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Exact holographic tensor networks for the Motzkin spin chain



Rafael N. Alexander1,2, Glen Evenbly3, and Israel Klich4

1Centre for Quantum Computation and Communication Technology, School of Science, RMIT University, Melbourne, VIC 3000, Australia
2Center for Quantum Information and Control, University of New Mexico, Albuquerque, NM 87131, USA
3School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA
4Department of Physics, University of Virginia, Charlottesville, Virginia 22903, USA

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The study of low-dimensional quantum systems has proven to be a particularly fertile field for discovering novel types of quantum matter. When studied numerically, low-energy states of low-dimensional quantum systems are often approximated via a tensor-network description. The tensor network’s utility in studying short range correlated states in 1D have been thoroughly investigated, with numerous examples where the treatment is essentially exact. Yet, despite the large number of works investigating these networks and their relations to physical models, examples of exact correspondence between the ground state of a quantum critical system and an appropriate scale-invariant tensor network have eluded us so far. Here we show that the features of the quantum-critical Motzkin model can be faithfully captured by an analytic tensor network that exactly represents the ground state of the physical Hamiltonian. In particular, our network offers a two-dimensional representation of this state by a correspondence between walks and a type of tiling of a square lattice. We discuss connections to renormalization and holography.

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► References

[1] Kenneth G. Wilson. The renormalization group and critical phenomena. Rev. Mod. Phys., 55: 583–600, Jul 1983. 10.1103/​RevModPhys.55.583.

[2] Michael E. Fisher. Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys., 70: 653–681, Apr 1998. 10.1103/​RevModPhys.70.653.

[3] Subir Sachdev. Quantum Phase Transitions. Cambridge University Press, 2 edition, 2011. 10.1017/​CBO9780511973765.

[4] H. v. Löhneysen, T. Pietrus, G. Portisch, H. G. Schlager, A. Schröder, M. Sieck, and T. Trappmann. Non-fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability. Phys. Rev. Lett., 72: 3262–3265, May 1994. 10.1103/​PhysRevLett.72.3262.

[5] S R Julian, C Pfleiderer, F M Grosche, N D Mathur, G J McMullan, A J Diver, I R Walker, and G G Lonzarich. The normal states of magnetic d and f transition metals. Journal of Physics: Condensed Matter, 8 (48): 9675–9688, nov 1996. 10.1088/​0953-8984/​8/​48/​002.

[6] S. A. Grigera, R. S. Perry, A. J. Schofield, M. Chiao, S. R. Julian, G. G. Lonzarich, S. I. Ikeda, Y. Maeno, A. J. Millis, and A. P. Mackenzie. Magnetic field-tuned quantum criticality in the metallic ruthenate $text{Sr}_3text{Ru}_2text{O}_7$. Science, 294 (5541): 329–332, 2001. 10.1126/​science.1063539.

[7] Román Orús. A practical introduction to tensor networks: Matrix product states and projected entangled pair states. Annals of Physics, 349: 117–158, 2014. 10.1016/​j.aop.2014.06.013.

[8] 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 (22): 223001, may 2017. 10.1088/​1751-8121/​aa6dc3.

[9] Mark Fannes, Bruno Nachtergaele, and Reinhard F Werner. Finitely correlated states on quantum spin chains. Communications in mathematical physics, 144 (3): 443–490, 1992. 10.1007/​BF02099178.

[10] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett., 69: 2863–2866, Nov 1992. 10.1103/​PhysRevLett.69.2863.

[11] G. Vidal. Class of quantum many-body states that can be efficiently simulated. Phys. Rev. Lett., 101: 110501, Sep 2008. 10.1103/​PhysRevLett.101.110501.

[12] G. Vidal. Entanglement renormalization. Phys. Rev. Lett., 99: 220405, Nov 2007. 10.1103/​PhysRevLett.99.220405.

[13] Ian Affleck, Tom Kennedy, Elliott H. Lieb, and Hal Tasaki. Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett., 59: 799–802, Aug 1987. 10.1103/​PhysRevLett.59.799.

[14] Glen Evenbly and Steven R. White. Entanglement renormalization and wavelets. Phys. Rev. Lett., 116: 140403, Apr 2016. 10.1103/​PhysRevLett.116.140403.

[15] Jutho Haegeman, Brian Swingle, Michael Walter, Jordan Cotler, Glen Evenbly, and Volkher B. Scholz. Rigorous free-fermion entanglement renormalization from wavelet theory. Phys. Rev. X, 8: 011003, Jan 2018. 10.1103/​PhysRevX.8.011003.

[16] Ramis Movassagh and Peter W. Shor. Supercritical entanglement in local systems: Counterexample to the area law for quantum matter. Proceedings of the National Academy of Sciences, 113 (47): 13278–13282, 2016. 10.1073/​pnas.1605716113.

[17] Sergey Bravyi, Libor Caha, Ramis Movassagh, Daniel Nagaj, and Peter W. Shor. Criticality without frustration for quantum spin-1 chains. Phys. Rev. Lett., 109: 207202, Nov 2012. 10.1103/​PhysRevLett.109.207202.

[18] A.Yu. Kitaev. Fault-tolerant quantum computation by anyons. Annals of Physics, 303 (1): 2–30, 2003. ISSN 0003-4916. https:/​/​​10.1016/​S0003-4916(02)00018-0.

[19] Daniel S. Rokhsar and Steven A. Kivelson. Superconductivity and the quantum hard-core dimer gas. Phys. Rev. Lett., 61: 2376–2379, Nov 1988. 10.1103/​PhysRevLett.61.2376.

[20] David Perez-García, Frank Verstraete, Michael M Wolf, and J Ignacio Cirac. $text{PEPS}$ as unique ground states of local $text{Hamiltonians}$. Quantum Information & Computation, 8 (6-7): 650–663, 2008.

[21] Norbert Schuch, Ignacio Cirac, and David Pérez-García. Peps as ground states: Degeneracy and topology. Annals of Physics, 325 (10): 2153–2192, 2010. https:/​/​​10.1016/​j.aop.2010.05.008.

[22] Norbert Schuch, David Pérez-García, and Ignacio Cirac. Classifying quantum phases using matrix product states and projected entangled pair states. Phys. Rev. B, 84: 165139, Oct 2011. 10.1103/​PhysRevB.84.165139.

[23] Zhao Zhang, Amr Ahmadain, and Israel Klich. Novel quantum phase transition from bounded to extensive entanglement. Proceedings of the National Academy of Sciences, 114 (20): 5142–5146, 2017. 10.1073/​pnas.1702029114.

[24] Olof Salberger and Vladimir Korepin. Entangled spin chain. Reviews in Mathematical Physics, 29 (10): 1750031, 2017. 10.1142/​S0129055X17500313.

[25] L. Dell’Anna, O. Salberger, L. Barbiero, A. Trombettoni, and V. E. Korepin. Violation of cluster decomposition and absence of light cones in local integer and half-integer spin chains. Phys. Rev. B, 94: 155140, Oct 2016. 10.1103/​PhysRevB.94.155140.

[26] Olof Salberger, Takuma Udagawa, Zhao Zhang, Hosho Katsura, Israel Klich, and Vladimir Korepin. Deformed fredkin spin chain with extensive entanglement. Journal of Statistical Mechanics: Theory and Experiment, 2017 (6): 063103, jun 2017. 10.1088/​1742-5468/​aa6b1f.

[27] Zhao Zhang and Israel Klich. Entropy, gap and a multi-parameter deformation of the fredkin spin chain. Journal of Physics A: Mathematical and Theoretical, 50 (42): 425201, sep 2017. 10.1088/​1751-8121/​aa866e.

[28] Takuma Udagawa and Hosho Katsura. Finite-size gap, magnetization, and entanglement of deformed fredkin spin chain. Journal of Physics A: Mathematical and Theoretical, 50 (40): 405002, sep 2017. 10.1088/​1751-8121/​aa85b5.

[29] Fumihiko Sugino and Pramod Padmanabhan. Area law violations and quantum phase transitions in modified motzkin walk spin chains. Journal of Statistical Mechanics: Theory and Experiment, 2018 (1): 013101, jan 2018. 10.1088/​1742-5468/​aa9dcb.

[30] M. Bal, M. M. Rams, V. Zauner, J. Haegeman, and F. Verstraete. Matrix product state renormalization. Phys. Rev. B, 94: 205122, Nov 2016. 10.1103/​PhysRevB.94.205122.

[31] Fernando G. S. L. Brandão, Elizabeth Crosson, M. Burak Şahinoğlu, and John Bowen. Quantum error correcting codes in eigenstates of translation-invariant spin chains. Phys. Rev. Lett., 123: 110502, Sep 2019. 10.1103/​PhysRevLett.123.110502.

[32] Xiao Chen, Eduardo Fradkin, and William Witczak-Krempa. Gapless quantum spin chains: multiple dynamics and conformal wavefunctions. Journal of Physics A: Mathematical and Theoretical, 50 (46): 464002, oct 2017. 10.1088/​1751-8121/​aa8dbc.

[33] Sukhwinder Singh, Robert N. C. Pfeifer, and Guifre Vidal. Tensor network states and algorithms in the presence of a global u(1) symmetry. Phys. Rev. B, 83: 115125, Mar 2011. 10.1103/​PhysRevB.83.115125.

[34] I Klich, S. H. Lee, and K Iida. Glassiness and exotic entropy scaling induced by quantum fluctuations in a disorder-free frustrated magnet. Nature communications, 5 (3497), 2014. 10.1038/​ncomms4497.

[35] G. Evenbly and G. Vidal. Algorithms for entanglement renormalization. Phys. Rev. B, 79: 144108, Apr 2009. 10.1103/​PhysRevB.79.144108.

[36] Andrew J. Ferris. Fourier transform for fermionic systems and the spectral tensor network. Phys. Rev. Lett., 113: 010401, Jul 2014. 10.1103/​PhysRevLett.113.010401.

