quantum-computing

Resourcefulness of non-classical continuous-variable quantum gates

Quantum Science and Technology (arXiv overlay)

24 min read

0 likes

⚡ Quantum Brief

AI Audio Summary

0:00 / 0:00

Click to play

Resourcefulness of non-classical continuous-variable quantum gates

Understand this faster with AI

Ask ChatGPT

Research with Perplexity

Explain with Claude

Discuss on Grok

AbstractIn continuous-variable quantum computation, identifying key elements that enable a quantum computational advantage is a long-standing issue. Starting from the standard results on the necessity of Wigner negativity, we develop a comprehensive and versatile approach in which the techniques of $(s)$-ordered quasiprobabilities are exploited to provide rigorous statements on the simulability of photonic quantum circuits consisting of previously characterized gates and thereby identifying the contribution of each quantum gate to the potential achievement of quantum computational advantage. This is achieved by means of an analysis of the so-called transfer function, allowing us to highlight the resourcefulness of a gate set. As such this technique can be straightforwardly applied to current continuous-variables quantum circuits, while also constraining the tolerable amount of losses above which any potential quantum advantage can be ruled out. We use $(s)$-ordered quasiprobability distributions on phase-space to capture the non-classical features in the protocol, and focus our technique entirely on the ordering parameter $s$. This allows us to highlight the resourcefulness and robustness to loss of a universal set of unitary gates comprising three distinct Gaussian gates and any non-Gaussian unitary gate, providing important insight on the role of non-Gaussianity.Featured image: Generic quantum-optical scheme depicted by $M$ input modes, described by a density operator $\rho_{\mathrm{in}}$ processed through a trace-preserving quantum channel $\mathcal{E}$, that can be decomposed into a sequence of trace-preserving quantum channels $\mathcal{E} = \mathcal{E}_1 \circ \mathcal{E}_2 \circ \dots \circ \mathcal{E}_k$. This produces the output state $\rho_{out} = \mathcal{E}(\rho_{in})$ and an output probability distribution $p({x}) = Tr[\rho_{out}\Pi_{{x}}]$ sampled by measuring the POVM $\Pi_{{x}}$.Popular summaryAmong the different platforms being explored for quantum computation, photonic systems are especially appealing. Photons can be manipulated with well-established optical components, can be used to generate entanglement, and are relatively robust against decoherence compared with many matter-based qubits, although they remain sensitive to losses. A central question is to understand which features of a photonic quantum circuit are genuinely responsible for a possible quantum advantage over classical computers. It has long been known that purely Gaussian optical ingredients, namely Gaussian states, Gaussian operations, and Gaussian measurements, can be efficiently simulated classically. Some form of non-Gaussianity, often associated with Wigner negativity, is therefore needed. However, Wigner negativity alone does not fully answer the question: it tells us that a circuit may contain a useful quantum resource, but not whether this resource is strong enough, placed in the right part of the circuit, or robust enough against experimental noise. In this work, we develop a general method to analyze photonic quantum circuits gate by gate. The idea is to represent each input state, optical operation, and measurement through a family of phase-space functions, and to ask whether the action of each gate can still be described using ordinary, non-negative probability distributions. When this is possible, the corresponding part of the circuit can be reproduced by classical sampling, much like in previous simulation methods based on positive Wigner functions. The main advantage of this approach is that it identifies the resourcefulness of each gate independently of the detailed input state. The relevant parameter is the nonclassical depth, which quantifies how strongly nonclassical the incoming state must be before a given gate can no longer be simulated in this classical way. This provides a sharper criterion than simply asking whether Wigner negativity is present somewhere in the circuit. The method also provides insights about robustness of a possible quantum advantage with respect to losses. For a given optical gate, it can determine how much loss is enough to wash out its useful nonclassical behavior and make that part of the computation classically simulable. In this sense, the results do not only say whether a circuit is potentially powerful, but also which components are most fragile and what experimental quality they must reach in order to remain computationally relevant. We apply this framework to important gates used in continuous-variable photonic quantum computation. For Gaussian gates such as squeezers and beam splitters, the method shows how nonclassicality is transformed, mixed between modes, or degraded by losses. For realistic photon subtraction, it clarifies when the operation can become genuinely resourceful, depending on the amount of nonclassicality (e.g. squeezing, in this instance) available at the input. Finally, the analysis is extended to non-Gaussian unitary gates, including the cubic phase gate, which is a key ingredient for universal continuous-variable quantum computation. Overall, the work provides a systematic way to draw the boundary between photonic circuits that can still be efficiently simulated on a classical computer and those that may contain the resources needed for a true quantum advantage. Importantly, this boundary can exclude quantum advantage even in some circuits where Wigner negativity is present, showing that the mere presence of non-Gaussianity is not always enough: what matters is how much nonclassicality is available, where it appears in the circuit, and whether it survives realistic losses.