HolographicControl.jl

Optimal-control synthesis of approximate holographic quantum error-correcting codes, built on Piccolo.jl.

The package bridges two mature subfields:

• Holographic codes — the HaPPY tensor-network construction (Pastawski et al. 2015) and its successors that realize the bulk-to-boundary encoding map of AdS/CFT as quantum error-correcting codes.
• Quantum optimal control — direct-collocation pulse optimization on physical Hamiltonians with bounded controls.

The novel contribution is to synthesize holographic codes dynamically — as the output of a physical Hamiltonian evolution under bounded controls — and to discover new approximate codes by optimizing the Petz recovery error directly.

Master's thesis, NYU

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Quick start

import Pkg
Pkg.activate(".")
Pkg.instantiate()

using HolographicControl

# The [[5,1,3]] perfect code from M1: 32×2 isometry built analytically
# from the stabilizer generators.
V_513 = five_qubit_isometry()

# Every weight-1 erasure recovers the bulk perfectly (M2 + AME(5,2)):
A_list = uniform_erasure_subregions(5, 1)
@show petz_recovery_objective(V_513; A_list = A_list)
# → ≈ 1e-16

To synthesize the [[5,1,3]] via Piccolo (M4):

julia --project=. examples/04g_synthesize_five_qubit_xy.jl

Expect ~4 hours wall on a laptop. The script prints rolled-out fidelity (0.999+) and the per-erasure Petz recovery error (all < 1e-3).

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What's in the box

Source modules (src/)

File	Purpose
HolographicControl.jl	Module entrypoint + exports
pauli.jl	Pauli matrices, pauli_string, single-qubit error sets
isometries.jl	is_isometry, polar_isometry, random_isometry, fidelity metrics, encoding input states
hamiltonians.jl	Drift Hamiltonians (nn_xx_yy_drift, nn_heisenberg_drift, etc.) + single-site drive matrices
recovery.jl	Partial trace, embed (HS-adjoint), Petz map, recovery error
objectives.jl	petz_recovery_objective + A_list constructors
entanglement.jl	Entropies, mutual information, is_reconstructable, entanglement_wedge_report
black_hole.jl	page_curve, page_time, Hayden–Preskill recoverability
holography.jl	RT minimal-surface weights, bulk–boundary MI, code_quality_summary
optimization.jl	discover_low_petz_isometry (Stage A), multistart, warm-start, curriculum
problems.jl	Piccolo SmoothPulseProblem builders + extractors
visualization.jl	CairoMakie plotters (page curve, wedge, Petz sweep, code comparison)
io.jl	save_isometry / load_isometry, save_pulse / load_pulse
reference_codes/five_qubit_code.jl	The [[5,1,3]] + Knill–Laflamme verification
reference_codes/repetition_code.jl	3-qubit classical repetition encoder
reference_codes/happy_pentagon.jl	Single-tile alias + multi-tile stub
reference_codes/four_one_two_code.jl	The [[4,1,2]] code + stabilizers
reference_codes/steane_code.jl	The [[7,1,3]] Steane code + stabilizers

Numbered examples (examples/)

Script	Milestone	What it does
00_piccolo_hello.jl	M0	1-qubit X gate via Piccolo (toolchain validation)
01_five_qubit_isometry.jl	M1	Build [[5,1,3]] analytically + verify KL conditions
02_petz_recovery.jl	M2	Sweep Petz recovery over all erasure weights on [[5,1,3]]
03_synthesize_repetition.jl	M3	Piccolo-synthesize the 3-qubit repetition encoder (~4 min)
04_synthesize_five_qubit.jl	M4a	First [[5,1,3]] attempt — STAGNATION case study at fid 0.91
04b_seed_diagnostic.jl	M4 escape	Phase 1 across seeds 1–4 (rules out seed luck)
04c_structure_probe.jl	M4 escape	Per-column overlap on best seed (rules out free_phase)
04d_drive_bounds_probe.jl	M4 escape	Bounds 1.0/2.0/4.0 (rules out drive amplitude)
04e_cubic_splines_m3_sanity.jl	M4 escape	Documents the BilinearIntegrator / cubic-spline mismatch
04f_structural_sweep.jl	M4 escape	5 (drift, control) configs — finds XY-only is the winner
04g_synthesize_five_qubit_xy.jl	M4 working	XY drift + seed=3 → fid 0.999 (PASSES)
04h_xy_seed_sweep.jl	M4 escape	Seed sweep at XY drift (seed=3 picks the winning basin)
05a_petz_objective_demo.jl	M5 L1	Petz objective discriminates good/bad codes on n=3
05b_petz_optimization_demo.jl	M5 L2	Standalone V optimization on n=3 (obj 0.067, beats baselines 2.47×)
05c_petz_optimization_n5.jl	M5 L2	n=5 — discovers the low-Petz manifold structure
05d_m3_vs_m5_link.jl	M5 L2	Fixed-target QOC vs discovery-objective QOC on the same system
05e_piccolo_m5_n3.jl	M5 L3	Monolithic Petz-objective-in-Piccolo at n=3 (tractable)
05f_piccolo_m5_n5.jl	M5 L3	Monolithic at n=5 — documents the exact-Hessian wall (> 6 h/eval)
06_page_curve_demo.jl	M5	Page curves and black-hole diagnostics across reference codes
07_decomposed_m5_n5.jl	M5 L3 working	Decomposed M5: Stage-A discovery of V* (~3 min) + optional Stage-B realization
08_visualization_demo.jl	M5	CairoMakie figure layer
09_multistart_demo.jl	M5	Multistart synthesis pattern
10_holographic_dashboard.jl	M5	RT weights, bulk–boundary MI, code-quality dashboard

Tests (test/)

3170 passing tests, ~18 min wall (includes one small Piccolo synthesis). Run with julia --project=. -e 'using Pkg; Pkg.test()'.