[37] Alexander Jahn, Zoltán Zimborás, and Jens Eisert. Central charges of aperiodic holographic tensor-network models. Phys. Rev. A, 102: 042407, Oct 2020. 10.1103/​PhysRevA.102.042407.

[38] Robert N. C. Pfeifer, Glen Evenbly, and Guifré Vidal. Entanglement renormalization, scale invariance, and quantum criticality. Phys. Rev. A, 79: 040301, Apr 2009. 10.1103/​PhysRevA.79.040301.

[39] J Ignacio Cirac, David Perez-Garcia, Norbert Schuch, and Frank Verstraete. Matrix product unitaries: structure, symmetries, and topological invariants. Journal of Statistical Mechanics: Theory and Experiment, 2017 (8): 083105, aug 2017. 10.1088/​1742-5468/​aa7e55.

[40] M. Burak Şahinoğlu, Sujeet K. Shukla, Feng Bi, and Xie Chen. Matrix product representation of locality preserving unitaries. Phys. Rev. B, 98: 245122, Dec 2018. 10.1103/​PhysRevB.98.245122.

[41] Brian Swingle. Entanglement renormalization and holography. Phys. Rev. D, 86: 065007, Sep 2012. 10.1103/​PhysRevD.86.065007.

[42] Marika Taylor. Non-relativistic holography. arXiv preprint arXiv:0812.0530, 2008. https:/​/​​abs/​0812.0530.

[43] Koushik Balasubramanian and John McGreevy. Gravity duals for nonrelativistic conformal field theories. Phys. Rev. Lett., 101: 061601, Aug 2008. 10.1103/​PhysRevLett.101.061601.

[44] D. T. Son. Toward an ads/​cold atoms correspondence: A geometric realization of the schrödinger symmetry. Phys. Rev. D, 78: 046003, Aug 2008. 10.1103/​PhysRevD.78.046003.

[45] Marika Taylor. Lifshitz holography. Classical and Quantum Gravity, 33 (3): 033001, jan 2016. 10.1088/​0264-9381/​33/​3/​033001.

[46] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich. Exact rainbow tensor networks for the colorful motzkin and fredkin spin chains. Phys. Rev. B, 100: 214430, Dec 2019. 10.1103/​PhysRevB.100.214430.

Cited by

[1] Thomas Schuster, Bryce Kobrin, Ping Gao, Iris Cong, Emil T. Khabiboulline, Norbert M. Linke, Mikhail D. Lukin, Christopher Monroe, Beni Yoshida, and Norman Y. Yao, “Many-body quantum teleportation via operator spreading in the traversable wormhole protocol”, arXiv:2102.00010.

[2] Rafael N. Alexander, Amr Ahmadain, Zhao Zhang, and Israel Klich, “Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains”, Physical Review B 100 21, 214430 (2019).

[3] Glen Evenbly, “Number-State Preserving Tensor Networks as Classifiers for Supervised Learning”, arXiv:1905.06352.

[4] Jacob Miller, Guillaume Rabusseau, and John Terilla, “Tensor Networks for Probabilistic Sequence Modeling”, arXiv:2003.01039.

[5] A. Ahmadain and I. Klich, “Emergent geometry and path integral optimization for a Lifshitz action”, Physical Review D 103 10, 105013 (2021).

[6] Fumihiko Sugino, “Highly Entangled Spin Chains and 2D Quantum Gravity”, Symmetry 12 6, 916 (2020).

The above citations are from SAO/NASA ADS (last updated successfully 2021-09-21 15:40:03). The list may be incomplete as not all publishers provide suitable and complete citation data.

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Simultaneous Perturbation Stochastic Approximation of the Quantum Fisher Information



Julien Gacon1,2, Christa Zoufal1,3, Giuseppe Carleo2, and Stefan Woerner1

1IBM Quantum, IBM Research – Zurich, CH-8803 Rüschlikon, Switzerland
2Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
3Institute for Theoretical Physics, ETH Zurich, CH-8092 Zürich, Switzerland

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The Quantum Fisher Information matrix (QFIM) is a central metric in promising algorithms, such as Quantum Natural Gradient Descent and Variational Quantum Imaginary Time Evolution. Computing the full QFIM for a model with $d$ parameters, however, is computationally expensive and generally requires $mathcal{O}(d^2)$ function evaluations. To remedy these increasing costs in high-dimensional parameter spaces, we propose using simultaneous perturbation stochastic approximation techniques to approximate the QFIM at a constant cost. We present the resulting algorithm and successfully apply it to prepare Hamiltonian ground states and train Variational Quantum Boltzmann Machines.

► BibTeX data

► References

[1] Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon. Simulated Quantum Computation of Molecular Energies. Science, 309 (5741): 1704–1707, September 2005. 10.1126/​science.1113479.

[2] Alberto Peruzzo et al. A variational eigenvalue solver on a photonic quantum processor. Nature Communications, 5: 4213, July 2014. 10.1038/​ncomms5213.

[3] Mari Carmen Bañuls et al. Simulating lattice gauge theories within quantum technologies. European Physical Journal D, 74 (8): 165, August 2020. 10.1140/​epjd/​e2020-100571-8.

[4] Alejandro Perdomo-Ortiz, Neil Dickson, Marshall Drew-Brook, Geordie Rose, and Alán Aspuru-Guzik. Finding low-energy conformations of lattice protein models by quantum annealing. Scientific Reports, 2: 571, August 2012. 10.1038/​srep00571.

[5] Mark Fingerhuth, Tomáš Babej, and Christopher Ing. A quantum alternating operator ansatz with hard and soft constraints for lattice protein folding. arXiv, October 2018. URL https:/​/​​abs/​1810.13411.

[6] Anton Robert, Panagiotis Kl. Barkoutsos, Stefan Woerner, and Ivano Tavernelli. Resource-efficient quantum algorithm for protein folding. npj Quantum Information, 7 (1): 38, February 2021. ISSN 2056-6387. 10.1038/​s41534-021-00368-4.

[7] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann. A Quantum Approximate Optimization Algorithm. arXiv, November 2014. URL https:/​/​​abs/​1411.4028.

[8] Austin Gilliam, Stefan Woerner, and Constantin Gonciulea. Grover Adaptive Search for Constrained Polynomial Binary Optimization. arXiv, December 2019. URL https:/​/​​abs/​1912.04088. 10.22331/​q-2021-04-08-428.

[9] Lee Braine, Daniel J. Egger, Jennifer Glick, and Stefan Woerner. Quantum Algorithms for Mixed Binary Optimization applied to Transaction Settlement. arXiv, October 2019. URL https:/​/​​abs/​1910.05788. 10.1109/​TQE.2021.3063635.

[10] J. Gacon, C. Zoufal, and S. Woerner. Quantum-enhanced simulation-based optimization. In 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), pages 47–55, 2020. 10.1109/​QCE49297.2020.00017.

[11] D. J. Egger et al. Quantum computing for finance: State-of-the-art and future prospects. IEEE Transactions on Quantum Engineering, 1: 1–24, 2020. 10.1109/​TQE.2020.3030314.

[12] J. S. Otterbach et al. Unsupervised Machine Learning on a Hybrid Quantum Computer. arXiv, December 2017. URL https:/​/​​abs/​1712.05771.

[13] Vojtěch Havlíček et al. Supervised learning with quantum-enhanced feature spaces. Nature, 567 (7747): 209–212, March 2019. 10.1038/​s41586-019-0980-2.

[14] Maria Schuld. Quantum machine learning models are kernel methods. arXiv, January 2021. URL https:/​/​​abs/​2101.11020.

[15] Nikolaj Moll et al. Quantum optimization using variational algorithms on near-term quantum devices. Quantum Science and Technology, 3 (3): 030503, July 2018. 10.1088/​2058-9565/​aab822.

[16] Sam McArdle et al. Variational ansatz-based quantum simulation of imaginary time evolution. npj Quantum Information, 5 (1), Sep 2019. ISSN 2056-6387. 10.1038/​s41534-019-0187-2.

[17] Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C. Benjamin. Theory of variational quantum simulation. Quantum, 3: 191, October 2019. ISSN 2521-327X. 10.22331/​q-2019-10-07-191.

[18] Christa Zoufal, Aurélien Lucchi, and Stefan Woerner. Variational quantum boltzmann machines. Quantum Machine Intelligence, 3: 7, 2020. ISSN 2524-4914. 10.1007/​s42484-020-00033-7.

[19] Taku Matsui. Quantum statistical mechanics and Feller semigroup. Quantum Probability Communications, 1998. 10.1142/​9789812816054_0004.

[20] Masoud Khalkhali and Matilde Marcolli. An Invitation to Noncommutative Geometry. World Scientific, 2008. 10.1142/​6422.

[21] J. Eisert, M. Friesdorf, and C. Gogolin. Quantum many-body systems out of equilibrium. Nature Physics, 11 (2), 2015. 10.1038/​nphys3215.

[22] Fernando G. S. L. Brandão et al. Quantum SDP Solvers: Large speed-ups, optimality, and applications to quantum learning. arXiv, 2017. URL https:/​/​​abs/​1710.02581.

[23] Mohammad H. Amin, Evgeny Andriyash, Jason Rolfe, Bohdan Kulchytskyy, and Roger Melko. Quantum Boltzmann Machine. Phys. Rev. X, 8, 2018. 10.1103/​PhysRevX.8.021050.

[24] James Stokes, Josh Izaac, Nathan Killoran, and Giuseppe Carleo. Quantum natural gradient. Quantum, 4: 269, May 2020. ISSN 2521-327X. 10.22331/​q-2020-05-25-269.

[25] S. Amari and S. C. Douglas. Why natural gradient? In Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP ’98 (Cat. No.98CH36181), volume 2, pages 1213–1216 vol.2, 1998. 10.1109/​ICASSP.1998.675489.

