Abstract

Quantum computation exploits superposition, entanglement, interference, and measurement to process information. Research has moved from theoretical algorithms to functional devices where the performance is not only limited by qubit physics, but also by the control, compilation, error mitigation, and error correction. This review considers quantum computing on various levels of abstraction, including physical qubits, quantum algorithms, system architecture, fault-tolerance, and emerging applications. Study-specific evidence regarding scale, operational quality, connectivity, control and suppression of errors is compared across the different types of superconducting circuits, trapped ions, semiconductor spin qubits, neutral atoms and photonic processors. The review of the fundamental algorithms, noisy intermediate-scale quantum computing, variational and hybrid quantum-classical approaches, quantum simulation, quantum machine learning, and security implications is also discussed. Measurable progress has been demonstrated in published experiments for logical-qubit operation, below-threshold surface-code memory, programmable photonic sampling and platform-specific scaling. However, the reported metrics are not directly interchangeable, and practical advantage remains task- and baseline-dependent. Further progress will need to be accompanied by coordinated developments in hardware reliability, error correction, classical control, estimates of algorithmic resources, and clear classical method comparisons.

References

• Y. Alexeev et al., "Quantum computer systems for scientific discovery," PRX Quantum, vol. 2, no. 1, Art. no. 017001, Feb. 2021, doi: 10.1103/PRXQuantum.2.017001. [Online]. Available: https://doi.org/10.1103/PRXQuantum.2.017001
• D. P. DiVincenzo, "The physical implementation of quantum computation," Fortschritte der Physik, vol. 48, no. 9-11, pp. 771-783, 2000, doi: 10.1002/1521-3978(200009)48:9/11<771::AID-PROP771>3.0.CO;2-E. [Online]. Available: https://doi.org/10.1002/1521-3978(200009)48:9/11%3C771::AID-PROP771%3E3.0.CO;2-E
• R. Acharya et al., "Quantum error correction below the surface code threshold," Nature, vol. 638, no. 8052, pp. 920-926, Feb. 2025, doi: 10.1038/s41586-024-08449-y. [Online]. Available: https://doi.org/10.1038/s41586-024-08449-y
• K. Bharti et al., "Noisy intermediate-scale quantum algorithms," Reviews of Modern Physics, vol. 94, no. 1, Art. no. 015004, Feb. 2022, doi: 10.1103/RevModPhys.94.015004. [Online]. Available: https://doi.org/10.1103/RevModPhys.94.015004
• R. Acharya et al., "Suppressing quantum errors by scaling a surface code logical qubit," Nature, vol. 614, pp. 676-681, Feb. 2023, doi: 10.1038/s41586-022-05434-1. [Online]. Available: https://doi.org/10.1038/s41586-022-05434-1
• Y. Wu et al., "Strong quantum computational advantage using a superconducting quantum processor," Physical Review Letters, vol. 127, no. 18, Art. no. 180501, Oct. 2021, doi: 10.1103/PhysRevLett.127.180501. [Online]. Available: https://doi.org/10.1103/PhysRevLett.127.180501
• L. Egan et al., "Fault-tolerant control of an error-corrected qubit," Nature, vol. 598, pp. 281-286, Oct. 2021, doi: 10.1038/s41586-021-03928-y. [Online]. Available: https://doi.org/10.1038/s41586-021-03928-y
• S. A. Moses et al., "A race-track trapped-ion quantum processor," Physical Review X, vol. 13, no. 4, Art. no. 041052, Dec. 2023, doi: 10.1103/PhysRevX.13.041052. [Online]. Available: https://doi.org/10.1103/PhysRevX.13.041052
• G. Burkard, T. D. Ladd, J. M. Nichol, A. Pan, and J. R. Petta, "Semiconductor spin qubits," Reviews of Modern Physics, vol. 95, no. 2, Art. no. 025003, Jun. 2023, doi: 10.1103/RevModPhys.95.025003. [Online]. Available: https://doi.org/10.1103/RevModPhys.95.025003
• S. G. J. Philips et al., "Universal control of a six-qubit quantum processor in silicon," Nature, vol. 609, pp. 919-924, Sep. 2022, doi: 10.1038/s41586-022-05117-x. [Online]. Available: https://doi.org/10.1038/s41586-022-05117-x
• X. Xue et al., "Quantum logic with spin qubits crossing the surface code threshold," Nature, vol. 601, pp. 343-347, Jan. 2022, doi: 10.1038/s41586-021-04273-w. [Online]. Available: https://doi.org/10.1038/s41586-021-04273-w
