Encoded Quantum Signal Processing

• Encoded Quantum Signal Processing is a framework that embeds quantum signals into unitary blocks, logical codes, or hybrid systems to enable structured transformations.
• It leverages methodologies like QSP and QSVT to implement polynomial or Laurent-polynomial transformations, precisely controlling operator spectra and error bounds.
• Applications include Hamiltonian simulation, metrology, and quantum arithmetic, offering tangible improvements in query complexity and algorithmic robustness.

Encoded Quantum Signal Processing denotes a family of quantum information-processing constructions in which the quantity to be transformed is first embedded into a unitary, a block-encoding, a logical code space, or a hybrid continuous-variable/discrete-variable coupling, and is then manipulated by a structured processing primitive. In the block-encoding literature, this typically means polynomial or Laurent-polynomial transformations of an encoded operator by QSP, QSVT, or their generalizations; in metrology, it denotes a framework in which syndrome measurements inside an error-detecting code play the role of the signal-processing primitive; and in several recent extensions it includes embedded arithmetic, hybrid CV–DV transforms, and parallel or multivariate variants (Laneve, 25 Jun 2025, Marrero et al., 24 Mar 2026).

1. Terminological scope and basic model

Recent work uses the expression in several closely related senses rather than as a single fixed formalism. The common element is that the “signal” is not processed as an unconstrained classical variable, but as an encoded quantum object whose accessible transformations are dictated by unitarity, block structure, or code constraints.

Usage	Encoded object	Representative papers
Block-encoding/QSVT sense	Matrix or unitary embedded in a unitary block	(Motlagh et al., 2023, Laneve, 25 Jun 2025, Lu et al., 2024)
Logical-code/metrology sense	Signal preserved in a logical subspace and processed by syndrome measurement	(Marrero et al., 24 Mar 2026)
Hybrid or embedded-data sense	Register-encoded, amplitude-encoded, or CV–DV signal coupled to a small processing system	(Ollive et al., 24 Mar 2025, Liu et al., 2024, Shukla et al., 9 Feb 2026)

In the standard encoded-operator convention, a unitary $U$ is a block-encoding of $A$ if

$$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$

or, equivalently, if the top-left block equals $A/\beta$ up to the stated normalization and approximation conventions. QSP and QSVT then use an ancilla-mediated sequence to implement polynomial transformations of the encoded spectrum or singular values, with query complexity proportional to polynomial degree (Motlagh et al., 2023).

This operator-centric usage has gradually broadened. One line of work reinterprets QSP itself as a state-conversion problem in an $L^2$ function space and uses adversary bounds to characterize feasible protocols (Laneve, 25 Jun 2025). Another line generalizes QSP from $U(2)$ to $U(N)$ so that multiple polynomials are realized simultaneously from one encoded input (Lu et al., 2024). Still others recast arithmetic, metrology, or CV–DV state transfer as instances of encoded signal processing (Ollive et al., 24 Mar 2025, Marrero et al., 24 Mar 2026, Liu et al., 2024).

2. Canonical block-encoded formulations

In the standard univariate SU(2) picture, QSP alternates signal-dependent rotations with phase rotations. A canonical form writes

$$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$

with matrix elements

$$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$

subject to

$|P(x)|^2+(1-x^2)|Q(x)|^2=1.$

Degree and parity constraints are part of the canonical feasibility conditions in this setting (Laneve, 25 Jun 2025).

Generalized Quantum Signal Processing replaces single-axis phase gadgets by arbitrary SU(2) rotations and uses the $A$0-controlled signal operator

$A$1

Its central theorem states that

$A$2

if and only if $A$3, $A$4, and $A$5 on the unit circle. In this formulation, the practical parity and realness restrictions of standard QSP are lifted, leaving only the unitarity bound $A$6 on $A$7; the same framework also treats negative powers through controlled use of $A$8 and provides a recursive angle-construction procedure when $A$9 and $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$0 are known (Motlagh et al., 2023).

A further refinement uses directional control between forward and reverse walk steps. When control between $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$1 and $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$2 is available at essentially the same cost as a standard controlled-$$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$3, generalized QSP can reduce Hamiltonian-simulation cost by a factor of $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$4. The mechanism is that controlling between $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$5 and $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$6 advances the attainable Fourier order by $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$7 per controlled step rather than $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$8, so achieving degree $$(\langle 0^a|\otimes I)\,U\,(|0^a\rangle\otimes I)=A/\beta,$$9 requires roughly $A/\beta$0 controls rather than $A/\beta$1 (Berry et al., 2024).

