Codes and Protocols for Distilling T, controlled-S, and Toffoli Gates

We present several different codes and protocols to distill $T$, controlled-$S$, and Toffoli (or $CCZ$) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal $T$. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency $\gamma\rightarrow 1$. We also present a Reed-Muller based construction of these codes which obtains a worse $\gamma$ but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of $CCZ$ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of 1703.07847. Several examples, including a Reed-Muller code for $T$-to-Toffoli distillation, punctured Reed-Muller codes for $T$-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a $512$ T-gate to $10$ Toffoli gate code with distance $8$ as well as triorthogonal codes with parameters $[[887,137,5]],[[912,112,6]],[[937,87,7]]$ with very low prefactors in front of the leading order error terms in those codes.

[1]  Adam Paetznick,et al.  Universal fault-tolerant quantum computation with only transversal gates and error correction. , 2013, Physical review letters.

[2]  Barbara M. Terhal,et al.  Majorana fermion codes , 2010, 1004.3791.

[3]  Emanuel Knill,et al.  Magic-state distillation with the four-qubit code , 2012, Quantum Inf. Comput..

[4]  Joseph M Renes,et al.  Efficient polar coding of quantum information. , 2011, Physical review letters.

[5]  Cody Jones,et al.  Distillation protocols for Fourier states in quantum computing , 2013, Quantum Inf. Comput..

[6]  Harold N. Ward.  Weight polarization and divisibility , 1990, Discret. Math..

[7]  E. Knill.  Fault-Tolerant Postselected Quantum Computation: Threshold Analysis , 2004 .

[8]  Andreas Klappenecker,et al.  Optimal realizations of simplified Toffoli gates , 2004, Quantum Inf. Comput..

[9]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[10]  A. Calderbank,et al.  Quantum Error Correction and Orthogonal Geometry , 1996, quant-ph/9605005.

[11]  Matthew B. Hastings.  Small Majorana fermion codes , 2017, Quantum Inf. Comput..

[12]  Mark Howard,et al.  Unifying Gate Synthesis and Magic State Distillation. , 2016, Physical review letters.

[13]  A. Kitaev,et al.  Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages) , 2004, quant-ph/0403025.

[14]  Gottesman.  Class of quantum error-correcting codes saturating the quantum Hamming bound. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[15]  T. Beth,et al.  Quantum BCH Codes , 1999, quant-ph/9910060.

[16]  Cody Jones,et al.  Low-overhead constructions for the fault-tolerant Toffoli gate , 2012, 1212.5069.

[17]  E. Knill.  Fault-Tolerant Postselected Quantum Computation: Schemes , 2004, quant-ph/0402171.

[18]  S. Bravyi,et al.  Magic-state distillation with low overhead , 2012, 1209.2426.

[19]  Liang Fu,et al.  Quantum Error Correction for Complex and Majorana Fermion Qubits , 2017, 1703.00459.

[20]  Jeongwan Haah,et al.  Magic state distillation with low space overhead and optimal asymptotic input count , 2017, 1703.07847.

[21]  Earl T. Campbell,et al.  Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost , 2016, 1606.01904.

[22]  N. J. A. Sloane,et al.  Restrictions on Weight Distribution of Reed-Muller Codes , 1969, Inf. Control..

[23]  Bryan Eastin,et al.  Distilling one-qubit magic states into Toffoli states , 2012, 1212.4872.

[24]  Cody Jones,et al.  Composite Toffoli gate with two-round error detection , 2013, 1303.6971.

[25]  Gábor Tardos,et al.  A constructive proof of the general lovász local lemma , 2009, JACM.

[26]  Erdal Arikan,et al.  Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input Memoryless Channels , 2008, IEEE Transactions on Information Theory.

[27]  N. J. A. Sloane,et al.  Weight enumerator for second-order Reed-Muller codes , 1970, IEEE Trans. Inf. Theory.

[28]  M. Mariantoni,et al.  Surface codes: Towards practical large-scale quantum computation , 2012, 1208.0928.

[29]  Cody Jones,et al.  Multilevel distillation of magic states for quantum computing , 2012, 1210.3388.

[30]  M. Freedman,et al.  Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes , 2016, 1610.05289.

[31]  Shor,et al.  Good quantum error-correcting codes exist. , 1995, Physical review. A, Atomic, molecular, and optical physics.