demonstrate that there exist thresholds for p and q which allow arbitrarily high fidelity for local error correction. We will assume that the classical processors and memory are very reliable. (In Section 5.8.3, we briefly discuss how this assumption could be relaxed.)
5.5 Processor memory
Following G´acs’s constructions [31, 33, 32], we impart a hierarchy on the layout of processors. We treat each processor as acell, which will update its state based on its memory contents along with the memory contents of its eight nearest neighbors (north, northeast, east, southeast, south, southwest, west, and northwest). Rect- angular groups ofQ×Qcells are identified as belonging to acolony. Rectangular groups of Q×Q colonies are identified as belonging to a super-colony. This hier- archy continues on to higher and higher levels (until the whole lattice is included).
We will assume that the linear lattice size L=Qk for some integerk. Errors will later be classified within a hierarchy of levels as well, and we will demonstrate how each level of errors will be handled by the corresponding level of colonies.
Several constants are hardcoded into the device, which will be explained below.
They includeQ(the colony size),U (the work period),fC (the threshold for a cell’s count), and fN (the threshold for a neighboring cell’s count). As a reference, we will later choose Q = 16, U = 400, fC = 4/5, and fN = 1/5 for deriving an accuracy threshold.
5.5.1 Memory fields
Every cell has the following fields of memory: Address, Age, and Syndromes.
Address has two subfields x and y that identify the horizontal and vertical ad- dress of the cell in its colony. Ageis a local “clock” that counts the number of time steps in a work period (of length U). Syndromes has nine subfields (each just a bit), which store the most recent syndrome measurements at this site and its eight
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Table 5.1: Processor memory fields for local error correction of toric codes
field name size description
Address 2 dlogQe horizontal and vertical address in colony Age dlogUe local clock running from 0 to (U - 1) Syndromes 9 syndrome at this site and its neighbors
Count 9 dlogUe coarse-grained count of syndrome history CountSignal 8 value of neighboring colony’s syndrome NewCountSignal 8 temporary storage of value forCountSignal
FlipSignal 8 signal for flipping at end of a work period NewFlipSignal 8 temporary storage of value for FlipSignal
neighboring sites. A cell’s most recent measurement is denotedSyndromes.C, and the other subfields are Syndromes.N, Syndromes.E, Syndromes.S,Syndromes.W, Syndromes.NE,Syndromes.SE,Syndromes.SW, andSyndromes.NW, referring to the cell’s northern, eastern, southern, western, and four diagonal neighbors, respec- tively.
The following fields of memory are also needed at some particular cells: Count, CountSignal, NewCountSignal, FlipSignal, and NewFlipSignal. The Count field has nine subfields (similar to Syndromes) which keep track of how often a syndrome is present at the center of a colony (either the current colony or one of its eight neighboring colonies) during a work period. The four “signal” fields provide a means of communication between colonies (traveling one cell per time step). They each have eight subfields corresponding to the eight directions making up the neighborhood of a cell. The CountSignal field contains the information of whether syndromes are present or absent at a particular time step in the cen- ters of neighboring colonies. The FlipSignal field contains a signal for control- ling flips across colonies at the end of a work period. The NewCountSignal and NewFlipSignal fields provide temporary storage of neighbors’ values, before up-
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datingCountSignalandFlipSignal. More specifically, during each time step the values inCountSignalfor each cell are copied into its neighbors’NewCountSignal fields, which then replace CountSignal, and similarly FlipSignal is copied into neighboring cells via the NewFlipSignalfield.
These additional fields could be included in every processor’s memory (so that the required resources remain homogenous), but only the colony centers and a few other cells actually use their contents for decision making in our model. These extra memory contents could instead be utilized for increasing the reliability of the center sites’ data through redundancy, if we considered faulty classical memory storage (which we mostly ignore in this present work).
The processor memory fields are summarized in Table 5.1. The size column reports the number of bits needed to store a field. For each level of the hierarchy, another set of all of these fields must be present (with Q and U replaced by Qn and Un for the nth level). Note that for typical values of the constants, the required memory for each processor is less than a kilobyte if only a few levels are implemented (as would likely be the case for any practical device). The device is initialized by setting all of the Addressfields to each cell’s relative position in its colony, and resetting all other fields to zero.
G´acs [33] (see also Gray’s description in [39]) avoids this increase in memory requirements for larger and larger lattices by implementing self-simulation in the classical processors. In his model, each level of the hierarchy of colonies (after the first) is simulated by the next lower level. In effect, each colony simulates a cell of a super-colony, and so on. However, an incredible amount of work needs to go into showing how such a simulation is possible with only local update rules and furthermore how it is robust to errors. Since the memory requirements to handle a few levels of colonies without self-simulation are very modest, we have decided to avoid these torturous details in order to focus on how our device handles quantum errors. We believe that it should be possible in principle to fully implement the ideas in his work so that the memory requirements become constant (independent of the size of the lattice or the amount of time desired to maintain memory), but
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we have not worked through the details explicitly.
5.5.2 Memory processing
We assume that the processors are synchronized in our model. That is, the local clocks represented by theAgefield march in lock step with one another. G´acs [33]
demonstrated that asynchronous classical processors can still robustly compute in a one-dimensional cellular array, but again we have decided to forego a lot of the burdensome details by not considering the most general model.
The general framework of the processing during each time step (and in parallel at each site) is as follows.
• The Agefield is incremented by one (modU).
• The syndromeXXXX is measured, and its value is stored inSyndromes.C.
• The neighbors’ syndrome measurements are copied into the respective sub- fields of Syndromes.
• The CountSignalsubfields from neighboring cells are copied into the corre- sponding NewCountSignal subfields.
• The FlipSignal subfields from neighboring cells are copied into the corre- sponding NewFlipSignalsubfields.
• At the center of a colony (which is determined from Address information), allCountSignal subfields are replaced with the value ofSyndromes.C. Else- where, CountSignal is replaced withNewCountSignal.
• At the center of a colony, theCount field is updated based onSyndromes.C and NewCountSignal (details below). Elsewhere, FlipSignal is replaced withNewFlipSignal.
• Finally, the processor follows the basic rules outlined in Section 5.6 to pos- sibly flip one of its neighboring spins or, if located at the center of a colony at the end of a work period, set itsFlipSignal field.