Quantum Block-Encoding Input Model

• Block-Encoding Input Model is a framework that encodes matrix or operator data into a unitary, enabling quantum algorithms like QSVT and qubitization.
• It details various access models—including oracle-based, state-preparation, and arithmetic methods—each with distinct normalization, ancilla, and circuit depth requirements.
• The approach emphasizes resource trade-offs and specialized constructions, highlighting the interplay between classical preprocessing, implementation cost, and success probability.

The block-encoding input model is the mechanism by which a quantum algorithm receives matrix or operator data as a unitary-access primitive. In its standard form, a unitary $U$ acting on ancillas and system is an $(\alpha,a,\varepsilon)$-block-encoding of an operator $A$ when $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$; in the exact case, the top-left block of $U$ is $A/\alpha$. In recent work, the phrase “input model” denotes not only this definition but also the concrete assumptions under which such a unitary is compiled: sparse-access and state-preparation oracles, explicit arithmetic circuits, Fourier-symbol oracles, tensor-network contractions, and variationally trained unitaries all instantiate different block-encoding input models with sharply different normalization factors, ancilla counts, gate depths, and classical preprocessing requirements (Shi et al., 2024, Li et al., 2023, Mahmud et al., 10 Apr 2026).

1. Formal role in quantum algorithms

Block encoding is the standard access primitive for QSVT, qubitization, and related polynomial-transform methods. In this framework, $\alpha$ is the subnormalization factor, and the success amplitude in a postselection picture is $1/\alpha$; when $U$ acts on $|0^a\rangle\otimes|\psi\rangle$, the ancilla-zero component is $(\alpha,a,\varepsilon)$0. In QSP- and QSVT-based simulation, the number of uses of the block-encoding scales linearly in $(\alpha,a,\varepsilon)$1 up to polylogarithmic factors in the target precision, so reducing $(\alpha,a,\varepsilon)$2 directly reduces query complexity (Liu et al., 9 Oct 2025).

The same formalism accommodates non-Hermitian inputs. One standard route is Hermitian dilation, $(\alpha,a,\varepsilon)$3, which allows QSVT to act on singular values while preserving a Hermitian signal operator. This is used explicitly in quantum linear-algebra settings such as Kalman filtering, where addition, multiplication, and inversion are all expressed through block-encodings and composed inside a unified framework (Shi et al., 2024).

A central distinction in the literature is therefore between the abstract definition of a block-encoding and the concrete way the unitary is realized. Some constructions assume oracle access to entries or sparse structure; others compile the unitary explicitly from problem structure, so that no qRAM, signed amplitude loading, or black-box entry oracle is required. The latter trend is especially pronounced in recent work on differential operators, tensor networks, and many-body Hamiltonians (Mahmud et al., 10 Apr 2026).

2. Oracle-based, state-preparation, and arithmetic access models

A large class of input models is oracle-based. In dense-matrix settings, a common assumption is access to state-preparation oracles $(\alpha,a,\varepsilon)$4 and $(\alpha,a,\varepsilon)$5 that prepare row- and column-weighted superpositions; their composition yields a $(\alpha,a,\varepsilon)$6-block-encoding of $(\alpha,a,\varepsilon)$7 with polylogarithmic query time in the matrix dimensions, provided the data are stored in a quantum-accessible structure (Shi et al., 2024). In sparse and second-quantized settings, the input model is instead phrased in terms of a sparsity oracle $(\alpha,a,\varepsilon)$8 and an amplitude oracle $(\alpha,a,\varepsilon)$9. For second-quantized Hamiltonians, SWAP-based implementations of $A$0 and SELECT-SWAP data lookup for $A$1 reduce the T-count per oracle invocation to $A$2 in the number of interaction terms $A$3 (Liu et al., 9 Oct 2025).

A different line of work replaces generic oracles by arithmetic descriptions of structure. For matrices with repeated values and patterned sparsity, one can specify nonzero entries by a value label $A$4, a multiplicity label $A$5, and reversible arithmetic maps between $A$6 and row/column coordinates. In that model, the dominant data-loading cost depends on the number $A$7 of distinct values rather than the matrix dimension, and different schemes produce different subnormalizations: a base scheme with $A$8, a preamplified scheme with $A$9, and a PREP/UNPREP scheme with $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$0 when the requisite commutation conditions hold (Sünderhauf et al., 2023).

