Block-Encoding of Operators
Embedding a non-unitary operator as a sub-block of a larger unitary, the interface that modern simulation algorithms are built on.
The construction
A unitary U is an (alpha, m, epsilon) block-encoding of an operator A on n qubits if the top-left block of U, accessed by the m ancilla qubits, equals A / alpha up to error epsilon. Formally, (<0|^m tensor I) U (|0>^m tensor I) is approximately A / alpha. The factor alpha is the subnormalization; A must be scaled down so its block fits inside a unitary.
Why embed operators this way
Quantum computers apply unitaries. Physically interesting operators, including Hamiltonians and projectors, are generally not unitary. Block-encoding gives a uniform way to access any such operator: prepare the ancillas in |0>, apply U, and post-select on measuring |0>. The success amplitude carries the action of A / alpha.
How to build one
- From an LCU: the PREPARE-SELECT-PREPARE construction is a block-encoding of sum_j a_j U_j with alpha = sum_j a_j.
- From sparse-access oracles: for a d-sparse Hamiltonian, oracles for entry positions and values yield a block-encoding with alpha proportional to d times the max entry.
- From density matrices, POVMs, or purified access, each with its own subnormalization.
Composing block-encodings
Block-encodings form an algebra. Products of block-encodings block-encode products of operators (with multiplied alphas). Linear combinations combine via LCU on the ancillas. This composability lets you build complicated operators from simple parts, which is the backbone of the quantum singular value transformation.
The gateway to QSP and qubitization
Once H is block-encoded, quantum signal processing and qubitization apply polynomial transformations to its eigenvalues (or singular values) by interleaving the block-encoding with single-qubit rotations. Simulating e^(-iHt) becomes the task of approximating the function e^(-ix t) as a polynomial and applying it through the block-encoding. Block-encoding is thus the standard interface between a Hamiltonian and the near-optimal algorithms that process it.