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Entangling logical qubits without physical operations

Jin Ming Koh, Anqi Gong, Andrei C. Diaconu, Daniel Bochen Tan, Alexandra A. Geim, Michael J. Gullans, Norman Y. Yao, Mikhail D. Lukin, Shayan Majidy

arXiv:2601.20927v1quant-ph

TL;DR

Fault-tolerant entangling gates are constrained by the errors and overheads of physical operations. This paper develops phantom codes, which implement logical entanglement through physical-qubit relabelling, and finds roughly 56× and 94× reductions in logical infidelity for GHZ preparation and Trotterized many-body simulation, respectively, at comparable resources and a 24% acceptance rate.

  • Problem

    Physical two-qubit gates and measurements limit scalable fault-tolerant logical entangling operations through their error rates and resource overheads.

    Source: C. Implications · p. 4

  • Method

    The authors systematically identify and construct phantom quantum error-correcting codes, characterize their logical operations, and benchmark them against surface codes using end-to-end noisy simulations.

    Source: §I · p. 2, B. Summary of results · p. 4, A. Exhaustive code enumeration · p. 5

  • Results

    Phantom codes reduce logical infidelity by approximately 56× for GHZ preparation and 94× for Trotterized many-body simulation at a 24% preselection acceptance rate and comparable physical resources.

    Source: B. Summary of results · p. 4

  • Takeaways & Limitations

    Phantom codes provide a zero-overhead route to logical entanglement and can outperform surface-code architectures for workloads with dense local entangling structure.

    Source: C. Implications · p. 4, §VI · p. 11

    On CSS phantom codes, a broad class of strictly transversal logical gates—including H, S, CZ, and magic gates—is ruled out, so other gates require non-uniform operations or qubit permutations.

    Source: c. Limitations on strictly transversal… · p. 30

Abstract

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Fault-tolerant logical entangling gates are essential for scalable quantum computing, but are limited by the error rates and overheads of physical two-qubit gates and measurements. To address this limitation, we introduce phantom codes-quantum error-correcting codes that realize entangling gates between all logical qubits in a code block purely through relabelling of physical qubits during compilation, yielding perfect fidelity with no spatial or temporal overhead. We present a systematic study of such codes. First, we identify phantom codes using complementary numerical and analytical approaches. We exhaustively enumerate all $2.71 \times 10^{10}$ inequivalent CSS codes up to $n=14$ and identify additional instances up to $n=21$ via SAT-based methods. We then construct higher-distance phantom-code families using quantum Reed-Muller codes and the binarization of qudit codes. Across all identified codes, we characterize other supported fault-tolerant logical Clifford and non-Clifford operations. Second, through end-to-end noisy simulations with state preparation, full QEC cycles, and realistic physical error rates, we demonstrate scalable advantages of phantom codes over the surface code across multiple tasks. We observe a one-to-two order-of-magnitude reduction in logical infidelity at comparable qubit overhead for GHZ-state preparation and Trotterized many-body simulation tasks, given a modest preselection acceptance rate. Our work establishes phantom codes as a viable architectural route to fault-tolerant quantum computation with scalable benefits for workloads with dense local entangling structure, and introduces general tools for systematically exploring the broader landscape of quantum error-correcting codes.

I. INTRODUCTION · II. PHANTOM CODES · A. Definitions and key properties

The paper introduces phantom codes, which eliminate in-block logical entangling gates through compilation-time relabelling of physical qubits, achieving zero overhead and perfect fidelity. It systematically expands this code family and establishes its structural properties, fault-tolerant capabilities, and practical advantages under realistic noise.

I. INTRODUCTION

  • The work addresses QEC’s computation overhead by designing codes around logical operations, motivated by the substantial error and space–time costs of conventional logical entangling gates.The broader program also develops improved decoding and fault-tolerant state-preparation tools for non-LDPC phantom codes and other non-LDPC codes.Source: §I · pp. 1–2
  • The study expands the phantom-code landscape from two previously known examples to over a hundred thousand new instances and multiple error-correcting families.It combines numerical searches with analytic constructions and also identifies fault-tolerant logical Clifford and non-Clifford gates across the discovered codes.Source: §I · pp. 1–2
  • One-to-two-order-of-magnitude logical-fidelity improvements over surface-code baselines were demonstrated for GHZ preparation and Trotterized many-body simulation using nearly identical physical-qubit counts.End-to-end noisy simulations included state preparation and full QEC cycles, covered 8–64 logical qubits, and used a preselection acceptance rate of approximately 24%.Source: §I · p. 2

A. Definitions and key properties

  • For CSS codes, phantomness is defined using a logical basis in which every ordered-pair CNOT is permutation-realizable, and the property is independent of the chosen logical basis.The CSS setting supplies transversal interblock CNOTs, while the phantom criterion concerns individually addressable in-block CNOTs rather than products of them.Source: A. Definitions and key properties · pp. 2–3

II. PHANTOM CODES

  • Phantom codes are stabilizer codes in which every ordered logical-qubit CNOT can be implemented solely by permuting physical qubits, with permutations absorbed as compilation-time relabelling.This gives zero overhead and perfect fidelity because no physical permutation is executed; all in-block entangling gates vanish from the compiled physical circuit.Source: §I · p. 1, A. Definitions and key properties · p. 2

A. Definitions and key properties

  • Phantom codes support zero-depth, all-to-all in-block entangling connectivity even when CNOTs are interleaved with non-Clifford gates, without operator spread.This distinguishes them from Pauli-frame tracking, permutation-invariant codes, and constructions requiring local physical Clifford gates.Source: A. Definitions and key properties · pp. 2–3
  • Arbitrary CNOT circuits across 2a CSS phantom codeblocks require physical depth at most 4(2a −1), or at most 2(2a −1) for unidirectional CNOTs.Residual logical-qubit permutations can be compiled away at zero cost.Source: A. Definitions and key properties · p. 3
  • A Hamming bound constrains the (n, k, d) parameters of CSS phantom codes through the number of minimum-weight logical operators and the combinatorial quantity B(n, d).Some phantom families saturate the bound, but computing its defining quantities is not generally known to be polynomial-time and the standard upper bound becomes loose for d > 2.Source: A. Definitions and key properties · pp. 3–4

B. Summary of results

The paper identifies phantom codes through exhaustive, SAT-based, and algebraic constructions, characterizes their logical operations, and benchmarks them against surface codes under realistic noise. These codes reduce logical infidelity by approximately 56× for GHZ preparation and 94× for Trotterized many-body simulation at a 24% acceptance rate.