[26] J.C. Spall. Multivariate stochastic approximation using a simultaneous perturbation gradient approximation. IEEE Transactions on Automatic Control, 37 (3): 332–341, 1992. 10.1109/​9.119632.

[27] Lingyao Meng and James C. Spall. Efficient computation of the fisher information matrix in the em algorithm. In 2017 51st Annual Conference on Information Sciences and Systems (CISS), pages 1–6, 2017. 10.1109/​CISS.2017.7926126.

[28] A. Cauchy. Methode generale pour la resolution des systemes d’equations simultanees. C.R. Acad. Sci. Paris, 25: 536–538, 1847. 10.1017/​cbo9780511702396.063.

[29] J. C. Spall. Accelerated second-order stochastic optimization using only function measurements. In Proceedings of the 36th IEEE Conference on Decision and Control, volume 2, pages 1417–1424 vol.2, December 1997. 10.1109/​CDC.1997.657661. ISSN: 0191-2216.

[30] Yuan Yao, Pierre Cussenot, Alex Vigneron, and Filippo M. Miatto. Natural Gradient Optimization for Optical Quantum Circuits. arXiv, June 2021. URL https:/​/​​abs/​2106.13660.

[31] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Phys. Rev. A, 99 (3): 032331, March 2019. 10.1103/​PhysRevA.99.032331.

[32] Johannes Jakob Meyer. Fisher Information in Noisy Intermediate-Scale Quantum Applications. Quantum, 5: 539, September 2021. ISSN 2521-327X. 10.22331/​q-2021-09-09-539.

[33] Andrea Mari, Thomas R. Bromley, and Nathan Killoran. Estimating the gradient and higher-order derivatives on quantum hardware. Phys. Rev. A, 103 (1): 012405, Jan 2021. 10.1103/​PhysRevA.103.012405.

[34] Harry Buhrman, Richard Cleve, John Watrous, and Ronald de Wolf. Quantum fingerprinting. Phys. Rev. Lett., 87 (16): 167902, Sep 2001. 10.1103/​PhysRevLett.87.167902.

[35] Lukasz Cincio, Yiğit Subaşı, Andrew T. Sornborger, and Patrick J. Coles. Learning the quantum algorithm for state overlap. arXiv, November 2018. URL http:/​/​​abs/​1803.04114. 10.1088/​1367-2630/​aae94a.

[36] A. Elben, B. Vermersch, C. F. Roos, and P. Zoller. Statistical correlations between locally randomized measurements: A toolbox for probing entanglement in many-body quantum states. Phys. Rev. A, 99 (5), May 2019. 10.1103/​PhysRevA.99.052323.

[37] Kristan Temme, Tobias J. Osborne, Karl Gerd H. Vollbrecht, David Poulin, and Frank Verstraete. Quantum Metropolis Sampling. Nature, 471, 2011. 10.1038/​nature09770.

[38] Man-Hong Yung and Alán Aspuru-Guzik. A quantum–quantum Metropolis algorithm. Proceedings of the National Academy of Sciences, 109 (3), 2012. 10.1073/​pnas.1111758109.

[39] David Poulin and Pawel Wocjan. Sampling from the Thermal Quantum Gibbs State and Evaluating Partition Functions with a Quantum Computer. Phys. Rev. Lett., 103 (22), 2009. 10.1103/​PhysRevLett.103.220502.

[40] Mario Motta and et al. Determining eigenstates and thermal states on a quantum computer using quantum imaginary time evolution. Nature Physics, 16 (2), 2020. 10.1038/​s41567-019-0704-4.

[41] Fernando G. S. L. Brandão and Michael J. Kastoryano. Finite Correlation Length Implies Efficient Preparation of Quantum Thermal States. Communications in Mathematical Physics, 365 (1), 2019. 10.1007/​s00220-018-3150-8.

[42] Michael J. Kastoryano and Fernando G. S. L. Brandão. Quantum Gibbs Samplers: The Commuting Case. Communications in Mathematical Physics, 344 (3), 2016. 10.1007/​s00220-016-2641-8.

[43] Jingxiang Wu and Timothy H. Hsieh. Variational Thermal Quantum Simulation via Thermofield Double States. Phys. Rev. Lett., 123 (22), 2019. 10.1103/​PhysRevLett.123.220502.

[44] Anirban Chowdhury, Guang Hao Low, and Nathan Wiebe. A Variational Quantum Algorithm for Preparing Quantum Gibbs States. arXiv, 2020. URL https:/​/​​abs/​2002.00055.

[45] A.D. McLachlan. A variational solution of the time-dependent Schrödinger equation. Molecular Physics, 8 (1), 1964. 10.1080/​00268976400100041.

[46] Héctor Abraham et al. Qiskit: An open-source framework for quantum computing. 2019. 10.5281/​zenodo.2562110.

[47] IBM Quantum, 2021. URL https:/​/​​services/​docs/​services/​runtime/​.

[48] Sergey Bravyi, Jay M. Gambetta, Antonio Mezzacapo, and Kristan Temme. Tapering off qubits to simulate fermionic hamiltonians. arXiv, 2017. URL https:/​/​​abs/​1701.08213.

[49] Abhinav Kandala et al. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. Nature, 549 (7671): 242–246, September 2017. 10.1038/​nature23879.

[50] Abhinav Kandala, Kristan Temme, Antonio D. Corcoles, Antonio Mezzacapo, Jerry M. Chow, and Jay M. Gambetta. Error mitigation extends the computational reach of a noisy quantum processor. Nature, 567 (7749): 491–495, March 2019. 10.1038/​s41586-019-1040-7.

[51] Jonas M. Kübler, Andrew Arrasmith, Lukasz Cincio, and Patrick J. Coles. An Adaptive Optimizer for Measurement-Frugal Variational Algorithms. Quantum, 4: 263, May 2020. ISSN 2521-327X. 10.22331/​q-2020-05-11-263.

Cited by

[1] Tobias Haug, Kishor Bharti, and M. S. Kim, “Capacity and quantum geometry of parametrized quantum circuits”, arXiv:2102.01659.

[2] Johannes Jakob Meyer, “Fisher Information in Noisy Intermediate-Scale Quantum Applications”, arXiv:2103.15191.

[3] Tobias Haug and M. S. Kim, “Optimal training of variational quantum algorithms without barren plateaus”, arXiv:2104.14543.

[4] Tobias Haug and M. S. Kim, “Natural parameterized quantum circuit”, arXiv:2107.14063.

[5] Martin Larocca, Nathan Ju, Diego García-Martín, Patrick J. Coles, and M. Cerezo, “Theory of overparametrization in quantum neural networks”, arXiv:2109.11676.

[6] Christa Zoufal, David Sutter, and Stefan Woerner, “Error Bounds for Variational Quantum Time Evolution”, arXiv:2108.00022.

[7] Anna Lopatnikova and Minh-Ngoc Tran, “Quantum Natural Gradient for Variational Bayes”, arXiv:2106.05807.

The above citations are from SAO/NASA ADS (last updated successfully 2021-10-23 12:31:38). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-10-23 12:31:36).

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Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation



Patrick Rall

Quantum Information Center, University of Texas at Austin

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We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quantum estimation algorithms make assumptions that make them unsuitable for this ‘coherent’ setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both conceptually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not require a quantum Fourier transform, and they do not require a quantum sorting network to compute the median of several estimates. Instead, they use block-encoding techniques to compute the estimate one bit at a time, performing all amplification via singular value transformation. These improved subroutines accelerate the performance of quantum Metropolis sampling and quantum Bayesian inference.

Presentation at TQC 2021

A fundamental objective of quantum computing is to help study physical systems. One of the earliest results in the area was a fast quantum algorithm for measuring the energy of a system, which can serve as a building block for other quantum algorithms. However this algorithm is very complicated and hard to analyze. In this paper we present a simpler method based on applying polynomials to the Hamiltonian that extract each of the bits of the estimate. This technique is up to 20x faster than the prior state of the art.

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► References

[1] Pawel Wocjan, Kristan Temme, Szegedy Walk Unitaries for Quantum Maps arXiv:2107.07365 (2021).

[2] John M. Martyn, Zane M. Rossi, Andrew K. Tan, Isaac L. Chuang, A Grand Unification of Quantum Algorithms arXiv:2105.02859 (2021).

[3] Lin Lin, Yu Tong, Heisenberg-limited ground state energy estimation for early fault-tolerant quantum computers arXiv:2102.11340 (2021).

[4] Earl T. Campbell, Early fault-tolerant simulations of the Hubbard model arXiv:2012.09238 (2020).

[5] Yuan Su, Hsin-Yuan Huang, Earl T. Campbell, Nearly tight Trotterization of interacting electrons arXiv:2012.09194 Quantum 5, 495 (2020).

[6] Alexander Engel, Graeme Smith, Scott E. Parker, A framework for applying quantum computation to nonlinear dynamical systems arXiv:2012.06681 Physics of Plasmas 28, 062305 (2020).

[7] Dong An, Noah Linden, Jin-Peng Liu, Ashley Montanaro, Changpeng Shao, Jiasu Wang, Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance arXiv:2012.06283 Quantum 5, 481 (2020).

[8] Isaac Chuang, Grand unification of quantum algorithms. Seminar presentation at IQC Waterloo. (2020).

[9] Lewis Wright, Fergus Barratt, James Dborin, George H. Booth, Andrew G. Green, Automatic Post-selection by Ancillae Thermalisation arXiv:2010.04173 Phys. Rev. Research 3, 033151 (2020).

[10] Srinivasan Arunachalam, Vojtech Havlicek, Giacomo Nannicini, Kristan Temme, Pawel Wocjan, Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions arXiv:2009.11270 (2020).

[11] András Gilyén, Zhao Song, Ewin Tang, An improved quantum-inspired algorithm for linear regression arXiv:2009.07268 (2020).

