We present an infinite family of protocols to distill magic states for $T$-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the $T$-gate depth of the circuit scale linearly in the logarithm of the target error $\delta$ (up to $\log \log 1/\delta$). Unlike other distillation protocols, this protocol achieves this performance without concatenation and the input magic states are injected at various steps in the circuit rather than all at the start of the circuit. The protocol can be modified to distill magic states for other gates at the third level of the Clifford hierarchy, with the same asymptotic performance. The protocol relies on the construction of weakly self-dual CSS codes with many logical qubits and large distance, allowing us to implement control-SWAPs on multiple qubits. We call this code the "inner code". The control-SWAPs are then used to measure properties of the magic state and detect errors, using another code that we call the "outer code". Alternatively, we use weakly-self dual CSS codes which implement controlled Hadamards for the inner code, reducing circuit depth. We present several specific small examples of this protocol.
 P. W. Shor, Fault-tolerant quantum computation, in Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on (1996) pp. 56–65, quant-ph/9605011.
 D. Aharonov and M. Ben-Or, Fault tolerant quantum computation with constant error, SIAM J. Comput. 38, 1207–1282 (2008), quant-ph/9611025v2.
 D. Gottesman, A class of quantum error-correcting codes saturating the quantum hamming bound, Phys. Rev. A 54, 1862 (1996), quant-ph/9604038.
 A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett. 78, 405–408 (1997), quant-ph/9605005.
 N. C. Jones, R. V. Meter, A. G. Fowler, P. L. McMahon, J. Kim, T. D. Ladd, and Y. Yamamoto, Layered architecture for quantum computing, Physical Review X 2, 031007 (2012), 1010.5022.
 S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005), quant-ph/0403025.
 A. M. Meier, B. Eastin, and E. Knill, Magic-state distillation with the four-qubit code, Quant. Inf. Comp. 13, 195 (2013), 1204.4221.
 G. Duclos-Cianci and D. Poulin, Reducing the quantum computing overhead with complex gate distillation, Phys. Rev. A 91, 042315 (2015), 1403.5280v1.
 I. L. Chuang and D. Gottesman, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature 402, 390–393 (1999).
 A. Paetznick and B. W. Reichardt, Universal fault-tolerant quantum computation with only transversal gates and error correction, Phys. Rev. Lett. 111, 090505 (2013), 1304.3709.
 T. Jochym-O'Connor and R. Laflamme, Using concatenated quantum codes for universal fault-tolerant quantum gates, Phys. Rev. Lett. 112, 010505 (2014), 1309.3310.
 J. T. Anderson, G. Duclos-Cianci, and D. Poulin, Fault-tolerant conversion between the steane and reed-muller quantum codes, Phys. Rev. Lett. 113, 080501 (2014), 1403.2734.
 H. Bombin, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J.Phys. 17, 083002 (2015), 1311.0879v6.
 T. Jochym-O'Connor and S. D. Bartlett, Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout, Phys. Rev. A 93, 022323 (2016), 1509.04255.
 H. Bombin, Dimensional jump in quantum error correction, New Journal of Physics 18, 043038 (2016), 1412.5079v3.
 T. J. Yoder, R. Takagi, and I. L. Chuang, Universal fault-tolerant gates on concatenated stabilizer codes, Physical Review X 6, 031039 (2016), 1603.03948.
 B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009), 0811.4262.
 S. Bravyi and R. König, Classification of topologically protected gates for local stabilizer codes, Phys. Rev. Lett 110, 170503 (2013), 1206.1609.
 E. T. Campbell and J. O'Gorman, An efficient magic state approach to small angle rotations, Quantum Science and Technology 1, 015007 (2016), 1603.04230v2.
 B. W. Reichardt, Improved magic states distillation for quantum universality, Quant. Inf. Proc. 4, 251–264 (2005), quant-ph/0411036v1.
 H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Physical Review Letters 97, 180501 (2006), quant-ph/0605138.
 E. T. Campbell and M. Howard, Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost, Phys. Rev. A 95, 022316 (2017), 1606.01904.
 S. Bravyi, B. Leemhuis, and B. M. Terhal, Majorana fermion codes, New J.Phys. 12, 083039 (2010), 1004.3791.
 A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098–1105 (1996), quant-ph/9512032.
 J. Preskill, Lecture notes on quantum computation, Caltech Ph219.
 Z. Furedi, F. Lazebnik, A. Seress, V. A. Ustimenko, and A. J. Woldar, Graphs of prescribed girth and bi-degree, Journal of Combinatorial Theory, Series B 64, 228–239 (1995).
 R. Raussendorf, J. Harrington, and K. Goyal, Topological fault-tolerance in cluster state quantum computation, New Journal of Physics 9, 199 (2007), quant-ph/0703143v1.
 A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Surface codes: Towards practical large-scale quantum computation, Phys. Rev. A 86, 032324 (2012), 1208.0928.
 J. O'Gorman and E. T. Campbell, Quantum computation with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017), 1605.07197.
 E. T. Campbell, H. Anwar, and D. E. Browne, Magic state distillation in all prime dimensions using quantum reed-muller codes, Phys. Rev. X 2, 041021 (2012), 1205.3104.
 E. T. Campbell, Enhanced fault-tolerant quantum computing in $d$-level systems, Phys. Rev. Lett. 113, 230501 (2014), 1406.3055.
