Towards practical classical processing for the surface code arXiv:1110.5133 Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg Centre for Quantum Computation and Communication Technology School of Physics, The University of Melbourne, Australia ?
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Towards practical classical processing for the surface code arXiv:1110.5133 Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg Centre for Quantum.
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Towards practical classical processing for the surface code
arXiv:1110.5133
Austin G. Fowler, Adam C. Whiteside, Lloyd C. L. Hollenberg
Centre for Quantum Computation and Communication TechnologySchool of Physics, The University of Melbourne, Australia
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Overview
• why study the surface code?• what is the surface code?• classical processing challenges
– correcting errors fast enough– interpreting logical measurements
• complexity optimal error processing– performance in detail– 3 second fault-tolerant distance 1000
• summary and further work
Why study the surface code?• no known parallel arbitrary
interaction quantum computer (QC) architecture
• many 2-D nearest neighbor (NN) architectures– ion traps, superconducting
computing is essentially a measurement based scheme
– logical gates, including the identity gate, introduce byproduct operators
– determining what these byproduct operators are is exceedingly difficult, especially after optimizing a braid pattern
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Correcting errors fast enough• consider the life cycle
of a single error• in the bulk of the
lattice, a single gate error is always detected at two space-time points
• given a detailed error model for each gate, the probability that any given pair of space-time points will be connected by a single error can be calculated
• these probabilities can be represented by two lattices of cylinders
• recently completed open source tool to perform such error analysis and visualisation
detect X error detect Z error
• the primal (Z errors) and dual (X errors) lattices are independent deterministically constructed objects
• terminology: dots and lines• weight of line –ln(pline)• stochastically detected
errors are represented by vertices associated with specific dots
• edges between vertices have weight equal to the minimum weight path through the lattice
• by choosing a minimum weight matching of vertices, corrections can be applied highly likely to preserve the logical state
Correcting errors fast enough
• minimum weight perfect matching was invented by Jack Edmonds in 1965
• very well studied algorithm, however best publicly available implementations have complexity O(n3) for complete graphs
• don’t support continuous processing
• don’t support parallel processing• actually quite slow
Correcting errors fast enough
• as recently as this year [PRA 83, 020302(R) (2011)], distance ~ 10 was the largest surface code that had been studied fault-tolerantly
• renormalization techniques have been used to study large non-fault-tolerant surface codes [PRL 104, 050504 (2010), arXiv:1111.0831]
• need something much better if surface code quantum computer is to be built
• to describe our fast matching algorithm, replace complex 3-D lattice with simple uniform weight 2-D lattice (grey lines)
• vertices (error chain endpoints) are represented by black dots
• matched edges are thick black lines
• shaded regions are space-time locations the algorithm has explored
Correcting errors fast enough
Complexity optimal error correction
Complexity optimal error correction
choose a vertex
Complexity optimal error correction
explore local space-time region until other objects encountered
Complexity optimal error correction
if unmatched vertices encountered, match with one
Complexity optimal error correction
choose another vertex
Complexity optimal error correction
expand until other objects are encountered, build alternating tree
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Complexity optimal error correction
it alternating tree outer space-time regions can’t be expanded, form blossom
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Complexity optimal error correction
uniformly expand space-time region around blossom until other objects encountered
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Complexity optimal error correction
two options in this case, unmatched vertex or boundary, choose vertex
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Complexity optimal error correction
choose another vertex
Complexity optimal error correction
form alternating tree
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Complexity optimal error correction
expand outer nodes, contract inner nodes
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Complexity optimal error correction
form blossom
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Complexity optimal error correction
grow alternating tree
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Complexity optimal error correction
expand outer nodes, contract inner nodes, forbidden region entered
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Complexity optimal error correction
undo expand outer, contract inner
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Complexity optimal error correction
undo grow alternating tree
Complexity optimal error correction
undo form blossom
Complexity optimal error correction
undo expand outer, contract inner... done until have additional data