Block Encoding of Linear Transformation (BELT)

• Block Encoding of Linear Transformation (BELT) is a framework to embed arbitrary linear maps, including non-unitary and non-CP maps, into unitary operators.
• It constructs composite unitaries using state preparation and Choi matrix block-encodings to simulate non-physical operations with improved resource efficiency.
• BELT finds practical applications in entanglement detection, quantum channel inversion, and simulation of pseudo-differential operators, broadening quantum algorithm capabilities.

Block Encoding of Linear Transformation (BELT) refers to a general framework and specific quantum protocols for embedding a (potentially non-unitary, non-Hermitian, and even non-completely positive) linear transformation or map into a block of an overall unitary operator. This enables the coherent manipulation and interrogation of the linear map's action using quantum circuits, even in cases where the map cannot be physically realized as a standalone operation. Block encodings, as a conceptual backbone of many quantum algorithms, are a central enabling technology for quantum linear algebra, simulation, information processing, and error mitigation.

1. General Framework and Definition

BELT formalizes the embedding of an arbitrary linear map $\mathcal{N}$ into a unitary extension $U$ such that the action of $\mathcal{N}$—for example, on a density matrix $\rho$—is represented by a principal (often upper-left) block of $U$: $$\|\mathcal{N}(\rho) - \alpha \left(\langle 0^m | \otimes I \right) U \left(|0^m\rangle \otimes I\right) \|_{\infty} \leq \epsilon,$$ where $\alpha$ is a scaling (subnormalization) factor, $m$ counts ancilla qubits, and $\epsilon$ is the accuracy. Unlike Gate-based block encodings for matrices (as in the QSVT paradigm), BELT is defined to accept arbitrary (not necessarily Hermitian- or even completely positive-preserving) linear maps, including morphisms outside the quantum channel class.

The operational significance stems from the observation that not all physically relevant maps—such as the transpose, inverses of channels, or Hermitian but non-completely positive maps—can act directly on a density matrix in quantum hardware; their images may not be positive semidefinite. BELT resolves this by embedding the output $\mathcal{N}(\rho)$ as a block of a larger unitary, so that quantum manipulations can proceed while bypassing the requirement of physical realizability for the intermediate object.

2. Technical Construction and Mathematical Foundation

The BELT protocol is constructed as follows for a linear map $\mathcal{N}$ and a density operator $\rho$:

• Assume access to a state preparation oracle $U^\rho$ such that $U^\rho |0^{r+n}\rangle = |\psi_\rho\rangle$ is a purification of $\rho$.
• Assume a unitary oracle $U_\mathcal{N}$ that provides an $(\alpha, m, \epsilon)$-block encoding of the partially transposed Choi matrix $\Lambda_\mathcal{N}^{T_1}$.
• The composite unitary implementing BELT is

$$V = (I_m \otimes U^{\rho\dagger} \otimes I_k) \cdot (U_\mathcal{N} \otimes I_r) \cdot (I_m \otimes U^\rho \otimes I_k)$$

where $r$ is the number of qubits used to purify $\rho$ and $k$ is an auxiliary register.

It can be proven that $V$ constitutes an $(\alpha, m + r + n, \epsilon)$-block encoding of $\mathcal{N}(\rho)$. The output is accessible by postselecting the ancilla register on $|0^{m+r+n}\rangle$. This circuit-level realization generalizes standard block encoding for matrices to arbitrary linear maps on density matrices.

The critical mathematical tool is the block encoding of the Choi matrix of $\mathcal{N}$. The protocol exploits tensor network properties: the action of $\mathcal{N}$ on $\rho$ can be represented via a contraction of Choi matrices and state vectors, and thus embedded within a unitary circuit by sequentially applying $U^\rho$, $U_\mathcal{N}$, and $U^{\rho\dagger}$.

3. Addressing Non-Physical and Non-CP Maps

Quantum information tasks frequently require simulations involving non-completely positive or even non-Hermitian maps. Examples include:

• Entanglement detection via positive but non-CP maps (e.g., partial transpose, reduction maps).
• Inversion of a quantum channel $\mathcal{E}$, where $\mathcal{E}^{-1}$ is linear and trace-preserving but not CP.
• Simulation of pseudo-differential operators in quantum simulation, where the operator (e.g., a discreetized $T$) acts as $T\rho T^\dagger$ but may not be physically implementable unless $T$ is unitary.

Direct application of such maps would, in general, render nonphysical matrices (i.e., not positive semidefinite), making them inaccessible via standard state preparation or quantum evolution. BELT's sub-block embedding circumvents this barrier, as no positivity or CP requirement is imposed on the block-encoded object. This enables quantum algorithms to manipulate, extract information from, or further process the result $\mathcal{N}(\rho)$ through coherent evolution.