Sec. III A

  • Exhaustive enumeration identifies every phantom CSS code among 2.71 × 10^10 codes with block length n ≤ 14.This is the Sec. III A search regime.Source: B. Summary of results · p. 4

Sec. III B

  • SAT-based search extends identification to n = 21 and establishes minimal-n realizations across a range of k and d.The Boolean-satisfiability formulation targets codes beyond the exhaustive n ≤ 14 range.Source: B. Summary of results · p. 4

Sec. III C

  • Quantum Reed–Muller constructions produce infinite families of phantom codes with higher k and distances up to d ≤ √n.These qRM families generalize hypercube codes to higher distances.Source: B. Summary of results · p. 4

Sec. III D

  • Binarization and concatenation provide a qudit-to-qubit construction for phantom codes with distances d > √n.The scheme extends the distance range beyond that reached by the qRM construction.Source: B. Summary of results · p. 4

Sec. IV

  • The identified phantom codes support additional Clifford and non-Clifford logical operations implemented by local Cliffords, permutations, fold-type interactions, and diagonal rotations.These include in-block two-qubit fold gates and magic gates from diagonal single-qubit rotations.Source: B. Summary of results · p. 4

Sec. V

  • 56× lower logical infidelity is achieved for 4–64 logical qubits in GHZ preparation at 24% acceptance and comparable physical-qubit counts.The benchmark compares a representative phantom code with surface-code baselines under realistic noise.Source: B. Summary of results · p. 4
  • 94× lower logical infidelity is observed for 8–64 logical qubits in 8-body-term Trotterized simulations using nearly identical physical resources and 24% acceptance.The comparison uses surface-code baselines under p = 10^-3 noise.Source: B. Summary of results · p. 4

C. Implications · III. CODE DISCOVERY & CONSTRUCTION

The paper positions phantom codes as zero-overhead logical-entangling codes with application-specific advantages, while introducing systematic resources for discovering and constructing codes across the broader design space. These results motivate co-optimizing storage efficiency and logical-gate efficiency, including tailored non-LDPC solutions.

C. Implications

  • Phantom codes reduce logical error rates and overhead for densely entangling circuits, improving prospects for fermionic simulation and correlated-phase preparation [24] [27].Their benefits apply to long-range-connected hardware, with especially pronounced expected impact on trapped-ion processors because neutral-atom platforms support parallel transversal entangling gates.Source: C. Implications · p. 4
  • Fault-tolerant codes should jointly optimize storage efficiency and logical-gate efficiency, rather than treating high rate or LDPC structure as sufficient.The work targets the zero-overhead logical-entangling extreme and finds the resulting codes practically competitive despite appearing to exclude high-rate qLDPC codes.Source: C. Implications · p. 4
  • The compiled database, SAT searches, and automated logical-operation pipeline enable systematic exploration of trade-offs among logical overhead, distance, stabilizer weight, and gate sets.Relaxing the zero-cost constraint can identify codes tailored to hardware connectivity or objectives such as reducing logical magic cost.Source: C. Implications · p. 4
  • Benchmarking suggests that appropriately decoded and state-prepared non-LDPC codes can outperform LDPC codes on tailored applications.This challenges the assumption that LDPC structure’s generic advantages determine practical viability for every workload.Source: C. Implications · p. 4
  • Table I catalogs representative phantom codes and their logical Clifford, fold-diagonal, and magic-gate sets, including codes found by enumeration, SAT search, qRM, and binarized-qudit constructions.The table includes codes that outperform prior or newly generated examples in parameters, gate sets, or both, and notes Hadamard-dual variants with conjugated gate sets.Source: Table I · p. 5

III. CODE DISCOVERY & CONSTRUCTION

  • Phantom codes are identified through complementary numerical searches and analytic constructions, with Table I summarizing representative codes that outperform earlier and newly generated examples.The full technical details are provided in the Appendices.Source: §III · p. 5, Table I · p. 5
  • SAT solvers formulate Boolean constraints to search for satisfying assignments or certify that none exists, supporting two major components of the numerical workflow.Appendix B provides a primer on the SAT-based approach.Source: §III · p. 5

A. Exhaustive code enumeration · B. SAT-based code discovery · C. Phantom quantum Reed–Muller codes

The paper combines exhaustive enumeration, SAT-based discovery, and analytic quantum Reed–Muller constructions to systematically identify phantom codes across increasing sizes and logical-qubit counts. These approaches reveal large code catalogues, near-optimal minimal lengths, scalable qRM families, and additional fault-tolerant operations.

A. Exhaustive code enumeration

  • The catalogue search defines equivalence under qubit permutations and global H^⊗n, then reduces enumeration using Hadamard duality and tripartite Tanner-graph canonical labelling.These reductions replace O(I^2) pairwise isomorphism checks with O(I) canonical-labelling operations, despite iterations exceeding 10^10 candidates.Source: A. Exhaustive code enumeration · p. 5
  • The enumeration identifies 2.71 × 10^10 inequivalent CSS codes with n ≤14, the largest complete stabilizer-code catalogue assembled to date.The search uses iterative stabilizer extensions, CSS-specific reductions, and canonical labelling of tripartite Tanner graphs to make equivalence testing tractable.Source: A. Exhaustive code enumeration · p. 5
  • Among these catalogued codes, 1.39 × 10^5 are CSS phantom codes, approximately one in every two hundred thousand CSS codes.Phantomness is tested by SAT constraints requiring qubit permutations to preserve the codespace and implement every pairwise logical CNOT.Source: A. Exhaustive code enumeration · p. 5

B. SAT-based code discovery

  • SAT-based code discovery extends phantom-code identification to n = 21 and determines minimal block lengths through unsatisfiability certificates.The solver treats stabilizer matrices and permutation constraints as variables while enforcing distance-d constraints through nonzero syndromes for all relevant errors.Source: B. SAT-based code discovery · p. 6
  • Compared with all CSS codes, phantom codes have near-identical minimal lengths for k = 2, small deviations for k = 3, and a larger deviation for k = 4.The increasing deviations reflect the progressively stronger constraints imposed by the required phantom gate set.Source: B. SAT-based code discovery · p. 6

C. Phantom quantum Reed–Muller codes

  • Quantum Reed–Muller constructions produce scalable phantom families by gauge-fixing selected logical qubits, with parameters J2^m, m − l + 1, min(2^m−l, 2^l)K.Reducing k by one doubles d at fixed n up to d = √n, while balancing stabilizer ranks can improve logical error rates when m = 2l.Source: C. Phantom quantum Reed–Muller codes · p. 6
  • Phantom qRM codes support preselection-based state preparation, the full logical Clifford group, and a d = 2 transversal magic-gate scheme.Their operations include fold-SiSj gates, teleported Hadamards from an all-|+⟩ state, and temporary projection into hypercube codes for transversal non-Clifford gates.Source: C. Phantom quantum Reed–Muller codes · pp. 6–7

D. Binarization and concatenation scheme · E. Other code constructions

The binarize-and-concatenate scheme converts high-distance self-dual GF(4) qudit codes into phantom qubit codes, while additional constructions extend the design space through concatenation, hypergraph products, and punctured hypercube codes.

