feat: Migrate the init kernel CPP tests to noir

[sirasistant]

Please provide a paragraph or two giving a summary of the change, including relevant motivation and context.

Checklist:

Remove the checklist to signal you've completed it. Enable auto-merge if the PR is ready to merge.

• If the pull request requires a cryptography review (e.g. cryptographic algorithm implementations) I have added the 'crypto' tag.
• I have reviewed my diff in github, line by line and removed unexpected formatting changes, testing logs, or commented-out code.
• Every change is related to the PR description.
• I have linked this pull request to relevant issues (if any exist).

[AztecBot]

Benchmark results

No metrics with a significant change found.

Detailed results

All benchmarks are run on txs on the Benchmarking contract on the repository. Each tx consists of a batch call to create_note and increment_balance, which guarantees that each tx has a private call, a nested private call, a public call, and a nested public call, as well as an emitted private note, an unencrypted log, and public storage read and write.

This benchmark source data is available in JSON format on S3 here.

Values are compared against data from master at commit 84f8db20 and shown if the difference exceeds 1%.

L2 block published to L1

Each column represents the number of txs on an L2 block published to L1.

Metric	8 txs	32 txs	128 txs
l1_rollup_calldata_size_in_bytes	45,444	179,588	716,132
l1_rollup_calldata_gas	222,888	868,112	3,449,468
l1_rollup_execution_gas	841,975	3,595,220	22,204,837
l2_block_processing_time_in_ms	2,020 (+3%)	7,608	29,754 (+1%)
note_successful_decrypting_time_in_ms	294 (-1%)	862	3,254 (+4%)
note_trial_decrypting_time_in_ms	95.0 (-4%)	70.0 (-4%)	134
l2_block_building_time_in_ms	13,173	52,365	210,304 (+1%)
l2_block_rollup_simulation_time_in_ms	11,854	47,132 (+1%)	188,930 (+1%)
l2_block_public_tx_process_time_in_ms	1,277 (-1%)	5,103	20,885 (+2%)

L2 chain processing

Each column represents the number of blocks on the L2 chain where each block has 16 txs.

Metric	5 blocks	10 blocks
node_history_sync_time_in_ms	21,563 (+1%)	42,876 (+1%)
note_history_successful_decrypting_time_in_ms	2,051 (+2%)	4,027 (+2%)
note_history_trial_decrypting_time_in_ms	121 (+1%)	141 (-1%)
node_database_size_in_bytes	1,632,726	1,099,041
pxe_database_size_in_bytes	27,188	54,187

Circuits stats

Stats on running time and I/O sizes collected for every circuit run across all benchmarks.

Circuit	circuit_simulation_time_in_ms	circuit_input_size_in_bytes	circuit_output_size_in_bytes
private-kernel-init	72.3	61,697	18,841
private-kernel-ordering	43.6	24,233	8,089
base-rollup	1,770	656,309	811
root-rollup	80.1 (+6%)	4,072	1,097
private-kernel-inner	53.6 (+1%)	81,504	18,841
public-kernel-private-input	41.8 (+1%)	41,455	18,841
public-kernel-non-first-iteration	26.6 (+1%)	41,497	18,841
merge-rollup	0.949 (+6%)	2,592	873

Miscellaneous

Transaction sizes based on how many contracts are deployed in the tx.

Metric	0 deployed contracts	1 deployed contracts
tx_size_in_bytes	8,723	27,240

[kevaundray]

Lets get this merged!