[12] Phillip W. K. Jensen, Lasse Bjørn Kristensen, Jakob S. Kottmann, Alán Aspuru-Guzik, Quantum Computation of Eigenvalues within Target Intervals Quantum Science and Technology 6, 015004 arXiv:2005.13434 (2020).

[13] Patrick Rall, Quantum Algorithms for Estimating Physical Quantities using Block-Encodings Phys. Rev. A 102, 022408 arXiv:2004.06832 (2020).

[14] Alessandro Roggero, Spectral density estimation with the Gaussian Integral Transform Phys. Rev. A 102, 022409 arXiv:2004.04889 (2020).

[15] Rui Chao, Dawei Ding, Andras Gilyen, Cupjin Huang, Mario Szegedy, Finding Angles for Quantum Signal Processing with Machine Precision arXiv:2003.02831 (2020).

[16] Lin Lin, Yu Tong, Near-optimal ground state preparation arXiv:2002.12508 Quantum 4, 372 (2020).

[17] Andrew M. Childs, Yuan Su, Minh C. Tran, Nathan Wiebe, Shuchen Zhu, A Theory of Trotter Error Phys. Rev. X 11, 011020 arXiv:1912.08854 (2019).

[18] Dmitry Grinko, Julien Gacon, Christa Zoufal, Stefan Woerner, Iterative Quantum Amplitude Estimation npj Quantum Inf 7, 52 arXiv:1912.05559 (2019).

[19] Jessica Lemieux, Bettina Heim, David Poulin, Krysta Svore, Matthias Troyer, Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm Quantum 4, 287 arXiv:1910.01659 (2019).

[20] Scott Aaronson, Patrick Rall, Quantum Approximate Counting,Simplified Symposium on Simplicity in Algorithms. 2020, 24-32 arXiv:1908.10846(2019).

[21] Aram W. Harrow, Annie Y. Wei, Adaptive Quantum Simulated Annealing for Bayesian Inference and Estimating Partition Functions Proc. of SODA 2020 arXiv:1907.09965 (2019).

[22] Iordanis Kerenidis, Jonas Landman, Alessandro Luongo, and Anupam Prakash, q-means: A quantum algorithm for unsupervised machine learning arXiv:1812.03584 NIPS 32 (2018).

[23] Yassine Hamoudi, Frédéric Magniez, Quantum Chebyshev’s Inequality and Applications ICALP, LIPIcs Vol 132, pages 69:1-99:16 arXiv:1807.06456 (2018).

[24] Jeongwan Haah, Product Decomposition of Periodic Functions in Quantum Signal Processing Quantum 3, 190. arXiv:1806.10236 (2018).

[25] András Gilyén, Yuan Su, Guang Hao Low, Nathan Wiebe, Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics arXiv:1806.01838 Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing (STOC 2019) Pages 193–204 (2018).

[26] David Poulin, Alexei Kitaev, Damian S. Steiger, Matthew B. Hastings, Matthias Troyer, Quantum Algorithm for Spectral Measurement with Lower Gate Count arXiv:1711.11025 Phys. Rev. Lett. 121, 010501 (2017).

[27] Guang Hao Low, Isaac L. Chuang, Hamiltonian Simulation by Uniform Spectral Amplification arXiv:1707.05391 (2017).

[28] Iordanis Kerenidis, Anupam Prakash, Quantum gradient descent for linear systems and least squares arXiv:1704.04992 Phys. Rev. A 101, 022316 (2017).

[29] Yosi Atia, Dorit Aharonov, Fast-forwarding of Hamiltonians and exponentially precise measurements Nature Communications volume 8, 1572 arXiv:1610.09619 (2016).

[30] Guang Hao Low, Isaac L. Chuang, Hamiltonian Simulation by Qubitization Quantum 3, 163 arXiv:1610.06546 (2016).

[31] Guang Hao Low, Isaac L. Chuang, Optimal Hamiltonian Simulation by Quantum Signal Processing Phys. Rev. Lett. 118, 010501 arXiv:1606.02685 (2016).

[32] Iordanis Kerenidis, Anupam Prakash, Quantum Recommendation Systems arXiv:1603.08675 ITCS 2017, p. 49:1–49:21 (2016).

[33] Andrew M. Childs, Robin Kothari, Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision SIAM Journal on Computing 46, 1920-1950 arXiv:1511.02306 (2015).

[34] Ashley Montanaro, Quantum speedup of Monte Carlo methods Proc. Roy. Soc. Ser. A, vol. 471 no. 2181, 20150301 arXiv:1504.06987 (2015).

[35] Shelby Kimmel, Guang Hao Low, Theodore J. Yoder, Robust Calibration of a Universal Single-Qubit Gate-Set via Robust Phase Estimation Phys. Rev. A 92, 062315 arXiv:1502.02677 (2015).

[36] Dominic W. Berry, Andrew M. Childs, Robin Kothari, Hamiltonian simulation with nearly optimal dependence on all parameters arXiv:1501.01715 Proc. FOCS, pp. 792-809 (2015).

[37] Amnon Ta-Shma, Inverting well conditioned matrices in quantum logspace STOC ’13, Pages 881–890 (2013).

[38] Robert Beals, Stephen Brierley, Oliver Gray, Aram Harrow, Samuel Kutin, Noah Linden, Dan Shepherd, Mark Stather, Efficient Distributed Quantum Computing Proc. R. Soc. A 2013 469, 20120686 arXiv:1207.2307 (2012).

[39] Maris Ozols, Martin Roetteler, Jérémie Roland, Quantum Rejection Sampling arXiv:1103.2774 IRCS’12 pages 290-308 (2011).

[40] Man-Hong Yung, Alán Aspuru-Guzik, A Quantum-Quantum Metropolis Algorithm arXiv:1011.1468 PNAS 109, 754-759 (2011).

[41] Andris Ambainis, Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations arXiv:1010.4458 STACS’12, 636-647 (2010).

[42] K. Temme, T.J. Osborne, K.G. Vollbrecht, D. Poulin, F. Verstraete, Quantum Metropolis Sampling arXiv:0911.3635 Nature volume 471, pages 87–90 (2009).

[43] Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco Servedio, Emanuele Viola, Bounded Independence Fools Halfspaces arXiv:0902.3757 FOCS ’09, Pages 171–180 (2009).

[44] Aram W. Harrow, Avinatan Hassidim, Seth Lloyd, Quantum algorithm for solving linear systems of equations Phys. Rev. Lett. 103, 150502 arXiv:0811.3171 (2008).

[45] B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, G. J. Pryde, Entanglement-free Heisenberg-limited phase estimation Nature.450:393-396 arXiv:0709.2996 (2007).

[46] Chris Marriott, John Watrous, Quantum Arthur-Merlin Games CC, 14(2): 122 – 152 arXiv:cs/​0506068 (2005).

[47] Mario Szegedy, Quantum speed-up of Markov chain based algorithms FOCS ’04, Pages 32-41 (2004).

[48] Hartmut Klauck, Quantum Time-Space Tradeoffs for Sorting STOC 03, Pages 69–76 arXiv:quant-ph/​0211174 (2002).

[49] Peter Hoyer, Jan Neerbek, Yaoyun Shi, Quantum complexities of ordered searching, sorting, and element distinctness 28th ICALP, LNCS 2076, pp. 346-357 arXiv:quant-ph/​0102078 (2001).

[50] Isaac Chuang and Michael Nielsen, Quantum Computation and Quantum Information Cambridge University Press. ISBN-13: 978-1107002173 (2000).

[51] Gilles Brassard, Peter Hoyer, Michele Mosca, Alain Tapp, Quantum Amplitude Amplification and Estimation Quantum Computation and Quantum Information, 305:53-74 arXiv:quant-ph/​0005055 (2000).

[52] Dorit Aharonov, Alexei Kitaev, Noam Nisan, Quantum Circuits with Mixed States STOC ’97, pages 20-30 arXiv:quant-ph/​9806029 (1998).

[53] Ashwin Nayak, Felix Wu, The quantum query complexity of approximating the median and related statistics arXiv:quant-ph/​9804066 STOC ’99 pp 384-393 (1998).

[54] Charles H. Bennett, Ethan Bernstein, Gilles Brassard, Umesh Vazirani, Strengths and Weaknesses of Quantum Computing arXiv:quant-ph/​9701001 SIAM Journal on Computing 26(5):1510-1523 (1997).

[55] A. Yu. Kitaev, Quantum measurements and the Abelian Stabilizer Problem arXiv:quant-ph/​9511026 (1995).

[56] Peter W. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer SIAM J.Sci.Statist.Comput. 26, 1484 arXiv:quant-ph/​9508027 (1995).

[57] Theodore J. Rivlin, An Introduction to the Approximation of Functions Dover Publications, Inc. New York. ISBN-13:978-0486640693 (1969).

Cited by

[1] Yuan Su, Hsin-Yuan Huang, and Earl T. Campbell, “Nearly tight Trotterization of interacting electrons”, arXiv:2012.09194.

[2] John M. Martyn, Zane M. Rossi, Andrew K. Tan, and Isaac L. Chuang, “A Grand Unification of Quantum Algorithms”, arXiv:2105.02859.

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On Crossref’s cited-by service no data on citing works was found (last attempt 2021-10-23 15:14:09).

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Multidimensional cluster states using a single spin-photon interface coupled strongly to an intrinsic nuclear register



Cathryn P. Michaels, Jesús Arjona Martínez, Romain Debroux, Ryan A. Parker, Alexander M. Stramma, Luca I. Huber, Carola M. Purser, Mete Atatüre, and Dorian A. Gangloff

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, UK

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Photonic cluster states are a powerful resource for measurement-based quantum computing and loss-tolerant quantum communication. Proposals to generate multi-dimensional lattice cluster states have identified coupled spin-photon interfaces, spin-ancilla systems, and optical feedback mechanisms as potential schemes. Following these, we propose the generation of multi-dimensional lattice cluster states using a single, efficient spin-photon interface coupled strongly to a nuclear register. Our scheme makes use of the contact hyperfine interaction to enable universal quantum gates between the interface spin and a local nuclear register and funnels the resulting entanglement to photons via the spin-photon interface. Among several quantum emitters, we identify the silicon-29 vacancy centre in diamond, coupled to a nanophotonic structure, as possessing the right combination of optical quality and spin coherence for this scheme. We show numerically that using this system a 2×5-sized cluster state with a lower-bound fidelity of 0.5 and repetition rate of 65 kHz is achievable under currently realised experimental performances and with feasible technical overhead. Realistic gate improvements put 100-photon cluster states within experimental reach.