D. Binarization and concatenation scheme

  • The construction starts from high-distance self-dual GF(4) qudit codes encoding one logical qudit, targeting distances beyond the phantom qRM family’s d ≤√n bound.The underlying GF(4) codes achieve extremal distances at small block lengths.Source: D. Binarization and concatenation scheme · p. 7
  • Binarization followed by concatenation with the J4, 2, 2K phantom code is essential for converting GF(4) qudit codes into phantom qubit codes.Binarization alone is insufficient because the J4, 2, 2K layer supplies required phantom-code substructure.Source: D. Binarization and concatenation scheme · p. 7
  • The B&C family preserves large distance and supports logical Hadamard and CZ gates through self-duality and Hermiticity of the starting qudit codes.Concatenation introduces many low-weight stabilizers, although some stabilizer generators still have weight at least the code distance.Source: D. Binarization and concatenation scheme · p. 7

E. Other code constructions

  • Simple concatenation combines a phantom outer code Jnout, k, doutK with a high-distance inner code Jnin, 1, dinK, yielding Jninnout, k, dindoutK and outer-level relabelling CNOTs.A surface code is given as an example of a high-distance inner code.Source: E. Other code constructions · p. 7
  • Hypergraph products preserve input-code permutation symmetries while increasing both k and d, enabling phantom constructions such as simplex–repetition products.The smallest listed families, J7, 2, 2K and J49, 3, 4K, have lower rates than phantom qRM codes.Source: E. Other code constructions · p. 7

D. Binarization and concatenation scheme

  • Quadratic-residue GF(4) codes map from Jn, k, dK4 to J2n, 2k, dK2 before concatenation with J4, 2, 2K produces a phantom code.The intermediate binarized code is not itself phantom but serves as the outer code in the concatenation.Source: Figure 4 · p. 7

E. Other code constructions

  • Puncturing J2D, D, 2K hypercube codes yields J2D −1, D, 2K phantom codes; both versions saturate the CSS phantom-code Hamming bound, though puncturing can reduce gates.For example, J15, 4, 2K supports CCZ but not CCCZ.Source: E. Other code constructions · p. 7

IV. LOGICAL GATES BEYOND PHANTOM

Phantom codes support logical operations beyond permutation CNOTs through automorphism Cliffords, non-uniform diagonal rotations, and fold-diagonal constructions. However, a no-go theorem limits strictly transversal gates that do not commute with the permutation logical gates, while combined mechanisms can generate full logical Clifford groups.

Fundamental constraint

  • Theorem 3 rules out strictly transversal logical gates that do not commute with a phantom code’s permutation logical gates, including H, S, CZ, and magic gates.This constraint applies to strictly transversal gates acting across any number of codeblocks.Source: §IV · p. 7

Automorphism Cliffords

  • Permutation symmetries of the extended symplectic stabilizer matrix identify automorphism Cliffords, including a J4, 2, 2K example where H⊗4 implements logical H⊗2 SWAP.The method analyzes column permutations preserving the stabilizer row span, with physical SWAP, H, and S combinations inducing logical operations.Source: §IV · p. 8

Diagonal magic gates

  • Non-uniform physical Z rotations yield logical diagonal magic gates by assigning integer powers that preserve computational-basis states up to state-dependent phases.The rotations can use distinct angles on different physical qubits and target a chosen diagonal Clifford-hierarchy level such as T.Source: §IV · p. 8

Fold-diagonal gates

  • Fold-diagonal gates arise by embedding a code into a larger code and restricting patterned one- and two-qubit diagonal interactions to depth one.In favorable cases, these fold gates preserve the X-sector code distance.Source: §IV · p. 8

Combined logical operations

  • Fold-SiSj gates combined with automorphism operations implementing H⊗k generate the full logical Clifford group in the J20, 2, 6K phantom code.Together, automorphisms, diagonal magic gates, and fold-diagonal constructions provide a broad class of logical operations beyond permutation CNOTs.Source: §IV · p. 8

V. NUMERICAL BENCHMARKING · A. Operation of codes and error correction · B. Single-codeblock repeated CNOT circuits

End-to-end noisy simulations compare the J64, 4, 8K phantom qRM code with optimized surface codes across increasingly complex benchmarks. In repeated single-codeblock CNOT circuits, phantom codes offset costly preparation through operation-free in-block CNOTs, producing major logical-failure reductions.

V. NUMERICAL BENCHMARKING

  • Phantom-code practicality is limited by high-weight stabilizers, non-LDPC structure, costly initialization and syndrome extraction, and the fact that generic circuits include CNOTs that are not in-block.The repeated-CNOT benchmark isolates the most favorable regime, where all CNOTs act within one codeblock and incur no compiled physical-operation cost.Source: §V · p. 8, B. Single-codeblock repeated CNOT… · p. 9
  • The benchmark compares J64, 4, 8K against rotated surface codes d = 5–8 at physical error rates 10^-3 and 5 × 10^-4 across CNOT, GHZ, and Trotterized-simulation tasks.These benchmarks progress from in-block entanglement to interblock operations and use circuit widths exceeding the per-codeblock logical dimension.Source: §V · p. 8, B. Single-codeblock repeated CNOT… · p. 9

A. Operation of codes and error correction

  • The optimized surface-code baseline combines LDPC operation, low-overhead preparation, bare-ancilla syndrome extraction, matching decoding, transversal interblock CNOTs, and correlated fault-tolerant decoding.Correlated decoding reduces syndrome-extraction rounds per transversal CNOT from O(d) to O(1), with one round used here.Source: A. Operation of codes and error… · p. 8
  • The non-LDPC phantom code uses Steane-style error correction, a short-depth four-to-one certified state factory with qRM automorphisms, and spatiotemporal sliding-window correlated list/MLE decoders.Constant-sized decoding windows make costs scale linearly with logical circuit size on a fixed code.Source: A. Operation of codes and error… · p. 9
  • Strict preselection accepts ∼24% and ∼49% of preparations at two-qubit error rates 10^-3 and 5 × 10^-4, respectively; allowing weight-one residual errors raises these to ∼40% and ∼64%.The comparisons in this section use strict syndrome-free preselection.Source: A. Operation of codes and error… · p. 9

B. Single-codeblock repeated CNOT circuits

  • ∼1300× (∼120×) improvement over d = 6 (d = 8) surface codes occurs at depth 10, while phantom logical gate time remains effectively zero.The phantom advantage remains similar at projected future physical error rates; surface-code failure rates grow linearly with depth, whereas the phantom rate remains unchanged.Source: Figure 5 · p. 9, B. Single-codeblock repeated CNOT… · p. 10
  • ∼55× lower logical failure than d = 6 and ∼3× lower than d = 8 are achieved during state preparation at comparable and roughly doubled physical-qubit footprints, respectively.The phantom preparation factory uses 64 data qubits and 64 × 3 ancilla qubits, with preselection overhead included.Source: B. Single-codeblock repeated CNOT… · p. 10

C. Multiple-codeblock GHZ state preparation · D. Trotterized many-body quantum simulation

Across multi-codeblock GHZ preparation and Trotterized many-body simulation, phantom codes retain substantial logical-fidelity advantages over surface codes despite interblock operations, QEC overhead, increasing system size, and projected error rates.

