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Survey on the Bounds of the Threshold For Quantum Decoherence

Survey on the Bounds of the Threshold For Quantum Decoherence. Chris Graves December 12, 2012. Goals For Studying Quantum Computation. (Experimentalists). A) Build a Large Scale Quantum Computer. B) Figure Out What We Can Do Once We Get One. (Theorists). Threshold Theorem.

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Survey on the Bounds of the Threshold For Quantum Decoherence

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  1. Survey on the Bounds of the Threshold For Quantum Decoherence Chris Graves December 12, 2012

  2. Goals For Studying Quantum Computation (Experimentalists) A) Build a Large Scale Quantum Computer B) Figure Out What We Can Do Once We Get One (Theorists)

  3. Threshold Theorem Theorem: There exists an error rate threshold ƞth > 0 such that any ideal polynomial sized quantum circuit can be accurately simulated by a robust polynomial time quantum circuit that is resistant to any error rate ƞ < ƞth Proven by Aharonov & Ben-Or (1996) Assumes: Ability to generate fresh ancilla qubits when needed Ability to perform operations in parallel

  4. 10-5 10-4 10-3 10-2 10-1 100 Threshold Bounds ƞth Lower Bounds Upper Bounds Universal quantum computing is possible if we can get the error rates below these bounds Any quantum computer subject to an error rate above these bounds will become useless *Shown on a pseudo-logarithmic scale

  5. Concatenated QEC Codes + Reasonable overhead - Relatively low thresholds - Ignores physical distance between qubits Threshold Lower Bounds • 7-qubit codes → 2.73 x 10-5 (Alferis, Gottesman, Preskill 2005) • Bacon-Shor codes → 1.9 x 10-4 (Alferis, Cross 2006) • Golay codes → 1.32 x 10-3 (Paetznick, Reichardt 2011)

  6. Quantum Error Detection + Relatively high thresholds - Prohibitively expensive overhead - Ignores physical distance between qubits Threshold Lower Bounds • Estimated 1%-3% (Knill 2004) • Rigorously Proved .1% (Alferis, Gottesman 2007)

  7. Surface Codes + More accurately deals with locality + High simulated thresholds - Harder to analyze rigorously - Seems to be more complicated to implement Threshold Lower Bounds • 1% simulated (Wang, Fowler, Hollenberg 2010) • 18.9%!!! simulated (Wootton, Loss, 2012)

  8. Can be simulated by classical computer • 74% entanglement between two and one qubit gates becomes impossible (Harrow, Nielsen 2003) • 45.3% for perfect Clifford gates and arbitrary noisy 1-qubit gates (Buhrman et al 2006) Threshold Upper Bounds Output becomes random after logarithmic depth

  9. References Gottesman (2009) arXiv:0904.2557v1 Aharonov, Ben-Or (1996) arXiv:quant-ph/9611025 Alferis, Gottesman, Preskill (2005) arXiv:quant-ph/0504218v3 Alferis, Cross (2006) arXiv:quant-ph/0610063 Paetznick, Reichardt (2011) arXiv:1106.2190v1 Knill (2004) arXiv:quant-ph/0410199v2 Alferis, Gottesman (2007) arXiv:quant-ph/0703264v2 Wang, Fowler, Hollenberg (2010) arXiv:1009.3686v1 Wootton, Loss (2012) arXiv:1202.4316v3 Harrow, Nielsen 2003) arXiv:quant-ph/0301108v1 Buhrman et al (2006) arXiv:quant-ph/0604141v2 Razborov (2003) arXiv:quant-ph/0310136v1 Kempe et al (2008) arXiv:0802.1464v1 Cleve, Watrous (2000) arXiv:quant-ph/0006004v1

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