quantum-computing

Quantum phase estimation with optimal confidence interval using three control qubits

Kaur Kristjuhan and Dominic W. Berry

28 min read

0 likes

⚡ Quantum Brief

We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. We show how to prepare the corresponding state in a far more efficient way than prior work. By utilizing mid-circuit measurements, no more than three qubits are required for the control register to achieve any desired precision. In the textbook version of QPE, the proposed control state is a uniform superpositi AbstractQuantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations.

AI Audio Summary

0:00 / 0:00

Click to play

Quantum News · Media Library

Understand this faster with AI

Ask ChatGPT

Research with Perplexity

Explain with Claude

Discuss on Grok

AbstractQuantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits.Featured image: Overview of the phase estimation procedure. A matrix product representation is used to prepare a DPSS state, which provides the phase estimate with optimal confidence interval. By utilizing mid-circuit measurements, no more than three qubits are required for the control register to achieve any desired precision. Popular summaryQuantum phase estimation (QPE) is an important and widely used quantum algorithm. The purpose of QPE is to perform a measurement on a quantum computer that determines the value of a phase, corresponding to the eigenvalue of a unitary operator. Like any estimation procedure, QPE's output comes with a confidence interval, a range in which the true phase is likely to lie. In QPE, the size of the confidence interval is determined by a control state prepared on an auxiliary register. In the textbook version of QPE, the proposed control state is a uniform superposition state, which is easy to prepare, but leaves a comparatively large likelihood of producing an estimate that is far from the true answer. In this paper, we discuss how to efficiently prepare a different control state; the discrete prolate spheroidal sequence (DPSS) state. It is well known that the DPSS state provides the optimal confidence interval, but the quantum circuits needed to prepare it have been far too costly. Our contribution is to provide efficient, explicit quantum circuits that prepare this state. We show how the circuit cost scales with the desired precision and demonstrate how mid-circuit measurements can be used to implement our method to any desired phase precision while allocating no more than three qubits to the control register. This produces a result not only optimal in a mathematical sense, but also practical for early-generation fault-tolerant quantum computers, where qubits are scarce.► BibTeX data@article{Kristjuhan2026quantumphase, doi = {10.22331/q-2026-08-19-2193}, url = {https://doi.org/10.22331/q-2026-08-19-2193}, title = {Quantum phase estimation with optimal confidence interval using three control qubits}, author = {Kristjuhan, Kaur and Berry, Dominic W.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2193}, month = aug, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Journal on Computing 26, 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172 [2] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. ``Simulated quantum computation of molecular energies''. Science 309, 1704–1707 (2005). https://doi.org/10.1126/science.1113479 [3] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Rev. Mod. Phys. 92, 015003 (2020). https://doi.org/10.1103/RevModPhys.92.015003 [4] Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. ``Quantum chemistry in the age of quantum computing''. Chemical Reviews 119, 10856–10915 (2019). https://doi.org/10.1021/acs.chemrev.8b00803 [5] 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 [6] Mario Motta and Julia E Rice. ``Emerging quantum computing algorithms for quantum chemistry''.

Wiley Interdisciplinary Reviews: Computational Molecular Science 12, e1580 (2022). https://doi.org/10.1002/wcms.1580 [7] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão. ``Quantum algorithms: A survey of applications and end-to-end complexities''.