Quantum states composed of multiple entangled photons are a key resource in quantum computing networks, both for robust communication and for implementing computational tasks. Photonic cluster states whose entanglement is multidimensional are required for universal quantum protocols. Such cluster states can be obtained from a highly efficient single-photon source, together with entangling gates between distinct emitters or between local spins. We propose to use the multidimensional entanglement naturally available to a single diamond colour center strongly coupled to an intrinsic nuclear spin to create multi-dimensional cluster states of photons. Our simulations show that 100-photon cluster states are realisable within achievable experimental parameters.

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[1] A. Aspect, P. Grangier, G. Roger, Experimental Tests of Realistic Local Theories via Bell’s Theorem, Phys. Rev. Lett. 47 (7) (1981) 460–463. doi:10.1103/​PhysRevLett.47.460.

[2] A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67 (6) (1991) 661–663. doi:10.1103/​PhysRevLett.67.661.

[3] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger, Experimental quantum teleportation, Nature 390 (6660) (1997) 575–579. doi:10.1038/​37539.

[4] R. Raussendorf, H. J. Briegel, A One-Way Quantum Computer, Phys. Rev. Lett. 86 (22) (2001) 5188–5191. doi:10.1103/​physrevlett.86.5188.

[5] R. Raussendorf, D. E. Browne, H. J. Briegel, Measurement-based quantum computation on cluster states, Phys. Rev. A 68 (2) (2003) 022312. doi:10.1103/​PhysRevA.68.022312.

[6] H. J. Briegel, D. E. Browne, W. Dür, R. Raussendorf, M. V. den Nest, Measurement-based quantum computation, Nat. Phys. 5 (1) (2009) 19–26. doi:10.1038/​nphys1157.

[7] H. J. Kimble, The quantum internet, Nature 453 (7198) (2008) 1023–1030. doi:10.1038/​nature07127.

[8] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, J. L. O’Brien, Quantum computers., Nature 464 (7285) (2010) 45–53. doi:10.1038/​nature08812.

[9] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74 (1) (2002) 145–195. doi:10.1103/​RevModPhys.74.145.

[10] H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. Lu, J.-W. Pan, Quantum computational advantage using photons, Science 370 (6523) (2020) 1460–1463. doi:10.1126/​science.abe8770.

[11] F. Xu, X. Ma, Q. Zhang, H.-K. Lo, J.-W. Pan, Secure quantum key distribution with realistic devices, Rev. Mod. Phys. 92 (2) (2020) 025002. doi:10.1103/​RevModPhys.92.025002.

[12] H. J. Briegel, R. Raussendorf, Persistent entanglement in arrays of interacting particles, Phys. Rev. Lett. 86 (5) (2001) 910–913. doi:10.1103/​PhysRevLett.86.910.

[13] M. Varnava, D. E. Browne, T. Rudolph, How Good Must Single Photon Sources and Detectors Be for Efficient Linear Optical Quantum Computation?, Phys. Rev. Lett. 100 (6) (2008) 060502. doi:10.1103/​PhysRevLett.100.060502.

[14] M. Zwerger, H. J. Briegel, W. Dür, Measurement-based quantum communication, Appl. Phys. B 122 (3) (2016) 50. doi:10.1007/​s00340-015-6285-8.

[15] K. Azuma, K. Tamaki, H.-K. Lo, All-photonic quantum repeaters, Nat. Commun. 6 (1) (2015) 6787. doi:10.1038/​ncomms7787.

[16] W. P. Grice, Arbitrarily complete Bell-state measurement using only linear optical elements, Phys. Rev. A 84 (4) (2011) 042331. doi:10.1103/​PhysRevA.84.042331.

[17] T. Kilmer, S. Guha, Boosting linear-optical Bell measurement success probability with predetection squeezing and imperfect photon-number-resolving detectors, Phys. Rev. A 99 (3) (2019) 032302. doi:10.1103/​PhysRevA.99.032302.

[18] F. Ewert, P. van Loock, 3/​4-Efficient Bell Measurement with Passive Linear Optics and Unentangled Ancillae, Phys. Rev. Lett. 113 (14) (2014) 140403. doi:10.1103/​PhysRevLett.113.140403.

[19] D. E. Browne, T. Rudolph, Resource-Efficient Linear Optical Quantum Computation, Phys. Rev. Lett. 95 (1) (2005) 010501. doi:10.1103/​PhysRevLett.95.010501.

[20] Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J. Briegel, J.-W. Pan, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature 430 (6995) (2004) 54–58. doi:10.1038/​nature02643.

[21] W. B. Gao, C. Y. Lu, X. C. Yao, P. Xu, O. Gühne, A. Goebel, Y. A. Chen, C. Z. Peng, Z. B. Chen, J. W. Pan, Experimental demonstration of a hyper-entangled ten-qubit Schrödinger cat state, Nat. Phys. 6 (5) (2010) 331–335. doi:10.1038/​nphys1603.

[22] X.-L. Wang, L.-K. Chen, W. Li, H.-L. Huang, C. Liu, C. Chen, Y.-H. Luo, Z.-E. Su, D. Wu, Z.-D. Li, H. Lu, Y. Hu, X. Jiang, C.-Z. Peng, L. Li, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, J.-W. Pan, Experimental Ten-Photon Entanglement, Phys. Rev. Lett. 117 (21) (2016) 210502. doi:10.1103/​PhysRevLett.117.210502.

[23] D. Istrati, Y. Pilnyak, J. C. Loredo, C. Antón, N. Somaschi, P. Hilaire, H. Ollivier, M. Esmann, L. Cohen, L. Vidro, C. Millet, A. Lemaı̂tre, I. Sagnes, A. Harouri, L. Lanco, P. Senellart, H. S. Eisenberg, Sequential generation of linear cluster states from a single photon emitter, Nat. Commun. 11 (1) (2020) 5501. doi:10.1038/​s41467-020-19341-4.

[24] W. Asavanant, Y. Shiozawa, S. Yokoyama, B. Charoensombutamon, H. Emura, R. N. Alexander, S. Takeda, J.-i. Yoshikawa, N. C. Menicucci, H. Yonezawa, A. Furusawa, Generation of time-domain-multiplexed two-dimensional cluster state, Science 366 (6463) (2019) 373–376. doi:10.1126/​science.aay2645.

[25] N. H. Lindner, T. Rudolph, Proposal for Pulsed On-Demand Sources of Photonic Cluster State Strings, Phys. Rev. Lett. 103 (11) (2009) 113602. doi:10.1103/​PhysRevLett.103.113602.

[26] I. Schwartz, D. Cogan, E. R. Schmidgall, Y. Don, L. Gantz, O. Kenneth, N. H. Lindner, D. Gershoni, Deterministic generation of a cluster state of entangled photons, Science 354 (6311) (2016) 434–437. doi:10.1126/​science.aah4758.

[27] D. Gonţa, T. Radtke, S. Fritzsche, Generation of two-dimensional cluster states by using high-finesse bimodal cavities, Phys. Rev. A 79 (6) (2009) 062319. doi:10.1103/​PhysRevA.79.062319.

[28] S. E. Economou, N. Lindner, T. Rudolph, Optically Generated 2-Dimensional Photonic Cluster State from Coupled Quantum Dots, Phys. Rev. Lett. 105 (9) (2010) 093601. doi:10.1103/​PhysRevLett.105.093601.

[29] A. Mantri, T. F. Demarie, J. F. Fitzsimons, Universality of quantum computation with cluster states and (X, Y)-plane measurements, Sci. Rep. 7 (1) (2017) 42861. doi:10.1038/​srep42861.

[30] M. Gimeno-Segovia, T. Rudolph, S. E. Economou, Deterministic Generation of Large-Scale Entangled Photonic Cluster State from Interacting Solid State Emitters, Phys. Rev. Lett. 123 (7) (2019) 070501. doi:10.1103/​PhysRevLett.123.070501.

[31] A. Russo, E. Barnes, S. E. Economou, Generation of arbitrary all-photonic graph states from quantum emitters, New J. Phys. 21 (5) (2019) 055002. doi:10.1088/​1367-2630/​ab193d.

[32] A. Russo, E. Barnes, S. E. Economou, Photonic graph state generation from quantum dots and color centers for quantum communications, Phys. Rev. B 98 (8) (2018) 085303. doi:10.1103/​PhysRevB.98.085303.

[33] D. Buterakos, E. Barnes, S. E. Economou, Deterministic Generation of All-Photonic Quantum Repeaters from Solid-State Emitters, Phys. Rev. X 7 (4) (2017) 041023. doi:10.1103/​PhysRevX.7.041023.

[34] G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbrüggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann, J. Wrachtrup, Quantum error correction in a solid-state hybrid spin register, Nature 506 (7487) (2014) 204–207. doi:10.1038/​nature12919.

[35] D. A. Gangloff, G. Éthier-Majcher, C. Lang, E. V. Denning, J. H. Bodey, D. M. Jackson, E. Clarke, M. Hugues, C. Le Gall, M. Atatüre, Quantum interface of an electron and a nuclear ensemble, Science 364 (6435) (2019) 62–66. doi:10.1126/​science.aaw2906.