3. Characterization, synthesis, and multivariate structure

A major conceptual shift is the recasting of QSP as a state-conversion problem over the Hilbert space $A/\beta$2 of square-integrable functions. In the univariate case, the target state is

$A/\beta$3

and the relevant adversary-bound instance is built from Gram operators of the initial and final functional states together with a filter linked to the oracle. The principal result is exact: in the univariate SU(2) case, the adversary-bound characterization identifies all and only the achievable QSP protocols. Equivalently, any feasible adversary solution yields an SU(2) QSP protocol of length equal to the degree, and every SU(2) QSP protocol yields a feasible catalyst. The same framework gives a unique catalyst Gram matrix, up to unitary freedom, through a displacement equation, and extends to multivariate QSP by reducing minimal-space synthesis to a rank-minimization problem over the feasible adversary set (Laneve, 25 Jun 2025).

The $A/\beta$4 generalization replaces a single-qubit signal register by an $A/\beta$5-dimensional ancilla with tunable ancilla unitaries and projectors. In this setting, one no longer asks for a single polynomial $A/\beta$6, but for a matrix polynomial $A/\beta$7 realized simultaneously in the encoded block. The central feasibility condition is contractivity:

$A/\beta$8

for $A/\beta$9-QSP, and

$L^2$0

for $L^2$1-QSVT. A recursive degree-reduction algorithm based on projectors onto leading-coefficient column spaces, together with polynomial matrix spectral factorization, provides constructive synthesis. This framework also supports separable realizations of bi-variate functions and yields an amplitude-estimation formulation with asymptotically optimal query complexity (Lu et al., 2024).

A distinct synthesis development removes classical angle finding altogether. Instead of solving for phase factors, one compiles a diagonal operator $L^2$2 that encodes sampled values of $L^2$3 on a Fourier register and inserts it into a phase-estimation–style circuit built from coherent accumulators of powers of the signal unitary. For degree-$L^2$4 Laurent polynomials the resulting circuit is exact up to the stated block-encoding scale, and for general $L^2$5 the error is bounded by

$L^2$6

where $L^2$7 is the best degree-$L^2$8 Laurent-approximation error. The same construction gives a $L^2$9 block-encoding of $U(2)$0 using $U(2)$1 controlled uses each of $U(2)$2 and $U(2)$3, one call to the function encoder, and $U(2)$4 additional gates (Alase, 13 Jan 2025).

4. Encoded architectures beyond standard block-encoding

In quantum metrology, “Encoded Quantum Signal Processing” is formalized as a code-based logical sensing framework. Given an $U(2)$5 code with logical space $U(2)$6, logical $U(2)$7, encoding or error channel $U(2)$8, and decoding channel $U(2)$9, one EQSP iterate is

$U(N)$0

For a repetition code under a single-qubit signal rotation $U(N)$1, syndrome measurement induces a logical rotation with angle

$U(N)$2

so syndrome measurement acts as the nonlinear signal transform. The framework proves that product-state sensing with syndrome post-processing remains limited to standard quantum limit scaling, with $U(N)$3 at $U(N)$4, and then presents four protocols that recover Heisenberg-limited precision through entanglement or sequential amplification while maintaining exponential suppression of undetected bit-flip errors in code distance (Marrero et al., 24 Mar 2026).

In embedded quantum arithmetic, the signal is encoded by a query operator whose eigenphases or block amplitudes carry numerical data. The paper’s “embedded QSP” viewpoint uses ancilla quantum encoding based on a multi-controlled $U(N)$5 rotation to generate a signal operator $U(N)$6, then applies QSP transforms and phase estimation. Two scalar functions mediate between amplitude and phase:

$U(N)$7

After the QSP-processed quantity is remapped into a phase-bearing unitary, QPE writes the result into a binary register. In this formulation, ancilla encoding, amplitude-to-phase conversion, and phase readout appear as native subroutines closely related to AQE, HHL-style ancilla rotations, and QAE (Ollive et al., 24 Mar 2025).

A hybrid CV–DV formulation embeds the signal into qubit-controlled phases generated by oscillator quadratures. The single-variable primitive uses

$U(N)$8

and the corresponding QSP sequence produces matrix entries that are Laurent polynomials in $U(N)$9 and $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$0. A non-Abelian extension alternates controlled phases of $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$1 and $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$2. These constructions support fully unitary AD/DA conversion between qubits and oscillators and an implementation of the $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$3-qubit QFT via free oscillator evolution, though the runtime is exponential in the number of qubits because the required analog phase-space area or bandwidth scales with the digital Hilbert-space dimension (Liu et al., 2024).

5. Algorithmic applications

Hamiltonian simulation remains the canonical application domain. In the generalized-QSP framework, trigonometric phase functions such as $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$4 and $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$5 are implemented with

$$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$6

controlled-$$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$7 and $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$8-qubit operations, and qubitization then yields an $$W_{\boldsymbol{\phi}}(x)=e^{i\phi_0 \sigma_z}\prod_{k=1}^d \big(e^{i\arccos(x)\sigma_x}e^{i\phi_k\sigma_z}\big),$$9-approximation of $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$0 with

$$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$1

applications of the walk operator. The same framework gives an $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$2-approximate solution to the fractional-query problem using

$$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$3

controlled-$$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$4 and $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$5-qubit gates, and also treats square-root phase functions and normal-matrix synthesis (Motlagh et al., 2023).