Cambridge University Press. (2025). arXiv:2310.03011. https://doi.org/10.1017/9781009639651 arXiv:2310.03011 [8] Michael A Nielsen and Isaac L Chuang. ``Quantum computation and quantum information''. Cambridge university press. (2010). https://doi.org/10.1017/CBO9780511976667 [9] Dominic W. Berry, Mária Kieferová, Artur Scherer, Yuval R. Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, and Ryan Babbush. ``Improved techniques for preparing eigenstates of fermionic Hamiltonians''. npj Quantum Information 4, 22 (2018). https://doi.org/10.1038/s41534-018-0071-5 [10] David Poulin, Alexei Kitaev, Damian S. Steiger, Matthew B. Hastings, and Matthias Troyer. ``Quantum algorithm for spectral measurement with a lower gate count''. Phys. Rev. Lett. 121, 010501 (2018). https://doi.org/10.1103/PhysRevLett.121.010501 [11] F.J. Harris. ``On the use of windows for harmonic analysis with the discrete Fourier transform''. Proceedings of the IEEE 66, 51–83 (1978). https://doi.org/10.1109/PROC.1978.10837 [12] Benjamin P Abbott, Rich Abbott, Thomas D Abbott, Fausto Acernese, Kendall Ackley, Carl Adams, Thomas Adams, Paolo Addesso, Rana X Adhikari, Vaishali B Adya, et al. ``GW170817: Observation of gravitational waves from a binary neutron star inspiral''. Phys. Rev. Lett. 119, 161101 (2017). https://doi.org/10.1103/PhysRevLett.119.161101 [13] Samantha A Usman, Alexander H Nitz, Ian W Harry, Christopher M Biwer, Duncan A Brown, Miriam Cabero, Collin D Capano, Tito Dal Canton, Thomas Dent, Stephen Fairhurst, Marcel S Kehl, Drew Keppel, Badri Krishnan, Amber Lenon, Andrew Lundgren, Alex B Nielsen, Larne P Pekowsky, Harald P Pfeiffer, Peter R Saulson, Matthew West, and Joshua L Willis. ``The PyCBC search for gravitational waves from compact binary coalescence''. Classical and Quantum Gravity 33, 215004 (2016). https://doi.org/10.1088/0264-9381/33/21/215004 [14] A. Luis and J. Peřina. ``Optimum phase-shift estimation and the quantum description of the phase difference''. Phys. Rev. A 54, 4564–4570 (1996). https://doi.org/10.1103/PhysRevA.54.4564 [15] Andrew M. Childs. ``On the relationship between continuous- and discrete-time quantum walk''. Commun. Math. Phys 294, 581–603 (2009). https://doi.org/10.1007/s00220-009-0930-1 [16] Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven. ``Encoding electronic spectra in quantum circuits with linear T complexity''. Phys. Rev. X 8, 041015 (2018). https://doi.org/10.1103/PhysRevX.8.041015 [17] Peyman Najafi, Pedro C. S. Costa, and Dominic W. Berry. ``Optimum phase estimation with two control qubits''. AVS Quantum Science 5, 023802 (2023). https://doi.org/10.1116/5.0147954 [18] Dominic W Berry, Yu Tong, Tanuj Khattar, Alec White, Tae In Kim, Guang Hao Low, Sergio Boixo, Zhiyan Ding, Lin Lin, Seunghoon Lee, et al. ``Rapid initial-state preparation for the quantum simulation of strongly correlated molecules''. PRX Quantum 6, 020327 (2025). https://doi.org/10.1103/PRXQuantum.6.020327 [19] Hiroshi Imai and Masahito Hayashi. ``Fourier analytic approach to phase estimation in quantum systems''. New Journal of Physics 11, 043034 (2009). https://doi.org/10.1088/1367-2630/11/4/043034 [20] Rei Sakuma, Kaito Wada, Shu Kanno, Kimberlee Keithley, Kenji Sugisaki, Takashi Abe, Hajime Nakamura, and Naoki Yamamoto. ``Quantum-phase-estimation-based filtering: Performance analysis and application to low-energy spectral calculations''. Phys. Rev. A 113, 012602 (2026). https://doi.org/10.1103/m85k-7q32 [21] David Slepian. ``Analytic solution of two apodization problems''. Journal of the Optical Society of America 55, 1110–1115 (1965). https://doi.org/10.1364/JOSA.55.001110 [22] Sandrine J Thomas, Rémi Soummer, Daren Dillon, Bruce Macintosh, Julia W Evans, Donald Gavel, Anand Sivaramakrishnan, Christian Marois, and Ben R Oppenheimer. ``Testing the APLC on the LAO ExAO testbed''.

In Adaptive Optics Systems. Volume 7015, pages 1845–1855. SPIE (2008). https://doi.org/10.1117/12.789691 [23] Changqing Xie, Xiaoli Zhu, Lina Shi, and Ming Liu. ``Spiral photon sieves apodized by digital prolate spheroidal window for the generation of hard-x-ray vortex''. Optics Letters 35, 1765–1767 (2010). https://doi.org/10.1364/OL.35.001765 [24] David Slepian. ``Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: The discrete case''.

