Survey on the Bounds of the Threshold For Quantum Decoherence

Chris Graves

December 12, 2012

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)

Threshold TheoremTheorem: 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

Threshold Bounds

100

Lower Bounds Upper Bounds

ƞth

10-110-5 10-3 10-210-4

*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

Threshold Lower BoundsConcatenated QEC Codes

+ Reasonable overhead- Relatively low thresholds- Ignores physical distance between qubits

• 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)

Threshold Lower BoundsQuantum Error Detection

• Estimated 1%-3% (Knill 2004)• Rigorously Proved .1% (Alferis, Gottesman 2007)

+ Relatively high thresholds- Prohibitively expensive overhead- Ignores physical distance between qubits

Threshold Lower BoundsSurface 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)

Threshold Upper BoundsCan 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

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