Transversality Versus Universality for Additive Quantum Codes

Logic gates can be performed on data encoded in quantum code blocks such that errors introduced by faulty gates can be corrected. The important class of transversal gates acts bitwise between corresponding qubits of code blocks and thus limits error propagation. If any quantum gate could be implemented using transversal gates, the set would be universal. We study the structure of GF(4)-additive quantum codes and prove that no universal set of transversal logic gates exists for these codes. This result is in stark contrast with the classical case, where universal transversal gate sets exist, and strongly supports the idea that additional quantum techniques, based, for example, on quantum teleportation or magic state distillation, are necessary to achieve universal fault-tolerant quantum computation on additive codes.

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

[2]  Maarten Van den Nest,et al.  The LU-LC conjecture, diagonal local operations and quadratic forms over GF(2) , 2007, Quantum Inf. Comput..

[3]  Andrew W. Cross,et al.  Local unitary versus local Clifford equivalence of stabilizer and graph states , 2007 .

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

[5]  Dorit Aharonov,et al.  Fault-tolerant quantum computation with constant error , 1997, STOC '97.

[6]  E. Knill,et al.  Resilient Quantum Computation , 1998 .

[7]  Yaoyun Shi.  Both Toffoli and controlled-NOT need little help to do universal quantum computing , 2003, Quantum Inf. Comput..

[8]  Ben Reichardt,et al.  Fault-Tolerant Quantum Computation , 2016, Encyclopedia of Algorithms.

[9]  E. Knill,et al.  Threshold Accuracy for Quantum Computation , 1996, quant-ph/9610011.

[10]  N. Sloane,et al.  Quantum Error Correction Via Codes Over GF , 1998 .

[11]  Andrew M. Steane,et al.  Fault-tolerant logical gate networks for Calderbank-Shor-Steane codes , 2003, quant-ph/0311014.

[12]  Daniel Gottesman,et al.  Stabilizer Codes and Quantum Error Correction , 1997, quant-ph/9705052.

[13]  Frederic T. Chong,et al.  Quantum Memory Hierarchies: Efficient Designs to Match Available Parallelism in Quantum Computing , 2006, 33rd International Symposium on Computer Architecture (ISCA'06).

[14]  B. Reichardt.  Improved magic states distillation for quantum universality , 2004, quant-ph/0411036.

[15]  D. Gottesman.  The Heisenberg Representation of Quantum Computers , 1998, quant-ph/9807006.

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

[17]  Dorit Aharonov,et al.  Fault-tolerant Quantum Computation with Constant Error Rate * , 1999 .

[18]  A. Steane.  Space, Time, Parallelism and Noise Requirements for Reliable Quantum Computing , 1997, quant-ph/9708021.

[19]  I. Chuang,et al.  Quantum Teleportation is a Universal Computational Primitive , 1999, quant-ph/9908010.

[20]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[21]  B. De Moor,et al.  Local unitary versus local Clifford equivalence of stabilizer states , 2005 .

[22]  R. Raussendorf,et al.  A fault-tolerant one-way quantum computer , 2005, quant-ph/0510135.

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

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

[25]  Frederic T. Chong,et al.  A general purpose architectural layout for arbitrary quantum computations , 2005, SPIE Defense + Commercial Sensing.

[26]  D. Gottesman.  Theory of fault-tolerant quantum computation , 1997, quant-ph/9702029.

[27]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[28]  Frederic T. Chong,et al.  A Practical Architecture for Reliable Quantum Computers , 2002, Computer.

[29]  Maarten Van den Nest,et al.  Local equivalence of stabilizer states and codes , 2005 .

[30]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[31]  Harold N. Ward.  An Introduction to Divisible Codes , 1999, Des. Codes Cryptogr..

[32]  Ben Reichardt,et al.  Quantum Universality from Magic States Distillation Applied to CSS Codes , 2005, Quantum Inf. Process..

[33]  Eric M. Rains.  Quantum Codes of Minimum Distance Two , 1999, IEEE Trans. Inf. Theory.

[34]  Finite Phase Space Methods in Quantum Information , 2006 .

[35]  N. J. A. Sloane,et al.  Quantum Error Correction Via Codes Over GF(4) , 1998, IEEE Trans. Inf. Theory.

[36]  J. Preskill.  Reliable quantum computers , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[37]  N. J. A. Sloane,et al.  The Invariants of the Cli ord , 1999 .

[38]  T. Tweed.  Space , 2011, STEM Education in the Primary School.