Quantum Signal Processing Framework

• Quantum Signal Processing (QSP) is a matrix-function synthesis framework that enables polynomial transformations of block-encoded operators in quantum algorithms.
• It extends the classical univariate QSP to multivariate, non-commutative, and inhomogeneous settings, offering compact and asymptotically optimal subroutines for simulations and eigenvalue transformations.
• Recent advances in QSP facilitate direct matrix function realizations in multi-parameter Hamiltonian simulations, though challenges remain in fully characterizing non-commutative cases.

Quantum Signal Processing (QSP) Framework

Quantum Signal Processing (QSP) is a matrix-function synthesis framework of central importance in quantum algorithms, particularly in the design and implementation of polynomial transformations of block-encoded operators. The QSP paradigm leverages the block-encoding formalism of quantum linear algebra to achieve conceptually compact and asymptotically optimal quantum subroutines for simulation, eigenvalue transformation, and more. The classical univariate QSP theory is now fully characterized, and recent research has made significant advances in understanding and extending QSP to the multivariate, non-commutative, and inhomogeneous settings (Németh et al., 2023).

1. Univariate QSP: Model and Characterization

QSP begins with access to a "signal" unitary operator. In the univariate case, with $W \in U(N)$ or $GL(N)$ and a single ancilla, the block-encoding is

$$V = |0\rangle\langle0| \otimes W + |1\rangle\langle1| \otimes W^\dagger$$

which acts diagonally as $\operatorname{diag}(t, t^{-1})$ on eigenstates $| \theta \rangle$ of $W$ with $t = e^{i\theta}$.

The protocol interleaves $d+1$ fixed $SU(2)$ “modulation” gates $U_0, ..., U_d$ and $d$ calls of $\operatorname{diag}(t, t^{-1})$:

$$F(t) = U_0 \cdot \operatorname{diag}(t, t^{-1}) \cdot U_1 \cdots \operatorname{diag}(t, t^{-1}) \cdot U_d \in SU(2)$$

Haah's theorem characterizes achievable univariate QSP polynomials:

• $F(t)$ is a $2 \times 2$ matrix Laurent polynomial in $t$ with:
  1. $\deg F \leq d$
  2. $F(e^{i\theta}) \in SU(2)$ for all $\theta$
  3. $F(-t) = (-1)^d F(t)$

Satisfaction of these conditions is necessary and sufficient for the existence of angles $\varphi_0, ..., \varphi_d$ such that

$$F(t) = e^{i\varphi_0 \sigma_X} \cdot \operatorname{diag}(t, t^{-1}) \cdot e^{i\varphi_1 \sigma_X} \cdots \operatorname{diag}(t, t^{-1}) \cdot e^{i\varphi_d \sigma_X}$$

This fully determines the class of real-scalar and matrix-valued polynomials implementable via univariate QSP (Németh et al., 2023).

2. Multivariate QSP: Homogeneous and Inhomogeneous Cases

When accessing multiple commuting signal unitaries $W_1, W_2$ ($[W_1, W_2]=0$), QSP generalizes to homogeneous bivariate protocols. Define

$$V = |0\rangle\langle0| \otimes W_1 + |1\rangle\langle1| \otimes W_2$$

which acts as $\operatorname{diag}(a, b)$ with $a=e^{i\theta}$, $b=e^{i\phi}$.

A homogeneous bivariate QSP protocol of degree $d$ constructs

$$F(a,b) = U_0 \operatorname{diag}(a, b) U_1 \cdots \operatorname{diag}(a, b) U_d, \quad U_i \in SU(2)$$

Németh et al. (Németh et al., 2023) show that such $F(a,b)$ is implementable if and only if:

1. $F(a,b)$ is a homogeneous polynomial of total degree $d$ in $a,b$
2. $F(e^{i\theta},e^{i\phi}) \in U(2)$ for all $\theta,\phi$
3. $\det F(a,b) = (ab)^d$

The construction reduces to the univariate case: setting $t = \sqrt{a/b}$ transforms the problem into a standard univariate QSP via $F(a,b)/(ab)^{d/2} = \tilde F(t,t^{-1})$.

Inhomogeneous bivariate QSP (Rossi–Chuang variant) allows more general “signal call” sequences of $d_a$ insertions of $a$ and $d_b$ insertions of $b$, leading to

$$F(a,b) = U_0 \prod_{i=1}^d \operatorname{diag}(c_i, c_i^{-1}) U_i, \quad c_i \in \{a, b\}$$

Necessary conditions (Theorem 4.1 (Németh et al., 2023)):

• $\deg_a F \leq d_a$, $\deg_b F \leq d_b$
• $F(e^{i\theta},e^{i\phi}) \in SU(2)$
• Parity: $F(-a,b) = (-1)^{d_a}F(a,b)$, $F(a,-b) = (-1)^{d_b}F(a,b)$

Sufficiency holds when either degree is at most 1 (Theorem 4.2), via reduction to univariate QSP for the fixed variable.