 G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the clifford groups, Designs, Codes and Cryptography 24, 99–122 (2001), math/0001038.
 E. F. Assmus and J. D. Key, Designs and their Codes, 103 (Cambridge University Press, 1992).
 S. Lang, Algebra, revised 3rd ed. (Springer, 2002).
 Jeongwan Haah, "Towers of generalized divisible quantum codes", Physical Review A 97 4, 042327 (2018).
 Kun Fang and Zi-Wen Liu, "No-Go Theorems for Quantum Resource Purification", Physical Review Letters 125 6, 060405 (2020).
 Christopher Chamberland and Kyungjoo Noh, "Very low overhead fault-tolerant magic state preparation using redundant ancilla encoding and flag qubits", npj Quantum Information 6 1, 91 (2020).
 Ilkwon Sohn, Jeongho Bang, and Jun Heo, "Dynamic Concatenation of Quantum Error Correction in Integrated Quantum Computing Architecture", Scientific Reports 9 1, 3302 (2019).
 Earl T. Campbell and Mark Howard, "Magic state parity-checker with pre-distilled components", Quantum 2, 56 (2018).
 Akalank Jain and Shiroman Prakash, "Qutrit and ququint magic states", Physical Review A 102 4, 042409 (2020).
 Jeongwan Haah and Matthew B. Hastings, "Codes and Protocols for Distilling T, controlled-S, and Toffoli Gates", Quantum 2, 71 (2018).
 Christina Knapp, Eric M. Spanton, Andrea F. Young, Chetan Nayak, and Michael P. Zaletel, "Fractional Chern insulator edges and layer-resolved lattice contacts", Physical Review B 99 8, 081114 (2019).
 Michael Beverland, Earl Campbell, Mark Howard, and Vadym Kliuchnikov, "Lower bounds on the non-Clifford resources for quantum computations", Quantum Science and Technology 5 3, 035009 (2020).
 Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, and Damian Markham, "Fault-tolerant quantum speedup from constant depth quantum circuits", Physical Review Research 2 3, 033444 (2020).
 Christopher Chamberland and Andrew W. Cross, "Fault-tolerant magic state preparation with flag qubits", Quantum 3, 143 (2019).
 David Poulin, Alexei Kitaev, Damian S. Steiger, Matthew B. Hastings, and Matthias Troyer, "Quantum Algorithm for Spectral Measurement with a Lower Gate Count", Physical Review Letters 121 1, 010501 (2018).
 Diego Ristè, Luke C. G. Govia, Brian Donovan, Spencer D. Fallek, William D. Kalfus, Markus Brink, Nicholas T. Bronn, and Thomas A. Ohki, "Real-time processing of stabilizer measurements in a bit-flip code", npj Quantum Information 6 1, 71 (2020).
 Daniel Litinski, "Magic State Distillation: Not as Costly as You Think", Quantum 3, 205 (2019).
 Hayata Yamasaki, Takaya Matsuura, and Masato Koashi, "Cost-reduced all-Gaussian universality with the Gottesman-Kitaev-Preskill code: Resource-theoretic approach to cost analysis", Physical Review Research 2 2, 023270 (2020).
 J. Eli Bourassa, Rafael N. Alexander, Michael Vasmer, Ashlesha Patil, Ilan Tzitrin, Takaya Matsuura, Daiqin Su, Ben Q. Baragiola, Saikat Guha, Guillaume Dauphinais, Krishna K. Sabapathy, Nicolas C. Menicucci, and Ish Dhand, "Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer", Quantum 5, 392 (2021).
 Torsten Karzig, Yuval Oreg, Gil Refael, and Michael H. Freedman, "Robust Majorana magic gates via measurements", Physical Review B 99 14, 144521 (2019).
 James R. Seddon and Earl T. Campbell, "Quantifying magic for multi-qubit operations", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 2227, 20190251 (2019).
 Kyungjoo Noh, Liang Jiang, and Bill Fefferman, "Efficient classical simulation of noisy random quantum circuits in one dimension", Quantum 4, 318 (2020).
 Jeongwan Haah, Lukasz Fidkowski, and Matthew B. Hastings, "Nontrivial Quantum Cellular Automata in Higher Dimensions", arXiv:1812.01625.
 Matthew B. Hastings and Jeongwan Haah, "Distillation with Sublogarithmic Overhead", Physical Review Letters 120 5, 050504 (2018).
 Jeongwan Haah, "Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D", arXiv:1907.02075.
 Patrick Rall, "Fractal Properties of Magic State Distillation", arXiv:1708.09256.
 Adam Holmes, Yongshan Ding, Ali Javadi-Abhari, Diana Franklin, Margaret Martonosi, and Frederic T. Chong, "Resource Optimized Quantum Architectures for Surface Code Implementations of Magic-State Distillation", arXiv:1904.11528.
 Narayanan Rengaswamy, Robert Calderbank, Swanand Kadhe, and Henry D. Pfister, "Logical Clifford Synthesis for Stabilizer Codes", arXiv:1907.00310.
The above citations are from Crossref's cited-by service (last updated successfully 2021-04-21 17:50:04) and SAO/NASA ADS (last updated successfully 2021-04-21 17:50:05). The list may be incomplete as not all publishers provide suitable and complete citation data.
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.