4. Applications: Entanglement Detection, Channel Inversion, Simulation

BELT demonstrates utility across several fundamental areas:

• Entanglement Detection: Applying a non-CP map (partial transpose, reduction) to one subsystem of a bipartite density matrix allows the detection of negative eigenvalues, certifying entanglement. The BELT protocol renders the output accessible for spectral estimation—using techniques like QSVT—without resorting to quasiprobability sampling or expectation-value-only measurements. The circuit presented in the paper achieves an exponential improvement in sample complexity compared to single-copy or measurement-only strategies.
• Quantum Channel Inversion: For a channel $\mathcal{E}$ (not necessarily unitary), BELT enables a block encoding of $\mathcal{E}^{-1}(\mathcal{E}(\psi))$ for an unknown state $|\psi\rangle$. Employing robust oblivious amplitude amplification in conjunction with BELT and standard QSVT-style blocks, one can physically recover the original $|\psi\rangle$ given copies of $\mathcal{E}(\psi)$. This stands in contrast to expectation-value-based error mitigation, which does not produce a corrected state.
• Simulation of Pseudo-Differential Operators: When simulating discrete versions of operators $T$, BELT allows encoding $T\rho T^\dagger$ even when $T$ is not unitary or physically implementable, provided its (sub-)block encoding is accessible.

These use-cases highlight the scope of BELT, extending simulation to non-CP, non-Hermitian, and even non-physical quantum maps.

5. Comparative Resource Analysis and Sample Complexity

Compared to prior strategies based on Hermitian-preserving map exponentiation (HME) and naive circuit constructions, BELT achieves improvements in sample complexity and resource efficiency:

• Sample Complexity: BELT reduces the required number of oracle calls to the purification unitary for $\rho$ (and its inverse) from exponential (in system size $n$) to a constant or logarithmic function of $n$. In the context of channel inversion, for example, BELT removes the dependence on the approximation parameter $\epsilon$ and improves the scaling in the norm-bound parameter $\|\Lambda_{E^{-1}}^{T_1}\|_\infty$.
• Circuit Depth: Because BELT leverages concatenation of state-preparation and Choi-matrix block-encoding oracles, the circuit depth inherits the efficiency of its components. This enables practical instantiations for both small- and moderate-dimension systems.
• Expressiveness: Unlike QSVT or HME, which are limited to Hermitian-preserving or CP maps, BELT generalizes to arbitrary linear maps, including the transpose and other physically inaccessible operations. This broadens the range of quantum protocols that can be simulated coherently in hardware.

A summary of the key resource advantages is organized below:

Protocol	Applicable Maps	Sample/Oracle Calls	Circuit Depth
HME	Hermitian-preserving	Exp($n$)	$>$ BELT
BELT	Arbitrary	const/log($n$)	As per block-encoders
Naive/Virtual	Limited, e.g. CP	Exp($n$)	Moderate/undefined

6. Future Directions and Open Challenges

BELT establishes a paradigm for quantum information processing with linear maps and operators that surpass the capabilities of previous algorithmic primitives. Open problems and active research directions include:

• Lower Bounds and Norm-Dominated Efficiency: Determining optimal sample and resource lower bounds for generic linear maps in block-encoding protocols, and characterizing the operational meaning of the norm $\|\Lambda_N^{T_1}\|_\infty$.
• Combination with Other Primitives: Integrating BELT with continuous block encoding (Schrödingerisation, LCHS) for simulating time-dependent or non-Hermitian evolution.
• Lindbladian and Stochastic Simulation: Adapting BELT for simulation of Lindblad-type (open system) dynamics or stochastic differential operators.
• Algorithmic and Hardware Integration: Applying BELT-based constructions as subroutines in broader quantum algorithms for error mitigation, decoherence analysis, or subsystem spectral estimation.

BELT's generality—applicability to kernelized maps, density matrix functionals, or non-CP transformations—positions it as a foundational tool for quantum algorithms requiring blockwise simulation of arbitrary linear transformations.

7. Summary

BELT provides a systematic, scalable protocol for simulating arbitrary linear maps, including those that are non-CP, by embedding their action on a density matrix within a block of a larger unitary. This enables quantum protocols to coherently process, estimate spectral properties, or even prepare outputs of non-physical operations without resorting to intractable sampling or approximation schemes. Through applications in entanglement detection and channel inversion, BELT demonstrates both improved resource scaling and substantive extension of the quantum algorithmic toolbox, with ongoing research likely to further its reach across quantum information science and simulation (Wei et al., 18 Aug 2025).