Quipper.Algorithms.GSE.GSE

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.

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 ()

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 ()

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 ()

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

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_maxτ2ke^-iHτ2k 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

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.