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Illustration of QSP
Source publication
Elucidating a connection with nonlinear Fourier analysis (NLFA), we extend a well known algorithm in quantum signal processing (QSP) to represent measurable signals by square summable sequences. Each coefficient of the sequence is Lipschitz continuous as a function of the signal.
Contexts in source publication
Context 1
... P be the space of sequences = (ψ k ) k∈N of numbers ψ k ∈ (− π 2 , π 2 ). We equip P with the metric induced by the L ∞ -norm Define u d ((, x) to be the upper left entry of U d ((, x Figure 1 is a simplified cartoon of QSP, conflating for illustrative purpose the group SO(3) with its double-cover SU (2) and ignoring for simplicity the reflection symmetry in the product (1.5). For a given signal f , Theorem 1 provides tuning parameters ψ j with which we can then evaluate f at x = cos θ as follows. ...
Context 2
... the map F j → F j F j is real analytic from 1 (Z) to itself and the − 1 2 -th power is real analytic from [1, ∞) to (0, 1], the function C extends to a real analytic map from 1 (Z) to (0, 1]. As for T n , taking all F j = F and summing (5.8) in absolute value over all permutations of the indices j 1 to j n , the sum separates into a product of sums and one can estimate for |z| = 1 ...
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Citations
... More recently, QSP was shown to have connections with the Non-Linear Fourier Transform (or NLFT) [27,28]. In particular it was shown that the inverse NLFT can be used to stably compute a QSP protocol for a desired target function, using the Riemann-Hilbert-Weiss algorithm [29]. ...
... The Non-Linear Fourier transform is a mathematical tool with its own history and line of research [28,34]. We define it briefly and state some useful properties, inviting the interested reader to check [27] for a more comprehensive overview. ...
... Definition 4 ( [34,27]). Let H ≥k be the space of measurable functions (a, b) such that: ...
Quantum signal processing (QSP) and quantum singular value transformation (QSVT) are powerful techniques for the development of quantum procedures. They allow to derive circuits preparing desired polynomial transformations. Recent research [Alexis et al. 2024] showed that Non-Linear Fourier Analysis (NLFA) can be employed to numerically compute a QSP protocol, with provable stability. In this work we extend their result, showing that GQSP and the Non-Linear Fourier Transform over SU(2) are the same object. This statement - proven by a simple argument - has a bunch of consequences: first, the Riemann-Hilbert-Weiss algorithm can be turned, with little modifications and no penalty in complexity, into a unified, provably stable algorithm for the computation of phase factors in any QSP variant, including GQSP. Secondly, we derive a uniqueness result for the existence of GQSP phase factors based on the bijectivity of the Non-Linear Fourier Transform. Furthermore, NLFA provides a complete theory of infinite generalized quantum signal processing, which characterizes the class of functions approximable by GQSP protocols.
... Alternative approaches to SU(2) approximations [106][107][108] have led to various extensions of the QSP algorithm [108][109][110]. Moreover, the cosine-sine decomposition [111], which provides a simultaneous singular value decomposition of different blocks in the block-encoding unitary, offers a SU(2) structure for the QSVT method. ...
Solving quantum molecular systems presents a significant challenge for classical computation. The advent of early fault-tolerant quantum computing (EFTQC) devices offers a promising avenue to address these challenges, leveraging advanced quantum algorithms with reduced hardware requirements. This review surveys the latest developments in EFTQC and fully fault-tolerant quantum computing (FFTQC) algorithms for quantum molecular systems, covering encoding schemes, advanced Hamiltonian simulation techniques, and ground-state energy estimation methods. We highlight recent progress in overcoming practical barriers, such as reducing circuit depth and minimizing the use of ancillary qubits. Special attention is given to the potential quantum advantages achievable through these algorithms, as well as the limitations imposed by dequantization and classical simulation techniques. The review concludes with a discussion of future directions, emphasizing the need for optimized algorithms and experimental validation to bridge the gap between theoretical developments and practical implementation in EFTQC and FFTQC for quantum molecular systems.
... A recent series of papers [76,77] showed that the nonlinear Fourier transform (NLFT) of desired signal f (x) can be mapped to a set of infinite QSP phase factors when x ∈ [0, 1] and f is purely real or imaginary. One additional condition is that f satisfy a Szegö condition, that is ...
Quantum signal processing (QSP) relies on a historically costly pre-processing step, "QSP-processing/phase-factor finding." QSP-processing is now a developed topic within quantum algorithms literature, and a beginner accessible review of QSP-processing is overdue. This work provides a whirlwind tour through QSP conventions and pre-processing methods, beginning from a pedagogically accessible QSP convention. We then review QSP conventions associated with three common polynomial types: real polynomials with definite parity, sums of reciprocal/anti-reciprocal Chebyshev polynomials, and complex polynomials. We demonstrate how the conventions perform with respect to three criteria: circuit length, polynomial conditions, and pre-processing methods. We then review the recently introduced Wilson method for QSP-processing and give conditions where it can succeed with bound error. Although the resulting bound is not computationally efficient, we demonstrate that the method succeeds with linear error propagation for relevant target polynomials and precision regimes, including the Jacobi-Anger expansion used in Hamiltonian simulation algorithms. We then apply our benchmarks to three QSP-processing methods for QSP circuits and show that a method introduced by Berntson and S\"underhauf outperforms both the Wilson method and the standard optimization strategy for complex polynomials.