► BibTeX data@article{Frigerio2026resourcefulnessof, doi = {10.22331/q-2026-07-08-2155}, url = {https://doi.org/10.22331/q-2026-07-08-2155}, title = {Resourcefulness of non-classical continuous-variable quantum gates}, author = {Frigerio, Massimo and Debray, Antoine and Treps, Nicolas and Walschaers, Mattia}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2155}, month = jul, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Review 41, 303–332 (1999). https://doi.org/10.1137/S0036144598347011 [2] 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 [3] Scott Aaronson and Alex Arkhipov. ``The computational complexity of linear optics''. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing. Page 333–342. STOC '11New York, NY, USA (2011). Association for Computing Machinery. https://doi.org/10.1145/1993636.1993682 [4] Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. ``Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, 459–472 (2011). https://doi.org/10.1098/rspa.2010.0301 [5] Sergio Boixo, Sergei V. Isakov, Vadim N. Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J. Bremner, John M. Martinis, and Hartmut Neven. ``Characterizing quantum supremacy in near-term devices''. Nature Physics 14, 595–600 (2018). https://doi.org/10.1038/s41567-018-0124-x [6] Aram W Harrow and Ashley Montanaro. ``Quantum computational supremacy''. Nature 549, 203–209 (2017). https://doi.org/10.1038/nature23458 [7] Dominik Hangleiter and Jens Eisert. ``Computational advantage of quantum random sampling''. Rev. Mod. Phys. 95, 035001 (2023). https://doi.org/10.1103/RevModPhys.95.035001 [8] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [9] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, et al. ``Quantum computational advantage using photons''. Science 370, 1460–1463 (2020). https://doi.org/10.1126/science.abe8770 [10] Lars S. Madsen, Fabian Laudenbach, Mohsen Falamarzi. Askarani, Fabien Rortais, Trevor Vincent, Jacob F. F. Bulmer, Filippo M. Miatto, Leonhard Neuhaus, Helt, et al. ``Quantum computational advantage with a programmable photonic processor''. Nature 606, 75–81 (2022). https://doi.org/10.1038/s41586-022-04725-x [11] M. DeCross, R. Haghshenas, M. Liu, E. Rinaldi, J. Gray, Y. Alexeev, C. H. Baldwin, J. P. Bartolotta, M. Bohn, E. Chertkov, J. Cline, J. Colina, D. DelVento, J. M. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, D. Gresh, A. Hall, A. Hankin, A. Hansen, N. Hewitt, I. Hoffman, C. Holliman, R. B. Hutson, T. Jacobs, J. Johansen, P. J. Lee, E. Lehman, D. Lucchetti, D. Lykov, I. S. Madjarov, B. Mathewson, K. Mayer, M. Mills, P. Niroula, J. M. Pino, C. Roman, M. Schecter, P. E. Siegfried, B. G. Tiemann, C. Volin, J. Walker, R. Shaydulin, M. Pistoia, S. A. Moses, D. Hayes, B. Neyenhuis, R. P. Stutz, and M. Foss-Feig. ``Computational power of random quantum circuits in arbitrary geometries''. Phys. Rev. X 15, 021052 (2025). https://doi.org/10.1103/PhysRevX.15.021052 [12] Craig S. Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. ``Gaussian boson sampling''. Phys. Rev. Lett. 119, 170501 (2017). https://doi.org/10.1103/PhysRevLett.119.170501 [13] A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O'Brien, and T. C. Ralph. ``Boson sampling from a gaussian state''. Phys. Rev. Lett. 113, 100502 (2014). https://doi.org/10.1103/PhysRevLett.113.100502 [14] U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi, and G. Ferrini. ``Continuous-variable sampling from photon-added or photon-subtracted squeezed states''. Phys. Rev. A 96, 062307 (2017). https://doi.org/10.1103/PhysRevA.96.062307 [15] L. Chakhmakhchyan and N. J. Cerf. ``Boson sampling with gaussian measurements''. Phys. Rev. A 96, 032326 (2017). https://doi.org/10.1103/PhysRevA.96.032326 [16] Nicolás Quesada, Juan Miguel Arrazola, and Nathan Killoran. ``Gaussian boson sampling using threshold detectors''. Physical Review A 98, 062322 (2018). https://doi.org/10.1103/PhysRevA.98.062322 [17] Raúl García-Patrón, Jelmer J Renema, and Valery Shchesnovich. ``Simulating boson sampling in lossy architectures''. Quantum 3, 169 (2019). https://doi.org/10.22331/q-2019-08-05-169 [18] Jelmer J Renema. ``Simulability of partially distinguishable superposition and gaussian boson sampling''. Physical Review A 101, 063840 (2020). https://doi.org/10.1103/PhysRevA.101.063840 [19] J. J. Renema, A. Menssen, W. R. Clements, G. Triginer, W. S. Kolthammer, and I. A. Walmsley. ``Efficient classical algorithm for boson sampling with partially distinguishable photons''. Phys. Rev. Lett. 120, 220502 (2018). https://doi.org/10.1103/PhysRevLett.120.220502 [20] Saleh Rahimi-Keshari, Timothy C Ralph, and Carlton M Caves. ``Sufficient conditions for efficient classical simulation of quantum optics''. Physical Review X 6, 021039 (2016). https://doi.org/10.1103/PhysRevX.6.021039 [21] Jacob F. F. Bulmer, Bryn A. Bell, Rachel S. Chadwick, Alex E. Jones, Diana Moise, Alessandro Rigazzi, Jan Thorbecke, Utz-Uwe Haus, Thomas Van Vaerenbergh, Raj B. Patel, Ian A. Walmsley, and Anthony Laing. ``The boundary for quantum advantage in gaussian boson sampling''. Science Advances 8, eabl9236 (2022). https://doi.org/10.1126/sciadv.abl9236 [22] Haoyu Qi, Daniel J. Brod, Nicolás Quesada, and Raúl García-Patrón. ``Regimes of classical simulability for noisy gaussian boson sampling''. Phys. Rev. Lett. 124, 100502 (2020). https://doi.org/10.1103/PhysRevLett.124.100502 [23] Nicolás Quesada, Rachel S. Chadwick, Bryn A. Bell, Juan Miguel Arrazola, Trevor Vincent, Haoyu Qi, and Raúl García-Patrón. ``Quadratic speed-up for simulating gaussian boson sampling''. PRX Quantum 3, 010306 (2022). https://doi.org/10.1103/PRXQuantum.3.010306 [24] Peter D. Drummond, Bogdan Opanchuk, A. Dellios, and M. D. Reid. ``Simulating complex networks in phase space: Gaussian boson sampling''. Phys. Rev. A 105, 012427 (2022). https://doi.org/10.1103/PhysRevA.105.012427 [25] Changhun Oh, Minzhao Liu, Yuri Alexeev, Bill Fefferman, and Liang Jiang. ``Classical algorithm for simulating experimental gaussian boson sampling''. Nature Physics (2024). https://doi.org/10.1038/s41567-024-02535-8 [26] Daniel Gottesman. ``The heisenberg representation of quantum computers'' (1998). arXiv:quant-ph/9807006. arXiv:quant-ph/9807006 [27] Stephen D. Bartlett, Barry C. Sanders, Samuel L. Braunstein, and Kae Nemoto. ``Efficient classical simulation of continuous variable quantum information processes''.