C. Multiple-codeblock GHZ state preparation

  • The GHZ benchmark varies K = 4–64, decreasing the permutation-to-transversal CNOT ratio from 3 : 0 to 3:61 as interblock entangling operations increase.The study tests whether phantom-code benefits diminish as transversal CNOTs contribute more, but the reported advantage remains for all K.Source: C. Multiple-codeblock GHZ state… · p. 10
  • GHZ preparation incurs active Steane QEC for interblock CNOTs, requiring two ancillary codeblocks per data codeblock and mixed computational- and Hadamard-basis logical-state preparation.The implementation prepares |0⟩⊗k and |+⟩⊗k blocks, teleports logical qubits with at most two transversal CNOTs, and transversally measures one block with preselection.Source: C. Multiple-codeblock GHZ state… · p. 10
  • At K = 64, phantom codes achieve a ∼56× logical-infidelity reduction versus d = 6 surface codes while using 4608 versus 4544 physical qubits.Against d = 8 surface codes, they retain a ∼4× infidelity advantage with ∼1.8× fewer physical qubits; the relative fidelity advantage persists for all K and projected future error rates.Source: C. Multiple-codeblock GHZ state… · p. 10

D. Trotterized many-body quantum simulation

  • The many-body benchmark simulates eight-body Ising interactions with transverse-field rotations for K = 8–64 logical qubits over eight Trotter steps.At K = 64, the circuit exceeds 2400 logical gates and reaches two-qubit logical-gate depth 96, using a hypercube layout to reduce interblock transversal CNOTs.Source: D. Trotterized many-body quantum… · p. 10
  • At K = 64, the phantom-code simulation uses 4608 qubits, compared with 4544 and 8128 qubits for d = 6 and d = 8 surface codes, respectively.The phantom-code footprint includes ancillary codeblocks for Steane QEC and state-preparation factories; simulations fix θ = ϕ = π/2 for tractability.Source: D. Trotterized many-body quantum… · p. 10
  • The simulation compares sliding-window most-likely-error decoding with a correlated list decoder that is ∼100× faster but less accurate.Both codes use one syndrome-extraction round per transversal CNOT, while surface codes use matching-based correlated decoding.Source: D. Trotterized many-body quantum… · pp. 10–11
  • At near-term error rates, phantom codes reduce logical infidelity by ∼94× versus d = 6 and ∼10× versus d = 8 surface codes in many-body simulation.The advantage persists at future error rates, under relaxed preselection against d = 8, and with less accurate correlated list decoding under strict or relaxed preselection.Source: D. Trotterized many-body quantum… · p. 11, Figure 6 · p. 11

VI. DISCUSSION AND OUTLOOK · APPENDICES

Phantom codes trade encoding rate and locality for zero-overhead logical entanglement, offering advantages for dense local workloads while motivating new code-design, compiler, decoding, and experimental directions. The work’s databases and automated tools enable systematic searches for codes optimized to hardware and logical-operation objectives.

VI. DISCUSSION AND OUTLOOK

  • Phantom codes enable logical entanglement without physical overhead or infidelity and can outperform leading fault-tolerant architectures on well-chosen dense-entangling workloads.They also broaden code discovery through systematic cataloguing of small-footprint CSS codes and solver-based searches for targeted properties.Source: §VI · p. 11
  • All identified phantom codes are non-LDPC and encode k = O(log n) logical qubits, requiring Steane-style QEC and limiting their use to dense local patches linked by transversal CNOTs.Practical deployment therefore also needs preselected logical-state factories and dedicated decoders.Source: §VI · p. 11
  • Whether permutation-implemented CNOTs can coexist with LDPC structure, higher encoding rates, or higher-distance transversal magic gates remains an open structural question.The observed absence of LDPC phantom codes, empirical k = O(log n) bound, and distance-two transversal magic gates motivate constructions or no-go theorems.Source: §VI · p. 11
  • An open design problem is whether a resource-optimal continuum connects low-rate, large-automorphism phantom codes with high-rate qLDPC constructions.Phantom codes minimize temporal entangling cost, whereas prominent qLDPC codes favor higher encoding rates.Source: §VI · p. 11
  • The work’s database of CSS codes up to n = 14, SAT-based searches, and automated logical-operation extraction enable systematic hardware- and objective-specific code optimization.Applications include tailoring codes to superconducting connectivity and reducing logical magic cost.Source: §VI · p. 12
  • Automorphism-aware compilers could treat permutations as free, pack CNOT-dense subroutines into phantom codeblocks, and schedule interblock gates to minimize noise.Finding additional algorithms with dense logical entangling structure would guide these compiler and decoding developments.Source: §VI · p. 12
  • The STAR protocol could enable near-term fermionic simulations with small-angle rotations, while phantom codes may support highly entangled states for measurement-based computation and quantum communication.Demonstrating such states with near-zero entangling-gate cost would test the phantom-code paradigm experimentally.Source: §VI · p. 12

G. Other code constructions … 5. Uniform weight distribution of logical operators on phantom codes and the Hamming bound

This section develops algebraic criteria and structural bounds for permutation-implemented logical operations in CSS phantom codes. It characterizes code equivalence and weight uniformity, and bounds the number of physical Pauli operators realizing logical Pauli types while examining refinements and the applicability of Hamming bounds.

2. Conditions for permutation logical Clifford gate sets and phantomness on stabilizer codes

  • Phantomness is characterized by whether qubit permutations implement every pairwise logical CNOT in some logical basis.For stabilizer codes, the criterion is Proposition 2; for CSS codes, phantomness can be tested in any arbitrarily chosen CSS logical basis.Source: §2 · p. 19, §3 · p. 21

3. Conditions for permutation CNOT gate sets and phantomness on CSS codes

  • In a CSS logical basis, qubit permutations implement only CNOT circuits modulo logical Paulis, while phantomness itself is independent of the CSS logical basis.This restriction follows because permutations preserve X- and Z-type operators; a smaller generating gate set can improve computational efficiency when testing phantomness.Source: §3 · pp. 20–21

4. Code equivalence under qubit permutation and global Hadamards

  • ΠH equivalence combines qubit permutations with an optional global Hadamard, preserving code parameters and the phantom property.This equivalence is preferred because arbitrary local Clifford deformations need not preserve phantomness.Source: §4 · p. 21

