.
-> Qubit	The control qubit b_k.
-> Circ ()

Apply the operator Û_k ≈ e^{iE_maxτ2^k}e^{-iHτ2^k} to |ψ〉.

Main circuit

The main circuit for the GSE Algorithm. This consists of the initial state preparation, the Trotterized phase estimation circuit, the Quantum Fourier Transform, and the final measurement.

gse Source #

Arguments

:: Int	The number of precision qubits b.
-> Int	The number of basis functions M.
-> Int	The number of occupied orbitals N.
-> GSEData	The integral data h_pq and h_pqrs.
-> Double	The Hamiltonian scaling parameter τ.
-> Double	The maximum energy E_max.
-> (Int -> Int)	The function k ↦ N_k.
-> Bool	Use the "orthodox" Coulomb operator?
-> Circ [Bit]

The main circuit for the GSE Algorithm.