Parallel Quantum Signal Processing Via Polynomial Factorization

John M. Martyn; Zane M. Rossi; Kevin Z. Cheng; Yuan Liu; Isaac L. Chuang

doi:10.22331/q-2025-08-27-1834

Parallel Quantum Signal Processing Via Polynomial Factorization

John M. Martyn¹, Zane M. Rossi², Kevin Z. Cheng², Yuan Liu^3,4,5, and Isaac L. Chuang^2,6

¹Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
²Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
³Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27606, USA
⁴Department of Computer Science, North Carolina State University, Raleigh, NC 27606, USA
⁵Department of Physics, North Carolina State University, Raleigh, NC 27606, USA
⁶Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Published:	2025-08-27, volume 9, page 1834
Editor:	Aleksandrs Belovs
Eprint:	arXiv:2409.19043v2
Doi:	https://doi.org/10.22331/q-2025-08-27-1834
Citation:	Quantum 9, 1834 (2025).

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Abstract

Quantum signal processing (QSP) is a methodology for constructing polynomial transformations of a linear operator encoded in a unitary. Applied to an encoding of a state $\rho$, QSP enables the evaluation of nonlinear functions of the form $\text{tr}(P(\rho))$ for a polynomial $P(x)$, which encompasses relevant properties like entropies and fidelity. However, QSP is a sequential algorithm: implementing a degree-$d$ polynomial necessitates $d$ queries to the encoding, equating to a query depth $d$. Here, we reduce the depth of these property estimation algorithms by developing Parallel Quantum Signal Processing. Our algorithm parallelizes the computation of $\text{tr} (P(\rho))$ over $k$ systems and reduces the query depth to $d/k$, thus enabling a family of time-space tradeoffs for QSP. This furnishes a property estimation algorithm suitable for distributed quantum computers, and is realized at the expense of increasing the number of measurements by a factor $O( \text{poly}(d) 2^{O(k)} )$. We achieve this result by factorizing $P(x)$ into a product of $k$ smaller polynomials of degree $O(d/k)$, which are each implemented in parallel with QSP, and subsequently multiplied together with a swap test to reconstruct $P(x)$. We characterize the achievable class of polynomials by appealing to the fundamental theorem of algebra, and demonstrate application to canonical problems including entropy estimation and partition function evaluation.

Featured image: An illustration of standard QSP vs. parallel QSP, highlighting how parallel QSP distributes the computation over multiple threads, thus decreasing the required circuit depth.

Popular summary

Quantum signal processing (QSP) has emerged as a unifying framework for quantum algorithms, based on polynomial approximations to functions (e.g., Taylor series). The catch is that QSP algorithms require a circuit depth that scales with the polynomial degree, often leading to circuits too deep for today’s quantum hardware. In this work, the authors alleviate this constraint by developing a method to parallelize QSP algorithms across multiple quantum computers, thus reducing the required circuit depth on each computer. As QSP is grounded in polynomials, the trick to achieve this is a classic concept from algebra: polynomial factorization.

► BibTeX data

@article{Martyn2025parallelquantum,
  doi = {10.22331/q-2025-08-27-1834},
  url = {https://doi.org/10.22331/q-2025-08-27-1834},
  title = {Parallel {Q}uantum {S}ignal {P}rocessing {V}ia {P}olynomial {F}actorization},
  author = {Martyn, John M. and Rossi, Zane M. and Cheng, Kevin Z. and Liu, Yuan and Chuang, Isaac L.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1834},
  month = aug,
  year = {2025}
}

► References

[1] Theodore J. Yoder, Guang Hao Low, and Isaac L. Chuang. ``Fixed-point quantum search with an optimal number of queries''. Phys. Rev. Lett. 113, 210501 (2014).
https://doi.org/10.1103/PhysRevLett.113.210501

[2] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. ``Methodology of resonant equiangular composite quantum gates''. Physical Review X 6 (2016).
https://doi.org/10.1103/PhysRevX.6.041067

