Magic State Distillation: Not as Costly as You Think

Daniel Litinski

Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany

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Abstract

Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases, it can cost less to distill a magic state than to perform a logical Clifford gate on full-distance logical qubits.

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► References

[1] J. Preskill, Reliable quantum computers, Proc. Roy. Soc. Lond. A 454, 385 (1998).
https:/​/​doi.org/​10.1098/​rspa.1998.0167

[2] B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015).
https:/​/​doi.org/​10.1103/​RevModPhys.87.307

[3] E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017).
https:/​/​doi.org/​10.1038/​nature23460

[4] A. Y. Kitaev, Fault-tolerant quantum computation by anyons, Ann. Phys. 303, 2 (2003).
https:/​/​doi.org/​10.1016/​S0003-4916(02)00018-0

[5] 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).
https:/​/​doi.org/​10.1103/​PhysRevA.86.032324

[6] C. Horsman, A. G. Fowler, S. Devitt, and R. V. Meter, Surface code quantum computing by lattice surgery, New J. Phys. 14, 123011 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​12/​123011

[7] D. Litinski and F. v. Oppen, Lattice Surgery with a Twist: Simplifying Clifford Gates of Surface Codes, Quantum 2, 62 (2018).
https:/​/​doi.org/​10.22331/​q-2018-05-04-62

[8] A. G. Fowler and C. Gidney, Low overhead quantum computation using lattice surgery, arXiv:1808.06709 (2018).
arXiv:1808.06709

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

[10] R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. McClean, A. Paler, A. Fowler, and H. Neven, Encoding electronic spectra in quantum circuits with linear T complexity, Phys. Rev. X 8, 041015 (2018).
https:/​/​doi.org/​10.1103/​PhysRevX.8.041015

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

[12] A. G. Fowler, S. J. Devitt, and C. Jones, Surface code implementation of block code state distillation, Scientific Rep. 3, 1939 (2013).
https:/​/​doi.org/​10.1038/​srep01939

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

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

[15] G. Duclos-Cianci and K. M. Svore, Distillation of nonstabilizer states for universal quantum computation, Phys. Rev. A 88, 042325 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.042325

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

[17] 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 (2017a).
https:/​/​doi.org/​10.1103/​PhysRevA.95.022316

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

[19] J. Haah and M. B. Hastings, Codes and Protocols for Distilling $T$, controlled-$S$, and Toffoli Gates, Quantum 2, 71 (2018).
https:/​/​doi.org/​10.22331/​q-2018-06-07-71

[20] E. T. Campbell and M. Howard, Magic state parity-checker with pre-distilled components, Quantum 2, 56 (2018).
https:/​/​doi.org/​10.22331/​q-2018-03-14-56

[21] C. Gidney and A. G. Fowler, Efficient magic state factories with a catalyzed $|CCZ\rangle$ to $2|T\rangle$ transformation, Quantum 3, 135 (2019).
https:/​/​doi.org/​10.22331/​q-2019-04-30-135

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

[23] S. Bravyi and A. Cross, Doubled color codes, arXiv:1509.03239 (2015).
arXiv:1509.03239

[24] 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).
https:/​/​doi.org/​10.1103/​PhysRevA.93.022323

[25] H. Bombin, 2D quantum computation with 3D topological codes, arXiv:1810.09571 (2018).
arXiv:1810.09571

[26] C. Chamberland and A. W. Cross, Fault-tolerant magic state preparation with flag qubits, Quantum 3, 143 (2019).
https:/​/​doi.org/​10.22331/​q-2019-05-20-143

[27] B. J. Brown, A fault-tolerant non-Clifford gate for the surface code in two dimensions, arXiv:1903.11634 (2019).
arXiv:1903.11634

[28] D. Litinski, A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery, Quantum 3, 128 (2019).
https:/​/​doi.org/​10.22331/​q-2019-03-05-128

[29] M. Amy and M. Mosca, T-count optimization and Reed-Muller codes, IEEE Transactions on Information Theory , 1 (2019).
https:/​/​doi.org/​10.1109/​TIT.2019.2906374

[30] J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, Magic state distillation with low space overhead and optimal asymptotic input count, Quantum 1, 31 (2017).
https:/​/​doi.org/​10.22331/​q-2017-10-03-31

[31] A. J. Landahl and C. Ryan-Anderson, Quantum computing by color-code lattice surgery, arXiv:1407.5103 (2014).
arXiv:1407.5103

[32] Y. Li, A magic state’s fidelity can be superior to the operations that created it, New J. Phys. 17, 023037 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​2/​023037

[33] J. Lodyga, P. Mazurek, A. Grudka, and M. Horodecki, Simple scheme for encoding and decoding a qubit in unknown state for various topological codes, Scientific Rep. 5, 8975 (2015).
https:/​/​doi.org/​10.1038/​srep08975

[34] L. Lao et al., Preparing high-fidelity magic states with low costs, in preparation.

