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

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

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Threshold Theorem Theorem: There exists an error rate threshold ƞ th > 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 < ƞ th Proven by Aharonov & Ben-Or (1996) Assumes: Ability to generate fresh ancilla qubits when needed Ability to perform operations in parallel

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Threshold Bounds 10 0 Lower BoundsUpper Bounds ƞ th *Shown on a pseudo-logarithmic scale 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

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

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

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

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Threshold Upper Bounds 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) Output becomes random after logarithmic depth

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References Gottesman (2009) arXiv: v1 Aharonov, Ben-Or (1996) arXiv:quant-ph/ Alferis, Gottesman, Preskill (2005) arXiv:quant-ph/ v3 Alferis, Cross (2006) arXiv:quant-ph/ Paetznick, Reichardt (2011) arXiv: v1 Knill (2004) arXiv:quant-ph/ v2 Alferis, Gottesman (2007) arXiv:quant-ph/ v2 Wang, Fowler, Hollenberg (2010) arXiv: v1 Wootton, Loss (2012) arXiv: v3 Harrow, Nielsen 2003) arXiv:quant-ph/ v1 Buhrman et al (2006) arXiv:quant-ph/ v2 Razborov (2003) arXiv:quant-ph/ v1 Kempe et al (2008) arXiv: v1 Cleve, Watrous (2000) arXiv:quant-ph/ v1

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