Understanding Defect Diagrams
Craig Gidney
Overview
Part 1:
Defects in�Space
Part 2:
Defects in Spacetime
Part 3:
Parity Sheets
WIP
Part 1
Defects in Space
A "Defect" is an exception to a background pattern
Background pattern of the surface code: the checkerboard
Z
X
LEGEND
X Stabilizer
Z Stabilizer
Examples of surface code defects
Z
X
LEGEND
X Stabilizer
Z Stabilizer
Mixed basis stabilizer (red=X, blue=Z, pink=Y)
a 5-body stabilizer,
not 5 stabilizers
Examples of surface code defects
Z
X
LEGEND
X Stabilizer
Z Stabilizer
Mixed basis stabilizer (red=X, blue=Z, pink=Y)
a 5-body stabilizer,
not 5 stabilizers
How do we classify defects?
These two defects are both�"X boundaries", but what makes them the same type?
Are these also X boundaries?
If not, what are they?
Key concept: Excitations
An excitation is a stabilizer whose value is incorrect.�Typically: -1 instead of +1.
�There are two types of stabilizers and two corresponding types of excitations: X type and Z type.
�CAUTION: the X type excitation goes on the Z type stabilizer. The type name comes from the error that causes the excitation.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Operating on excitations
Excitations can be affected by injecting data qubit Pauli errors.�
Pauli errors add an excitation to each anticommuting neighboring stabilizers.�
Two excitations on the same stabilizer cancel out.
X
Z
(move back and forth between this slide and previous slide to see errors affect excitations)
Y
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
You can't locally remove an excitation in the bulk
In this diagram, there is no set of local Pauli errors which can be injected which result in a state with no excitations.
This inability to change the parity of the number of Z excitations (or X excitations) defines the bulk.
Terminology:�The bulk conserves X excitations.�The bulk conserves Z excitations.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
X boundaries absorb X excitations
An X boundary differs from the bulk in that it does not preserve X excitations.
There is a set of local Paulis which, if injected, would remove a lone X excitation.
�Terminology:�X boundaries absorb X excitations.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
X
X
Z boundaries absorb Z excitations
A Z boundary differs from the bulk in that it does not preserve Z excitations.
There is a set of local Paulis which, if injected, would remove a lone Z excitation.
�Terminology:�Z boundaries absorb Z excitations.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
Z
Crossing domain walls turns X into Z
A domain wall differs from the bulk in that it can locally turn an X excitation into a Z excitation (by crossing the wall).
Also turns Z into X.
Terminology:�Domain walls crosslink X/Z excitations.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
Z
Z
X
X
Domain walls can't locally cancel X + Z
Note that the domain wall's conversion involves crossing. A domain wall doesn't enable cancelling adjacent X and Z excitations.
In the diagram to the right, no set of injected Pauli errors can reduce the number of excitations present to be less than 2.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
Twists allow local cancellation of X + Z
A twist allows an X excitation next to a Z excitation to be locally cancelled.��This is strictly more powerful than a domain wall. For example, an X can be turned into a Z by introducing (unabsorbing) X+Z and cancelling the two Xs leaving a Z.���Terminology:�Twists absorb X+Z excitations.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Can't inject error here
X
X
Z
Z
Z
Summary of Types of Defect
No Defect
"Bulk"
Type X
"X Boundary"
Type Z
"Z Boundary"
Type H
"Domain Wall"
Type Y
"Twist"
Absorbs X�
Absorbs Z
Absorbs X+Z
Crosslinks X/Z
Conserves X
Conserves Z
Identifying defects
X type defect is nearby iff X excitations can be absorbed locally.
Z type defect is nearby iff Z excitations can be absorbed locally.
Y type defect is nearby iff X+Z excitations can be absorbed locally.
H type defect is nearby iff X excitations crosslink into Z excitations locally.
It doesn't matter how the defects are implemented.�
All that matters is categorizing what X and Z excitations can do.
Example: corners have twists
Can you locally cancel an X+Z excitation at a corner?
Yes.