The dictionary-based sparse model pushes this idea further. There, a sparse matrix is organized into classes of repeated values $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$1, together with injective maps $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$2 that specify where those values occur. The resulting unitary

$$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$3

block-encodes $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$4 with subnormalization $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$5, depth $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$6, and ancilla count $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$7, where $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$8 is the number of nonzeros and $$\|A-\alpha(\langle 0^a|\otimes I)U(|0^a\rangle\otimes I)\|\le \varepsilon$$9 is the number of dictionary items (Yang et al., 2024).

Approximate state-preparation approaches remain relevant for unstructured sparse data. S-FABLE block-encodes $U$0, then conjugates by outer Hadamards to recover a block-encoding of $U$1; LS-FABLE avoids the quadratic classical overhead by directly inserting scaled sparse entries into the rotation angles. For unstructured sparse matrices with $U$2 nonzeros, the reported empirical behavior is approximately $U$3 rotation gates and $U$4 CNOT gates after compression (Kuklinski et al., 2024).

Model	Access assumption	Characteristic feature
State-preparation oracles (Shi et al., 2024)	$U$5 for row/column amplitudes	Frobenius-norm normalization
Sparse oracle + amplitude oracle (Liu et al., 9 Oct 2025)	$U$6	$U$7 T-count
Arithmetic structured matrices (Sünderhauf et al., 2023)	reversible maps $U$8	loading depends on distinct values $U$9
Dictionary sparse (Yang et al., 2024)	value classes and injective maps $A/\alpha$0	$A/\alpha$1, depth $A/\alpha$2
S-FABLE / LS-FABLE (Kuklinski et al., 2024)	classical angle preprocessing or sparse-entry access	aggressive circuit compression for sparse $A/\alpha$3

3. Explicit structure-exploiting encodings for differential and Fourier operators

A prominent current direction is to eliminate generic data oracles entirely by exploiting operator structure. For the Difference-of-Gaussian operator on a periodic grid, the coefficients split into two normalized discrete Gaussian distributions $A/\alpha$4 and $A/\alpha$5. Exact Gaussian state-preparation circuits $A/\alpha$6 and $A/\alpha$7, a one-qubit branch indicator, a single Pauli-$A/\alpha$8 gate to encode the minus sign, and controlled cyclic shifts together yield an exact block-encoding of

$A/\alpha$9

with $\alpha$0, independent of grid size $\alpha$1, spatial dimension $\alpha$2, and stencil width $\alpha$3. The same work derives an exact success probability

$\alpha$4

and shows $\alpha$5 for smooth inputs as the periodic grid is refined (Mahmud et al., 10 Apr 2026).

For finite-difference discretizations of the Laplacian on periodic grids, the input model is fully explicit: ancillas prepare fixed superpositions, and controlled cyclic shifts realize the stencil. In one dimension this gives $\alpha$6; in $\alpha$7 dimensions the subnormalization is $\alpha$8, with $\alpha$9 T-gate complexity and $1/\alpha$0 under $1/\alpha$1 regularity assumptions (Sturm et al., 2 Sep 2025).

QFT-based models access operators through their Fourier symbols. For bounded-domain fractional Laplacians with open, zero-extension boundary conditions, the native QFT implements a periodic circulant surrogate rather than the Toeplitz truncation. Zero-padding into an $1/\alpha$2-point periodic register and compressing back to the physical $1/\alpha$3-point subspace produces

$1/\alpha$4

where the residual $1/\alpha$5 is controlled by the tail of the semi-discrete kernel and $1/\alpha$6 decays with $1/\alpha$7 according to the kernel decay exponent $1/\alpha$8 (Javanmard et al., 16 May 2026).

Pseudo-differential operators supply a broader Fourier-structured class. Generic PDOs can be block-encoded via QFT, phase multiplication, and arithmetic evaluation of the symbol $1/\alpha$9, but this yields normalization $U$0 and exponentially small success probability. Separable symbols $U$1 reduce the normalization to $U$2 and achieve $U$3 success probability, while dimension-wise fully separable symbols admit explicit QET constructions with $U$4 ancillas and gate complexity $U$5 (Li et al., 2023).