Physical Review Letters 88, 097904 (2002). https://doi.org/10.1103/PhysRevLett.88.097904 [28] Robin L Hudson. ``When is the Wigner quasi-probability density non-negative?''. Reports on Mathematical Physics 6, 249–252 (1974). https://doi.org/10.1016/0034-4877(74)90007-X [29] Francisco Soto and Pierre Claverie. ``When is the Wigner function of multidimensional systems nonnegative?''. Journal of Mathematical Physics 24, 97–100 (1983). https://doi.org/10.1063/1.525607 [30] Kevin E Cahill and Roy J Glauber. ``Density operators and quasiprobability distributions''. Physical Review 177, 1882 (1969). https://doi.org/10.1103/PhysRev.177.1882 [31] Andrea Mari and Jens Eisert. ``Positive wigner functions render classical simulation of quantum computation efficient''. Physical review letters 109, 230503 (2012). https://doi.org/10.1103/PhysRevLett.109.230503 [32] Victor Veitch, Christopher Ferrie, David Gross, and Joseph Emerson. ``Negative quasi-probability as a resource for quantum computation''. New Journal of Physics 14, 113011 (2012). https://doi.org/10.1088/1367-2630/14/11/113011 [33] Laura García-Álvarez, Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini. ``Efficient simulatability of continuous-variable circuits with large Wigner negativity''.