[3] Guang Hao Low and Isaac L. Chuang. ``Optimal hamiltonian simulation by quantum signal processing''. Physical Review Letters 118 (2017).
https://doi.org/10.1103/PhysRevLett.118.010501

[4] Guang Hao Low and Isaac L. Chuang. ``Hamiltonian simulation by qubitization''. Quantum 3, 163 (2019).
https://doi.org/10.22331/q-2019-07-12-163

[5] András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. ``Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics''. Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (2019).
https://doi.org/10.1145/3313276.3316366

[6] John M Martyn, Zane M Rossi, Andrew K Tan, and Isaac L Chuang. ``Grand unification of quantum algorithms''. PRX Quantum 2, 040203 (2021).
https://doi.org/10.1103/PRXQuantum.2.040203

[7] Jeongwan Haah. ``Product decomposition of periodic functions in quantum signal processing''. Quantum 3, 190 (2019).
https://doi.org/10.22331/q-2019-10-07-190

[8] Rui Chao, Dawei Ding, Andras Gilyén, Cupjin Huang, and Mario Szegedy. ``Finding angles for quantum signal processing with machine precision'' (2020). arXiv:2003.02831.
arXiv:2003.02831

[9] Yulong Dong, Xiang Meng, K. Birgitta Whaley, and Lin Lin. ``Efficient phase-factor evaluation in quantum signal processing''. Physical Review A 103 (2021).
https://doi.org/10.1103/physreva.103.042419

[10] Yulong Dong, Lin Lin, Hongkang Ni, and Jiasu Wang. ``Infinite quantum signal processing''. Quantum 8, 1558 (2024).
https://doi.org/10.22331/q-2024-12-10-1558

[11] Ashley Montanaro and Changpeng Shao. ``Quantum and classical query complexities of functions of matrices''. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing. Page 573–584. STOC 2024 New York, NY, USA (2024). Association for Computing Machinery.
https://doi.org/10.1145/3618260.3649665

[12] Ewin Tang and Kevin Tian. ``A CS guide to the quantum singular value transformation''. Chapter : 2024 Symposium on Simplicity in Algorithms (SOSA), pages 121–143. SIAM. (2024).
https://doi.org/10.1137/1.9781611977936.13

[13] Michel Alexis, Lin Lin, Gevorg Mnatsakanyan, Christoph Thiele, and Jiasu Wang. ``Infinite quantum signal processing for arbitrary Szegö functions'' (2024). arXiv:2407.05634.
arXiv:2407.05634

[14] Bjorn K. Berntson and Christoph Sünderhauf. ``Complementary polynomials in quantum signal processing'' (2024). arXiv:2406.04246.
https://doi.org/10.1007/s00220-025-05302-9
arXiv:2406.04246

[15] Kyle DeBry, Jasmine Sinanan-Singh, Colin D. Bruzewicz, David Reens, May E. Kim, Matthew P. Roychowdhury, Robert McConnell, Isaac L. Chuang, and John Chiaverini. ``Experimental quantum channel discrimination using metastable states of a trapped ion''. Phys. Rev. Lett. 131, 170602 (2023).
https://doi.org/10.1103/PhysRevLett.131.170602

[16] Yuta Kikuchi, Conor Mc Keever, Luuk Coopmans, Michael Lubasch, and Marcello Benedetti. ``Realization of quantum signal processing on a noisy quantum computer''. npj Quantum Information 9, 93 (2023).
https://doi.org/10.1038/s41534-023-00762-0

[17] David A Patterson, Frederick P Brooks Jr, Ivan E Sutherland, and Charles P Thacker. ``Computer architecture''. Elsevier Science. (2011).