Bell System Technical Journal 57, 1371–1430 (1978). https://doi.org/10.1002/j.1538-7305.1978.tb02104.x [25] J Kaiser and R Schafer. ``On the use of the $\text{I}_0$-sinh window for spectrum analysis''. IEEE Transactions on Acoustics, Speech, and Signal Processing 28, 105–107 (1980). https://doi.org/10.1109/TASSP.1980.1163349 [26] Dominic W Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, and Ryan Babbush. ``Analyzing prospects for quantum advantage in topological data analysis''. PRX Quantum 5, 010319 (2024). https://doi.org/10.1103/PRXQuantum.5.010319 [27] Lov Grover and Terry Rudolph. ``Creating superpositions that correspond to efficiently integrable probability distributions'' (2002). arXiv:quant-ph/0208112. arXiv:quant-ph/0208112 [28] Guang Hao Low, Vadym Kliuchnikov, and Luke Schaeffer. ``Trading T gates for dirty qubits in state preparation and unitary synthesis''. Quantum 8, 1375 (2024). https://doi.org/10.22331/q-2024-06-17-1375 [29] Sean Greenaway, William Pol, and Sukin Sim. ``A case study against QSVT: assessment of quantum phase estimation improved by signal processing techniques'' (2024). url: https://doi.org/10.48550/arXiv.2404.01396. https://doi.org/10.48550/arXiv.2404.01396 [30] Oliver O'Brien and Christoph Sünderhauf. ``Quantum state preparation via piecewise QSVT''. Quantum 9, 1786 (2025). https://doi.org/10.22331/q-2025-07-03-1786 [31] Sam McArdle, András Gilyén, and Mario Berta. ``Quantum state preparation without coherent arithmetic'' (2025). arXiv:2210.14892. arXiv:2210.14892 [32] Dhrumil Patel, Shi Jie Samuel Tan, Yiğit Subaşı, and Andrew T. Sornborger. ``Optimal coherent quantum phase estimation via tapering''. PRX Quantum 7, 020302 (2026). https://doi.org/10.1103/l5y6-6zxv [33] Guang Hao Low and Isaac L Chuang. ``Optimal Hamiltonian simulation by quantum signal processing''. Phys. Rev. Lett. 118, 010501 (2017). https://doi.org/10.1103/PhysRevLett.118.010501 [34] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush. ``Even more efficient quantum computations of chemistry through tensor hypercontraction''. PRX Quantum 2, 030305 (2021). https://doi.org/10.1103/PRXQuantum.2.030305 [35] Dominic W. Berry, Michael J. W. Hall, Marcin Zwierz, and Howard M. Wiseman. ``Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates''. Phys. Rev. A 86, 053813 (2012). https://doi.org/10.1103/PhysRevA.86.053813 [36] Frederik J. Simons. ``Slepian functions and their use in signal estimation and spectral analysis''. Page 891–923 in Handbook of Geomathematics.