4. Structured matrices, tensor networks, and compressed linear algebra

Rank-structured matrix classes give rise to distinct block-encoding input models. For one-pair semiseparable matrices $U$6, an exact factorization

$U$7

supports a block-encoding assembled from diagonal, inverse-diagonal, lower-triangular, and difference-diagonal pieces. The final semiseparable construction uses $U$8 ancillas, has polylogarithmic depth, and normalization

$U$9

with additive spectral-norm error controlled by fixed-point and arcsin approximation errors (Antonioli et al., 19 Mar 2026).

Matrix product operators define another major input model. One compiler dilates each MPO tensor into a $|0^a\rangle\otimes|\psi\rangle$0-qubit unitary, with $|0^a\rangle\otimes|\psi\rangle$1 determined by the bond dimension $|0^a\rangle\otimes|\psi\rangle$2. The full chain uses $|0^a\rangle\otimes|\psi\rangle$3 ancillas and $|0^a\rangle\otimes|\psi\rangle$4 one- and two-qubit gates, while the global normalization is the product of per-tensor normalizations $|0^a\rangle\otimes|\psi\rangle$5. The block is exact inside the designated ancilla subspace, although the postselection success probability typically decays exponentially with $|0^a\rangle\otimes|\psi\rangle$6 because $|0^a\rangle\otimes|\psi\rangle$7 grows multiplicatively (Nibbi et al., 2023).

A more recent MPO perspective treats tensor networks as compressed virtual-path LCU programs. Expanding each local MPO tensor into a unitary operator basis induces a path sum

$|0^a\rangle\otimes|\psi\rangle$8

with path normalization $|0^a\rangle\otimes|\psi\rangle$9. Conditional PREP and SELECT stages can then be compiled directly from the parent MPO, with cost $(\alpha,a,\varepsilon)$00 rather than explicit $(\alpha,a,\varepsilon)$01 Pauli-string growth for a degree-$(\alpha,a,\varepsilon)$02 polynomial expansion, provided the bond dimension and path normalization remain mild (Dumitrescu, 17 Jun 2026).

For dense classical matrices without sparsity or low-rank structure, BITBLE organizes state-preparation unitaries in binary trees and decouples multiplexors by Walsh–Hadamard/Gray-code linear algebra. Its exact normalization can be either $(\alpha,a,\varepsilon)$03 or $(\alpha,a,\varepsilon)$04, the classical preprocessing time is $(\alpha,a,\varepsilon)$05 with memory $(\alpha,a,\varepsilon)$06, and the ancilla count is only $(\alpha,a,\varepsilon)$07 or $(\alpha,a,\varepsilon)$08 depending on the variant (Li et al., 8 Apr 2025).

5. Many-body operator models and direct algebraic constructions

In second quantization, the input model is often built around the algebra of creation and annihilation operators rather than around matrix entries. One recent construction for general second-quantized Hamiltonians combines a SWAP-based sparsity oracle $(\alpha,a,\varepsilon)$09 with SELECT-SWAP data lookup for $(\alpha,a,\varepsilon)$10, giving per-oracle T-count $(\alpha,a,\varepsilon)$11. The same framework targets the $(\alpha,a,\varepsilon)$12-particle sector directly through an occupation-detection oracle $(\alpha,a,\varepsilon)$13, reducing the subnormalization from $(\alpha,a,\varepsilon)$14 to $(\alpha,a,\varepsilon)$15 for general one- and two-body Hamiltonians, with corresponding reductions to $(\alpha,a,\varepsilon)$16 for one-body terms and $(\alpha,a,\varepsilon)$17 for number-operator products (Liu et al., 9 Oct 2025).

LOBE block-encodes fermionic and bosonic ladder operators directly, avoiding Pauli-basis expansion. In that framework, single fermionic ladder operators and fermionic products have $(\alpha,a,\varepsilon)$18, while bosonic single-mode ladder operators have $(\alpha,a,\varepsilon)$19 and products $(\alpha,a,\varepsilon)$20 have $(\alpha,a,\varepsilon)$21, where $(\alpha,a,\varepsilon)$22 is the bosonic cutoff. The reported T-counts scale linearly with locality and as $(\alpha,a,\varepsilon)$23 in the bosonic register width, while benchmarks on quartic oscillator, $(\alpha,a,\varepsilon)$24, and Yukawa models show fewer non-Clifford gates, fewer ancillas, and lower rescaling factors than Pauli-expansion approaches (Simon et al., 14 Mar 2025).

