quantum-computing

All You Need is pi: Quantum Computing with Hermitian Gates

Ben Zindorf and Sougato Bose

14 min read

0 likes

⚡ Quantum Brief

Researchers demonstrated that quantum computation can be achieved using only Hermitian gates—self-inverting operations equivalent to π-pulses—combined with CNOT gates, creating a fully universal gate set. Any single-qubit operation can be decomposed into just two Hermitian gates, enabling high-fidelity state preparation even with amplitude errors, while four Hermitian gates achieve optimal gate decomposition. The study shows π-rotations about two fixed axes plus CNOT form a universal set, simplifying circuit design since non-identity Hermitian gates act as π-rotations up to a global phase. Two π-rotations can reorient any multi-controlled unitary’s axis, reducing complexity; a single CNOT suffices for controlled π-rotations, cutting resource demands in constrained architectures. This approach streamlines circuit compilation, reduces CNOT counts (e.g., for Toffoli gates in linear nearest-neighbor qubits), and enables potential "clocked" quantum computers with uniform gate durations.

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

AbstractUniversal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to $\pi$ rotations up to a global phase. We show that a gate set comprised of $\pi$ rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two $\pi$ rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled $\pi$ rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.Featured image: Implementation of an arbitrary Bloch sphere rotation as two $\pi$-rotations.Popular summaryThis paper presents a fundamental simplification of quantum computing by identifying a novel universal gate set composed entirely of Hermitian gates—operations that are their own inverses. These are gates enacted on a qubit by electromagnetic $\pi$-pulses (qubit flips about any given axis). Experimentalists regularly use them for noise cancellations. This work shows that the entire complexity of quantum computation can be built from these $\pi$-pulses in combination with CNOT gates. This presents a number of advantages: the self-inverting nature of Hermitian gates inherently simplifies the complex task of circuit optimization and compilation. Furthermore, their $\pi$-pulse nature helps suppressing errors and lead directly to high-fidelity gate implementation. Looking futuristically, as all $\pi$-pulses, as well as the CNOT (which can also be realized as a conditional pi-pulse), can be made of equal duration, it can help to make a "clocked" quantum computer.► BibTeX data@article{Zindorf2025allyouneedispi, doi = {10.22331/q-2025-12-02-1925}, url = {https://doi.org/10.22331/q-2025-12-02-1925}, title = {All {Y}ou {N}eed is pi: {Q}uantum {C}omputing with {H}ermitian {G}ates}, author = {Zindorf, Ben and Bose, Sougato}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1925}, month = dec, year = {2025} }► References [1] Michael A. Nielsen and Isaac Chuang. ``Quantum Computation and Quantum Information''. American Journal of Physics 70, 558–559 (2002). https://doi.org/10.1119/1.1463744 [2] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth. ``Solid-state electronic spin coherence time approaching one second''. Nature Communications 4, 1743 (2013). https://doi.org/10.1038/ncomms2771 [3] Serge Haroche. ``Collge de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities'' (2012). Centre for Quantum Technologies, National University of Singapore. https://www.cqt.sg/highlight/2012-03-college-de-france-lectures-serge-haroche/. https://www.cqt.sg/highlight/2012-03-college-de-france-lectures-serge-haroche/ [4] Sami Husain, Minaru Kawamura, and Jonathan A. Jones. ``Further analysis of some symmetric and antisymmetric composite pulses for tackling pulse strength errors''. Journal of Magnetic Resonance 230, 145–154 (2013). https://doi.org/10.1016/j.jmr.2013.02.007 [5] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. ``Optimal arbitrarily accurate composite pulse sequences''. Physical Review A 89, 022341 (2014). https://doi.org/10.1103/PhysRevA.89.022341 [6] R. Tycko, A. Pines, and J. Guckenheimer. ``Fixed point theory of iterative excitation schemes in NMR''. The Journal of Chemical Physics 83, 2775–2802 (1985). https://doi.org/10.1063/1.449228 [7] Jonathan A. Jones. ``Nested composite NOT gates for quantum computation''. Physics Letters A 377, 2860–2862 (2013). https://doi.org/10.1016/j.physleta.2013.08.040 [8] Lorenza Viola, Emanuel Knill, and Seth Lloyd. ``Dynamical Decoupling of Open Quantum Systems''.

Physical Review Letters 82, 2417–2421 (1999). https://doi.org/10.1103/PhysRevLett.82.2417 [9] Götz S. Uhrig. ``Keeping a Quantum Bit Alive by Optimized pi -Pulse Sequences''.

Physical Review Letters 98, 100504 (2007). https://doi.org/10.1103/PhysRevLett.98.100504 [10] Alexandre M. Souza, Gonzalo A. Álvarez, and Dieter Suter. ``Robust Dynamical Decoupling for Quantum Computing and Quantum Memory''.