[18] Sonika Johri, Damian S. Steiger, and Matthias Troyer. ``Entanglement spectroscopy on a quantum computer''. Physical Review B 96 (2017).
https://doi.org/10.1103/physrevb.96.195136

[19] A. Elben, B. Vermersch, M. Dalmonte, J.I. Cirac, and P. Zoller. ``Rényi entropies from random quenches in atomic hubbard and spin models''. Physical Review Letters 120 (2018).
https://doi.org/10.1103/physrevlett.120.050406

[20] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. ``Quantum fingerprinting''. Physical Review Letters 87 (2001).
https://doi.org/10.1103/physrevlett.87.167902

[21] Michał Oszmaniec, Daniel J Brod, and Ernesto F Galvão. ``Measuring relational information between quantum states, and applications''. New Journal of Physics 26, 013053 (2024).
https://doi.org/10.1088/1367-2630/ad1a27

[22] Guang Hao Low. ``Quantum signal processing by single-qubit dynamics''. PhD thesis. Massachusetts Institute of Technology. (2017). url: http://hdl.handle.net/1721.1/115025.
http://hdl.handle.net/1721.1/115025

[23] Yihui Quek, Eneet Kaur, and Mark M. Wilde. ``Multivariate trace estimation in constant quantum depth''. Quantum 8, 1220 (2024).
https://doi.org/10.22331/q-2024-01-10-1220

[24] Andrew K. Tan, Yuan Liu, Minh C. Tran, and Isaac L. Chuang. ``Perturbative model of noisy quantum signal processing''. Phys. Rev. A 107, 042429 (2023).
https://doi.org/10.1103/PhysRevA.107.042429

[25] Zane M Rossi, Jeffery Yu, Isaac L Chuang, and Sho Sugiura. ``Quantum advantage for noisy channel discrimination''. Physical Review A 105, 032401 (2022).
https://doi.org/10.1103/PhysRevA.105.032401

[26] Marcello Caleffi, Michele Amoretti, Davide Ferrari, Jessica Illiano, Antonio Manzalini, and Angela Sara Cacciapuoti. ``Distributed quantum computing: A survey''. Computer Networks 254, 110672 (2024).
https://doi.org/10.1016/j.comnet.2024.110672

[27] Yihui Quek, Daniel Stilck França, Sumeet Khatri, Johannes Jakob Meyer, and Jens Eisert. ``Exponentially tighter bounds on limitations of quantum error mitigation''. Nature Physics 20, 1648–1658 (2024).
https://doi.org/10.1038/s41567-024-02536-7

[28] Tianyi Peng, Aram W. Harrow, Maris Ozols, and Xiaodi Wu. ``Simulating large quantum circuits on a small quantum computer''. Physical Review Letters 125 (2020).
https://doi.org/10.1103/physrevlett.125.150504

[29] Danial Motlagh and Nathan Wiebe. ``Generalized quantum signal processing''. PRX Quantum 5, 020368 (2024).
https://doi.org/10.1103/PRXQuantum.5.020368

[30] Yulong Dong, Xiang Meng, K Birgitta Whaley, and Lin Lin. ``Efficient phase-factor evaluation in quantum signal processing''. Physical Review A 103, 042419 (2021).
https://doi.org/10.1103/PhysRevA.103.042419

[31] Lexing Ying. ``Stable factorization for phase factors of quantum signal processing''. Quantum 6, 842 (2022).
https://doi.org/10.22331/q-2022-10-20-842

[32] Shuntaro Yamamoto and Nobuyuki Yoshioka. ``Robust angle finding for generalized quantum signal processing'' (2024). arXiv:2402.03016.
arXiv:2402.03016

[33] Sathyawageeswar Subramanian and Min-Hsiu Hsieh. ``Quantum algorithm for estimating $\alpha$ -renyi entropies of quantum states''. Physical Review A 104 (2021).
https://doi.org/10.1103/physreva.104.022428

[34] András Gilyén and Tongyang Li. ``Distributional property testing in a quantum world'' (2019). arXiv:1902.00814.
arXiv:1902.00814

[35] Dorit Aharonov, Vaughan Jones, and Zeph Landau. ``A polynomial quantum algorithm for approximating the jones polynomial''. In Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing. Page 427–436. STOC '06 New York, NY, USA (2006). Association for Computing Machinery.
https://doi.org/10.1145/1132516.1132579

[36] András Gilyén and Alexander Poremba. ``Improved quantum algorithms for fidelity estimation'' (2022). arXiv:2203.15993.
arXiv:2203.15993

[37] Qisheng Wang, Ji Guan, Junyi Liu, Zhicheng Zhang, and Mingsheng Ying. ``New quantum algorithms for computing quantum entropies and distances''. IEEE Transactions on Information Theory 70, 5653–5680 (2024).
https://doi.org/10.1109/TIT.2024.3399014

[38] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge University Press. (2010).