Bosonic lattice Hamiltonians can also be block-encoded by signal-processing methods themselves. QSVT-based encodings use diagonal block-encodings of $(\alpha,a,\varepsilon)$25, $(\alpha,a,\varepsilon)$26, and $(\alpha,a,\varepsilon)$27 as primitives; QETU-based constructions work instead from controlled exponentials; LOVE-LCU realizes diagonal functions exactly via

$(\alpha,a,\varepsilon)$28

The reported conclusion is that QSVT has the best asymptotic scaling in qubits per site, whereas LOVE-LCU outperforms the alternatives for operators acting on up to $(\alpha,a,\varepsilon)$29 qubits (Kane et al., 2024).

Linear combinations of Pauli strings admit yet another algebraic input model. A stabilizer-based construction first transforms the Pauli strings into a pairwise anti-commuting set, making the normalized linear combination unitary, and then uses a correction transformation on an ancilla register to restore the original strings. The ancilla requirement scales logarithmically with the number of Pauli terms in the basic version, and larger ancilla registers can reduce circuit complexity further (Schillo et al., 9 Jan 2026).

6. Resource trade-offs, ambiguities, and current directions

Recent work makes clear that “the” block-encoding input model is not a single model but a family of access assumptions with different bottlenecks. One recurrent ambiguity is the relationship between subnormalization and practical cost. Smaller $(\alpha,a,\varepsilon)$30 is algorithmically advantageous, but it does not by itself imply a cheaper circuit. Single-ancilla exact dense-matrix block-encoding based on diagonal matrix migration attains spectral-norm subnormalization and a leading C-NOT count $(\alpha,a,\varepsilon)$31; for rank-$(\alpha,a,\varepsilon)$32 matrices this drops to $(\alpha,a,\varepsilon)$33. These bounds improve on earlier exact synthesis constants, yet they still scale exponentially in $(\alpha,a,\varepsilon)$34, reflecting the intrinsic cost of dense unstructured inputs (Li et al., 17 Mar 2026).

A second ambiguity concerns postselection versus normalization. Constant subnormalization does not guarantee constant success probability. The DoG construction has $(\alpha,a,\varepsilon)$35 independent of $(\alpha,a,\varepsilon)$36, $(\alpha,a,\varepsilon)$37, and $(\alpha,a,\varepsilon)$38, but its exact success probability is power-spectrum weighted and scales as $(\alpha,a,\varepsilon)$39 for smooth inputs on finer periodic grids (Mahmud et al., 10 Apr 2026). The explicit finite-difference Laplacian encoding exhibits the same $(\alpha,a,\varepsilon)$40 behavior under the stated regularity assumptions (Sturm et al., 2 Sep 2025). This shows that success probability may be controlled by the input state and operator spectrum even when the block-encoding normalization is structurally optimal.

A third trade-off is the balance between quantum resources and classical preprocessing. BITBLE uses only a few ancillas but requires $(\alpha,a,\varepsilon)$41 classical preprocessing time and $(\alpha,a,\varepsilon)$42 memory (Li et al., 8 Apr 2025). Dictionary-based sparse block encoding achieves logarithmic circuit depth, but only by spending $(\alpha,a,\varepsilon)$43 ancillas (Yang et al., 2024). Variational block-encoding can produce exact single-ancilla encodings with parameter counts close to the degrees of freedom of the target matrix, and symmetry-aware ansätze made optimization possible up to $(\alpha,a,\varepsilon)$44 qubits under permutation symmetry; however, the classical optimization itself ceases to be computationally feasible for large system sizes (Rullkötter et al., 23 Jul 2025).

Open directions in the literature are correspondingly diverse. Explicit structured encodings invite extensions to nonperiodic boundary conditions through modified shift encodings (Mahmud et al., 10 Apr 2026). MPO compilers suggest higher-dimensional PEPO generalizations, but that extension is left for future work (Nibbi et al., 2023). Semiseparable constructions point toward multi-pair semiseparable, HSS, and HODLR variants (Antonioli et al., 19 Mar 2026). This suggests that the evolution of block-encoding input models is likely to proceed less through a single universal oracle and more through increasingly specialized compilations that preserve the algebraic, geometric, or tensor-network structure of the target operator.