Block Encoding Interface

High level bloqs for defining bloq encodings and operations on block encodings.

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

BlockEncoding

Abstract interface for an arbitrary block encoding.

In general, given an \(s\)-qubit operator \(H\) then the \((s+a)\)-qubit unitary \(B[H] = U\) is a \((\alpha, a, \epsilon)\)-block encoding of \(H\) if it satisfies:

\[ \lVert H - \alpha (\langle G|_a\otimes I_s U |G\rangle_a \otimes I_s) \rVert \le \epsilon, \]

where \(a\) is an ancilla register and \(s\) is the system register, \(U\) is a unitary sometimes called a signal oracle, \(\alpha\) is a normalization constant chosen such that \(\alpha \ge \lVert H\rVert\) (where \(\lVert \cdot \rVert\) denotes the spectral norm), and \(\epsilon\) is the precision to which the block encoding is prepared. The state \(|G\rangle_a\) is sometimes called the signal state, and its form depends on the details of the block encoding.

For LCU based block encodings with \(H = \sum_l w_l U_l\) we have

\[ U = \sum_l |l\rangle\langle l| \otimes U_l \]

and \(|G\rangle = \sum_l \sqrt{\frac{w_l}{\alpha}}|l\rangle_a\), which define the usual SELECT and PREPARE oracles.

Other ways of building block encodings exist so we define the abstract base class BlockEncoding bloq, which expects values for \(\alpha\), \(\epsilon\), system and ancilla registers and a bloq which prepares the state \(|G\rangle\).

Users must specify:

1. the normalization constant \(\alpha \ge \lVert A \rVert\), where \(\lVert \cdot \rVert\) denotes the spectral norm.

2. the precision to which the block encoding is to be prepared (\(\epsilon\)).

Developers must provide a method to return a bloq to prepare \(|G\rangle\).

References

• Hamiltonian Simulation by Qubitization. Low et al. 2019. Sec 2 and 3 for introduction and definition of terms.

• The power of block-encoded matrix powers: improved regression techniques via faster Hamiltonian simulation. Chakraborty et al. 2018. Definition 3 page 8.

from qualtran.bloqs.block_encoding import BlockEncoding