[36] M. H. Metsch, K. Senkalla, B. Tratzmiller, J. Scheuer, M. Kern, J. Achard, A. Tallaire, M. B. Plenio, P. Siyushev, F. Jelezko, Initialization and Readout of Nuclear Spins via a Negatively Charged Silicon-Vacancy Center in Diamond, Phys. Rev. Lett. 122 (19) (2019) 190503. doi:10.1103/​PhysRevLett.122.190503.

[37] M. Atatüre, D. Englund, N. Vamivakas, S.-Y. Lee, J. Wrachtrup, Material platforms for spin-based photonic quantum technologies, Nat. Rev. Mater. 3 (5) (2018) 38–51. doi:10.1038/​s41578-018-0008-9.

[38] E. Janitz, M. K. Bhaskar, L. Childress, Cavity quantum electrodynamics with color centers in diamond, Optica 7 (10) (2020) 1232. doi:10.1364/​OPTICA.398628.

[39] J. L. O’Brien, A. Furusawa, J. Vučković, Photonic quantum technologies, Nat. Photonics 3 (12) (2009) 687–695. doi:10.1038/​nphoton.2009.229.

[40] M. Paillard, X. Marie, E. Vanelle, T. Amand, V. K. Kalevich, A. R. Kovsh, A. E. Zhukov, V. M. Ustinov, Time-resolved photoluminescence in self-assembled InAs/​GaAs quantum dots under strictly resonant excitation, Appl. Phys. Lett. 76 (1) (2000) 76–78. doi:10.1063/​1.125661.

[41] D. Najer, I. Söllner, P. Sekatski, V. Dolique, M. C. Löbl, D. Riedel, R. Schott, S. Starosielec, S. R. Valentin, A. D. Wieck, N. Sangouard, A. Ludwig, R. J. Warburton, A gated quantum dot strongly coupled to an optical microcavity, Nature 575 (7784) (2019) 622–627. doi:10.1038/​s41586-019-1709-y.

[42] P. Senellart, G. Solomon, A. White, High-performance semiconductor quantum-dot single-photon sources, Nat. Nanotechnol. 12 (11) (2017) 1026–1039. doi:10.1038/​nnano.2017.218.

[43] E. Peter, J. Hours, P. Senellart, A. Vasanelli, A. Cavanna, J. Bloch, J. M. Gérard, Phonon sidebands in exciton and biexciton emission from single GaAs quantum dots, Phys. Rev. B 69 (4) (2004) 041307. doi:10.1103/​PhysRevB.69.041307.

[44] C. Matthiesen, M. Geller, C. H. H. Schulte, C. Le Gall, J. Hansom, Z. Li, M. Hugues, E. Clarke, M. Atatüre, Phase-locked indistinguishable photons with synthesized waveforms from a solid-state source, Nat. Commun. 4 (1) (2013) 1600. doi:10.1038/​ncomms2601.

[45] K. Konthasinghe, J. Walker, M. Peiris, C. K. Shih, Y. Yu, M. F. Li, J. F. He, L. J. Wang, H. Q. Ni, Z. C. Niu, A. Muller, Coherent versus incoherent light scattering from a quantum dot, Phys. Rev. B 85 (23) (2012) 235315. doi:10.1103/​PhysRevB.85.235315.

[46] A. Bechtold, D. Rauch, F. Li, T. Simmet, P.-L. Ardelt, A. Regler, K. Müller, N. A. Sinitsyn, J. J. Finley, Three-stage decoherence dynamics of an electron spin qubit in an optically active quantum dot, Nat. Phys. 11 (12) (2015) 1005–1008. doi:10.1038/​nphys3470.

[47] R. Stockill, C. Le Gall, C. Matthiesen, L. Huthmacher, E. Clarke, M. Hugues, M. Atatüre, Quantum dot spin coherence governed by a strained nuclear environment, Nat. Commun. 7 (1) (2016) 12745. doi:10.1038/​ncomms12745.

[48] A. Högele, M. Kroner, C. Latta, M. Claassen, I. Carusotto, C. Bulutay, A. Imamoglu, Dynamic Nuclear Spin Polarization in the Resonant Laser Excitation of an InGaAs Quantum Dot, Phys. Rev. Lett. 108 (19) (2012) 197403. doi:10.1103/​PhysRevLett.108.197403.

[49] D. J. Christle, P. V. Klimov, C. F. de las Casas, K. Szász, V. Ivády, V. Jokubavicius, J. Ul Hassan, M. Syväjärvi, W. F. Koehl, T. Ohshima, N. T. Son, E. Janzén, Á. Gali, D. D. Awschalom, Isolated Spin Qubits in SiC with a High-Fidelity Infrared Spin-to-Photon Interface, Phys. Rev. X 7 (2) (2017) 021046. doi:10.1103/​PhysRevX.7.021046.

[50] G. Calusine, A. Politi, D. D. Awschalom, Silicon carbide photonic crystal cavities with integrated color centers, Appl. Phys. Lett. 105 (1) (2014) 011123. doi:10.1063/​1.4890083.

[51] A. Bourassa, C. P. Anderson, K. C. Miao, M. Onizhuk, H. Ma, A. L. Crook, H. Abe, J. Ul-Hassan, T. Ohshima, N. T. Son, G. Galli, D. D. Awschalom, Entanglement and control of single nuclear spins in isotopically engineered silicon carbide, Nat. Mater. 19 (12) (2020) 1319–1325. doi:10.1038/​s41563-020-00802-6.

[52] L. Spindlberger, A. Csóré, G. Thiering, S. Putz, R. Karhu, J. U. Hassan, N. T. Son, T. Fromherz, A. Gali, M. Trupke, Optical Properties of Vanadium in 4 H Silicon Carbide for Quantum Technology, Phys. Rev. Applied 12 (1) (2019) 014015. doi:10.1103/​PhysRevApplied.12.014015.

[53] G. Wolfowicz, C. P. Anderson, B. Diler, O. G. Poluektov, F. J. Heremans, D. D. Awschalom, Vanadium spin qubits as telecom quantum emitters in silicon carbide, Sci. Adv. 6 (18) (2020) eaaz1192. doi:10.1126/​sciadv.aaz1192.

[54] N. B. Manson, J. P. Harrison, M. J. Sellars, Nitrogen-vacancy center in diamond: Model of the electronic structure and associated dynamics, Phys. Rev. B 74 (10) (2006) 104303. doi:10.1103/​PhysRevB.74.104303.

[55] D. Riedel, I. Söllner, B. J. Shields, S. Starosielec, P. Appel, E. Neu, P. Maletinsky, R. J. Warburton, Deterministic Enhancement of Coherent Photon Generation from a Nitrogen-Vacancy Center in Ultrapure Diamond, Phys. Rev. X 7 (3) (2017) 031040. doi:10.1103/​PhysRevX.7.031040.

[56] M. Berthel, O. Mollet, G. Dantelle, T. Gacoin, S. Huant, A. Drezet, Photophysics of single nitrogen-vacancy centers in diamond nanocrystals, Phys. Rev. B 91 (3) (2015) 035308. doi:10.1103/​PhysRevB.91.035308.

[57] R. N. Patel, T. Schröder, N. Wan, L. Li, S. L. Mouradian, E. H. Chen, D. R. Englund, Efficient photon coupling from a diamond nitrogen vacancy center by integration with silica fiber, Light Sci. Appl. 5 (2) (2016) e16032–e16032. doi:10.1038/​lsa.2016.32.

[58] I. Aharonovich, S. Castelletto, D. A. Simpson, C.-H. Su, A. D. Greentree, S. Prawer, Diamond-based single-photon emitters, Reports Prog. Phys. 74 (7) (2011) 076501. doi:10.1088/​0034-4885/​74/​7/​076501.

[59] P. C. Humphreys, N. Kalb, J. P. Morits, R. N. Schouten, R. F. Vermeulen, D. J. Twitchen, M. Markham, R. Hanson, Deterministic delivery of remote entanglement on a quantum network, Nature 558 (7709) (2018) 268–273. doi:10.1038/​s41586-018-0200-5.

[60] W. Pfaff, T. H. Taminiau, L. Robledo, H. Bernien, M. Markham, D. J. Twitchen, R. Hanson, Demonstration of entanglement-by-measurement of solid-state qubits, Nat. Phys. 9 (1) (2013) 29–33. doi:10.1038/​nphys2444.

[61] J. N. Becker, B. Pingault, D. Groß, M. Gündoğan, N. Kukharchyk, M. Markham, A. Edmonds, M. Atatüre, P. Bushev, C. Becher, All-Optical Control of the Silicon-Vacancy Spin in Diamond at Millikelvin Temperatures, Phys. Rev. Lett. 120 (5) (2018) 053603. doi:10.1103/​PhysRevLett.120.053603.

[62] M. K. Bhaskar, R. Riedinger, B. Machielse, D. S. Levonian, C. T. Nguyen, E. N. Knall, H. Park, D. Englund, M. Lončar, D. D. Sukachev, M. D. Lukin, Experimental demonstration of memory-enhanced quantum communication, Nature 580 (7801) (2020) 60–64. doi:10.1038/​s41586-020-2103-5.

[63] D. D. Sukachev, A. Sipahigil, C. T. Nguyen, M. K. Bhaskar, R. E. Evans, F. Jelezko, M. D. Lukin, Silicon-Vacancy Spin Qubit in Diamond: A Quantum Memory Exceeding 10 ms with Single-Shot State Readout, Phys. Rev. Lett. 119 (22) (2017) 223602. doi:10.1103/​PhysRevLett.119.223602.

[64] E. Neu, M. Fischer, S. Gsell, M. Schreck, C. Becher, Fluorescence and polarization spectroscopy of single silicon vacancy centers in heteroepitaxial nanodiamonds on iridium, Phys. Rev. B 84 (20) (2011) 205211. doi:10.1103/​PhysRevB.84.205211.