Documentation

• LOG.md — milestone-by-milestone development narrative
• docs/M5_DESIGN.md — Layer 3 Piccolo-integration plan

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Milestone status

Milestone	Status	Reference
M0 — Piccolo toolchain	DONE	1Q X gate at fid 0.9999918
M1 — Analytic [[5,1,3]]	DONE	KL matrix exactly I_16
M2 — Petz recovery	DONE	AME(5,2) erasure thresholds confirmed
M3 — 3Q repetition via Piccolo	DONE	NLP & rollout both at fid 0.9999
M4 — [[5,1,3]] via Piccolo	DONE (XY drift variant)	Rollout fid 0.999, Petz Δ < 2e-4
M5 — Petz objective optimization	DONE via decomposition	Stage A discovers V*; see below

M5 result — the discovered code V*

Stage-A discovery on the single-erasure objective at n=5, n_bulk=1 (examples/07_decomposed_m5_n5.jl, seed 20260527) converges to an isometry V* that lies outside the stabilizer formalism:

Quantity	[[5,1,3]]	V*
Petz w=1 (single erasure)	~1e-16	6.76e-10
Petz w=2 (mean)	0.0	0.184
Page plateau S(|R|=2)	2.000 (AME)	1.7178
Wedge threshold k*	3	4
Reconstructable counts |A|=1..5	[0,0,10,5,1]	[0,0,0,5,1]
Largest nontrivial Pauli expectation	1.0 (16 of them)	0.357 (XZZYY)

A 15-seed sweep (seeds 1–15) lands in the same basin every time: k* = 4 in 15/15, S(|R|=2) median 1.748 over [1.688, 1.827], Petz w=1 median 8.1e-10.

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Conventions

Big-endian qubit ordering. A state |q_1 q_2 ... q_n⟩ maps to the 1-indexed Julia computational-basis index `1 + q_1·2^(n-1) + q_2·2^(n-2)

• ... + q_n. Equivalently, pauli_string("XZZXI")putsXon qubit 1,Ion qubit 5. The convention is documented insrc/HolographicControl.jl`'s module docstring.

ComplexF64 throughout. All operators are dense Matrix{ComplexF64}. The package is designed for n ≤ 7 qubits; multi-pentagon HaPPY tilings will need a tensor-network library (e.g., ITensors.jl).

Phase-coherent vs basis-invariant fidelity. Two distinct metrics serve different M-levels:

• subspace_fidelity(V_target, V_opt) = |tr(V_target' V_opt) / d_bulk|² — phase-coherent across columns, the right metric for M4 reproduction.
• code_subspace_fidelity(V_target, V_opt) = tr(P_target · P_opt) / d_bulk — invariant under logical unitaries, the right metric for M5 manifold questions.

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Dependencies

Direct (declared in Project.toml):

• Piccolo 1.16.0 (pinned for reproducibility)
• CairoMakie, ForwardDiff
• LinearAlgebra, Random, Serialization (stdlib)

Test-only:

• Test (stdlib)

A pinned manifest is committed as Manifest-v1.12.toml. On Julia 1.12, Pkg.instantiate() reproduces the exact dependency graph used for every result in this repository. On other Julia versions the file is ignored and the environment resolves fresh from Project.toml, which is what the CI matrix does on the 1.10 LTS leg.

Version-suffixed manifests are a Julia 1.11+ feature: Pkg prefers Manifest-v{major}.{minor}.toml over a plain Manifest.toml when the version matches. A single unsuffixed Manifest.toml cannot be shared across Julia minor versions, because stdlib versions differ between them.

Cubic-spline synthesis requires Piccolissimo

isometry_synthesis_problem_cubic (the CubicSplinePulse / SplinePulseProblem path) must not be run against Piccolo alone. Piccolo's default BilinearIntegrator samples only the :u knot values, while CubicSplinePulse also carries :du Hermite tangents as independent NLP variables that never enter the dynamics constraint. The optimizer is then free to set :du arbitrarily, and the NLP's reported fidelity diverges from the true rollout:

Phase 1 NLP fidelity = 1.000001
Phase 2 NLP fidelity = 1.000000
Rollout fidelity     = 0.776678     <- the real number

(reproduce with examples/04e_cubic_splines_m3_sanity.jl, which is included precisely to document this failure and exits with a FAIL banner.)

The correct integrator is SplineIntegrator from Piccolissimo.jl, which is not a dependency of this package. Cubic-spline Stage-B results were produced in a separate environment declaring both HolographicControl and Piccolissimo. Anyone reproducing that work needs Piccolissimo access; without it, use the bilinear isometry_synthesis_problem path instead.

Whichever path you take, report rolled_out_isometry(qcp) — the true dynamics — rather than synthesized_isometry(qcp), which is the NLP's own estimate.

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Citing this work

Citation TBD upon thesis submission. For early reference, cite the HaPPY paper (arXiv:1503.06237) and the Piccolo.jl repository.

License

MIT