Physical Review Research 2, 043322 (2020). https://doi.org/10.1103/PhysRevResearch.2.043322 [34] Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini. ``Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits''. Quantum 6, 867 (2022). https://doi.org/10.22331/q-2022-12-01-867 [35] Cameron Calcluth, Alessandro Ferraro, and Giulia Ferrini. ``Vacuum provides quantum advantage to otherwise simulatable architectures''. Phys. Rev. A 107, 062414 (2023). https://doi.org/10.1103/PhysRevA.107.062414 [36] Cameron Calcluth, Nicolas Reichel, Alessandro Ferraro, and Giulia Ferrini. ``Sufficient condition for universal quantum computation using bosonic circuits''. PRX Quantum 5, 020337 (2024). https://doi.org/10.1103/PRXQuantum.5.020337 [37] Ulysse Chabaud and Mattia Walschaers. ``Resources for bosonic quantum computational advantage''. Phys. Rev. Lett. 130, 090602 (2023). https://doi.org/10.1103/PhysRevLett.130.090602 [38] Ulysse Chabaud, Roohollah Ghobadi, Salman Beigi, and Saleh Rahimi-Keshari. ``Phase-space negativity as a computational resource for quantum kernel methods'' (2024). arXiv:2405.12378. https://doi.org/10.22331/q-2024-11-07-1519 arXiv:2405.12378 [39] Kevin E Cahill and Roy J Glauber. ``Ordered expansions in boson amplitude operators''. Physical Review 177, 1857 (1969). https://doi.org/10.1103/PhysRev.177.1857 [40] N. Lütkenhaus and Stephen M. Barnett. ``Nonclassical effects in phase space''. Phys. Rev. A 51, 3340–3342 (1995). https://doi.org/10.1103/PhysRevA.51.3340 [41] Ching Tsung Lee. ``Measure of the nonclassicality of nonclassical states''. Physical Review A 44, R2775 (1991). https://doi.org/10.1103/PhysRevA.44.R2775 [42] Frigerio, Massimo and Debray, Antoine. ``(s)-QPDs optimization tool''. https://github.com/EQ15T/Resourcefulness-s-QPDs. https://github.com/EQ15T/Resourcefulness-s-QPDs. [43] Seth Lloyd and Samuel L. Braunstein. ``Quantum computation over continuous variables''. Phys. Rev. Lett. 82, 1784–1787 (1999). https://doi.org/10.1103/PhysRevLett.82.1784 [44] Timo Hillmann, Fernando Quijandría, Göran Johansson, Alessandro Ferraro, Simone Gasparinetti, and Giulia Ferrini. ``Universal gate set for continuous-variable quantum computation with microwave circuits''. Phys. Rev. Lett. 125, 160501 (2020). https://doi.org/10.1103/PhysRevLett.125.160501 [45] Hakop Pashayan, Joel J Wallman, and Stephen D Bartlett. ``Estimating outcome probabilities of quantum circuits using quasiprobabilities''.