[65] E. Neu, D. Steinmetz, J. Riedrich-Möller, S. Gsell, M. Fischer, M. Schreck, C. Becher, Single photon emission from silicon-vacancy colour centres in chemical vapour deposition nano-diamonds on iridium, New J. Phys. 13 (2) (2011) 025012. doi:10.1088/​1367-2630/​13/​2/​025012.

[66] B. Pingault, D.-D. Jarausch, C. Hepp, L. Klintberg, J. N. Becker, M. Markham, C. Becher, M. Atatüre, Coherent control of the silicon-vacancy spin in diamond, Nat. Commun. 8 (1) (2017) 15579. doi:10.1038/​ncomms15579.

[67] A. M. Edmonds, M. E. Newton, P. M. Martineau, D. J. Twitchen, S. D. Williams, Electron paramagnetic resonance studies of silicon-related defects in diamond, Phys. Rev. B 77 (24) (2008) 245205. doi:10.1103/​PhysRevB.77.245205.

[68] T. Iwasaki, F. Ishibashi, Y. Miyamoto, Y. Doi, S. Kobayashi, T. Miyazaki, K. Tahara, K. D. Jahnke, L. J. Rogers, B. Naydenov, F. Jelezko, S. Yamasaki, S. Nagamachi, T. Inubushi, N. Mizuochi, M. Hatano, Germanium-Vacancy Single Color Centers in Diamond, Sci. Rep. 5 (1) (2015) 12882. doi:10.1038/​srep12882.

[69] M. K. Bhaskar, D. D. Sukachev, A. Sipahigil, R. E. Evans, M. J. Burek, C. T. Nguyen, L. J. Rogers, P. Siyushev, M. H. Metsch, H. Park, F. Jelezko, M. Lončar, M. D. Lukin, Quantum nonlinear optics with a germanium-vacancy color center in a nanoscale diamond waveguide, Phys. Rev. Lett. 118 (2017) 223603. doi:10.1103/​PhysRevLett.118.223603.

[70] Y. N. Palyanov, I. N. Kupriyanov, Y. M. Borzdov, N. V. Surovtsev, Germanium: a new catalyst for diamond synthesis and a new optically active impurity in diamond, Sci. Rep. 5 (1) (2015) 14789. doi:10.1038/​srep14789.

[71] M. E. Trusheim, B. Pingault, N. H. Wan, M. Gündoğan, L. De Santis, R. Debroux, D. Gangloff, C. Purser, K. C. Chen, M. Walsh, J. J. Rose, J. N. Becker, B. Lienhard, E. Bersin, I. Paradeisanos, G. Wang, D. Lyzwa, A. R.-P. Montblanch, G. Malladi, H. Bakhru, A. C. Ferrari, I. A. Walmsley, M. Atatüre, D. Englund, Transform-Limited Photons From a Coherent Tin-Vacancy Spin in Diamond, Phys. Rev. Lett. 124 (2) (2020) 023602. doi:10.1103/​PhysRevLett.124.023602.

[72] A. E. Rugar, S. Aghaeimeibodi, D. Riedel, C. Dory, H. Lu, P. J. McQuade, Z.-X. Shen, N. A. Melosh, J. Vučković, Quantum Photonic Interface for Tin-Vacancy Centers in Diamond, Phys. Rev. X 11 (3) (2021) 031021. doi:10.1103/​PhysRevX.11.031021.

[73] T. Iwasaki, Y. Miyamoto, T. Taniguchi, P. Siyushev, M. H. Metsch, F. Jelezko, M. Hatano, Tin-Vacancy Quantum Emitters in Diamond, Phys. Rev. Lett. 119 (25) (2017) 253601. doi:10.1103/​PhysRevLett.119.253601.

[74] J. Görlitz, D. Herrmann, G. Thiering, P. Fuchs, M. Gandil, T. Iwasaki, T. Taniguchi, M. Kieschnick, J. Meijer, M. Hatano, A. Gali, C. Becher, Spectroscopic investigations of negatively charged tin-vacancy centres in diamond, New J. Phys. 22 (1) (2020) 013048. doi:10.1088/​1367-2630/​ab6631.

[75] R. Debroux, C. P. Michaels, C. M. Purser, N. Wan, M. E. Trusheim, J. A. Martínez, R. A. Parker, A. M. Stramma, K. C. Chen, L. de Santis, E. M. Alexeev, A. C. Ferrari, D. Englund, D. A. Gangloff, M. Atatüre, Quantum control of the tin-vacancy spin qubit in diamond, arXiv:2106.00723 (2021).

[76] N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. Löbl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, R. J. Warburton, A bright and fast source of coherent single photons, Nat. Nanotechnol. 16 (4) (2021) 399–403. doi:10.1038/​s41565-020-00831-x.

[77] D. Kim, S. G. Carter, A. Greilich, A. S. Bracker, D. Gammon, Ultrafast optical control of entanglement between two quantum-dot spins, Nat. Phys. 7 (3) (2011) 223–229. doi:10.1038/​nphys1863.

[78] D. Ding, M. H. Appel, A. Javadi, X. Zhou, M. C. Löbl, I. Söllner, R. Schott, C. Papon, T. Pregnolato, L. Midolo, A. D. Wieck, A. Ludwig, R. J. Warburton, T. Schröder, P. Lodahl, Coherent Optical Control of a Quantum-Dot Spin-Qubit in a Waveguide-Based Spin-Photon Interface, Phys. Rev. Applied 11 (3) (2019) 031002. doi:10.1103/​PhysRevApplied.11.031002.

[79] J. H. Bodey, R. Stockill, E. V. Denning, D. A. Gangloff, G. Éthier-Majcher, D. M. Jackson, E. Clarke, M. Hugues, C. L. Gall, M. Atatüre, Optical spin locking of a solid-state qubit, npj Quantum Inf. 5 (1) (2019) 95. doi:10.1038/​s41534-019-0206-3.

[80] E. V. Denning, D. A. Gangloff, M. Atatüre, J. Mørk, C. Le Gall, Collective Quantum Memory Activated by a Driven Central Spin, Phys. Rev. Lett. 123 (14) (2019) 140502. doi:10.1103/​PhysRevLett.123.140502.

[81] C. F. De Las Casas, D. J. Christle, J. Ul Hassan, T. Ohshima, N. T. Son, D. D. Awschalom, Stark tuning and electrical charge state control of single divacancies in silicon carbide, Appl. Phys. Lett. 111 (26) (2017) 262403. doi:10.1063/​1.5004174.

[82] T. T. Tran, K. Bray, M. J. Ford, M. Toth, I. Aharonovich, Quantum emission from hexagonal boron nitride monolayers, Nat. Nanotechnol. 11 (1) (2016) 37–41. doi:10.1038/​nnano.2015.242.

[83] T. Zhong, J. M. Kindem, J. Rochman, A. Faraon, Interfacing broadband photonic qubits to on-chip cavity-protected rare-earth ensembles, Nat. Commun. 8 (1) (2017) 14107. doi:10.1038/​ncomms14107.

[84] I. Aharonovich, A. D. Greentree, S. Prawer, Diamond photonics, Nat. Photonics 5 (7) (2011) 397–405. doi:10.1038/​nphoton.2011.54.

[85] I. Aharonovich, E. Neu, Diamond Nanophotonics, Adv. Opt. Mater. 2 (10) (2014) 911–928. doi:10.1002/​adom.201400189.

[86] I. Aharonovich, D. Englund, M. Toth, Solid-state single-photon emitters, Nat. Photonics 10 (10) (2016) 631–641. doi:10.1038/​nphoton.2016.186.

[87] G. D. Fuchs, G. Burkard, P. V. Klimov, D. D. Awschalom, A quantum memory intrinsic to single nitrogen-vacancy centres in diamond, Nat. Phys. 7 (10) (2011) 789–793. doi:10.1038/​nphys2026.

[88] J. Holzgrafe, J. Beitner, D. Kara, H. S. Knowles, M. Atatüre, Error corrected spin-state readout in a nanodiamond, npj Quantum Inf. 5 (1) (2019) 13. doi:10.1038/​s41534-019-0126-2.

[89] E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, M. D. Lukin, Quantum entanglement between an optical photon and a solid-state spin qubit, Nature 466 (7307) (2010) 730–734. doi:10.1038/​nature09256.

[90] C. Bradac, W. Gao, J. Forneris, M. E. Trusheim, I. Aharonovich, Quantum nanophotonics with group IV defects in diamond, Nat. Commun. 10 (1) (2019) 5625. doi:10.1038/​s41467-019-13332-w.

[91] M. E. Trusheim, N. H. Wan, K. C. Chen, C. J. Ciccarino, J. Flick, R. Sundararaman, G. Malladi, E. Bersin, M. Walsh, B. Lienhard, H. Bakhru, P. Narang, D. Englund, Lead-related quantum emitters in diamond, Phys. Rev. B 99 (7) (2019) 075430. doi:10.1103/​PhysRevB.99.075430.

[92] N. H. Wan, T. J. Lu, K. C. Chen, M. P. Walsh, M. E. Trusheim, L. De Santis, E. A. Bersin, I. B. Harris, S. L. Mouradian, I. R. Christen, E. S. Bielejec, D. Englund, Large-scale integration of artificial atoms in hybrid photonic circuits, Nature 583 (7815) (2020) 226–231. doi:10.1038/​s41586-020-2441-3.

[93] K. Kuruma, B. Pingault, C. Chia, D. Renaud, P. Hoffmann, S. Iwamoto, C. Ronning, M. Lončar, Coupling of a single tin-vacancy center to a photonic crystal cavity in diamond, Applied Physics Letters 118 (23) (2021) 230601. doi:10.1063/​5.0051675.

[94] P. Fuchs, T. Jung, M. Kieschnick, J. Meijer, C. Becher, A cavity-based optical antenna for color centers in diamond, APL Photonics 6 (8) (2021) 086102. doi:10.1063/​5.0057161.

