Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count

Jeongwan Haah1, Matthew B. Hastings2,1, D. Poulin3, and D. Wecker1

1Quantum Architectures and Computation, Microsoft Research, Redmond, WA 98052, USA
2Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA
3Département de Physique & Institut Quantique, Université de Sherbrooke, Quebec, Canada

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.

Share

► BibTeX data

► References

[1] 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.
https://doi.org/10.1109/SFCS.1996.548464
arXiv:quant-ph/9605011

[2] D. Aharonov and M. Ben-Or, Fault tolerant quantum computation with constant error, SIAM J. Comput. 38, 1207-1282 (2008), quant-ph/​9611025v2.
https://doi.org/10.1137/S0097539799359385
arXiv:quant-ph/9611025v2

[3] E. Knill, R. Laflamme, and W. Zurek, Threshold accuracy for quantum computation, (1996), quant-ph/​9610011.
arXiv:quant-ph/9610011

[4] D. Gottesman, A class of quantum error-correcting codes saturating the quantum hamming bound, Phys. Rev. A 54, 1862 (1996), quant-ph/​9604038.
https://doi.org/10.1103/PhysRevA.54.1862
arXiv:quant-ph/9604038

[5] 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.
https://doi.org/10.1103/PhysRevLett.78.405
arXiv:quant-ph/9605005

[6] 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.
https://doi.org/10.1103/physrevx.2.031007
arXiv:1010.5022

[7] J. O'Gorman and E. T. Campbell, Quantum computation with realistic magic state factories, 1605.07197v2.
arXiv:1605.07197v2

[8] E. Knill, Fault-tolerant postselected quantum computation: Schemes, (2004a), quant-ph/​0402171v1.
arXiv:quant-ph/0402171v1

[9] E. Knill, Fault-tolerant postselected quantum computation: Threshold analysis, (2004b), quant-ph/​0404104v1.
arXiv:quant-ph/0404104v1

[10] S. Bravyi and A. Kitaev, Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005), quant-ph/​0403025.
https://doi.org/10.1103/PhysRevA.71.022316
arXiv:quant-ph/0403025

[11] A. M. Meier, B. Eastin, and E. Knill, Magic-state distillation with the four-qubit code, Quant. Inf. Comp. 13, 195 (2013), 1204.4221.
arXiv:1204.4221

[12] S. Bravyi and J. Haah, Magic state distillation with low overhead, Phys. Rev. A 86, 052329 (2012), 1209.2426.
https://doi.org/10.1103/PhysRevA.86.052329
arXiv:1209.2426

[13] C. Jones, Multilevel distillation of magic states for quantum computing, Phys. Rev. A 87, 042305 (2013a), 1210.3388v2.
https://doi.org/10.1103/PhysRevA.87.042305
arXiv:1210.3388v2

[14] G. Duclos-Cianci and D. Poulin, Reducing the quantum computing overhead with complex gate distillation, Phys. Rev. A 91, 042315 (2015), 1403.5280v1.
https://doi.org/10.1103/PhysRevA.91.042315
arXiv:1403.5280v1

[15] I. L. Chuang and D. Gottesman, Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, Nature 402, 390-393 (1999).
https://doi.org/10.1038/46503

[16] A. J. Landahl and C. Cesare, Complex instruction set computing architecture for performing accurate quantum $Z$ rotations with less magic, (2013), 1302.3240.
arXiv:1302.3240

[17] 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.
https://doi.org/10.1103/PhysRevLett.111.090505
arXiv:1304.3709

[18] 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.
https://doi.org/10.1103/PhysRevLett.112.010505
arXiv:1309.3310

[19] 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.
https://doi.org/10.1103/PhysRevLett.113.080501
arXiv:1403.2734

[20] H. Bombin, Gauge color codes: Optimal transversal gates and gauge fixing in topological stabilizer codes, New J.Phys. 17, 083002 (2015), 1311.0879v6.
https://doi.org/10.1088/1367-2630/17/8/083002
arXiv:1311.0879v6

[21] S. Bravyi and A. Cross, Doubled color codes, 1509.03239v1.
arXiv:1509.03239v1

[22] 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.
https://doi.org/10.1103/PhysRevA.93.022323
arXiv:1509.04255

[23] C. Jones, P. Brooks, and J. Harrington, Gauge color codes in two dimensions, Phys. Rev. A 93, 052332 (2016), 1512.04193.
https://doi.org/10.1103/PhysRevA.93.052332
arXiv:1512.04193

[24] H. Bombin, Dimensional jump in quantum error correction, New Journal of Physics 18, 043038 (2016), 1412.5079v3.
https://doi.org/10.1088/1367-2630/18/4/043038
arXiv:1412.5079v3