Physical Review Letters 115, 070501 (2015). https://doi.org/10.1103/PhysRevLett.115.070501 [46] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Physical Review A 64, 012310 (2001). https://doi.org/10.1103/PhysRevA.64.012310 [47] Mile Gu, Christian Weedbrook, Nicolas C. Menicucci, Timothy C. Ralph, and Peter van Loock. ``Quantum computing with continuous-variable clusters''. Phys. Rev. A 79, 062318 (2009). https://doi.org/10.1103/PhysRevA.79.062318 [48] Youngrong Lim and Changhun Oh. ``Approximating outcome probabilities of linear optical circuits''. npj Quantum Information 9, 124 (2023). https://doi.org/10.1038/s41534-023-00791-9 [49] Ulysse Chabaud, Roohollah Ghobadi, Salman Beigi, and Saleh Rahimi-Keshari. ``Phase-space negativity as a computational resource for quantum kernel methods''. Quantum 8, 1519 (2024). https://doi.org/10.22331/q-2024-11-07-1519Cited by[1] G. Bizzarri, S. Gherardini, M. Manrique, F. Bruni, I. Gianani, and M. Barbieri, "Quasiprobability distributions with weak measurements", Quantum Science and Technology 10 4, 045008 (2025). [2] Varun Upreti, Ulysse Chabaud, Zoë Holmes, and Armando Angrisani, "When quantum resources backfire: Non-gaussianity and symplectic coherence in noisy bosonic circuits", arXiv:2510.07264, (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-07-08 19:09:12). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-07-08 19:09:10: Could not fetch cited-by data for 10.22331/q-2026-07-08-2155 from Crossref. This is normal if the DOI was registered recently.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. AbstractIn continuous-variable quantum computation, identifying key elements that enable a quantum computational advantage is a long-standing issue. Starting from the standard results on the necessity of Wigner negativity, we develop a comprehensive and versatile approach in which the techniques of $(s)$-ordered quasiprobabilities are exploited to provide rigorous statements on the simulability of photonic quantum circuits consisting of previously characterized gates and thereby identifying the contribution of each quantum gate to the potential achievement of quantum computational advantage. This is achieved by means of an analysis of the so-called transfer function, allowing us to highlight the resourcefulness of a gate set. As such this technique can be straightforwardly applied to current continuous-variables quantum circuits, while also constraining the tolerable amount of losses above which any potential quantum advantage can be ruled out. We use $(s)$-ordered quasiprobability distributions on phase-space to capture the non-classical features in the protocol, and focus our technique entirely on the ordering parameter $s$. This allows us to highlight the resourcefulness and robustness to loss of a universal set of unitary gates comprising three distinct Gaussian gates and any non-Gaussian unitary gate, providing important insight on the role of non-Gaussianity.Featured image: Generic quantum-optical scheme depicted by $M$ input modes, described by a density operator $\rho_{\mathrm{in}}$ processed through a trace-preserving quantum channel $\mathcal{E}$, that can be decomposed into a sequence of trace-preserving quantum channels $\mathcal{E} = \mathcal{E}_1 \circ \mathcal{E}_2 \circ \dots \circ \mathcal{E}_k$. This produces the output state $\rho_{out} = \mathcal{E}(\rho_{in})$ and an output probability distribution $p({x}) = Tr[\rho_{out}\Pi_{{x}}]$ sampled by measuring the POVM $\Pi_{{x}}$.Popular summaryAmong the different platforms being explored for quantum computation, photonic systems are especially appealing. Photons can be manipulated with well-established optical components, can be used to generate entanglement, and are relatively robust against decoherence compared with many matter-based qubits, although they remain sensitive to losses. A central question is to understand which features of a photonic quantum circuit are genuinely responsible for a possible quantum advantage over classical computers. It has long been known that purely Gaussian optical ingredients, namely Gaussian states, Gaussian operations, and Gaussian measurements, can be efficiently simulated classically. Some form of non-Gaussianity, often associated with Wigner negativity, is therefore needed. However, Wigner negativity alone does not fully answer the question: it tells us that a circuit may contain a useful quantum resource, but not whether this resource is strong enough, placed in the right part of the circuit, or robust enough against experimental noise. In this work, we develop a general method to analyze photonic quantum circuits gate by gate. The idea is to represent each input