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
(Experimentalists)
B) Figure Out What We Can Do
Once We Get One
(Theorists)
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
Threshold Bounds
-5
10
ƞth
-4
10
-3
10
-2
10
Lower Bounds
Universal quantum computing is
possible if we can get the error
rates below these bounds
*Shown on a pseudo-logarithmic scale
-1
10
0
10
Upper Bounds
Any quantum computer subject
to an error rate above these
bounds will become useless
Threshold Lower Bounds
Concatenated 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 Bounds
Quantum Error Detection
+ Relatively high thresholds
- Prohibitively expensive overhead
- Ignores physical distance between
qubits
• Estimated 1%-3% (Knill 2004)
• Rigorously Proved .1% (Alferis, Gottesman 2007)
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)
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 1qubit gates (Buhrman et al 2006)
Output becomes random after logarithmic depth
References
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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