Fault-tolerant logical gate networks for Calderbank-Shor-Steane codes

Fault-tolerant logical operations for qubits encoded by Calderbank-Shor-Steane codes are discussed, with emphasis on methods that apply to codes of high rate, encoding k qubits per block with k>1. It is shown that the logical qubits within a given block can be prepared by a single recovery operation in any state whose stabilizer generator separates into X and Z parts. Optimized methods to move logical qubits around and to achieve controlled-NOT and Toffoli gates are discussed. It is found that the number of time steps required to complete a fault-tolerant quantum computation is the same when k>1 as when k=1.

[1]  E. Knill,et al.  Accuracy threshold for quantum computation , 1996 .

[2]  Barenco,et al.  Quantum networks for elementary arithmetic operations. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[3]  D. Aharonov.  A Simple Proof that Toffoli and Hadamard are Quantum Universal , 2003, quant-ph/0301040.

[4]  Andrew M. Steane.  Efficient fault-tolerant quantum computing , 1999, Nature.

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

[6]  E. Knill,et al.  Resilient quantum computation: error models and thresholds , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  DiVincenzo,et al.  Fault-Tolerant Error Correction with Efficient Quantum Codes. , 1996, Physical review letters.

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

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

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

[11]  Mário Ziman,et al.  Programmable Quantum Gate Arrays , 2001 .

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

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

[14]  A. Steane.  Overhead and noise threshold of fault-tolerant quantum error correction , 2002, quant-ph/0207119.

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

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

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

[18]  Preskill,et al.  Efficient networks for quantum factoring. , 1996, Physical review. A, Atomic, molecular, and optical physics.

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

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

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

[22]  Andrew Steane,et al.  Active Stabilization, Quantum Computation, and Quantum State Synthesis , 1997 .

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