Sparse Matrix Hermitian

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

SparseMatrixHermitian

Hermitian Block encoding of a sparse-access Hermitian matrix.

Given column and entry oracles \(O_c\) and \(O_A\) for an \(s\)-sparse Hermitian matrix \(A \in \mathbb{C}^{2^n \times 2^n}\), i.e. one where each row / column has exactly \(s\) non-zero entries, computes a \((s, n+1, \epsilon)\)-block encoding of \(A\) as follows:

           ┌────┐
a     |0> ─┤    ├─     |0> ───────────────────────X────────────────────
           │    │           ┌──┐                  |               ┌──┐
           │ U  │  =        │ n│ ┌────┐ ┌────┐    | ┌────┐ ┌────┐ │ n│
l   |0^n> ─┤  A ├─   |0^n> ─┤H ├─┤ O  ├─┤    ├─X──|─┤    ├─┤ O* ├─┤H ├─
           │    │           └──┘ |  c | │    │ |  | │    │ |  c | └──┘
           │    │                └────┘ │ O  │ │  | │ O* │ └────┘
b     |0> ─┤    ├─     |0> ────────|────┤  A ├─|──X─┤  A ├───|─────────
           |    |                ┌────┐ |    | |    |    | ┌────┐
           |    |                | O  | |    | |    |    | | O* |
j   |Psi> ─┤    ├─   |Psi> ──────┤  c ├─┤    ├─X────┤    ├─┤  c ├──────
           └────┘                └────┘ └────┘      └────┘ └────┘

To encode a matrix of irregular dimension, the matrix should first be embedded into one of dimension \(2^n \times 2^n\) for suitable \(n\). To encode a matrix where each row / column has at most \(s\) non-zero entries, some zeroes should be treated as if they were non-zero so that each row / column has exactly \(s\) non-zero entries.

For encoding a non-hermitian matrix, or a slightly more efficient (but non Hermitian-encoding) of a matrix, use :class:SparseMatrix instead.

Parameters

• col_oracle: The column oracle \(O_c\). See RowColumnOracle for definition.

• entry_oracle: The entry oracle \(O_A\). See EntryOracle for definition.

• eps: The precision of the block encoding.

• is_controlled: if True, returns the controlled block-encoding.

Registers

• ctrl: The single qubit control register. (present only if is_controlled is True)

• system: The system register.

• ancilla: The ancilla register.

• resource: The resource register (present only if bitsize > 0).

References

• Lecture Notes on Quantum Algorithms for Scientific Computation. Lin Lin (2022). Ch. 6.5. Proposition 6.8, Fig 6.7.

from qualtran.bloqs.block_encoding import SparseMatrixHermitian

Example Instances

from qualtran.bloqs.block_encoding.sparse_matrix import TopLeftRowColumnOracle
from qualtran.bloqs.block_encoding.sparse_matrix_hermitian import UniformSqrtEntryOracle

n = sympy.Symbol('n', positive=True, integer=True)
col_oracle = TopLeftRowColumnOracle(system_bitsize=n)
entry_oracle = UniformSqrtEntryOracle(system_bitsize=n, entry=0.3)
sparse_matrix_symb_hermitian_block_encoding = SparseMatrixHermitian(
    col_oracle, entry_oracle, eps=0
)

from qualtran.bloqs.block_encoding.sparse_matrix import TopLeftRowColumnOracle
from qualtran.bloqs.block_encoding.sparse_matrix_hermitian import UniformSqrtEntryOracle

col_oracle = TopLeftRowColumnOracle(system_bitsize=2)
entry_oracle = UniformSqrtEntryOracle(system_bitsize=2, entry=0.3)
sparse_matrix_hermitian_block_encoding = SparseMatrixHermitian(col_oracle, entry_oracle, eps=0)

Graphical Signature

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

Call Graph

from qualtran.resource_counting.generalizers import ignore_split_join
sparse_matrix_symb_hermitian_block_encoding_g, sparse_matrix_symb_hermitian_block_encoding_sigma = sparse_matrix_symb_hermitian_block_encoding.call_graph(max_depth=1, generalizer=ignore_split_join)
show_call_graph(sparse_matrix_symb_hermitian_block_encoding_g)
show_counts_sigma(sparse_matrix_symb_hermitian_block_encoding_sigma)

Counts totals:

• PrepareUniformSuperposition: 1

• PrepareUniformSuperposition†: 1

• Swap: 1

• Swap: 1

• TopLeftRowColumnOracle: 1

• TopLeftRowColumnOracle†: 1

• UniformSqrtEntryOracle: 1

• UniformSqrtEntryOracle†: 1