Double Factorization

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

Recall that for the single factorized Hamiltonian we have

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

One arrives at the double factorized (DF) Hamiltonian by further factorizing the \(W_{pq}^{(l)}\) terms as

\[ W^{(l)}_{pq} = \sum_{k} U^{(l)}_{pk} f_k^{(l)} U^{(l)*}_{qk}, \]

so that

\[ H = \sum_{pq}T'_{pq} a_p^\dagger a_q + \frac{1}{2} \sum_l U^{(l)}\left(\sum_{k}^{\Xi^{(l)}} f_k^{(l)} n_k\right)^2 U^{(l)\dagger} \]

where \(\Xi^{(l)} \) is the rank of second factorization.

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

DoubleFactorizationOneBody

Block encoding of double 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 double factorized Hamiltonian. Call L in Ref[1].

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

• num_eig: The total number of eigenvalues.

• 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 single qubit rotation for amplitude amplification. Called \(b_r\) in the reference.

• num_bits_rot: Number of bits of precision for rotations for amplitude amplification in uniform state preparation. Called \(\beth\) in Ref[1].

Registers

• succ_l: control for success for outer state preparation.

• succ_p: control for success for inner state preparation, this is reused in second application.

• l_ne_zero: control for one-body part of Hamiltonian.

• xi: data register for number storing \(\Xi^{(l)}\).

• p: Register for inner state preparation.

• rot_aa: A qubit to be rotated for amplitude amplification.

• spin: A single qubit register for spin superpositions.

• xi: Register for rank parameter.

• offset: Offset for p register.

• rot: Amplitude amplification angles for inner preparations.

• rotations: Rotations for basis rotations.

• sys: The system register.

Refererences: Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction

from qualtran.bloqs.chemistry.df.double_factorization import DoubleFactorizationOneBody

Example Instances

num_bits_state_prep = 12
num_bits_rot = 7
num_spin_orb = 10
num_aux = 50
num_eig = num_aux * (num_spin_orb // 2)
df_one_body = DoubleFactorizationOneBody(
    num_aux=num_aux,
    num_spin_orb=num_spin_orb,
    num_eig=num_eig,
    num_bits_state_prep=num_bits_state_prep,
    num_bits_rot=num_bits_rot,
)

Graphical Signature

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

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
df_one_body_g, df_one_body_sigma = df_one_body.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(df_one_body_g)
show_counts_sigma(df_one_body_sigma)

Counts totals:

• ApplyControlledZs: 1

• CSwap: 2

• H: 2

• In-Prep: 1

• In-Prep†: 1

• ProgRotGateArray: 1

• ProgRotGateArray†: 1

DoubleFactorizationBlockEncoding

Block encoding of double 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 double factorized Hamiltonian. Typically called \(L\).

• num_eig: The total number of eigenvalues.

• num_bits_state_prep: The number of bits of precision for coherent alias sampling. Called \(\aleph\) in Ref[1]. We assume this is the same for both outer and inner state preparations.

• num_bits_rot: Number of bits of precision for rotations amplification in uniform state preparation. Called \(\beth\) in Ref[1].

• num_bits_rot_aa_outer: Number of bits of precision for single qubit rotation for amplitude amplification in outer state preparation. Called \(b_r\) in the reference. Keeping inner and outer separate for consistency with openfermion.

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

Registers

• ctrl: Registers used to control application of Hamiltonian terms / preparation.

• l: Register for outer state preparation.

• p: Register for inner state preparation.

• rot_aa: A qubit to be rotated for amplitude amplification.

• spin: A single qubit register for spin superpositions.

• xi: Register for rank parameter.

• offset: Offset for p register.

• rot: Amplitude amplification angles for inner preparations.

• rotations: Rotations for basis rotations.

• sys: The system register.

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

from qualtran.bloqs.chemistry.df.double_factorization import DoubleFactorizationBlockEncoding

Example Instances

num_spin_orb = 10
num_aux = 50
num_eig = num_aux * (num_spin_orb // 2)
num_bits_state_prep = 12
num_bits_rot = 7
df_block_encoding = DoubleFactorizationBlockEncoding(
    num_spin_orb=num_spin_orb,
    num_aux=num_aux,
    num_eig=num_eig,
    num_bits_state_prep=num_bits_state_prep,
    num_bits_rot_aa_outer=1,
    num_bits_rot_aa_inner=7,
    num_bits_rot=num_bits_rot,
)

Graphical Signature

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

from qualtran.drawing import get_musical_score_data, draw_musical_score
msd = get_musical_score_data(df_block_encoding.decompose_bloq())
fig, ax = draw_musical_score(msd)
fig.set_size_inches(14, 10)

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
df_block_encoding_g, df_block_encoding_sigma = df_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(df_block_encoding_g)
show_counts_sigma(df_block_encoding_sigma)

Counts totals:

• B[H_1]: 2

• In_l-data_l: 1

• In_l-data_l†: 1

• OuterPrep: 1

• OuterPrep†: 1

• ReflectionUsingPrepare: 1