Physical Review Letters 106, 240501 (2011). https://doi.org/10.1103/PhysRevLett.106.240501 [11] Alexandre M. Souza, Gonzalo A. Álvarez, and Dieter Suter. ``Robust dynamical decoupling''. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, 4748–4769 (2012). https://doi.org/10.1098/rsta.2011.0355 [12] Nikita A. Nemkov, Evgeniy O. Kiktenko, Ilia A. Luchnikov, and Aleksey K. Fedorov. ``Efficient variational synthesis of quantum circuits with coherent multi-start optimization''. Quantum 7, 993 (2023). https://doi.org/10.22331/q-2023-05-04-993 [13] Alston S. Householder. ``Unitary Triangularization of a Nonsymmetric Matrix''. Journal of the ACM 5, 339–342 (1958). https://doi.org/10.1145/320941.320947 [14] Adriano Barenco, Charles H. Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter. ``Elementary gates for quantum computation''. Physical Review A 52, 3457–3467 (1995). https://doi.org/10.1103/PhysRevA.52.3457 [15] Danail Brezov, Clementina Mladenova, and Ivaïlo Mladenov. ``Vector Decompositions of Rotations''. Journal of Geometry and Symmetry in Physics 28, 67–103 (2012). https://doi.org/10.7546/jgsp-28-2012-67-103 [16] V. D. Donchev, C. D. Mladenova, and I. M. Mladenov. ``On the compositions of rotations''. AIP Conference Proceedings 1684, 080004 (2015). https://doi.org/10.1063/1.4934315 [17] Sergey Bravyi and Alexei Kitaev. ``Universal quantum computation with ideal Clifford gates and noisy ancillas''. Physical Review A 71, 022316 (2005). https://doi.org/10.1103/PhysRevA.71.022316 [18] Ben Zindorf and Sougato Bose. ``Multi-Controlled Quantum Gates in Linear Nearest Neighbor'' (2025). arXiv:2506.00695. https://doi.org/10.48550/arXiv.2506.00695 arXiv:2506.00695 [19] Stephen Wimperis. ``Iterative schemes for phase-distortionless composite 180° pulses''. Journal of Magnetic Resonance (1969) 93, 199–206 (1991). https://doi.org/10.1016/0022-2364(91)90043-S [20] S. Wimperis. ``Broadband, Narrowband, and Passband Composite Pulses for Use in Advanced NMR Experiments''. Journal of Magnetic Resonance, Series A 109, 221–231 (1994). https://doi.org/10.1006/jmra.1994.1159 [21] Hayk L. Gevorgyan and Nikolay V. Vitanov. ``Ultrahigh-fidelity composite rotational quantum gates''. Physical Review A 104, 012609 (2021). https://doi.org/10.1103/PhysRevA.104.012609 [22] Yun-Pil Shim, Jianjia Fei, Sangchul Oh, Xuedong Hu, and Mark Friesen. ``Single-qubit gates in two steps with rotation axes in a single plane'' (2013). arXiv:1303.0297. https://doi.org/10.48550/arXiv.1303.0297 arXiv:1303.0297 [23] Rafaella Vale, Thiago Melo D. Azevedo, Ismael C. S. Araújo, Israel F. Araujo, and Adenilton J. da Silva. ``Circuit Decomposition of Multicontrolled Special Unitary Single-Qubit Gates''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 43, 802–811 (2024). https://doi.org/10.1109/TCAD.2023.3327102 [24] Evandro C. R. Rosa, Eduardo I. Duzzioni, and Rafael de Santiago. ``Optimizing Gate Decomposition for High-Level Quantum Programming''. Quantum 9, 1659 (2025). https://doi.org/10.22331/q-2025-03-12-1659 [25] Ben Zindorf and Sougato Bose. ``Efficient implementation of multicontrolled quantum gates''.

Physical Review Applied 24, 044030 (2025). https://doi.org/10.1103/8blx-nfcr [26] Jan Gwinner, Marcin Briański, Wojciech Burkot, Lukasz Czerwiński, and Vladyslav Hlembotskyi. ``Benchmarking 16-element quantum search algorithms on superconducting quantum processors'' (2021). arXiv:2007.06539. https://doi.org/10.48550/arXiv.2007.06539 arXiv:2007.06539 [27] Ken M. Nakanishi, Takahiko Satoh, and Synge Todo. ``Quantum-gate decomposer'' (2021). arXiv:2109.13223. https://doi.org/10.48550/arXiv.2109.13223 arXiv:2109.13223 [28] Ken M. Nakanishi, Takahiko Satoh, and Synge Todo. ``Decompositions of multiple controlled- Z gates on various qubit-coupling graphs''. Physical Review A 110, 012604 (2024). https://doi.org/10.1103/PhysRevA.110.012604 [29] Pedro M. Q. Cruz and Bruno Murta. ``Shallow unitary decompositions of quantum Fredkin and Toffoli gates for connectivity-aware equivalent circuit averaging''. APL Quantum 1, 016105 (2024). https://doi.org/10.1063/5.0187026 [30] Dmitri Maslov and Ben Zindorf. ``Depth Optimization of CZ, CNOT, and Clifford Circuits''. IEEE Transactions on Quantum Engineering 3, 1–8 (2022). https://doi.org/10.1109/TQE.2022.3180900 [31] Carolina Allende, André Fonseca de Olivera, and Efrain Buksman. ``Synthesis of quantum circuits based on supervised learning and correlations''.

