Mitigation of Decoherence-Induced Quantum-Bit Errors and Quantum-Gate Errors Using Steane’s Code

Quantum processors require Quantum Error Correction Codes (QECC’s) for improving the fidelity of quantum logic gates. Fault tolerant QECC’s are capable of providing error rate improvements in quantum processors as long as the components are operating below a certain gate error probability. In this contribution, we quantify the depolarization probability bound, below which transversal QECC’s would give a better error probability than an uncoded gate. Both a low-complexity repetition code and Steane’s 7-bit QECC are characterized.

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

[2]  Lajos Hanzo,et al.  Quantum Error Correction Protects Quantum Search Algorithms Against Decoherence , 2016, Scientific Reports.

[3]  Andrew W. Cross,et al.  Validating quantum computers using randomized model circuits , 2018, Physical Review A.

[4]  Lajos Hanzo,et al.  The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure , 2015, IEEE Access.

[5]  L. Landau.  Fault-tolerant quantum computation by anyons , 2003 .

[6]  A. Kitaev.  Quantum Error Correction with Imperfect Gates , 1997 .

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

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

[9]  Lajos Hanzo,et al.  Quantum Coding Bounds and a Closed-Form Approximation of the Minimum Distance Versus Quantum Coding Rate , 2017, IEEE Access.

[10]  Laflamme,et al.  Perfect Quantum Error Correcting Code. , 1996, Physical review letters.

[11]  Laflamme,et al.  Quantum Logical Operations on Encoded Qubits. , 1996, Physical review letters.

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

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

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

[15]  J. Preskill,et al.  Topological quantum memory , 2001, quant-ph/0110143.

[16]  Scott Aaronson,et al.  Improved Simulation of Stabilizer Circuits , 2004, ArXiv.

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

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

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

[20]  Lajos Hanzo,et al.  Duality of Quantum and Classical Error Correction Codes: Design Principles and Examples , 2019, IEEE Communications Surveys & Tutorials.

[21]  N. Linke,et al.  High-Fidelity Quantum Logic Gates Using Trapped-Ion Hyperfine Qubits. , 2015, Physical review letters.

[22]  Lajos Hanzo,et al.  Quantum Topological Error Correction Codes are Capable of Improving the Performance of Clifford Gates , 2019, IEEE Access.

[23]  W. Munro,et al.  Quantum error correction for beginners , 2009, Reports on progress in physics. Physical Society.

[24]  A. Kitaev.  Quantum computations: algorithms and error correction , 1997 .

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

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

[27]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[28]  J. I. Cirac,et al.  Enforcing Coherent Evolution in Dissipative Quantum Dynamics , 1996, Science.

[29]  Lajos Hanzo,et al.  EXIT-Chart Aided Quantum Code Design Improves the Normalised Throughput of Realistic Quantum Devices , 2016, IEEE Access.

[30]  J. von Neumann,et al.  Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components , 1956 .

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

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

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

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

[35]  Vlatko Vedral,et al.  Basics of quantum computation , 1998, quant-ph/9802065.

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

[37]  John M. Martinis,et al.  Logic gates at the surface code threshold: Superconducting qubits poised for fault-tolerant quantum computing , 2014 .

[38]  John Preskill,et al.  Quantum Computing in the NISQ era and beyond , 2018, Quantum.

[39]  Olof Heden,et al.  A survey of perfect codes , 2008, Adv. Math. Commun..

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

[41]  D. Gottesman.  An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation , 2009, 0904.2557.

[42]  B. Terhal,et al.  Roads towards fault-tolerant universal quantum computation , 2016, Nature.

[43]  H. Bombin,et al.  Topological computation without braiding. , 2007, Physical review letters.

[44]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

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

[46]  Robert Raussendorf,et al.  Topological fault-tolerance in cluster state quantum computation , 2007 .