[39] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. ``Quantum amplitude amplification and estimation''. Quantum Computation and Information Page 53–74 (2002).
https://doi.org/10.1090/conm/305/05215

[40] Artur K. Ekert, Carolina Moura Alves, Daniel K. L. Oi, Michał Horodecki, Paweł Horodecki, and L. C. Kwek. ``Direct estimations of linear and nonlinear functionals of a quantum state''. Physical Review Letters 88 (2002).
https://doi.org/10.1103/physrevlett.88.217901

[41] Matthew B. Hastings, Iván González, Ann B. Kallin, and Roger G. Melko. ``Measuring renyi entanglement entropy in quantum monte carlo simulations''. Physical Review Letters 104 (2010).
https://doi.org/10.1103/physrevlett.104.157201

[42] Todd A. Brun. ``Measuring polynomial functions of states'' (2004). arXiv:quant-ph/0401067.
arXiv:quant-ph/0401067

[43] Paweł Horodecki and Artur Ekert. ``Method for direct detection of quantum entanglement''. Physical Review Letters 89 (2002).
https://doi.org/10.1103/physrevlett.89.127902

[44] Yiğit Subaşı, Lukasz Cincio, and Patrick J Coles. ``Entanglement spectroscopy with a depth-two quantum circuit''. Journal of Physics A: Mathematical and Theoretical 52, 044001 (2019).
https://doi.org/10.1088/1751-8121/aaf54d

[45] Justin Yirka and Yiğit Subaşı. ``Qubit-efficient entanglement spectroscopy using qubit resets''. Quantum 5, 535 (2021).
https://doi.org/10.22331/q-2021-09-02-535

[46] Alan Edelman and Hiroshi Murakami. ``Polynomial roots from companion matrix eigenvalues''. Mathematics of Computation 64, 763–776 (1995).
https://doi.org/10.1090/S0025-5718-1995-1262279-2

[47] Angus Lowe, Matija Medvidović, Anthony Hayes, Lee J. O'Riordan, Thomas R. Bromley, Juan Miguel Arrazola, and Nathan Killoran. ``Fast quantum circuit cutting with randomized measurements''. Quantum 7, 934 (2023).
https://doi.org/10.22331/q-2023-03-02-934

[48] Allan Tosta, Thais de Lima Silva, Giancarlo Camilo, and Leandro Aolita. ``Randomized semi-quantum matrix processing''. npj Quantum Information 10 (2024).
https://doi.org/10.1038/s41534-024-00883-0

[49] Yue Wang and Qi Zhao. ``Faster quantum algorithms with "fractional"-truncated series'' (2024). arXiv:2402.05595.
arXiv:2402.05595

[50] John M. Martyn and Patrick Rall. ``Halving the cost of quantum algorithms with randomization''. npj Quantum Information 11 (2025).
https://doi.org/10.1038/s41534-025-01003-2

[51] Dominic W. Berry, Graeme Ahokas, Richard Cleve, and Barry C. Sanders. ``Efficient quantum algorithms for simulating sparse hamiltonians''. Communications in Mathematical Physics 270, 359–371 (2006).
https://doi.org/10.1007/s00220-006-0150-x

[52] Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen. ``On the impossibility of general parallel fast-forwarding of hamiltonian simulation''. In Proceedings of the Conference on Proceedings of the 38th Computational Complexity Conference. CCC '23 Dagstuhl, DEU (2023). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik.
https://doi.org/10.4230/LIPIcs.CCC.2023.33

[53] Shelby Kimmel, Cedric Yen-Yu Lin, Guang Hao Low, Maris Ozols, and Theodore J. Yoder. ``Hamiltonian simulation with optimal sample complexity''. npj Quantum Information 3 (2017).
https://doi.org/10.1038/s41534-017-0013-7