5. Uniform weight distribution of logical operators on phantom codes and the Hamming bound

  • Phantom codes have uniform physical-Pauli weight distributions across all nontrivial X-type logical operators at every fixed weight vector, and likewise for Z-type operators.The uniformity follows by mapping logical operators through CNOT circuits realized as qubit permutations, which preserve physical Pauli weight vectors.Source: §5 · p. 22
  • The resulting Hamming bound limits the possible ⟦n, k, d⟧ parameters of CSS phantom codes when d equals the limiting X- or Z-sector distance.Its counting argument uses the common size η of weight-d logical-Pauli equivalence classes and the number 2^k − 1 of nontrivial sector logical operators.Source: §5 · pp. 22–23
  • B(n, d) upper-bounds the number of physical weight-d Pauli-X operators implementing a logical Pauli because pairwise products must have weight at least d; the Z case is analogous.Source: §5 · p. 23
  • If η is unknown, it can be replaced by any lower bound on the size of the physical weight-d Pauli equivalence class implementing a type-µ logical Pauli.Source: §5 · p. 23
  • Exact B(n, d) values can be computed with the Z3 SMT solver, while d = 2 has the closed form F(n, 2) = n(n −1)/2.A combinatorial counting argument gives the stated upper bound by requiring that no t-subset, with t = ⌊d/2⌋+1, appears in more than one support block.Source: §5 · p. 23
  • The bound resembles the classical Hamming bound but constrains strings to have exactly weight d; the quantum Hamming bound for non-degenerate stabilizer codes [54] does not directly apply to degenerate phantom codes such as the J20, 2, 6K code.Source: §5 · p. 23

6. Efficient addressable CNOT gates and arbitrary CNOT circuits between CSS phantom codeblocks … b. Permutation CNOT gate-set-preserving logical basis changes on phantom codes

This section characterizes efficient addressable CNOT operations between CSS phantom codeblocks, including arbitrary CNOT circuits, permutation constraints, and logical basis changes that preserve the full gate set. It also gives a specific condition under which a single-CZ logical basis change preserves the permutation-based logical CNOT gate set of the J4, 2, 2K phantom code.

6. Efficient addressable CNOT gates and arbitrary CNOT circuits between CSS phantom codeblocks

  • Between two CSS phantom codeblocks, arbitrary CNOT circuits use depth at most four up to a residual permutation, or at most two for unidirectional circuits.The construction interleaves in-block permutation CNOTs with depth-one transversal CNOTs, which need not be physically performed.Source: §6 · p. 23
  • Arbitrary CNOT circuits across 2^a CSS phantom codeblocks require physical depth at most 4(2^a −1), reduced to 2(2^a −1) for unidirectional circuits.The general bound allows a residual logical-qubit permutation; unidirectional circuits preserve logical ordering.Source: §6 · pp. 24–25
  • Residual logical-qubit permutations cost zero when compiled away and require physical depth at most 8k + 8 otherwise, independent of the number of codeblocks.Arbitrary SWAP circuits between two codeblocks have depth at most four, enabling the permutation bound.Source: §6 · pp. 25–26

a. Structure of permutations implementing logical gates on stabilizer codes

  • Any permutation implementing a nontrivial involutory logical gate on a stabilizer code has even period; without nontrivial identity permutations, such gates reduce to parallelizable period-two swaps.These structural results apply to permutation CNOT and SWAP gates, although phantom-code compilation can usually eliminate the physical permutations.Source: a. Structure of permutations… · p. 27

b. Permutation CNOT gate-set-preserving logical basis changes on phantom codes

  • Phantom-gate-set-preserving logical basis changes are generated by H⊗2, CZ, and CNOT for k = 2, or H⊗k and CNOT for k ≥3.These basis changes preserve the complete individually addressable CNOT set using products of the original permutation group.Source: b. Permutation CNOT… · pp. 28–29
  • The J4, 2, 2K code admits a logical basis change implemented by a single CZ gate while retaining all permutation-realized logical CNOT circuits.The transformed operators are X′_1 = X_1Z_2, X′_2 = X_2Z_1, Z′_1 = Z_1, and Z′_2 = Z_2.Source: b. Permutation CNOT… · p. 29
  • For binary k × k matrices with k ≥ 3, full rank of [A | B] and AEB^T + BE^-T A^T = 0 for every E ∈ GL(k, F_2) imply A = 0 or B = 0.The proof derives a contradiction because if both matrices were nonzero, their columns would span at most a two-dimensional subspace, violating full rank.Source: b. Permutation CNOT… · p. 29

c. Limitations on strictly transversal logical gate sets on stabilizer codes … 1. Review of standard form of stabilizer generator matrices

The paper combines a no-go theorem for strictly transversal gates on permutation-supported stabilizer codes with scalable SAT- and enumeration-based methods for discovering and classifying phantom codes. Exhaustive CSS-code results and higher-size SAT searches establish the computational scope and structure of the discovered code families.

c. Limitations on strictly transversal logical gate sets on stabilizer codes

  • A permutation-supported logical gate forbids any strictly transversal logical gate that fails to commute with its replicated action across codeblocks.The proof uses the commutation of identical qubit permutations across blocks with the tensor-product form of a strictly transversal operation, producing a contradiction for noncommuting logical actions.Source: c. Limitations on strictly transversal… · p. 30
  • On CSS phantom codes, this theorem excludes transversal H, S, CZ, and magic gates, except for H⊗2SWAP on some two-logical-qubit codes.Any other available logical gates must use nonuniform physical operations, additional qubit permutations, or both.Source: c. Limitations on strictly transversal… · p. 30

Appendix B: SAT preliminaries

  • SAT encodes F2 matrix constraints as Boolean XOR/AND clauses, converts them to conjunctive normal form, and returns satisfiable, unsatisfiable, or unfinished outcomes.The study used kissat and PySAT, with a 14-day limit per SAT instance; cryptominisat provided no performance advantage.Source: Appendix B · pp. 30–31

Appendix C: Exhaustive code enumeration

  • 2.71 × 10^10 inequivalent CSS codes with n ≤14 yielded 132,305 phantom codes with distance d ≥2, while code counts grow super-exponentially with n.The enumeration results are stratified by logical-qubit number and distances in Tables V and VI, with the growth visualized in Fig. 7.Source: Appendix C · p. 31, §8 · p. 33, Figure 7 · p. 33

1. Iteratively generating codes

  • The enumeration constructs every code by iteratively adding independent commuting stabilizers, deduplicating ΠH-equivalent codes through canonical Tanner-graph forms.Canonical-form comparison was empirically negligible relative to other enumeration costs, avoiding massive redundancy during generation.Source: §1 · p. 31, §2 · p. 31

3. Optimized enumeration of CSS codes

  • CSS structure reduces enumeration branching, permits restricting to r_x ≤ r_z, and replaces the general Tanner graph with a smaller tripartite representation.The computation used bliss [59], roughly 10,000 cores for 1.5 months, and more than 80 TB of temporary storage.Source: §3 · pp. 31–32, §4 · p. 32