Springer Berlin Heidelberg. (2010). https://doi.org/10.1007/978-3-642-01546-5_30 [37] Guifré Vidal. ``Efficient classical simulation of slightly entangled quantum computations''. Phys. Rev. Lett. 91, 147902 (2003). https://doi.org/10.1103/PhysRevLett.91.147902 [38] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96–192 (2011). https://doi.org/10.1016/j.aop.2010.09.012 [39] Christopher M. Dawson and Michael A. Nielsen. ``The Solovay-Kitaev algorithm'' (2005). arXiv:quant-ph/0505030. arXiv:quant-ph/0505030 [40] Tianyi Hao, Amanda Xu, and Swamit Tannu. ``Reducing T gates with unitary synthesis''. In Proceedings of the 31st ACM International Conference on Architectural Support for Programming Languages and Operating Systems, Volume 2. Page 1589–1604. ASPLOS '26New York, NY, USA (2026). Association for Computing Machinery. https://doi.org/10.1145/3779212.3790210 [41] Matthew Amy, Dmitri Maslov, Michele Mosca, and Martin Roetteler. ``A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 32, 818–830 (2013). https://doi.org/10.1109/TCAD.2013.2244643 [42] Vlad Gheorghiu, Michele Mosca, and Priyanka Mukhopadhyay. ``T-count and T-depth of any multi-qubit unitary''. npj Quantum Information 8, 141 (2022). https://doi.org/10.1038/s41534-022-00651-y [43] Anouk Paradis, Jasper Dekoninck, Benjamin Bichsel, and Martin Vechev. ``Synthetiq: Fast and versatile quantum circuit synthesis''. Proceedings of the ACM on Programming Languages 8, 55–82 (2024). https://doi.org/10.1145/3649813 [44] Mathias Weiden, Justin Kalloor, John Kubiatowicz, Ed Younis, and Costin Iancu. ``High-precision multi-qubit Clifford+T synthesis by unitary diagonalization''. Electronic Proceedings in Theoretical Computer Science 426, 215–230 (2025). https://doi.org/10.4204/eptcs.426.8 [45] Raban Iten, Roger Colbeck, Ivan Kukuljan, Jonathan Home, and Matthias Christandl. ``Quantum circuits for isometries''. Phys. Rev. A 93, 032318 (2016). https://doi.org/10.1103/PhysRevA.93.032318 [46] Aaron Szasz, Ed Younis, and Wibe De Jong. ``Numerical circuit synthesis and compilation for multi-state preparation''. In 2023 IEEE International Conference on Quantum Computing and Engineering (QCE). Volume 01, pages 768–778. (2023). https://doi.org/10.1109/QCE57702.2023.00092 [47] Stepan Fomichev, Kasra Hejazi, Modjtaba Shokrian Zini, Matthew Kiser, Joana Fraxanet, Pablo Antonio Moreno Casares, Alain Delgado, Joonsuk Huh, Arne-Christian Voigt, Jonathan E Mueller, et al. ``Initial state preparation for quantum chemistry on quantum computers''. PRX Quantum 5, 040339 (2024). https://doi.org/10.1103/PRXQuantum.5.040339 [48] William J Huggins, Oskar Leimkuhler, Torin F Stetina, and K Birgitta Whaley. ``Efficient state preparation for the quantum simulation of molecules in first quantization''. PRX Quantum 6, 020319 (2025). https://doi.org/10.1103/PRXQuantum.6.020319 [49] Robert B Griffiths and Chi-Sheng Niu. ``Semiclassical Fourier transform for quantum computation''. Phys. Rev. Lett. 76, 3228 (1996). https://doi.org/10.1103/PhysRevLett.76.3228 [50] Wolfram Research (2007), SpheroidalPS, Wolfram Language function, https://reference.wolfram.com/language/ref/SpheroidalPS.html. https://reference.wolfram.com/language/ref/SpheroidalPS.html [51] `` NIST Digital Library of Mathematical Functions''. https://dlmf.nist.gov/30.4, Release 1.2.3 of 2024-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. https://dlmf.nist.gov/30.4 [52] Wolfram Research, Inc. ``Mathematica, Version 14.3''. Champaign, IL, 2025. [53] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, et al. ``SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python''. Nature Methods 17, 261–272 (2020). https://doi.org/10.1038/s41592-019-0686-2 [54] Robert M. Gray. ``Toeplitz and circulant matrices: A review''. Foundations and Trends in Communications and Information Theory 2, 155–239 (2006). https://doi.org/10.1561/0100000006 [55] A.V. Oppenheim, R.W. Schafer, and J.R. Buck. ``Discrete-time signal processing''.

Prentice Hall International Editions Series. Prentice Hall. (1999). https://dl.acm.org/doi/10.5555/294797 [56] G.H. Golub and C.F. Van Loan. ``Matrix computations''.

Johns Hopkins Studies in the Mathematical Sciences.

