Gauge color codes in two dimensions

We present a family of quantum error-correcting codes that support a universal set of transversal logic gates using only local operations on a two-dimensional array of physical qubits. The construction is a subsystem version of color codes where gauge fixing through local measurements dynamically determines which gates are transversal. Although the operations are local, the underlying code is not topological in structure, which is how the construction circumvents no-go constraints imposed by the Bravyi-K\"onig and Pastawski-Yoshida theorems. We provide strong evidence that the encoding has no error threshold in the conventional sense, though it is still possible to have logical gates with error probability much lower than that of physical gates.

[1]  D. Leung,et al.  Methodology for quantum logic gate construction , 2000, quant-ph/0002039.

[2]  Ashley M. Stephens,et al.  Efficient fault-tolerant decoding of topological color codes , 2014, 1402.3037.

[3]  David Poulin,et al.  Fault-tolerant conversion between the Steane and Reed-Muller quantum codes. , 2014, Physical review letters.

[4]  Akihiro Munemasa,et al.  On triply even binary codes , 2010, J. Lond. Math. Soc..

[5]  Martin Suchara,et al.  Efficient Algorithms for Maximum Likelihood Decoding in the Surface Code , 2014, 1405.4883.

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

[7]  H. Bombin.  Gauge Color Codes: Optimal Transversal Gates and Gauge Fixing in Topological Stabilizer Codes , 2013, 1311.0879.

[8]  J. Preskill.  Work: Quantum Information and Computation , 1998 .

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

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

[11]  Stephen D. Bartlett,et al.  Stacked codes: Universal fault-tolerant quantum computation in a two-dimensional layout , 2015, 1509.04255.

[12]  Andrew J. Landahl,et al.  Quantum computing by color-code lattice surgery , 2014, 1407.5103.

[13]  Robert Raussendorf,et al.  Fault-tolerant quantum computation with high threshold in two dimensions. , 2007, Physical review letters.

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

[15]  Shor,et al.  Scheme for reducing decoherence in quantum computer memory. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[16]  D. Poulin.  Stabilizer formalism for operator quantum error correction. , 2005, Physical review letters.

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

[18]  Andrew J. Landahl,et al.  Fault-tolerant quantum computing with color codes , 2011, 1108.5738.

[19]  Ashley M. Stephens,et al.  Fault-tolerant thresholds for quantum error correction with the surface code , 2013, 1311.5003.

[20]  Simon J. Devitt,et al.  Surface code implementation of block code state distillation , 2013, Scientific Reports.

[21]  Andrew W. Cross,et al.  Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit , 2008, 0801.2360.

[22]  Andrew W. Cross,et al.  A comparative code study for quantum fault tolerance , 2007, Quantum Inf. Comput..

[23]  Michael E. Beverland,et al.  Universal transversal gates with color codes: A simplified approach , 2014, 1410.0069.

[24]  Panos Aliferis,et al.  Subsystem fault tolerance with the Bacon-Shor code. , 2007, Physical review letters.

[25]  Bryan Eastin,et al.  Restrictions on transversal encoded quantum gate sets. , 2008, Physical review letters.

[26]  A. Steane.  Multiple-particle interference and quantum error correction , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[27]  Nicolas Delfosse,et al.  Decoding color codes by projection onto surface codes , 2013, ArXiv.

[28]  D. Bacon.  Operator quantum error-correcting subsystems for self-correcting quantum memories , 2005, quant-ph/0506023.

[29]  N. C. Jones,et al.  Logic Synthesis for Fault-Tolerant Quantum Computers , 2013, 1310.7290.

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

[31]  Fernando Pastawski,et al.  Fault-tolerant logical gates in quantum error-correcting codes , 2014, 1408.1720.

[32]  Steane,et al.  Simple quantum error-correcting codes. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[33]  Sergey Bravyi,et al.  Classification of topologically protected gates for local stabilizer codes. , 2012, Physical review letters.

[34]  H. Bombin,et al.  Single-Shot Fault-Tolerant Quantum Error Correction , 2014, 1404.5504.

[35]  Andrew W. Cross,et al.  Doubled Color Codes , 2015, 1509.03239.

[36]  Steane,et al.  Error Correcting Codes in Quantum Theory. , 1996, Physical review letters.

[37]  Benjamin J. Brown,et al.  Fault-tolerant error correction with the gauge color code , 2015, Nature Communications.

[38]  David Poulin,et al.  Unified and generalized approach to quantum error correction. , 2004, Physical review letters.

[39]  Isaac L. Chuang,et al.  Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations , 1999, Nature.

[40]  Austin G. Fowler,et al.  Surface code quantum computing by lattice surgery , 2011, 1111.4022.

[41]  E. Knill.  Quantum computing with realistically noisy devices , 2005, Nature.