state, optical operation, and measurement through a family of phase-space functions, and to ask whether the action of each gate can still be described using ordinary, non-negative probability distributions. When this is possible, the corresponding part of the circuit can be reproduced by classical sampling, much like in previous simulation methods based on positive Wigner functions. The main advantage of this approach is that it identifies the resourcefulness of each gate independently of the detailed input state. The relevant parameter is the nonclassical depth, which quantifies how strongly nonclassical the incoming state must be before a given gate can no longer be simulated in this classical way. This provides a sharper criterion than simply asking whether Wigner negativity is present somewhere in the circuit. The method also provides insights about robustness of a possible quantum advantage with respect to losses. For a given optical gate, it can determine how much loss is enough to wash out its useful nonclassical behavior and make that part of the computation classically simulable. In this sense, the results do not only say whether a circuit is potentially powerful, but also which components are most fragile and what experimental quality they must reach in order to remain computationally relevant. We apply this framework to important gates used in continuous-variable photonic quantum computation. For Gaussian gates such as squeezers and beam splitters, the method shows how nonclassicality is transformed, mixed between modes, or degraded by losses. For realistic photon subtraction, it clarifies when the operation can become genuinely resourceful, depending on the amount of nonclassicality (e.g. squeezing, in this instance) available at the input. Finally, the analysis is extended to non-Gaussian unitary gates, including the cubic phase gate, which is a key ingredient for universal continuous-variable quantum computation. Overall, the work provides a systematic way to draw the boundary between photonic circuits that can still be efficiently simulated on a classical computer and those that may contain the resources needed for a true quantum advantage. Importantly, this boundary can exclude quantum advantage even in some circuits where Wigner negativity is present, showing that the mere presence of non-Gaussianity is not always enough: what matters is how much nonclassicality is available, where it appears in the circuit, and whether it survives realistic losses.► BibTeX data@article{Frigerio2026resourcefulnessof, doi = {10.22331/q-2026-07-08-2155}, url = {https://doi.org/10.22331/q-2026-07-08-2155}, title = {Resourcefulness of non-classical continuous-variable quantum gates}, author = {Frigerio, Massimo and Debray, Antoine and Treps, Nicolas and Walschaers, Mattia}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2155}, month = jul, year = {2026} }► References [1] Peter W. Shor. ``Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer''. SIAM Review 41, 303–332 (1999). https://doi.org/10.1137/S0036144598347011 [2] 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 [3] Scott Aaronson and Alex Arkhipov. ``The computational complexity of linear optics''. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing. Page 333–342. STOC '11New York, NY, USA (2011). Association for Computing Machinery. https://doi.org/10.1145/1993636.1993682 [4] Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd. ``Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, 459–472 (2011). https://doi.org/10.1098/rspa.2010.0301 [5] Sergio Boixo, Sergei V. Isakov, Vadim N. Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J. Bremner, John M. Martinis, and Hartmut Neven. ``Characterizing quantum supremacy in near-term devices''. Nature Physics 14, 595–600 (2018). https://doi.org/10.1038/s41567-018-0124-x [6] Aram W Harrow and Ashley Montanaro. ``Quantum computational supremacy''. Nature 549, 203–209 (2017). https://doi.org/10.1038/nature23458 [7] Dominik Hangleiter and Jens Eisert. ``Computational advantage of quantum random sampling''. Rev. Mod. Phys. 95, 035001 (2023). https://doi.org/10.1103/RevModPhys.95.035001 [8] Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C. Bardin, Rami Barends, et al. ``Quantum supremacy using a programmable superconducting processor''. Nature 574, 505–510 (2019). https://doi.org/10.1038/s41586-019-1666-5 [9] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, et al. ``Quantum computational advantage using photons''. Science 370, 1460–1463 (2020). https://doi.org/10.1126/science.abe8770 [10] Lars S. Madsen, Fabian Laudenbach, Mohsen Falamarzi. Askarani, Fabien Rortais, Trevor Vincent, Jacob