Sparse

SELECT and PREPARE for the second quantized sparse chemistry Hamiltonian.

Starting from the traditional second quantized chemistry Hamiltonian

\[ H = \sum_\sigma \sum_{pq} T_{pq} a_{p\sigma}^\dagger a_{q\sigma} + \frac{1}{2}\sum_{\alpha\beta} \sum_{pqrs} V_{pqrs} a_{p\sigma}^\dagger a_{r\beta}^\dagger a_{s\beta} a_{q\alpha}, \]

where \(a_{p\sigma}\) (\(a_{q\sigma}^\dagger\)) annihilate (create) an electron in the \(p\)-th orbital of spin \(\sigma\). We can rewrite this expression using the Jordan-Wigner transformation as

\[ H = T' + V', \]

where

\[ T' = \frac{1}{2} \sum_\sigma \sum_{pq} T_{pq}'Q_{pq\sigma}, \]

\[ V' = \sum_{\alpha\beta}\sum_{pqrs}V_{pqrs}Q_{pq\alpha}Q_{rs\beta}, \]

and \(V = (pq|rs)\) are the usual two-electron integrals in chemist’s notation,

\[ T'_{pq} = T_{pq} - \sum_r V_{pqrr}, \]

and

\[\begin{split} Q_{pq\sigma} = \begin{cases} X_{p\sigma}\vec{Z}X_{q\sigma} & p < q \\ Y_{p\sigma}\vec{Z}Y_{q\sigma} & p > q \\ -Z_{p\sigma} & p = q \end{cases}. \end{split}\]

The sparse Hamiltonian simply sets to zero any term in the Hamiltonian where \(|V_{pqrs}|\) is less than some threshold. This reduces the amount of data that is required to be loaded during state preparation as only non-zero symmetry inequivalent terms are required (the two electron integrals exhibit 8-fold permutational symmetry). Symmetries are restored by initially appropriately weighting these non-zero terms and then using \(|+\rangle\) states to perform control swaps between the \(pqrs\) registers.

from qualtran import Bloq, CompositeBloq, BloqBuilder, Signature, Register
from qualtran import QBit, QInt, QUInt, QAny
from qualtran.drawing import show_bloq, show_call_graph, show_counts_sigma
from typing import *
import numpy as np
import sympy
import cirq

PrepareSparse

Prepare oracle for the sparse chemistry Hamiltonian

Prepare the state:

\[ |0\rangle|+\rangle|0\rangle|0\rangle \sum_{\sigma}\sum_{pq} \sqrt{\frac{T_{pq}'}{2\lambda}} |\theta_{pq}^T\rangle|pq\sigma\rangle|000\rangle +|1\rangle|+\rangle|+\rangle|+\rangle \sum_{\alpha\beta} \sum_{pqrs} \sqrt{\frac{\tilde{V}_{pqrs}'}{2\lambda}} |\theta_{pqrs}^V\rangle|pq\alpha\rangle|rs\beta\rangle \]

Parameters

• num_spin_orb: The number of spin orbitals.

• num_non_zero: The number of non-zero matrix elements.

• num_bits_state_prep: the number of bits of precision for state preparation. This will control the size of the keep register.

• num_bits_rot_aa: The number of bits of precision for the single-qubit rotation for amplitude amplification during the uniform state preparation. Default 8.

• is_adjoint: Whether we are apply PREPARE or PREPARE^dag

• log_block_size: qroam (log) block size.

Registers

• d: the register indexing non-zero matrix elements.

• pqrs: the register to store the spatial orbital index.

• sigma: the register prepared for alias sampling.

• alpha: spin for (pq) indicies.

• beta: spin for (rs) indicies.

• rot_aa: the qubit rotated for amplitude amplification.

• swap_pq: a |+> state to restore the symmetries of the p and q indices.

• swap_rs: a |+> state to restore the symmetries of the r and s indices.

• swap_pqrs: a |+> state to restore the symmetries of between (pq) and (rs).

• theta: sign qubit.

• alt_pqrs: the register to store the alternate values for the spatial orbital indices.

• theta: A two qubit register for the sign bit and it’s alternate value.

• keep: The register containing the keep values for alias sampling.

• less_than: A single qubit for the result of the inequality test during alias sampling.

• flag_1b: a single qubit register indicating whether to apply only the one-body SELECT.

• alt_flag_1b: alternate value for flag_1b

Refererences: Even More Efficient Quantum Computations of Chemistry Through Tensor hypercontraction Eq. A11.

from qualtran.bloqs.chemistry.sparse import PrepareSparse

Example Instances

from qualtran.bloqs.chemistry.sparse.prepare_test import build_random_test_integrals

num_spin_orb = 6
tpq, eris = build_random_test_integrals(num_spin_orb // 2)
prep_sparse = PrepareSparse.from_hamiltonian_coeffs(
    num_spin_orb, tpq, eris, num_bits_state_prep=4, log_block_size=1
)

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([prep_sparse],
           ['`prep_sparse`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
prep_sparse_g, prep_sparse_sigma = prep_sparse.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(prep_sparse_g)
show_counts_sigma(prep_sparse_sigma)

Counts totals:

• CSwap: 1

• CSwap: 8

• H: 3

• H⨂4: 1

• LessThanEqual: 1

• PrepareUniformSuperposition: 1

• QROAMClean: 1

SelectSparse

SELECT oracle for the sparse Hamiltonian.

Implements the two applications of Fig. 13.

Parameters

• num_spin_orb: The number of spin orbitals. Typically called N.

• control_val: The control value. If None, then the bloq is not controlled.

Registers

• flag_1b: a single qubit to flag whether the one-body Hamiltonian is to be applied or not during SELECT.

• swap_pq: a |+> state to restore the symmetries of the p and q indices.

• swap_rs: a |+> state to restore the symmetries of the r and s indices.

• swap_pqrs: a |+> state to restore the symmetries of between (pq) and (rs).

• theta: sign qubit.

• pqrs: the register to store the spatial orbital index.

• alpha: spin for (pq) indicies.

• beta: spin for (rs) indicies.

Refererences: Even More Efficient Quantum Computations of Chemistry Through Tensor hypercontraction Fig 13.

from qualtran.bloqs.chemistry.sparse import SelectSparse

Example Instances

num_spin_orb = 4
sel_sparse = SelectSparse(num_spin_orb)

Graphical Signature

from qualtran.drawing import show_bloqs
show_bloqs([sel_sparse],
           ['`sel_sparse`'])

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
sel_sparse_g, sel_sparse_sigma = sel_sparse.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(sel_sparse_g)
show_counts_sigma(sel_sparse_sigma)

Counts totals:

• S: 1

• SelectedMajoranaFermion(X): 1

• SelectedMajoranaFermion(X): 1

• SelectedMajoranaFermion(Y): 1

• SelectedMajoranaFermion(Y): 1