• D. Bluvstein et al., "Logical quantum processor based on reconfigurable atom arrays," Nature, vol. 626, pp. 58-65, Feb. 2024, doi: 10.1038/s41586-023-06927-3. [Online]. Available: https://doi.org/10.1038/s41586-023-06927-3
• L. S. Madsen et al., "Quantum computational advantage with a programmable photonic processor," Nature, vol. 606, pp. 75-81, Jun. 2022, doi: 10.1038/s41586-022-04725-x. [Online]. Available: https://doi.org/10.1038/s41586-022-04725-x
• J. Preskill, "Quantum computing in the NISQ era and beyond," Quantum, vol. 2, Art. no. 79, Aug. 2018, doi: 10.22331/q-2018-08-06-79. [Online]. Available: https://doi.org/10.22331/q-2018-08-06-79
• P. W. Shor, "Polynomial-time algorithms for prime factorisation and discrete logarithms on a quantum computer," SIAM Journal on Computing, vol. 26, no. 5, pp. 1484-1509, Oct. 1997, doi: 10.1137/S0097539795293172. [Online]. Available: https://doi.org/10.1137/S0097539795293172
• L. K. Grover, "A fast quantum mechanical algorithm for database search," in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 212-219, doi: 10.1145/237814.237866. [Online]. Available: https://doi.org/10.1145/237814.237866
• S. Lloyd, "Universal quantum simulators," Science, vol. 273, no. 5278, pp. 1073-1078, Aug. 1996, doi: 10.1126/science.273.5278.1073. [Online]. Available: https://doi.org/10.1126/science.273.5278.1073
• M. Cerezo et al., "Variational quantum algorithms," Nature Reviews Physics, vol. 3, pp. 625-644, Aug. 2021, doi: 10.1038/s42254-021-00348-9. [Online]. Available: https://doi.org/10.1038/s42254-021-00348-9
• D. Claudino, "The basics of quantum computing for chemists," International Journal of Quantum Chemistry, vol. 122, no. 23, Art. no. e26990, 2022, doi: 10.1002/qua.26990. [Online]. Available: https://doi.org/10.1002/qua.26990
• M. Schuld and N. Killoran, "Is quantum advantage the right goal for quantum machine learning?" PRX Quantum, vol. 3, no. 3, Art. no. 030101, Jul. 2022, doi: 10.1103/PRXQuantum.3.030101. [Online]. Available: https://doi.org/10.1103/PRXQuantum.3.030101
• A. Melnikov, M. Kordzanganeh, A. Alodjants, and R.-K. Lee, "Quantum machine learning: From physics to software engineering," Advances in Physics: X, vol. 8, no. 1, Art. no. 2165452, 2023, doi: 10.1080/23746149.2023.2165452. [Online]. Available: https://doi.org/10.1080/23746149.2023.2165452
• S. Krinner et al., "Realising repeated quantum error correction in a distance-three surface code," Nature, vol. 605, pp. 669-674, May 2022, doi: 10.1038/s41586-022-04566-8. [Online]. Available: https://doi.org/10.1038/s41586-022-04566-8
• V. V. Sivak et al., "Real-time quantum error correction beyond break-even," Nature, vol. 616, pp. 50-55, Apr. 2023, doi: 10.1038/s41586-023-05782-6. [Online]. Available: https://doi.org/10.1038/s41586-023-05782-6
• B. L. Brock et al., "Quantum error correction of qudits beyond break-even," Nature, 2025, doi: 10.1038/s41586-025-08899-y. [Online]. Available: https://doi.org/10.1038/s41586-025-08899-y
• C. Gidney and M. Ekerå, "How to factor 2048-bit RSA integers in 8 hours using 20 million noisy qubits," Quantum, vol. 5, Art. no. 433, Apr. 2021, doi: 10.22331/q-2021-04-15-433. [Online]. Available: https://doi.org/10.22331/q-2021-04-15-433
• D. Gao et al., "Establishing a new benchmark in quantum computational advantage with a 105-qubit Zuchongzhi 3.0 processor," Physical Review Letters, vol. 134, no. 9, Art. no. 090601, Mar. 2025, doi: 10.1103/PhysRevLett.134.090601. [Online]. Available: https://doi.org/10.1103/PhysRevLett.134.090601
• National Institute of Standards and Technology, Module-Lattice-Based Key-Encapsulation Mechanism Standard, FIPS 203. Gaithersburg, MD, USA: U.S. Department of Commerce, 2024, doi: 10.6028/NIST.FIPS.203. [Online]. Available: https://doi.org/10.6028/NIST.FIPS.203
• National Institute of Standards and Technology, Module-Lattice-Based Digital Signature Standard, FIPS 204. Gaithersburg, MD, USA: U.S. Department of Commerce, 2024, doi: 10.6028/NIST.FIPS.204. [Online]. Available: https://doi.org/10.6028/NIST.FIPS.204
• National Institute of Standards and Technology, Stateless Hash-Based Digital Signature Standard, FIPS 205. Gaithersburg, MD, USA: U.S. Department of Commerce, 2024, doi: 10.6028/NIST.FIPS.205. [Online]. Available: https://doi.org/10.6028/NIST.FIPS.205