F. F. Bulmer, Filippo M. Miatto, Leonhard Neuhaus, Helt, et al. ``Quantum computational advantage with a programmable photonic processor''. Nature 606, 75–81 (2022). https://doi.org/10.1038/s41586-022-04725-x [11] M. DeCross, R. Haghshenas, M. Liu, E. Rinaldi, J. Gray, Y. Alexeev, C. H. Baldwin, J. P. Bartolotta, M. Bohn, E. Chertkov, J. Cline, J. Colina, D. DelVento, J. M. Dreiling, C. Foltz, J. P. Gaebler, T. M. Gatterman, C. N. Gilbreth, J. Giles, D. Gresh, A. Hall, A. Hankin, A. Hansen, N. Hewitt, I. Hoffman, C. Holliman, R. B. Hutson, T. Jacobs, J. Johansen, P. J. Lee, E. Lehman, D. Lucchetti, D. Lykov, I. S. Madjarov, B. Mathewson, K. Mayer, M. Mills, P. Niroula, J. M. Pino, C. Roman, M. Schecter, P. E. Siegfried, B. G. Tiemann, C. Volin, J. Walker, R. Shaydulin, M. Pistoia, S. A. Moses, D. Hayes, B. Neyenhuis, R. P. Stutz, and M. Foss-Feig. ``Computational power of random quantum circuits in arbitrary geometries''. Phys. Rev. X 15, 021052 (2025). https://doi.org/10.1103/PhysRevX.15.021052 [12] Craig S. Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. ``Gaussian boson sampling''. Phys. Rev. Lett. 119, 170501 (2017). https://doi.org/10.1103/PhysRevLett.119.170501 [13] A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. O'Brien, and T. C. Ralph. ``Boson sampling from a gaussian state''. Phys. Rev. Lett. 113, 100502 (2014). https://doi.org/10.1103/PhysRevLett.113.100502 [14] U. Chabaud, T. Douce, D. Markham, P. van Loock, E. Kashefi, and G. Ferrini. ``Continuous-variable sampling from photon-added or photon-subtracted squeezed states''. Phys. Rev. A 96, 062307 (2017). https://doi.org/10.1103/PhysRevA.96.062307 [15] L. Chakhmakhchyan and N. J. Cerf. ``Boson sampling with gaussian measurements''. Phys. Rev. A 96, 032326 (2017). https://doi.org/10.1103/PhysRevA.96.032326 [16] Nicolás Quesada, Juan Miguel Arrazola, and Nathan Killoran. ``Gaussian boson sampling using threshold detectors''. Physical Review A 98, 062322 (2018). https://doi.org/10.1103/PhysRevA.98.062322 [17] Raúl García-Patrón, Jelmer J Renema, and Valery Shchesnovich. ``Simulating boson sampling in lossy architectures''. Quantum 3, 169 (2019). https://doi.org/10.22331/q-2019-08-05-169 [18] Jelmer J Renema. ``Simulability of partially distinguishable superposition and gaussian boson sampling''. Physical Review A 101, 063840 (2020). https://doi.org/10.1103/PhysRevA.101.063840 [19] J. J. Renema, A. Menssen, W. R. Clements, G. Triginer, W. S. Kolthammer, and I. A. Walmsley. ``Efficient classical algorithm for boson sampling with partially distinguishable photons''. Phys. Rev. Lett. 120, 220502 (2018). https://doi.org/10.1103/PhysRevLett.120.220502 [20] Saleh Rahimi-Keshari, Timothy C Ralph, and Carlton M Caves. ``Sufficient conditions for efficient classical simulation of quantum optics''. Physical Review X 6, 021039 (2016). https://doi.org/10.1103/PhysRevX.6.021039 [21] Jacob F. F. Bulmer, Bryn A. Bell, Rachel S. Chadwick, Alex E. Jones, Diana Moise, Alessandro Rigazzi, Jan Thorbecke, Utz-Uwe Haus, Thomas Van Vaerenbergh, Raj B. Patel, Ian A. Walmsley, and Anthony Laing. ``The boundary for quantum advantage in gaussian boson sampling''. Science Advances 8, eabl9236 (2022). https://doi.org/10.1126/sciadv.abl9236 [22] Haoyu Qi, Daniel J. Brod, Nicolás Quesada, and Raúl García-Patrón. ``Regimes of classical simulability for noisy gaussian boson sampling''. Phys. Rev. Lett. 124, 100502 (2020). https://doi.org/10.1103/PhysRevLett.124.100502 [23] Nicolás Quesada, Rachel S. Chadwick, Bryn A. Bell, Juan Miguel Arrazola, Trevor Vincent, Haoyu Qi, and Raúl García-Patrón. ``Quadratic speed-up for simulating gaussian boson sampling''. PRX Quantum 3, 010306 (2022). https://doi.org/10.1103/PRXQuantum.3.010306 [24] Peter D. Drummond, Bogdan Opanchuk, A. Dellios, and M. D. Reid. ``Simulating complex networks in phase space: Gaussian boson sampling''. Phys. Rev. A 105, 012427 (2022). https://doi.org/10.1103/PhysRevA.105.012427 [25] Changhun Oh, Minzhao Liu, Yuri Alexeev, Bill Fefferman, and Liang Jiang. ``Classical algorithm for simulating experimental gaussian boson sampling''. Nature Physics (2024). https://doi.org/10.1038/s41567-024-02535-8 [26] Daniel Gottesman. ``The heisenberg representation of quantum computers'' (1998). arXiv:quant-ph/9807006. arXiv:quant-ph/9807006 [27] Stephen D. Bartlett, Barry C. Sanders, Samuel L. Braunstein, and Kae Nemoto. ``Efficient classical simulation of continuous variable quantum information processes''.

Read Original

Source Information

Source: Quantum Science and Technology (arXiv overlay)

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

Resourcefulness of non-classical continuous-variable quantum gates

Tags

Source Information