[54] Kevin Cheng. ``Parallel QSP''. https://github.com/kevinchengg/parallelQSP (2024).
https://github.com/kevinchengg/parallelQSP

[55] Tiff Brydges, Andreas Elben, Petar Jurcevic, Benoît Vermersch, Christine Maier, Ben P. Lanyon, Peter Zoller, Rainer Blatt, and Christian F. Roos. ``Probing rényi entanglement entropy via randomized measurements''. Science 364, 260–263 (2019).
https://doi.org/10.1126/science.aau4963

[56] N. M. Linke, S. Johri, C. Figgatt, K. A. Landsman, A. Y. Matsuura, and C. Monroe. ``Measuring the rényi entropy of a two-site fermi-hubbard model on a trapped ion quantum computer''. Physical Review A 98 (2018).
https://doi.org/10.1103/physreva.98.052334

[57] Mohamed Hibat-Allah, Martin Ganahl, Lauren E. Hayward, Roger G. Melko, and Juan Carrasquilla. ``Recurrent neural network wave functions''. Physical Review Research 2 (2020).
https://doi.org/10.1103/physrevresearch.2.023358

[58] Youle Wang, Lei Zhang, Zhan Yu, and Xin Wang. ``Quantum phase processing and its applications in estimating phase and entropies''. Physical Review A 108 (2023).
https://doi.org/10.1103/physreva.108.062413

[59] Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. ``Quantum principal component analysis''. Nature Physics 10, 631–633 (2014).
https://doi.org/10.1038/nphys3029

[60] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Simulating hamiltonian dynamics with a truncated Taylor series''. Physical Review Letters 114 (2015).
https://doi.org/10.1103/PhysRevLett.114.090502

[61] Youle Wang, Benchi Zhao, and Xin Wang. ``Quantum algorithms for estimating quantum entropies''. Phys. Rev. Appl. 19, 044041 (2023).
https://doi.org/10.1103/PhysRevApplied.19.044041

[62] David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. ``Quantum state tomography via compressed sensing''. Physical Review Letters 105 (2010).
https://doi.org/10.1103/physrevlett.105.150401

[63] Cristina Butucea, Mădălin Guţă, and Theodore Kypraios. ``Spectral thresholding quantum tomography for low rank states''. New Journal of Physics 17, 113050 (2015).
https://doi.org/10.1088/1367-2630/17/11/113050

[64] Israel F. Araujo, Carsten Blank, Ismael C. S. Araújo, and Adenilton J. da Silva. ``Low-rank quantum state preparation''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 43, 161–170 (2024).
https://doi.org/10.1109/tcad.2023.3297972

[65] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017).
https://doi.org/10.1088/1751-8121/aa6dc3

[66] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac. ``Matrix product state representations'' (2007). arXiv:quant-ph/0608197.
arXiv:quant-ph/0608197

[67] Marcin Jarzyna and Rafał Demkowicz-Dobrzański. ``Matrix product states for quantum metrology''. Physical Review Letters 110 (2013).
https://doi.org/10.1103/physrevlett.110.240405

[68] Nic Ezzell, Zoë Holmes, and Patrick J. Coles. ``The quantum low-rank approximation problem'' (2022). arXiv:2203.00811.
arXiv:2203.00811

[69] Anirban N. Chowdhury, Guang Hao Low, and Nathan Wiebe. ``A variational quantum algorithm for preparing quantum gibbs states'' (2020). arXiv:2002.00055.
arXiv:2002.00055

[70] Brian Skinner, Jonathan Ruhman, and Adam Nahum. ``Measurement-induced phase transitions in the dynamics of entanglement''. Physical Review X 9 (2019).
https://doi.org/10.1103/physrevx.9.031009

[71] Benjamin Schumacher. ``Quantum coding''. Phys. Rev. A 51, 2738–2747 (1995).
https://doi.org/10.1103/PhysRevA.51.2738