3. Non-Commutative and General Multivariate QSP

For $n$ potentially non-commuting signals $X_1, ..., X_n$, the homogeneous multivariate QSP protocol is

$$F(x_1,\dots,x_n) = U_0 \operatorname{diag}(x_1,\dots,x_n) U_1 \cdots \operatorname{diag}(x_1,\dots,x_n) U_d, \quad U_i \in SU(n)$$

with $F$ interpreted as a non-commutative polynomial of degree $d$.

Necessary conditions for achievable $F$ (Theorem 3.2 (Németh et al., 2023)):

• Homogeneity: $F$ is homogeneous degree $d$
• $$F(X_1,...,X_n) \in U(\mathbb{C}^n \otimes \mathcal{H})$$ for any Hilbert space $\mathcal{H}$, $X_i \in U(\mathcal{H})$
• Determinant: $\det F(X_1,...,X_n) = \prod_{i=1}^n \det X_i^d$

Unlike the commuting case, sufficiency is not proven for $n>2$ or in the general non-commutative class. Standard reductions to the univariate theory are invalid when no single variable appears with maximal weight, and new algebraic tools are required.

4. Achievability, Counterexamples, and Characterization

Explicit counterexamples show that necessary conditions are insufficient in full generality. Németh et al. construct a bivariate, even-parity polynomial $F_{2,2}(a,b)$ of degree $(2,2)$ satisfying all necessary conditions, but for which no inhomogeneous (slotwise) decomposition exists (Theorem 4.2). This disproves the conjecture of Rossi and Chuang for restricted classes of MQSP even when extra symmetric structures are imposed.

Verification uses a refined necessary lemma: for any decomposition, at least one of the matrix products among the four extreme coefficients must vanish. Direct computation on $F_{2,2}(a,b)$ confirms non-vanishing, hence non-implementability. These explicit constructions demarcate the limits of both slot-based and classical-choice implementations (Németh et al., 2023).

5. Protocol Synthesis and Reduction Techniques

The bivariate homogeneous case is algorithmically reduced to a univariate QSP construction by identifying suitable variable transformations (e.g., $t = \sqrt{a/b}$), synthesizing the univariate QSP, and subsequently reconstructing the multivariate protocol. In the inhomogeneous degree-restricted case, explicit factorization is possible by recursively reducing the polynomial as a function in one variable and treating the remaining variable parametrically, followed by spectral-factorization (as for Laurent polynomials) with control unitaries depending on the secondary variable.

For general multivariate QSP, no general synthesis exists beyond the commutative two-variable homogeneous case. Constructive sufficiency for non-commuting multivariate QSP is an unresolved question, and numerical algorithms for extracting modulation unitaries in $SU(n)$ remain an open direction.

6. Significance, Applications, and Open Directions

These advances unify the algebraic structure of classical univariate QSP (Haah's theorem) with multivariate and non-commutative variants, extending the class of quantum algorithms accessible by circuit-based polynomial transformations. Crucially, the multivariate, homogeneous setting allows direct realization of multivariate matrix polynomials within a single $SU(n)$-valued quantum circuit, rather than incurring overhead from linear-combination-of-unitary (LCU) gadgets. This offers practical gains for:

• Direct polynomial transformations for time-dependent and multi-parameter Hamiltonian simulation
• Multi-parameter eigenvalue transformation problems
• Efficient implementation of matrix functions $f(A,B)$ for $A,B$ Hermitian (with or without commutation)
• Quantum optimization algorithms with several continuous optimization parameters

The identification of the limits of classical slot-based and inhomogeneous variants, along with the explicit counterexamples, guides both the formulation of future conjectures and the design of new algebraic and algorithmic tools. Research priorities include:

• Resolving sufficiency for general non-commutative, homogeneous multivariate QSP protocols
• Developing efficient, practical algorithms for extracting modulation unitaries in high-dimensional $SU(n)$ implementations
• Exploiting inhomogeneous or degree-restricted MQSP for targeted quantum algorithmic primitives

These results constitute a significant step toward a full algebraic and algorithmic understanding of the multivariate QSP landscape, anchoring further progress in both theoretical and practical quantum algorithm design (Németh et al., 2023).