Johns Hopkins University Press. (2013). https://doi.org/10.56021/9781421407944 [57] C.C. Paige and M. Wei. ``History and generality of the CS decomposition''. Linear Algebra and its Applications 208-209, 303–326 (1994). https://doi.org/10.1016/0024-3795(94)90446-4 [58] Brian D Sutton. ``Computing the complete CS decomposition''. Numerical Algorithms 50, 33–65 (2009). https://doi.org/10.1007/s11075-008-9215-6 [59] David Wierichs, Maxwell West, Roy T. Forestano, M. Cerezo, and Nathan Killoran. ``Recursive Cartan decompositions for unitary synthesis'' (2025). arXiv:2503.19014. arXiv:2503.19014 [60] Hai-Rui Wei and Yao-Min Di. ``Decomposition of orthogonal matrix and synthesis of two-qubit and three-qubit orthogonal gates''. Quantum Info. Comput. 12, 262–270 (2012). https://doi.org/10.26421/QIC12.3-4-6 [61] Vivek V. Shende, Stephen S. Bullock, and Igor L. Markov. ``Synthesis of Quantum Logic Circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). arXiv:quant-ph/0406176. https://doi.org/10.1109/TCAD.2005.855930 arXiv:quant-ph/0406176 [62] Aleks Kissinger and John van de Wetering. ``Pyzx: Large scale automated diagrammatic reasoning''. Electronic Proceedings in Theoretical Computer Science 318, 229–241 (2020). https://doi.org/10.4204/eptcs.318.14 [63] Ross Duncan, Aleks Kissinger, Simon Perdrix, and John van de Wetering. ``Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus''. Quantum 4, 279 (2020). https://doi.org/10.22331/q-2020-06-04-279 [64] https://github.com/zxcalc/pyzx. https://github.com/zxcalc/pyzx [65] Oscar Perdomo, Nelson Castaneda, and Roger Vogeler. ``Preparation of 3-qubit states'' (2022). arXiv:2201.03724. arXiv:2201.03724 [66] Yonghae Lee and Taewan Kim. ``Three-qubit-state preparation: Classification and explicit circuits''. Phys. Rev. A 113, 062444 (2026). https://doi.org/10.1103/5bw6-339b [67] Vadym Kliuchnikov, Kristin Lauter, Romy Minko, Adam Paetznick, and Christophe Petit. ``Shorter quantum circuits via single-qubit gate approximation''. Quantum 7, 1208 (2023). https://doi.org/10.22331/q-2023-12-18-1208 [68] Craig Gidney. ``Halving the cost of quantum addition''. Quantum 2, 74 (2018). https://doi.org/10.22331/q-2018-06-18-74 [69] Vivien Vandaele. ``Asymptotically optimal quantum circuits for comparators and incrementers'' (2026). arXiv:2603.12917. arXiv:2603.12917 [70] Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush. ``Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization''. Quantum 3, 208 (2019). https://doi.org/10.22331/q-2019-12-02-208 [71] Austin G. Fowler, Matteo Mariantoni, John M. Martinis, and Andrew N. Cleland. ``Surface codes: Towards practical large-scale quantum computation''. Phys. Rev. A 86, 032324 (2012). https://doi.org/10.1103/PhysRevA.86.032324 [72] Sungyeon Kook, Yujin Kang, Ilkwon Sohn, and Jun Heo. ``Low spatial cost CCZ magic state factory'' (2026). arXiv:2606.24170. arXiv:2606.24170 [73] Craig Gidney and Austin G. Fowler. ``Efficient magic state factories with a catalyzed $|CCZ\rangle$ to $2|T\rangle$ transformation''. Quantum 3, 135 (2019). https://doi.org/10.22331/q-2019-04-30-135 [74] Jeongwan Haah and Matthew B. Hastings. ``Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates''. Quantum 2, 71 (2018). https://doi.org/10.22331/q-2018-06-07-71 [75] Craig Gidney. ``How to factor 2048 bit RSA integers with less than a million noisy qubits'' (2025). arXiv:2505.15917. arXiv:2505.15917 [76] Guang Hao Low, William J. Huggins, Dominic W. Berry, Tanuj Khattar, Alec F. White, Nicholas C. Rubin, and Ryan Babbush. ``A denser planar surface code'' (2026). arXiv:2605.30455. arXiv:2605.30455 [77] Craig Gidney, Noah Shutty, and Cody Jones. ``Magic state cultivation: growing T states as cheap as CNOT gates'' (2024). arXiv:2409.17595. arXiv:2409.17595 [78] Jahan Claes. ``Cultivating T states on the surface code with only two-qubit gates'' (2025). arXiv:2509.05232. arXiv:2509.05232 [79] Emma Rosenfeld, Craig Gidney, Gabrielle Roberts, Alexis Morvan, Nathan Lacroix, Dvir Kafri, Jeffrey Marshall, Ming Li, Volodymyr Sivak, et al. ``Magic state cultivation on a superconducting quantum processor'' (2025). arXiv:2512.13908. arXiv:2512.13908 [80] Cody Jones. ``Low-overhead constructions for the fault-tolerant Toffoli gate''. Phys. Rev. A 87, 022328 (2013). https://doi.org/10.1103/PhysRevA.87.022328 [81] Vladyslav Bohun, Illia Lukin, Mykola Luhanko, Georgios Korpas, Philippe JS De Brouwer, Mykola Maksymenko, and Maciej Koch-Janusz. ``Entanglement scaling in matrix product state representation of smooth functions and their shallow quantum circuit approximations''. Phys. Rev. Res. 8, 023062 (2026). https://doi.org/10.1103/yyvr-dtsb [82] https://github.com/MQSdk/DPSS-QPE. https://github.com/MQSdk/DPSS-QPE [83] John Watrous. ``The theory of quantum information''.