5. Alternative approaches

  • Canonical forms scale as O(ET_can), outperforming pairwise equivalence checks at O(E^2T_iso), while standard-form matrix enumeration remains astronomical, exceeding 10^17 cases for J14, 2K CSS codes.The iterative Jn,nK → Jn,1K direction was preferred over graph-state enumeration because complete graph-state databases were unavailable.Source: §5 · p. 32

7. SAT solving for CNOT permutation gate sets and phantomness

  • SAT tests complete individually addressable permutation CNOT sets for p = 2,…,k, identifying weak phantom codes for p < k and phantom codes for p = k.Distances are computed exactly by brute force, or separately as d_x and d_z for CSS codes, with d = min(d_x,d_z); SAT-based discovery extends beyond exhaustive n ≤14 enumeration.Source: §6 · p. 32, §7 · p. 33, Appendix D · p. 36

2. SAT problem formulation for discovery of stabilizer phantom codes … 2. Quantum Reed–Muller code construction

This section develops systematic methods for discovering and constructing stabilizer and CSS phantom codes, using SAT formulations and quantum Reed–Muller and hypercube structures, including punctured variants. It also examines associated logical gates, code-switching options, and trade-offs in logical magic capabilities.

2. SAT problem formulation for discovery of stabilizer phantom codes

  • SAT instances encode stabilizer commutation, phantomness, logical basis changes, qubit permutations, and simultaneous lower- and upper-bound constraints enforcing exact distance d.For CSS codes, the formulation simplifies by fixing B = C = 0, using half-symplectic representations, restricting variables to permutations, and independently constraining d_x and d_z.Source: §2 · pp. 36–37, §3 · p. 37

3. SAT problem formulation for discovery of CSS phantom codes

  • SAT certifies minimal block lengths for k = 2, 3 CSS phantom codes, while removing gate-set constraints turns the same formulation into a general code generator.Table VII combines exhaustive enumeration for n ≤14 with SAT discovery beyond n = 14, including ranges or lower bounds when searches cannot certify minimality.Source: §3 · p. 37, Table VII · p. 38

Appendix E: Phantom quantum Reed–Muller codes

  • Because numerical searches are tractable only for k ≤4, the paper constructs an infinite CSS phantom-code family by fixing selected qRM logical qubits to |0⟩ or |+⟩.The resulting family has parameters ⟦2^m, m−l+1, min(2^(m−l), 2^l)⟧.Source: Appendix E · p. 37

1. Review of Reed–Muller codes and polynomial formalism

  • Reed–Muller codes are evaluation-vector codes generated by degree-bounded square-free polynomials, with RM(r, m) having length 2^m, dimension Σ_{j=0}^r C(m,j), and distance 2^(m−r).Affine variable transformations induce coordinate permutations and act as automorphisms, yielding Aut(RM(r, m)) = AGL(m, F2).Source: §1 · p. 38, a. Polynomial formalism · p. 38

2. Quantum Reed–Muller code construction

  • Quantum Reed–Muller codes use RM(l−1, m) and RM(m−l−1, m) as X- and Z-type stabilizers, while degree-l monomials define X logicals and complemented monomials define Z logicals.These CSS codes are not automatically phantom, motivating the constructions that follow.Source: §2 · p. 40
  • All l = 1 qRM codes are phantom hypercube codes with parameters ⟦2^D, D, 2⟧, because affine coordinate permutations realize every logical CNOT circuit.Concatenating hypercube codes with a phase-flip repetition code produces higher-distance phantom codes with parameters ⟦2^(2D−2), D, 2^(D−1)⟧, though sector stabilizer imbalance affects logical error rates.Source: §2 · p. 41
  • Promoting selected qRM logicals to X- or Z-type stabilizers yields phantom codes with m−l+1 logical qubits, unchanged X-distance 2^(m−l), and Z-distance 2^l.A GL(m−l+1, F2) subgroup acts on the retained logicals to implement arbitrary permutation CNOT circuits, while orbit-wise promotion enables more balanced variants.Source: §2 · p. 41
  • The balanced ⟦64, 4, 8⟧ phantom qRM code used for benchmarking has 32 X-type and 28 Z-type stabilizers, avoiding the theorem construction’s approximately 50× worse Z-sector error rate.The balanced code’s two logical-sector error rates are similar and approximately the geometric mean of the less balanced construction’s rates.Source: §2 · p. 42
  • Setting all but one stabilizer orbit to Z type enables addressable diagonal logical S_iS_j and CZ_ij gates through physical fold circuits.The fold circuit uses S gates on the fold line and CZ gates between mirrored physical-qubit pairs.Source: §2 · p. 42
  • Different balancing choices are gauge fixings of the parent J2^m code, enabling transversal Steane-EC code switching before using balanced versions for CNOTs and less-balanced versions for diagonal gates.Fault-tolerant state-preparation schemes are required for both code versions.Source: §2 · p. 42
  • Coupling the J64, 4, 8K construction with its dual yields a non-phantom code that retains pairwise-CNOT products, permutation logical Hadamards, and transversal-S-induced logical CZ.Choosing the first and last orbits as logicals realizes this dual coupling.Source: §2 · p. 42
  • Puncturing the coordinate 00···0 from hypercube codes yields phantom codes with parameters J2^D−1, D, (d_x = 2^D−1−1, d_z = 2)K.The punctured family retains a single global X-type stabilizer and inherits phantomness through the automorphism group GL(D, F_2).Source: §2 · p. 42
  • The punctured construction is a shortened RM(D−1, D) code whose Z-type logicals and stabilizers are codewords vanishing at 00···0, with paired logicals given by complement monomials.The paired Z-type logical of x_i is x_1···x_{i−1}x_{i+1}···x_D without negation.Source: §2 · p. 42
  • The punctured family loses the highest-level transversal magic gate but retains magic gates through level D−1 and fold-implemented logical S_iS_j operations.The original hypercube code supports Dth-level magic via transversal Z rotations, whereas puncturing preserves lower-level gates by addressing subcubes.Source: §2 · p. 42
  • Punctured hypercube codes also include the J7, 3, (d_x = 3, d_z = 2)K phantom code, while its fold construction can be compiled as CNOT_12S_2CNOT_12 = CZ S_1S_2.The fold-diagonal gate follows by encoding, applying S on a subcube, and undoing the encoding.Source: §2 · pp. 42–43
  • Concatenating the J7, 3, (d_x = 3, d_z = 2)K punctured phantom code with a phase-flip repetition code produces a J14, 3, (d_x = 3, d_z = 4)K phantom code with fold-diagonal gates.This construction has 8 X-type and 3 Z-type stabilizers and was also found by exhaustive enumeration; two additional J14, 3, (d_x = 3, d_z = 4)K phantom codes likewise admit fold-diagonal gates.Source: §2 · pp. 42–43

