Single Factorization

Bloqs for the single-factorized chemistry Hamiltonian in second quantization.

Recall that for the single factorized Hamiltonian we have

\[ H = \sum_{pq}T'_{pq} a_p^\dagger a_q + \frac{1}{2} \sum_l \left(\sum_{pq} W_{pq}^{(l)} a_p^\dagger a_q\right)^2. \]

where \(\sum_l^L W_{pq}^{(l)} W_{rs}^{(l)} = (pq|rs)\) are the standard chemist’s electron repulsion integrals.

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

SingleFactorizationOneBody

Block encoding of single factorization one-body Hamiltonian.

Implements inner “half” of Fig. 15 in the reference. This block encoding is applied twice (with a reflection around the inner state preparation registers) to implement (roughly) the square of this one-body operator.

Parameters

• num_aux: Dimension of auxiliary index for single factorized Hamiltonian. Called \(L\) in the reference.

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

• num_bits_state_prep: The number of bits of precision for coherent alias sampling. Called \(\aleph\) in the reference.

• num_bits_rot_aa: Number of bits of precision for rotations for amplitude amplification in uniform state preparation. Called \(b_r\) in the reference.

• is_adjoint: Whether this bloq is daggered or not. This affects the QROM cost.

• kp1: QROAM blocking factor for data prepared over l (auxiliary) index. Defaults to 1 (i.e. QROM).

• kp1: QROAM blocking factor for data prepared over pq indicies. Defaults to 1 (i.e.) QROM.

• kp1_inv: QROAM blocking factor for inverting data prepared over l (auxiliary) index.

• kp1_inv: QROAM blocking factor for inverting of data prepared over pq.

Registers

• succ_l: Flag for success of l state preparation.

• l_ne_zero: Flag for l != 0.

• succ_pq: Flag for success of pq.

• l: register to store L values for auxiliary index.

• p: spatial orbital index. range(0, num_spin_orb // 2)

• q: spatial orbital index. range(0, num_spin_orb // 2)

• rot_aa: Qubit to rotate for amplitude amplification for state preparation.

• swap_pq: Qubit for controlling swaps over p and q registers.

• spin: Qubit for controlling swaps over system registers.

• sys: The system registers.

Refererences: Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction Fig. 15 page 43.

from qualtran.bloqs.chemistry.sf.single_factorization import SingleFactorizationOneBody

Example Instances

num_aux = 50
num_bits_state_prep = 12
num_bits_rot_aa = 7
num_spin_orb = 10
sf_one_body = SingleFactorizationOneBody(
    num_aux=num_aux,
    num_spin_orb=num_spin_orb,
    num_bits_state_prep=num_bits_state_prep,
    num_bits_rot_aa=num_bits_rot_aa,
    is_adjoint=False,
)

Graphical Signature

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

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
sf_one_body_g, sf_one_body_sigma = sf_one_body.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(sf_one_body_g)
show_counts_sigma(sf_one_body_sigma)

Counts totals:

• CSwap: 2

• H: 4

• InnerPrepareSingleFactorization: 1

• InnerPrepareSingleFactorization†: 1

• SelectSingleFactorization: 1

SingleFactorizationBlockEncoding

Block encoding of single factorization Hamiltonian.

Implements Fig. 15 in the reference.

Parameters

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

• num_aux: Dimension of auxiliary index for single factorized Hamiltonian. Called \(L\) in the reference.

• num_bits_state_prep: The number of bits of precision for coherent alias sampling. Called \(\aleph\) in the reference.

• num_bits_rot_aa_outer: Number of bits of precision for rotations for amplitude amplification in outer uniform state preparation. Called \(b_r\) in the reference.

• num_bits_rot_aa_inner: Number of bits of precision for rotations for amplitude amplification in inner uniform state preparation. Called \(b_r\) in the reference.

• kp1: QROAM blocking factor for data prepared over l (auxiliary) index. Defaults to 1 (i.e. QROM).

• kp1: QROAM blocking factor for data prepared over pq indicies. Defaults to 1 (i.e.) QROM.

• kp1_inv: QROAM blocking factor for inverting the data prepared over l (auxiliary) index.

• kp1_inv: QROAM blocking factor for inverting the data prepared over pq.

Registers

• ctrl: Control registers for state preparation.

• l: register to store L values for auxiliary index.

• p: spatial orbital index. range(0, num_spin_orb // 2)

• q: spatial orbital index. range(0, num_spin_orb // 2)

• rot_aa: Qubit to rotate for amplitude amplification for state preparation.

• swap_pq: Qubit for controlling swaps over p and q registers.

• spin: Qubit for controlling swaps over system registers.

• sys: The system registers.

Refererences: Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction Fig 15, page 43.

from qualtran.bloqs.chemistry.sf.single_factorization import SingleFactorizationBlockEncoding

Example Instances

num_spin_orb = 10
num_aux = 50
num_bits_state_prep = 12
sf_block_encoding = SingleFactorizationBlockEncoding(
    num_spin_orb=num_spin_orb,
    num_aux=num_aux,
    num_bits_state_prep=num_bits_state_prep,
    num_bits_rot_aa_outer=7,
    num_bits_rot_aa_inner=7,
)

Graphical Signature

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

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
sf_block_encoding_g, sf_block_encoding_sigma = sf_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(sf_block_encoding_g)
show_counts_sigma(sf_block_encoding_sigma)

Counts totals:

• B[H]: 2

• OuterPrepareSingleFactorization: 1

• OuterPrepareSingleFactorization†: 1

• ReflectionUsingPrepare: 1