[95] C. Hepp, T. Müller, V. Waselowski, J. N. Becker, B. Pingault, H. Sternschulte, D. Steinmüller-Nethl, A. Gali, J. R. Maze, M. Atatüre, C. Becher, Electronic Structure of the Silicon Vacancy Color Center in Diamond, Phys. Rev. Lett. 112 (3) (2014) 036405. doi:10.1103/​PhysRevLett.112.036405.

[96] L. J. Rogers, K. D. Jahnke, M. W. Doherty, A. Dietrich, L. P. McGuinness, C. Müller, T. Teraji, H. Sumiya, J. Isoya, N. B. Manson, F. Jelezko, Electronic structure of the negatively charged silicon-vacancy center in diamond, Phys. Rev. B 89 (23) (2014) 235101. doi:10.1103/​PhysRevB.89.235101.

[97] S. Meesala, Y.-I. Sohn, B. Pingault, L. Shao, H. A. Atikian, J. Holzgrafe, M. Gündoğan, C. Stavrakas, A. Sipahigil, C. Chia, R. Evans, M. J. Burek, M. Zhang, L. Wu, J. L. Pacheco, J. Abraham, E. Bielejec, M. D. Lukin, M. Atatüre, M. Lončar, Strain engineering of the silicon-vacancy center in diamond, Phys. Rev. B 97 (20) (2018) 205444. doi:10.1103/​PhysRevB.97.205444.

[98] Y.-I. Sohn, S. Meesala, B. Pingault, H. A. Atikian, J. Holzgrafe, M. Gündoğan, C. Stavrakas, M. J. Stanley, A. Sipahigil, J. Choi, M. Zhang, J. L. Pacheco, J. Abraham, E. Bielejec, M. D. Lukin, M. Atatüre, M. Lončar, Controlling the coherence of a diamond spin qubit through its strain environment, Nat. Commun. 9 (1) (2018) 2012. doi:10.1038/​s41467-018-04340-3.

[99] A. Gali, J. R. Maze, Ab initio study of the split silicon-vacancy defect in diamond: Electronic structure and related properties, Phys. Rev. B 88 (23) (2013) 235205. doi:10.1103/​PhysRevB.88.235205.

[100] B. Pingault, The silicon-vacancy centre in diamond for quantum information processing, Ph.D. thesis, Cambridge (2017). doi:10.17863/​CAM.15577.

[101] T. H. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, R. Hanson, Universal control and error correction in multi-qubit spin registers in diamond, Nat. Nanotechnol. 9 (3) (2014) 171–176. doi:10.1038/​nnano.2014.2.

[102] I. Schwartz, J. Scheuer, B. Tratzmiller, S. Müller, Q. Chen, I. Dhand, Z.-Y. Wang, C. Müller, B. Naydenov, F. Jelezko, M. B. Plenio, Robust optical polarization of nuclear spin baths using Hamiltonian engineering of nitrogen-vacancy center quantum dynamics, Sci. Adv. 4 (8) (2018) eaat8978. doi:10.1126/​sciadv.aat8978.

[103] K. De Greve, L. Yu, P. L. McMahon, J. S. Pelc, C. M. Natarajan, N. Y. Kim, E. Abe, S. Maier, C. Schneider, M. Kamp, et al., Quantum-dot spin–photon entanglement via frequency downconversion to telecom wavelength, Nature 491 (7424) (2012) 421–425. doi:10.1038/​nature11577.

[104] W. Gao, P. Fallahi, E. Togan, J. Miguel-Sánchez, A. Imamoglu, Observation of entanglement between a quantum dot spin and a single photon, Nature 491 (7424) (2012) 426–430. doi:10.1038/​nature11573.

[105] J. R. Schaibley, A. P. Burgers, G. A. McCracken, L.-M. Duan, P. R. Berman, D. G. Steel, A. S. Bracker, D. Gammon, L. J. Sham, Demonstration of Quantum Entanglement between a Single Electron Spin Confined to an InAs Quantum Dot and a Photon, Phys. Rev. Lett. 110 (16) (2013) 167401. doi:10.1103/​PhysRevLett.110.167401.

[106] R. Vasconcelos, S. Reisenbauer, C. Salter, G. Wachter, D. Wirtitsch, J. Schmiedmayer, P. Walther, M. Trupke, Scalable spin–photon entanglement by time-to-polarization conversion, npj Quantum Inf. 6 (1) (2020) 9. doi:10.1038/​s41534-019-0236-x.

[107] E. A. Chekhovich, S. F. C. da Silva, A. Rastelli, Nuclear spin quantum register in an optically active semiconductor quantum dot, Nat. Nanotechnol. 15 (12) (2020) 999–1004. doi:10.1038/​s41565-020-0769-3.

[108] Z.-H. Wang, G. de Lange, D. Ristè, R. Hanson, V. V. Dobrovitski, Comparison of dynamical decoupling protocols for a nitrogen-vacancy center in diamond, Phys. Rev. B 85 (15) (2012) 155204. doi:10.1103/​PhysRevB.85.155204.

Cited by

[1] Bikun Li, Sophia E. Economou, and Edwin Barnes, “Entangled photon factory: How to generate quantum resource states from a minimal number of quantum emitters”, arXiv:2108.12466.

The above citations are from SAO/NASA ADS (last updated successfully 2021-10-23 14:31:01). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-10-23 14:31:00).

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Dynamically Generated Logical Qubits



Matthew B. Hastings1,2 and Jeongwan Haah2

1Station Q, Microsoft Quantum, Santa Barbara, CA 93106-6105, USA
2Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA

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We present a quantum error correcting code with $textit{dynamically generated logical qubits}$. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a $two$-qubit Pauli measurement.

► BibTeX data

► References

[1] A. Kitaev, “Fault-tolerant quantum computation by anyons,” Annals of Physics 303, 2–30 (2003), arXiv:quant-ph/​9707021.

[2] D. Poulin, “Stabilizer formalism for operator quantum error correction,” Physical Review Letters 95, 230504 (2005), arXiv:quant-ph/​0508131.

[3] S. Bravyi, G. Duclos-Cianci, D. Poulin, and M. Suchara, “Subsystem surface codes with three-qubit check operators,” Quantum Information and Computation 13, 963–985 (2013), arXiv:1207.1443.

[4] H. Bombin, “Topological subsystem codes,” Physical Review A 81, 032301 (2010), arXiv:0908.4246.

[5] D. Bacon, “Operator quantum error-correcting subsystems for self-correcting quantum memories,” Physical Review A 73, 012340 (2006), arXiv:quant-ph/​0506023.

[6] T. Karzig, C. Knapp, R. M. Lutchyn, P. Bonderson, M. B. Hastings, C. Nayak, J. Alicea, K. Flensberg, S. Plugge, Y. Oreg, C. M. Marcus, and M. H. Freedman, “Scalable designs for quasiparticle-poisoning-protected topological quantum computation with majorana zero modes,” Physical Review B 95, 235305 (2017), arXiv:1610.05289.

[7] Y. Li, X. Chen, and M. P. A. Fisher, “Quantum zeno effect and the many-body entanglement transition,” Phys. Rev. B 98, 205136 (2018), arXiv:1808.06134.

[8] B. Skinner, J. Ruhman, and A. Nahum, “Measurement-induced phase transitions in the dynamics of entanglement,” Phys. Rev. X 9, 031009 (2019), arXiv:1808.05953.

[9] M. J. Gullans and D. A. Huse, “Dynamical purification phase transition induced by quantum measurements,” Physical Review X 10, 041020 (2020), arXiv:1905.05195.

[10] A. Kitaev, “Anyons in an exactly solved model and beyond,” Annals of Physics 321, 2–111 (2006), arXiv:cond-mat/​0506438.

[11] K. Kawagoe and M. Levin, “Microscopic definitions of anyon data,” Physical Review B 101, 1910.11353 (2020), arXiv:115113.

[12] S. A. Kivelson, D. S. Rokhsar, and J. P. Sethna, “2e or not 2e : Flux quantization in the resonating valence bond state,” Europhysics Letters (EPL) 6, 353–358 (1988).

[13] L. Fidkowski, J. Haah, and M. B. Hastings, “How dynamical quantum memories forget,” Quantum 5, 382 (2021), arXiv:2008.10611.

Cited by

[1] Craig Gidney, Michael Newman, Austin Fowler, and Michael Broughton, “A Fault-Tolerant Honeycomb Memory”, arXiv:2108.10457.

[2] James R. Wootton, “Hexagonal matching codes with 2-body measurements”, arXiv:2109.13308.

[3] Yaodong Li and Matthew P. A. Fisher, “Robust decoding in monitored dynamics of open quantum systems with Z_2 symmetry”, arXiv:2108.04274.

[4] Edward H. Chen, Theodore J. Yoder, Youngseok Kim, Neereja Sundaresan, Srikanth Srinivasan, Muyuan Li, Antonio D. Córcoles, Andrew W. Cross, and Maika Takita, “Calibrated decoders for experimental quantum error correction”, arXiv:2110.04285.

[5] Christopher A. Pattison, Michael E. Beverland, Marcus P. da Silva, and Nicolas Delfosse, “Improved quantum error correction using soft information”, arXiv:2107.13589.

[6] Christophe Vuillot, “Planar Floquet Codes”, arXiv:2110.05348.

[7] Julia Wildeboer, Thomas Iadecola, and Dominic J. Williamson, “Symmetry-Protected Infinite-Temperature Quantum Memory from Subsystem Codes”, arXiv:2110.05710.

[8] Andrew J. Landahl and Benjamin C. A. Morrison, “Logical Majorana fermions for fault-tolerant quantum simulation”, arXiv:2110.10280.

[9] Jeongwan Haah and Matthew B. Hastings, “Boundaries for the Honeycomb Code”, arXiv:2110.09545.

The above citations are from SAO/NASA ADS (last updated successfully 2021-10-23 13:49:03). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2021-10-23 13:49:01).

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