Encoded-QSP methods have also been used to build block-encodings rather than merely transform them. For lattice bosons, QSVT and QETU can synthesize block-encodings of diagonal bosonic operators and functions of differences, while the exact LOVE-LCU method constructs diagonal block-encodings from exponentials of $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$6. In that setting, QSVT has the best asymptotic gate-count scaling with the number of qubits per site, whereas LOVE-LCU outperforms the other constructions for operators acting on up to $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$7 qubits; the resulting block-encodings are then fed into generalized QSP to simulate real-time evolution in lattice $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$8 theory (Kane et al., 2024).

The $$W_{00}(x)=P(x),\qquad W_{10}(x)=i\sqrt{1-x^2}\,Q(x),$$9 framework extends encoded signal processing to simultaneous multi-polynomial tasks. One application is amplitude estimation, where the outcome distribution of any valid degree-$|P(x)|^2+(1-x^2)|Q(x)|^2=1.$0 amplitude-estimation circuit is polynomial in the target amplitude, and the paper proves an exact equivalence between such distributions and a suitable $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$1-QSVT realization. Reported asymptotic lower bounds for window error are

$|P(x)|^2+(1-x^2)|Q(x)|^2=1.$2

with matching constructions up to small constants (Lu et al., 2024).

Property estimation furnishes a different application pattern. Parallel QSP factorizes a degree-$|P(x)|^2+(1-x^2)|Q(x)|^2=1.$3 polynomial $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$4 into $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$5 lower-degree factors, implements each factor in parallel, and reconstructs $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$6 with a generalized SWAP test. This reduces query depth from $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$7 to $|P(x)|^2+(1-x^2)|Q(x)|^2=1.$8, making the method natural for distributed quantum computers, at the expense of a measurement overhead

$|P(x)|^2+(1-x^2)|Q(x)|^2=1.$9

and the width required for $A$00 parallel encoded instances (Martyn et al., 2024).

Encoded-signal-processing modules also appear in signal-analysis settings outside block-encoding/QSVT proper. A sequency-band detector applies a sequency-ordered QWHT to an amplitude-encoded input state, marks a user-specified band with a reversible comparator-based oracle, and estimates the associated probability mass by QAE. The sequency-ordered QWHT has circuit depth $A$01 on an amplitude-encoded quantum state, and the whole subroutine remains fully quantum so that it can serve as a coherent upstream or downstream module in larger algorithms (Shukla et al., 9 Feb 2026).

6. Misconceptions, limitations, and open directions

A persistent misconception is that every normalized target transform should admit a simple single-qubit QSP realization. The multivariate literature explicitly shows otherwise. Classical layer stripping and univariate Fejér–Riesz completion do not directly generalize, and not all normalized multivariate pairs $A$02 are decomposable into a single-qubit M-QSP protocol. In the adversary-bound formulation, multivariate synthesis becomes a convex feasibility problem with a residual-rank objective, but identifying minimal rank is NP-hard in general; multivariate Fejér–Riesz completion typically requires strict inequality $A$03 and does not guarantee a single complementary polynomial or degree bounds for the complements (Laneve, 25 Jun 2025).

A second misconception is that encoding alone automatically upgrades metrological scaling. The code-based EQSP analysis proves a strict barrier: product-state sensing with syndrome post-processing cannot beat SQL scaling. Heisenberg scaling is recovered only through entanglement or sequential amplification, and it remains contingent on device quality conditions such as

$A$04

for longitudinal inhomogeneities. The same work identifies $A$05 noise as a major unresolved limitation because $A$06 creates syndrome ambiguity, and it notes that the post-selected “activation” channel has exponentially small success probability $A$07 near the metrologically relevant point $A$08 (Marrero et al., 24 Mar 2026).

Hybrid encoded-QSP architectures are powerful but not asymptotically free. In the CV–DV AD/DA and QFT constructions, the required displacement amplitude or reciprocal bandwidth scales as $A$09, so the runtime is exponential in the number of qubits even though some gate counts are only linear in $A$10. This is an information-theoretic limitation rather than a compilation artifact (Liu et al., 2024).

Several open directions recur across the literature. Generalized QSP explicitly calls for multivariable extensions and analogous treatments of QSVT for block-encoded non-square matrices (Motlagh et al., 2023). The $A$11 formulation leaves sharper characterizations of noncommuting multivariate targets and efficient polynomial-matrix factorization as practical open problems (Lu et al., 2024). Parallel QSP exposes a time–space–measurement tradeoff whose usefulness depends on lowering the $A$12 measurement overhead, suggesting further work on factorization strategies and lower-variance reconstruction schemes (Martyn et al., 2024). Taken together, these results suggest that “Encoded Quantum Signal Processing” is best understood not as a single algorithm, but as an expanding design paradigm for transforming quantum information that is accessible only through an encoding.