Using surface code experimental output correctly and effectively Austin Fowler Google Inc.
Overview Conclusions Gate and state definitions Error propagation Error detection Classical processing Fast feedforward Parameter tuning Error model tuning
Conclusions Classical processing for error correction cannot be broken into independent rounds as this is not fault-tolerant Corrections should not be applied to the quantum system, as they are constantly revised Fast feedforward primarily required in a large quantum computer rapidly executing a nontrivial quantum algorithm, and not for error correction Errors that have already occurred do not become more dangerous as additional errors occur
Gate and state definitions Computational basis: Initialization to Unitary gates: Measurement in computational basis:
Error propagation Above identities can be proven via matrix multiplication of the definitions on the previous slide
Error propagation Surface code data qubits (circles) constrained to be eigenstate of certain operators (stabilizers) Z-stabilizers enable detection of X errors Keep things simple and focus on a slice of the surface code
Error propagation
Error propagation X 1 1 Change in measured value indicates endpoint of error chain Most likely error chain simply connects to nearest boundary Record in software that we believe the top data qubit is associated with an X error
Error propagation Two error chain endpoints observed 1 1 X 1 1 Two error chain endpoints observed Most likely pattern of error chains is a single chain connecting the two endpoints Record in software that we believe the middle data qubit is associated with an X error
Error propagation Two error chain endpoints observed 1 X Two error chain endpoints observed Most likely pattern of error chains is a single chain connecting the two endpoints Flip the classical measurement value
Error propagation Two error chain endpoints observed 1 1 X 1 Two error chain endpoints observed Most likely pattern of error chains is a single chain connecting the two endpoints Record in software that we believe the middle data qubit is associated with an X error and flip the second classical measurement value on the lower measurement qubit Round by round processing fails to correctly identify the above error
Classical processing 10 data qubits One detection event
Classical processing 10 data qubits One detection event Explore uniformly, boundary found
Classical processing 10 data qubits One detection event X 10 data qubits One detection event Explore uniformly, boundary found Match detection event to boundary, record belief that X error present
Classical processing X Two more detection events
Classical processing Two more detection events X Two more detection events Pick one, explore, current time boundary encountered
Classical processing Two more detection events X Two more detection events Pick one, explore, current time boundary encountered Explore around other, exploratory regions touch
Classical processing Two more detection events X Two more detection events Pick one, explore, current time boundary encountered Explore around other, exploratory regions touch Match, record belief that two more X errors present X X
Classical processing X One more detection event X X
Classical processing One more detection event X One more detection event Explore, current time boundary encountered, must wait for more data X X
Classical processing One more detection event X One more detection event Explore, current time boundary encountered, must wait for more data Explore further, boundary encountered X X
Classical processing One more detection event X X X One more detection event Explore, current time boundary encountered, must wait for more data Explore further, boundary encountered Match, record belief that two more X errors present X X
Classical processing One more detection event X One more detection event Explore, current time boundary encountered, must wait for more data Explore further, boundary encountered Match, record belief that two more X errors present Cancel double error Don’t apply physical corrections X X
Classical processing X X X
Classical processing X X X
Classical processing X X X X X
Classical processing X X X
Classical processing X X X
Classical processing X X X
Classical processing X X X X X
Classical processing X X X
Classical processing X X X
Classical processing X X X X X X X
Classical processing X X X X X
Classical processing X X X X X
Classical processing X X X X X
Classical processing X X X X X
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Classical processing X
Fast feedforward Could be useful if measurement is QND and the |0> state is more robust than |1> and the qubits really are qubits and no better reset gate is available Really need good reset gate in any system with leakage
Fast feedforward Fast feedforward is necessary only at the error corrected logical gate level where it is a critical ingredient in fast implementations of quantum algorithms T gates are probabilistic and 50% of the time require S gate corrections Prior logical measurements determine the basis of future logical measurements At the moment the speed at which this can occur is entirely limited by the speed classical processing
Parameter tuning Suppose have continuously running surface code quantum computer Every measurement qubit can be associated with the detection event rate Choose array of parameters Adjust parameter If detection event rate increases reverse direction of adjustment Designed to keep gates tuned up while long algorithm runs J. Kelly et al. in preparation
Error model tuning Error model is essentially a weighted graph Can track the relative number of times each geometrically distinct edge is observed Enables better matching
Conclusions Classical processing for error correction cannot be broken into independent rounds as this is not fault-tolerant Corrections should not be applied to the quantum system, as they are constantly revised Fast feedforward primarily required in a large quantum computer rapidly executing a nontrivial quantum algorithm, and not for error correction Errors that have already occurred do not become more dangerous as additional errors occur