[35] Cramming more power into a quantum device, https:/​/​www.ibm.com/​blogs/​research/​2019/​03/​ power-quantum-device/​, accessed: 2019-05-09.
https:/​/​www.ibm.com/​blogs/​research/​2019/​03/​power-quantum-device/​

[36] K. Wright, K. Beck, S. Debnath, J. Amini, Y. Nam, N. Grzesiak, J.-S. Chen, N. Pisenti, M. Chmielewski, C. Collins, et al., Benchmarking an 11-qubit quantum computer, arXiv:1903.08181 (2019).
arXiv:1903.08181

[37] The Python script and Mathematica notebook can be found on GitHub, see https:/​/​github.com/​litinski/​magicstates.
https:/​/​github.com/​litinski/​magicstates

[38] E. T. Campbell and M. Howard, Unifying gate synthesis and magic state distillation, Phys. Rev. Lett. 118, 060501 (2017b).
https:/​/​doi.org/​10.1103/​PhysRevLett.118.060501

[39] P. Selinger, Quantum circuits of $T$-depth one, Phys. Rev. A 87, 042302 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.042302

[40] C. Jones, Low-overhead constructions for the fault-tolerant Toffoli gate, Phys. Rev. A 87, 022328 (2013b).
https:/​/​doi.org/​10.1103/​PhysRevA.87.022328

[41] C. Gidney, Halving the cost of quantum addition, Quantum 2, 74 (2018).
https:/​/​doi.org/​10.22331/​q-2018-06-18-74

[42] B. J. Brown and S. Roberts, Universal fault-tolerant measurement-based quantum computation, arXiv:1811.11780 (2018).
arXiv:1811.11780

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[1] Michael Hanks, Marta P. Estarellas, William J. Munro, and Kae Nemoto, "Effective Compression of Quantum Braided Circuits Aided by ZX-Calculus", Physical Review X 10 4, 041030 (2020).

[2] Craig Gidney and Martin Ekerå, "How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits", arXiv:1905.09749, Quantum 5, 433 (2021).

[3] M. Lostaglio and A. Ciani, "Error Mitigation and Quantum-Assisted Simulation in the Error Corrected Regime", Physical Review Letters 127 20, 200506 (2021).

[4] T. R. Scruby, D. E. Browne, P. Webster, and M. Vasmer, "Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code", Quantum 6, 721 (2022).

[5] Lingling Lao and Ben Criger, Proceedings of the 19th ACM International Conference on Computing Frontiers 113 (2022) ISBN:9781450393386.

[6] Casey Duckering, Jonathan M. Baker, David I. Schuster, and Frederic T. Chong, 2020 53rd Annual IEEE/ACM International Symposium on Microarchitecture (MICRO) 173 (2020) ISBN:978-1-7281-7383-2.

[7] Jeongwan Haah and Matthew B. Hastings, "Measurement sequences for magic state distillation", arXiv:2007.07929, Quantum 5, 383 (2021).

[8] Mikkel V. Larsen, Christopher Chamberland, Kyungjoo Noh, Jonas S. Neergaard-Nielsen, and Ulrik L. Andersen, "Fault-Tolerant Continuous-Variable Measurement-based Quantum Computation Architecture", PRX Quantum 2 3, 030325 (2021).

[9] Principles of Superconducting Quantum Computers 327 (2022) ISBN:9781119750727.

[10] Paul Webster and Stephen D. Bartlett, "Fault-tolerant quantum gates with defects in topological stabilizer codes", Physical Review A 102 2, 022403 (2020).

[11] Thomas Häner, Damian S. Steiger, Torsten Hoefler, and Matthias Troyer, Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis 1 (2021) ISBN:9781450384421.