[25] 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.
https://doi.org/10.1103/physrevx.6.031039
arXiv:1603.03948

[26] B. Eastin and E. Knill, Restrictions on transversal encoded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009), 0811.4262.
https://doi.org/10.1103/PhysRevLett.102.110502
arXiv:0811.4262

[27] S. Bravyi and R. König, Classification of topologically protected gates for local stabilizer codes, Phys. Rev. Lett 110, 170503 (2013), 1206.1609.
https://doi.org/10.1103/PhysRevLett.110.170503
arXiv:1206.1609

[28] 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.
https://doi.org/10.1088/2058-9565/1/1/015007
arXiv:1603.04230v2

[29] B. W. Reichardt, Improved magic states distillation for quantum universality, Quant. Inf. Proc. 4, 251-264 (2005), quant-ph/​0411036v1.
https://doi.org/10.1007/s11128-005-7654-8
arXiv:quant-ph/0411036v1

[30] A. M. Steane, Error correcting codes in quantum theory, Physical Review Letters 77, 793-797 (1996).
https://doi.org/10.1103/physrevlett.77.793

[31] H. Bombin and M. A. Martin-Delgado, Topological quantum distillation, Physical Review Letters 97, 180501 (2006), quant-ph/​0605138.
https://doi.org/10.1103/physrevlett.97.180501
arXiv:quant-ph/0605138

[32] B. Eastin, Distilling one-qubit magic states into toffoli states, Phys. Rev. A 87, 032321 (2013), 1212.4872v2.
https://doi.org/10.1103/PhysRevA.87.032321
arXiv:1212.4872v2

[33] C. Jones, Low-overhead constructions for the fault-tolerant toffoli gate, Physical Review A 87, 022328 (2013b), 1212.5069.
https://doi.org/10.1103/physreva.87.022328
arXiv:1212.5069

[34] 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.
https://doi.org/10.1103/PhysRevA.95.022316
arXiv:1606.01904

[35] S. Bravyi, B. Leemhuis, and B. M. Terhal, Majorana fermion codes, New J.Phys. 12, 083039 (2010), 1004.3791.
https://doi.org/10.1088/1367-2630/12/8/083039
arXiv:1004.3791

[36] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Phys. Rev. A 54, 1098-1105 (1996), quant-ph/​9512032.
https://doi.org/10.1103/PhysRevA.54.1098
arXiv:quant-ph/9512032

[37] D. Gottesman, Stabilizer codes and quantum error correction, (1997), quant-ph/​9705052.
arXiv:quant-ph/9705052

[38] J. Preskill, Lecture notes on quantum computation, Caltech Ph219.
http:/​/​www.theory.caltech.edu/​people/​preskill/​ph229/​notes/​chap7.pdf

[39] M. Sipser and D. A. Spielman, Expander codes, IEEE Transactions on Information Theory 42, 1710-1722 (1996).
https://doi.org/10.1109/18.556667

[40] 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).
https://doi.org/10.1006/jctb.1995.1033

[41] 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.
https://doi.org/10.1088/1367-2630/9/6/199
arXiv:quant-ph/0703143v1

[42] 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.
https://doi.org/10.1103/PhysRevA.86.032324
arXiv:1208.0928

[43] J. O'Gorman and E. T. Campbell, Quantum computation with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017), 1605.07197.
https://doi.org/10.1103/PhysRevA.95.032338
arXiv:1605.07197

[44] M. Grassl and T. Beth, Quantum bch codes, in Proceedings X. International Symposium on Theoretical Electrical Engineering, Magdeburg (1999) pp. 207-212, quant-ph/​9910060.
arXiv:quant-ph/9910060

[45] J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, Magic state distillation at intermediate size, 1709.02789.
arXiv:1709.02789

[46] M. B. Hastings, Small majorana fermion codes, 1703.00612.
arXiv:1703.00612

[47] 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.
https://doi.org/10.1103/PhysRevX.2.041021
arXiv:1205.3104

[48] E. T. Campbell, Enhanced fault-tolerant quantum computing in $d$-level systems, Phys. Rev. Lett. 113, 230501 (2014), 1406.3055.
https://doi.org/10.1103/PhysRevLett.113.230501
arXiv:1406.3055

[49] 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.
https://doi.org/10.1023/a:1011233615437
arXiv:math/0001038

[50] G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory (Springer-Verlag, 2006).
https://doi.org/10.1007/3-540-30731-1

[51] E. F. Assmus and J. D. Key, Designs and their Codes, 103 (Cambridge University Press, 1992).

[52] S. Vijay and L. Fu, Quantum error correction for complex and majorana fermion qubits, 1703.00459.
arXiv:1703.00459

[53] S. Lang, Algebra, revised 3rd ed. (Springer, 2002).