3. Addressable diagonal logical gates via folding … 1. GF(4) CSS quantum codes and binarization

This section develops fold-based logical gates for phantom qRM codes and constructs high-distance phantom codes by binarizing GF(4)-linear CSS codes into valid CSS qubit codes. It covers supported logical operations, associated distance and rotation-basis limitations, and explicit distance-preserving code examples.

a. Involutions and associated fold-type circuits on qRM phantom codes

  • Fold-type circuits implement SS and CZ between selected qRM logical qubits, and phantom-code permutations extend these gates to any logical pair.The circuits associate involutions with single-qubit S operations on fixed points and CZ operations on paired coordinates; targeted CNOTs enable addressable S gates with an intervening Steane error-correction round.Source: a. Involutions and associated… · p. 43

b. Distilling SS states on qRM and self-dual binarization-and-concatenation phantom codes

  • Fold-diagonal gates are not generally distance-preserving: on the J64, 4, 8K code, τSS has (d_x, d_z) = (7, 4) and τCZ has (d_x, d_z) = (6, 4).Although the circuit-level distance remains d = 4 > 3, the reduced distance can worsen logical-error performance relative to distance-preserving gates.Source: b. Distilling SS states on qRM and… · p. 44
  • An outer J6, 2, 2K code distills SS resource states from phantom-code blocks, using interblock CNOTs, fold-SS operations, and postselected measurements.The protocol is advantageous when folding is substantially noisier than transversal CNOTs, such as when fold logical errors are one to two orders of magnitude worse.Source: b. Distilling SS states on qRM and… · p. 45, Figure 12 · p. 45

4. Full logical Clifford group for qRM phantom codes

  • Folding, targeted S injection, and teleportation implement addressable Hadamards, completing the full logical Clifford group on m > 2 phantom qRM codes.The construction combines Hadamard teleportation, CZ substitution with S and S†, CNOT, and transversal X-basis measurements.Source: §4 · p. 45

5. Magic gates on qRM phantom codes

  • qRM phantom codes support distance-limited magic gates by decoupling into hypercube-code subblocks and applying transversal T or T† rotations to realize CCZ and higher controlled-Z gates.The intermediate decoupled codes can have distance two, limiting fault tolerance; alternative distance-four constructions may require high-depth monomial swaps.Source: §5 · p. 46

a. STAR for the J64, 4, 8K qRM phantom code

  • For the J64, 4, 8K qRM phantom code, STAR prepares small-angle logical rotations through weight-8 physical RZ rotations, error detection, and teleportation.The induced angle satisfies tan(γ/2) = tan^8(θ/2), and the probabilistic correction sequence requires two STAR circuits on average for one logical Z rotation.Source: a. STAR for the J64, 4, 8K qRM phantom… · pp. 47–48

b. Limitations of STAR for CSS codes with even-weight stabilizer generators

  • For CSS codes whose pure-X and pure-Z logicals and stabilizers all have even weight, the STAR protocol cannot implement logical RZ after general physical gates and error detection.The restriction follows from even-weight Z-type combinations and odd-weight Y-type combinations, which impose incompatible reality properties on the relevant coefficients.Source: b. Limitations of STAR for CSS codes… · p. 48

Appendix F: Binarization and concatenation scheme for CSS phantom codes

  • The binarization-and-concatenation construction turns a GF(4) CSS code with parameters Jn, 1, dK4 into a phantom qubit code with parameters J4n, 2, ≥2dK2, using the J4, 2, 2K2 inner code.The construction uses GF(4)-linear X- and Z-type stabilizer codes, with conjugation and trace providing the underlying F2-linear binarization framework.Source: Appendix F · p. 49, §1 · p. 49

1. GF(4) CSS quantum codes and binarization

  • GF(4) elements map bijectively to two F2 components in a self-dual normal basis, allowing each GF(4) qudit to be implemented with two qubits.Pauli operators and X- and Z-type stabilizers are expanded componentwise under this mapping.Source: §1 · p. 49
  • The GF(4) stabilizer commutation relation translates through trace linearity, proving that binarization produces a valid CSS qubit code.The commutation condition is expanded over the two binary components and traced into F2.Source: §1 · p. 49
  • Binarization converts the J3,1,2K4 and J5,1,3K4 examples into J6,2,2K2 and J10,2,3K2 qubit codes, respectively.For GF(4)-linear codes, stabilizers multiplied by ω or ω2 are also stabilizers; linearly independent rows are retained after binarization.Source: §1 · pp. 49–50

2. Concatenation with the J4, 2, 2K code · 3. Small cyclic codes · a. Proof of phantom property

Concatenating GF(4) codes with J4, 2, 2K2 doubles distance and produces phantom qubit codes when coordinate transformations satisfy a permutation criterion. Small cyclic and quadratic-residue families meet this criterion, with phantomness proved by stabilizer-preserving permutations implementing the Frobenius transform.

2. Concatenation with the J4, 2, 2K code

  • The concatenated mapping sends GF(4)-qudit Paulis to four-qubit Paulis and introduces weight-four X- and Z-type stabilizer generators from the inner blocks.The mapping includes Xω 7→XXII, Zω2 7→IIZZ, Xω2 7→XIXI, Zω 7→IZIZ, X1 7→IXXI, and Z1 7→IZZI.Source: §2 · p. 50
  • A GF(4) CSS code of distance d yields a concatenated CSS qubit code with distance at least 2d.Each nontrivial GF(4)-qudit Pauli doubles in weight during binarization and concatenation.Source: §2 · p. 50
  • Concatenation turns the non-phantom J6, 2, 2K2 and J10, 2, 3K2 codes into the phantom J12, 2, 4K2 and J20, 2, 6K2 codes.The procedure encodes each qubit pair representing a GF(4)-qudit through an inner J4, 2, 2K2 block and independently reproduces codes found by SAT.Source: §2 · p. 50

3. Small cyclic codes

  • A concrete family of GF(4)-linear codes yields good distances and satisfies Proposition 14’s sufficient condition, guaranteeing that the corresponding B&C qubit codes are phantom.Source: §3 · p. 51
  • Binary quadratic-residue codes have the same parameters, so B&C offers no advantage over the usual K = 1 concatenation; encoding each qubit of the J4, 2, 2K code with this binary QR code yields a phantom code.Source: §3 · p. 52
  • The resulting B&C qubit distances are at least twice the corresponding GF(4) qudit distances, matching the actual distances numerically for the Table I constructions.The qudit-distance lower bounds come from [34, Table VI] minus one.Source: §2 · p. 50, §3 · p. 52

a. Proof of phantom property

  • For cyclic-code B&C constructions, coordinate-wise conjugation followed by j 7→2j mod p is implemented by within-block and across-block permutations, respectively.The transformed stabilizer zeros are preserved, so the stabilizers remain unchanged.Source: a. Proof of phantom property · pp. 52–53
  • The same permutation maps the logical operator γl_x to α γ^2 l_x for α ∈ {1, ω, ω2}, proving phantomness after composing with an internal α^-1 transformation.The constant is α = l(1) = sum_i l_i and belongs to {1, ω, ω2}.Source: a. Proof of phantom property · pp. 52–53

b. Logical gates … a. Connecting a CSS code and its Hadamard-dual

This section extends phantom-code constructions by identifying additional logical Clifford gates and related code families, including codes formed by connecting CSS codes with their Hadamard-duals and paired logical operations derived from permutation-based CNOT constructions.