[72] Aaron Somoroff, Quentin Ficheux, Raymond A. Mencia, Haonan Xiong, Roman Kuzmin, and Vladimir E. Manucharyan. ``Millisecond coherence in a superconducting qubit''. Phys. Rev. Lett. 130, 267001 (2023).
https://doi.org/10.1103/PhysRevLett.130.267001

[73] Yulong Dong, Jonathan A. Gross, and Murphy Yuezhen Niu. ``Optimal low-depth quantum signal-processing phase estimation''. Nature Communications 16 (2025).
https://doi.org/10.1038/s41467-025-56724-x

[74] Lorenzo Laneve. ``Quantum signal processing over SU(N)'' (2024). arXiv:2311.03949.
arXiv:2311.03949

[75] Xi Lu, Yuan Liu, and Hongwei Lin. ``Quantum signal processing and quantum singular value transformation on U(N)'' (2024). arXiv:2408.01439.
arXiv:2408.01439

[76] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller. ``The randomized measurement toolbox''. Nature Reviews Physics 5, 9–24 (2022).
https://doi.org/10.1038/s42254-022-00535-2

[77] Zane M. Rossi and Isaac L. Chuang. ``Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle''. Quantum 6, 811 (2022).
https://doi.org/10.22331/q-2022-09-20-811

[78] Balázs Németh, Blanka Kövér, Boglárka Kulcsár, Roland Botond Miklósi, and András Gilyén. ``On variants of multivariate quantum signal processing and their characterizations'' (2023). arXiv:2312.09072.
arXiv:2312.09072

[79] Niladri Gomes, Hokiat Lim, and Nathan Wiebe. ``Multivariable qsp and bosonic quantum simulation using iterated quantum signal processing'' (2024). arXiv:2408.03254.
arXiv:2408.03254

[80] Yuri Alexeev, Maximilian Amsler, Marco Antonio Barroca, Sanzio Bassini, Torey Battelle, Daan Camps, David Casanova, Young jai Choi, Frederic T Chong, Charles Chung, et al. ``Quantum-centric supercomputing for materials science: A perspective on challenges and future directions''. Future Generation Computer Systems (2024).
https://doi.org/10.1016/j.future.2024.04.060

[81] Yuan Liu, Oinam R Meitei, Zachary E Chin, Arkopal Dutt, Max Tao, Isaac L Chuang, and Troy Van Voorhis. ``Bootstrap embedding on a quantum computer''. Journal of Chemical Theory and Computation 19, 2230–2247 (2023).
https://doi.org/10.1021/acs.jctc.3c00012

[82] Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan. ``Quantum algorithms for quantum chemistry and quantum materials science''. Chemical Reviews 120, 12685–12717 (2020).
https://doi.org/10.1021/acs.chemrev.9b00829

[83] Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma. ``Exponential improvement in precision for simulating sparse hamiltonians''. Proceedings of the 46th Annual ACM Symposium on Theory of Computing (2014).
https://doi.org/10.1145/2591796.2591854

[84] Dominic W. Berry, Andrew M. Childs, and Robin Kothari. ``Hamiltonian simulation with nearly optimal dependence on all parameters''. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science. Page 792–809. IEEE (2015).
https://doi.org/10.1109/focs.2015.54

[85] Isaac Chuang, Andrew Tan, and John M Martyn. ``PyQSP: Python Quantum Signal Processing''. https://github.com/ichuang/pyqsp (2021).
https://github.com/ichuang/pyqsp

[86] Theodore J Rivlin. ``Chebyshev polynomials''. Courier Dover Publications. (2020).

[87] John P Boyd. ``Chebyshev and fourier spectral methods''. Courier Corporation. (2001).

[88] Q. I. Rahman and G. Schmiesser. ``Analytic theory of polynomials''. Clarendon Press, Oxford. (2002).

[89] George B Arfken, Hans J Weber, and Frank E Harris. ``Mathematical methods for physicists: a comprehensive guide''. Academic press. (2011).

[90] J Oliver. ``A note on the signs of truncated chebyshev polynomials''. BIT Numerical Mathematics 18, 233–235 (1978).
https://doi.org/10.1007/BF01931700

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