Cambridge University Press. (2018). https://doi.org/10.1017/9781316848142Cited byCould not fetch Crossref cited-by data during last attempt 2026-08-19 07:09:19: Could not fetch cited-by data for 10.22331/q-2026-08-19-2193 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-08-19 07:09:20: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractQuantum phase estimation is an important routine in many quantum algorithms, particularly for estimating the ground state energy in quantum chemistry simulations. This estimation involves applying powers of a unitary to the ground state, controlled by an auxiliary state prepared on a control register. In many applications the goal is to provide a confidence interval for the phase estimate, and optimal performance is provided by a discrete prolate spheroidal sequence. We show how to prepare the corresponding state in a far more efficient way than prior work. We find that a matrix product state representation with a bond dimension of 4 is sufficient to give a highly accurate approximation for all dimensions tested, up to $2^{24}$. This matrix product state can be efficiently prepared using a sequence of simple three-qubit operations. When the dimension is a power of 2, the phase estimation can be performed with only three qubits for the control register, making it suitable for early-generation fault-tolerant quantum computers with a limited number of logical qubits.Featured image: Overview of the phase estimation procedure. A matrix product representation is used to prepare a DPSS state, which provides the phase estimate with optimal confidence interval. By utilizing mid-circuit measurements, no more than three qubits are required for the control register to achieve any desired precision. Popular summaryQuantum phase estimation (QPE) is an important and widely used quantum algorithm. The purpose of QPE is to perform a measurement on a quantum computer that determines the value of a phase, corresponding to the eigenvalue of a unitary operator. Like any estimation procedure, QPE's output comes with a confidence interval, a range in which the true phase is likely to lie. In QPE, the size of the confidence interval is determined by a control state prepared on an auxiliary register. In the textbook version of QPE, the proposed control state is a uniform superposition state, which is easy to prepare, but leaves a comparatively large likelihood of producing an estimate that is far from the true answer. In this paper, we discuss how to efficiently prepare a different control state; the discrete prolate spheroidal sequence (DPSS) state. It is well known that the DPSS state provides the optimal confidence interval, but the quantum circuits needed to prepare it have been far too costly. Our contribution is to provide efficient, explicit quantum circuits that prepare this state. We show how the circuit cost scales with the desired precision and demonstrate how mid-circuit measurements can be used to implement our method to any desired phase precision while allocating no more than three qubits to the control register. This produces a result not only optimal in a mathematical sense, but also practical for early-generation fault-tolerant quantum computers, where qubits are scarce.► BibTeX data@article{Kristjuhan2026quantumphase, doi = {10.22331/q-2026-08-19-2193}, url = {https://doi.org/10.22331/q-2026-08-19-2193}, title = {Quantum phase estimation with optimal confidence interval using three control qubits}, author = {Kristjuhan, Kaur and Berry, Dominic W.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2193}, month = aug, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Journal on Computing 26, 1484–1509 (1997). https://doi.org/10.1137/S0097539795293172 [2] Alán Aspuru-Guzik, Anthony D Dutoi, Peter J Love, and Martin Head-Gordon. ``Simulated quantum computation of molecular energies''. Science 309, 1704–1707 (2005). https://doi.org/10.1126/science.1113479 [3] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan. ``Quantum computational chemistry''. Rev. Mod. Phys. 92, 015003 (2020). https://doi.org/10.1103/RevModPhys.92.015003 [4] Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. ``Quantum chemistry in the age of quantum computing''. Chemical Reviews 119, 10856–10915 (2019). https://doi.org/10.1021/acs.chemrev.8b00803 [5] 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 [6] Mario Motta and Julia E Rice. ``Emerging quantum computing algorithms for quantum chemistry''.

Prentice Hall International Editions Series. Prentice Hall. (1999). https://dl.acm.org/doi/10.5555/294797 [56] G.H. Golub and C.F. Van Loan. ``Matrix computations''.

Johns Hopkins Studies in the Mathematical Sciences.

Read Original

Source Information

Source: Quantum Journal

Website: https://quantum-journal.org/feed/

Discussion

0 professional contributions

Be the first to add a constructive contribution.

Quantum phase estimation with optimal confidence interval using three control qubits

Tags

Source Information