... One can verify that the top-left entry of U (x, Ψ) is a complex polynomial in x. Moreover, for any target polynomial f ∈ R[x] satisfying (1) deg(f ) = d, (2) the parity of f is d mod 2, (3) ∥f ∥ ∞ := max x∈ [−1,1] |f (x)| ≤ 1, we can find phase factors Ψ ∈ R d+1 such that f (x) is equal to the real (or imaginary) part of the top-left entry of U (x, Ψ) for all x ∈ [−1, 1] [12]. By setting x = z+z −1 2 with z = e iθ , θ ∈ [0, 2π), the representation in Eq. (1) can be viewed as a 2 × 2 matrix Laurent polynomial, and we are interested in its values on the unit circle. ...
... For fixed k, Φ (1) and Φ (2) , applying mean value inequality to the function ...
... where n is the effective length of Φ (1) − Φ (2) . The last inequality follows Lemma 13, and notice that ∥Φ ...
Quantum signal processing (QSP) represents a real scalar polynomial of degree d using a product of unitary matrices of size 2 × 2 , parameterized by ( d + 1 ) real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree d → ∞ . While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the ℓ 1 space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the ℓ 1 space. The algorithm uses only double precision arithmetic operations, and provably converges when the ℓ 1 norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of d . This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit d → ∞ .
... A signal is typically a time-dependent function representing physical quantities, such as sound, images, or other types of data. Integrals on the semicircle have a significant application in signal processing (see, e.g., [1]). Let the weight function be Gegenbauer weight function, w(z) = (1 − λ) λ−1/2 , λ > −1/2. ...
Let D+ be defined as D+={z∈C:|z|<1,Imz>0} and Γ be a unit semicircle Γ={z=eiθ:0≤θ≤π}=∂D+. Let w(z) be a weight function which is positive and integrable on the open interval (-1,1), though possibly singularity at the endpoints, and which can be extended to a function w(z) holomorphic in the half disc D+. Orthogonal polynomials on the semicircle with respect to the complex-valued inner product (f,g)=∫Γf(z)g(z)w(z)(iz)-1dz=∫0πf(eiθ)g(eiθ)w(eiθ)dθwas introduced by Gautschi and Milovanović in ( J. Approx. Theory 46, 230-250, 1986) (for w(z)=1w(z)=1), where the certain basic properties were proved. Such orthogonality as well as the applications involving Gauss-Christoffel quadrature rules were further studied in Gautschi et al. (Constr. Approx. 3, 389-404, 1987) and Milovanović (2019). Inspired with Laurie’s paper (Math. Comp. 65(214), 739-747, 1996), this article introduces anti-Gaussian quadrature rules related to orthogonality on the semicircle, presents some of their properties, and suggests a numerical method for their construction. We demonstrate how these rules can be used to estimate the error of the corresponding Gaussian quadrature rules on the semicircle. Additionally, we introduce averaged Gaussian rules related to orthogonality on the semicircle to reduce the error of the corresponding Gaussian rules. Several numerical examples are provided.
... Quantum signal processing (QSP) [1][2][3][4][5][6][7] has emerged as a highly effective algorithmic technique within quantum computing. The central idea of quantum signal processing is to provide a method that gives a polynomial transformation of a blackbox input rotation by interspersing sequences of this rotation with predefined rotations. ...
We provide in this work a form of Modular Quantum Signal Processing that we call iterated quantum signal processing. This method recursively applies quantum signal processing to the outputs of other quantum signal processing steps, allowing polynomials to be easily achieved that would otherwise be difficult to find analytically. We specifically show by using a squaring quantum signal processing routine, that multiplication of phase angles can be approximated and in turn that any bounded degree multi-variate polynomial function of a set of phase angles can be implemented using traditional QSP ideas. We then discuss how these ideas can be used to construct phase functions relevant for quantum simulation such as the Coulomb potential and also discuss how to use these ideas to obviate the need for reversible arithmetic to compute square-root functions needed for simulations of bosonic Hamiltonians.
... Our strategy is based on a recent work [43], that discusses the intriguing relation between su(2) and su(1, 1) QSP and the nonlinear Fourier transform. Let us start by considering the QSP sequence in Eq. (3) and define ...
... whereÛ 0 ⃗ ϕ (w) = e iψ 0 Z . Amusingly, by using Lemma 1 of Ref. [43], we can establish the intimate relation between our QSP sequence and a truncated version of the nonlinear Fourier transform for sl(2, C) [44,45] ...
... Motivated by Ref. [43], we use the canonical form of the nonlinear Fourier transform ...
In recent years there has been an increasing interest on the theoretical and experimental investigation of space-time dual quantum circuits. They exhibit unique properties and have applications to diverse fields. Periodic space-time dual quantum circuits are of special interest, due to their iterative structure defined by the Floquet operator. A very similar iterative structure naturally appears in Quantum Signal processing (QSP), which has emerged as a framework that embodies all the known quantum algorithms. However, it is yet unclear whether there is deeper relation between these two apparently different concepts. In this work, we establish a relation between a circuit defining a Floquet operator in a single period and its space-time dual defining QSP sequences for the Lie algebra sl, which is the complexification of su(2). First, we show that our complexified QSP sequences can be interpreted in terms of action of the Lorentz group on density matrices and that they can be interpreted as hybrid circuits involving unitaries and measurements. We also show that unitary representations of our QSP sequences exist, although they are infinite-dimensional and are defined for bosonic operators in the Heisenberg picture. Finally, we also show the relation between our complexified QSP and the nonlinear Fourier transform for sl, which is a generalization of the previous results on su(2) QSP.