[12] Gökhan Torun, Hüseyin Talha Şenyaşa, and Ali Yildiz, "Resource theory of superposition: State transformations", Physical Review A 103 3, 032416 (2021).

[13] Isaac H. Kim, Ye-Hua Liu, Sam Pallister, William Pol, Sam Roberts, and Eunseok Lee, "Fault-tolerant resource estimate for quantum chemical simulations: Case study on Li-ion battery electrolyte molecules", Physical Review Research 4 2, 023019 (2022).

[14] Lan Luo, Zhi Ma, Dongdai Lin, and Hong Wang, "Fault-tolerance thresholds for code conversion schemes with quantum Reed–Muller codes", Quantum Science and Technology 5 4, 045022 (2020).

[15] 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).

[16] Mark Webber, Vincent Elfving, Sebastian Weidt, and Winfried K. Hensinger, "The impact of hardware specifications on reaching quantum advantage in the fault tolerant regime", AVS Quantum Science 4 1, 013801 (2022).

[17] Christopher Chamberland and Earl T. Campbell, "Universal Quantum Computing with Twist-Free and Temporally Encoded Lattice Surgery", PRX Quantum 3 1, 010331 (2022).

[18] Joonho Lee, Dominic W. Berry, Craig Gidney, William J. Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan Babbush, "Even More Efficient Quantum Computations of Chemistry Through Tensor Hypercontraction", PRX Quantum 2 3, 030305 (2021).

[19] Nikolaos Koukoulekidis and David Jennings, "Constraints on magic state protocols from the statistical mechanics of Wigner negativity", npj Quantum Information 8 1, 42 (2022).

[20] 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).

[21] Hao Dai, Shuangshuang Fu, and Shunlong Luo, "Detecting Magic States via Characteristic Functions", International Journal of Theoretical Physics 61 2, 35 (2022).

[22] Campbell K. McLauchlan and Benjamin Béri, "Fermion-Parity-Based Computation and Its Majorana-Zero-Mode Implementation", Physical Review Letters 128 18, 180504 (2022).

[23] Jérémie Guillaud and Mazyar Mirrahimi, "Error rates and resource overheads of repetition cat qubits", Physical Review A 103 4, 042413 (2021).

[24] Manoj G. Gowda and Pradeep Kiran Sarvepalli, "Color codes with twists: Construction and universal-gate-set implementation", Physical Review A 104 1, 012603 (2021).

[25] Michael E. Beverland, Aleksander Kubica, and Krysta M. Svore, "Cost of Universality: A Comparative Study of the Overhead of State Distillation and Code Switching with Color Codes", PRX Quantum 2 2, 020341 (2021).

[26] Y. Herasymenko and T.E. O'Brien, "A diagrammatic approach to variational quantum ansatz construction", Quantum 5, 596 (2021).

[27] Sam McArdle, Suguru Endo, Alán Aspuru-Guzik, Simon C. Benjamin, and Xiao Yuan, "Quantum computational chemistry", arXiv:1808.10402, Reviews of Modern Physics 92 1, 015003 (2020).

[28] Julio Carlos Magdalena de la Fuente, Nicolas Tarantino, and Jens Eisert, "Non-Pauli topological stabilizer codes from twisted quantum doubles", Quantum 5, 398 (2021).

[29] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T. Campbell, Connor T. Hann, Joseph Iverson, Harald Putterman, Thomas C. Bohdanowicz, Steven T. Flammia, Andrew Keller, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão, "Building a Fault-Tolerant Quantum Computer Using Concatenated Cat Codes", PRX Quantum 3 1, 010329 (2022).

[30] Christophe Piveteau, David Sutter, Sergey Bravyi, Jay M. Gambetta, and Kristan Temme, "Error Mitigation for Universal Gates on Encoded Qubits", Physical Review Letters 127 20, 200505 (2021).

[31] Shraddha Singh, Andrew S. Darmawan, Benjamin J. Brown, and Shruti Puri, "High-fidelity magic-state preparation with a biased-noise architecture", Physical Review A 105 5, 052410 (2022).

[32] Paul Webster, Michael Vasmer, Thomas R. Scruby, and Stephen D. Bartlett, "Universal fault-tolerant quantum computing with stabilizer codes", Physical Review Research 4 1, 013092 (2022).

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