Therefore there is a Y type defect at that corner.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
P
Pauli Error
Z Excitation
Y
Z
Z
Summary of Types of Defect (redux)
No Defect
"Bulk"
Type X
"X Boundary"
Type Z
"Z Boundary"
Type H
"Domain Wall"
Type Y
"Twist"
Conserves X
Conserves Z
Absorbs X�
Absorbs Z
Crosslinks X/Z
Absorbs X+Z
Defect Diagrams: only draw locations of defects
=
Z
X
LEGEND
Y
H
Stabilizer Configuration Diagram
Defect Diagram
Exercise: why is this square Y type instead of H type?
bulk
Implementing defects
Given a defect diagram, you can attempt to find a stabilizer configuration which implements it.
Not all diagrams can be realized. For example, it's impossible to end a domain wall in the bulk without creating a twist defect.
Implementable
Not Implementable
Z
X
LEGEND
Y
H
bulk
Part 2
Defects in Spacetime
Accounting for time dynamics
So far we have only discussed�static stabilizer configurations.
�Computation requires dynamic�stabilizer configurations.
�To account for this we must add�a dimension and consider defects�that may span across time.
time
Going up a dimension
Defects in space:
X boundaries are 1D paths�Z boundaries are 1D paths�Domain walls are 1D paths�Twists are 0D points
Excitations are 0D points�
Excitations are stabilizers with an incorrect value (typically -1 instead of +1).
Defects in spacetime:
X boundaries are 2D surfaces�Z boundaries are 2D surfaces�Domain walls are 2D surfaces�Twists are 1D paths
Excitations are still 0D points� (not 1D, didn't gain a dimension)
Excitations are detection events; places where a measurement set's parity was predictable and seen to be wrong.
Summary of Types of Defect (spacetime redux)
No defect
"Bulk"
Type X
"X Boundary"
Type Z
"Z Boundary"
Type H
"Domain Wall"
Type Y
"Twist"
Conserves X
Conserves Z
�Absorbs X
Crosslinks X/Z
�Absorbs X+Z
�Absorbs Z
Static defects across spacetime
time
time
Stabilizer configurations
over time:
Equivalent spacetime�defect diagram:
X,Y,Z defects around a memory patch
Domain wall ending in a twist via long stabilizers
Z
X
LEGEND
Y
H
Identifying defects (spacetime redux)
X type defect is nearby iff X excitations can be absorbed locally.
Z type defect is nearby iff Z excitations can be absorbed locally.
Y type defect is nearby iff X+Z excitations can be absorbed locally.
H type defect is nearby iff X excitations crosslink into Z excitations locally.
It doesn't matter how the defects are implemented. It doesn't matter whether they span across space, across time, or across both.
All that matters is categorizing what X and Z excitations can do.
What boundary is introduced by transversal resets?
apply Rx to
all data qubits
...
Clearly the resets are an exception to the normal pattern.
They're probably introducing some type of defect. But which type?
...
Getting rid of a detection event by adding errors (1/3)
Reminder: spacetime excitations are detection events. They are stabilizer measurements changing.
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
M
Measure Error
Z Excitation
+1
+1
-1
-1
Getting rid of a detection event by adding errors (2/3)
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
M
Measure Error
Z Excitation
M
Flipping previous measurement moves where the change was seen.�It moves the detection event.
+1
-1
-1
-1
Getting rid of a detection event by adding errors (3/3)
Z
X
LEGEND
X Stabilizer
Z Stabilizer
X Excitation
M
Measure Error
Z Excitation
M
First stabilizer measurement is random because of resets. Can't notice anything wrong, so no detection�event generated.
M
We locally absorbed an X excitation. This is an X boundary!
-1
-1
-1
-1
Transversal X resets introduce an X boundary in time
Stabilizer configurations
over time
Z
X
LEGEND
Y
H
Defect diagram
=
time
Example of a domain wall + twist in time
Z
X
LEGEND
Y
H
Stabilizer configurations
over time
Defect diagram
time
Ry row above Hadamards
X memory
experiment
Z memory
experiment
as stabilizer configurations
over time
as defect diagrams
X memory
experiment
Z memory
experiment
Showing all defects
Only showing Y defects
S gate as a series of stabilizer configurations
Input
Output
Grow
Y Defect Moves Across Patch
Shrink
Swaps
X Resets
Y Resets above Hadamards
X Measures
S gate as a defect diagram
=
(Cut out windows are to show interior. Actual defects have no windows)
S gate as braiding Y defects
Show all defects
X and Z defects hidden
in middle
Show Y Defects Only
S Gate = = exchange Y defects