Quantum Information Processing 23, 204 (2024). https://doi.org/10.1007/s11128-024-04426-6 [32] V.V. Shende, S.S. Bullock, and I.L. Markov. ``Synthesis of quantum-logic circuits''. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 25, 1000–1010 (2006). https://doi.org/10.1109/TCAD.2005.855930 [33] Vivek V. Shende, Igor L. Markov, and Stephen S. Bullock. ``Minimal universal two-qubit controlled-NOT-based circuits''. Physical Review A 69, 062321 (2004). https://doi.org/10.1103/PhysRevA.69.062321 [34] Farrokh Vatan and Colin Williams. ``Optimal Quantum Circuits for General Two-Qubit Gates''. Physical Review A 69, 032315 (2004). https://doi.org/10.1103/PhysRevA.69.032315 [35] G. Vidal and C. M. Dawson. ``Universal quantum circuit for two-qubit transformations with three controlled-NOT gates''. Physical Review A 69, 010301 (2004). https://doi.org/10.1103/PhysRevA.69.010301Cited byCould not fetch Crossref cited-by data during last attempt 2025-12-02 14:47:39: Could not fetch cited-by data for 10.22331/q-2025-12-02-1925 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-12-02 14:47:39: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractUniversal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to $\pi$ rotations up to a global phase. We show that a gate set comprised of $\pi$ rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two $\pi$ rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled $\pi$ rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.Featured image: Implementation of an arbitrary Bloch sphere rotation as two $\pi$-rotations.Popular summaryThis paper presents a fundamental simplification of quantum computing by identifying a novel universal gate set composed entirely of Hermitian gates—operations that are their own inverses. These are gates enacted on a qubit by electromagnetic $\pi$-pulses (qubit flips about any given axis). Experimentalists regularly use them for noise cancellations. This work shows that the entire complexity of quantum computation can be built from these $\pi$-pulses in combination with CNOT gates. This presents a number of advantages: the self-inverting nature of Hermitian gates inherently simplifies the complex task of circuit optimization and compilation. Furthermore, their $\pi$-pulse nature helps suppressing errors and lead directly to high-fidelity gate implementation. Looking futuristically, as all $\pi$-pulses, as well as the CNOT (which can also be realized as a conditional pi-pulse), can be made of equal duration, it can help to make a "clocked" quantum computer.► BibTeX data@article{Zindorf2025allyouneedispi, doi = {10.22331/q-2025-12-02-1925}, url = {https://doi.org/10.22331/q-2025-12-02-1925}, title = {All {Y}ou {N}eed is pi: {Q}uantum {C}omputing with {H}ermitian {G}ates}, author = {Zindorf, Ben and Bose, Sougato}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1925}, month = dec, year = {2025} }► References [1] Michael A. Nielsen and Isaac Chuang. ``Quantum Computation and Quantum Information''. American Journal of Physics 70, 558–559 (2002). https://doi.org/10.1119/1.1463744 [2] N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth. ``Solid-state electronic spin coherence time approaching one second''. Nature Communications 4, 1743 (2013). https://doi.org/10.1038/ncomms2771 [3] Serge Haroche. ``Collge de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities'' (2012). Centre for Quantum Technologies, National University of Singapore. https://www.cqt.sg/highlight/2012-03-college-de-france-lectures-serge-haroche/. https://www.cqt.sg/highlight/2012-03-college-de-france-lectures-serge-haroche/ [4] Sami Husain, Minaru Kawamura, and Jonathan A. Jones. ``Further analysis of some symmetric and antisymmetric composite pulses for tackling pulse strength errors''. Journal of Magnetic Resonance 230, 145–154 (2013). https://doi.org/10.1016/j.jmr.2013.02.007 [5] Guang Hao Low, Theodore J. Yoder, and Isaac L. Chuang. ``Optimal arbitrarily accurate composite pulse sequences''. Physical Review A 89, 022341 (2014). https://doi.org/10.1103/PhysRevA.89.022341 [6] R. Tycko, A. Pines, and J. Guckenheimer. ``Fixed point theory of iterative excitation schemes in NMR''. The Journal of Chemical Physics 83, 2775–2802 (1985). https://doi.org/10.1063/1.449228 [7] Jonathan A. Jones. ``Nested composite NOT gates for quantum computation''. Physics Letters A 377, 2860–2862 (2013). https://doi.org/10.1016/j.physleta.2013.08.040 [8] Lorenza Viola, Emanuel Knill, and Seth Lloyd. ``Dynamical Decoupling of Open Quantum Systems''.

Physical Review Letters 82, 2417–2421 (1999). https://doi.org/10.1103/PhysRevLett.82.2417 [9] Götz S. Uhrig. ``Keeping a Quantum Bit Alive by Optimized pi -Pulse Sequences''.

Read Original

Source Information

Source: Quantum Science and Technology (arXiv overlay)

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

Discussion

0 professional contributions

Be the first to add a constructive contribution.

All You Need is pi: Quantum Computing with Hermitian Gates

Tags

Source Information