b. Logical gates

  • For hermitian self-dual GF(4) B&C codes, transversal H implements logical HH up to SWAP12, while transversal S implements logical CZ up to Pauli corrections.For self-dual GF(4) QR codes of length p = 8k + 3, coordinate permutation i 7→2i mod p is additionally required for transversal H, and folding realizes logical SS and CZ up to Pauli corrections.Source: b. Logical gates · pp. 53–54

Appendix G: Other code constructions

  • The paper also surveys hypergraph-product, glued, and phantom-derived constructions, noting that the latter support products rather than individually addressable CNOTs and therefore are not phantom codes.Quantum Reed–Muller codes with r > 1 are cited as related phantom-like examples because they implement CNOT products through qubit permutations.Source: Appendix G · p. 55, §3 · p. 56

1. Hypergraph product phantom codes

  • The hypergraph product of an [n1, k, d1] code with full GL(k, F2) Tanner automorphisms and an [n2, 1, d2] code yields a CSS phantom code with parameters Jn1n2 + m1m2, k, min(d1, d2)K.The induced qubit permutations implement individually addressable logical CNOT gates; simplex–repetition instances include J7, 2, 2K and J49, 3, 4K codes.Source: §1 · pp. 55–56

2. Glued J4, 2, 2K codes as a family of d = 2 phantom codes

  • Gluing m copies of J4, 2, 2K produces CSS phantom families with parameters J4m, 2, 2, 2mK and J4m −1, 2, 2, 2m −1K for every m ≥1.The construction uses X-type stabilizers to form a tube, with logical CNOTs implemented by specified qubit swaps; removing one qubit gives the odd-length family.Source: §2 · p. 56

a. Connecting a CSS code and its Hadamard-dual

  • Connecting an Jn, k, (dx, dz)K CSS code to its Hadamard-dual with transversal CNOTs produces a J2n, 2kK code whose distance depends on logical–stabilizer overlaps, not solely on the primal parameters.Thus no general distance guarantee follows from (n, k, dx, dz) alone; productivity requires dx < dz, since dx ≥ dz implies d′ ≤ dz = d.Source: a. Connecting a CSS code and its… · pp. 57–58
  • For the J8, 3, (dx = 2, dz = 4)K hypercube code, the connection gives d′ = min(dz, 2dx) = 4 and reproduces the J16, 6, 4K tesseract code.The result follows from the cube geometry, which bounds the overlap between X-logicals and opposite-type operators by two.Source: a. Connecting a CSS code and its… · p. 58
  • The relevant operator-weight bound is 2|lx| + |gz| − 2|lx ∩ gz| ≥ |gz| + 2(|lx| − |lx ∩ gz|) ≥ |gz| ≥ 3.Equality occurs when lx lies entirely within gz and gz is a weight-three logical Z operator.Source: a. Connecting a CSS code and its… · p. 58
  • The construction produces a J14, 6, (dx = 3, dz = 3)K code when lx lies within gz and gz has weight three.This achieves equality in the stated bound, with dz = 3.Source: a. Connecting a CSS code and its… · p. 58
  • A permutation π implementing CNOTij on the primal code yields π ⊕ π implementing CNOTijCNOT(j+k)(i+k) on the connected code.The same permutation acts on the two n-qubit sets from the primal and dual codes.Source: a. Connecting a CSS code and its… · p. 58

b. Two-sub-lattice CSS codes by doubling a non-CSS code … c. Logical diagonal fold gates

The paper extends phantom-code gate analysis from CSS doubling and automorphism Cliffords to transversal and fold-diagonal logical gates. It provides systematic algebraic searches, basis-independence results, embedded-code constructions, and limitations from implementation depth and search complexity.

b. Two-sub-lattice CSS codes by doubling a non-CSS code

  • Doubling an ⟦n,k,d⟧ non-CSS phantom code yields a CSS ⟦2n,2k,d′⟧ code with d ≤ d′ ≤ 2d, while inducing paired permutation CNOTs.The resulting CSS code is not itself phantom, and applying the same permutation to both qubit blocks implements paired logical CNOT gates.Source: b. Two-sub-lattice CSS codes by… · p. 58

1. Automorphism logical Clifford gates

  • Automorphism logical Clifford gates are found by intersecting automorphism groups of a three-block binary representation, yielding circuits generated by H, S, and SWAP plus Pauli corrections.MAGMA computes the preserving automorphisms; physical H, S, and SWAP correspond to specified coordinate permutations, while tableau-derived Pauli corrections restore positive stabilizer signs.Source: §1 · p. 59
  • For CSS phantom codes, automorphism logical-gate existence is independent of the chosen CSS logical basis, and any supported gate pattern extends to all corresponding distinct logical-qubit choices.Logical CNOT basis changes can be implemented by qubit permutations, while logical swaps imply uniform support for single-qubit and pairwise gates across logical qubits.Source: §1 · pp. 59–60

a. Review of phase polynomials

  • Phase polynomials represent diagonal Clifford-hierarchy gates modulo N = 2^l, with XOR-to-product identities enabling systematic manipulation of single- and multi-qubit diagonal operations.Examples include x mod 2 → Z, x mod 4 → S, x mod 8 → T, and higher-order controlled gates represented by phase monomials.Source: a. Review of phase polynomials · p. 60

b. Logical diagonal gates via products of single-qubit Z-basis rotations

  • Products of physical single-qubit Z rotations implement valid logical diagonal gates exactly when phase-weight invariance holds over X-type stabilizer cosets, reducible to a kernel computation over Z_N.Kernel generators enumerate all codespace-preserving combinations of T gates and powers, and evaluating the resulting polynomial recovers the logical phase action at arbitrary hierarchy levels.Source: b. Logical diagonal gates via products… · pp. 61–62

c. Logical diagonal fold gates

  • Embedded-code constructions generalize transversal diagonal searches to fold gates using physical single- and multi-qubit diagonal operations, especially depth-one S and CZ implementations.Ancillas are added for data-qubit pairs, entangled through CNOTs, stabilized with Z_xZ_iZ_j, and mapped back to data-qubit operations through the embedding identity.Source: c. Logical diagonal fold gates · pp. 62–63, Figure 17 · p. 62
  • Restricting fold constructions to depth-one physical implementations gives fault-distance at least half the code distance, while a logical phase gate with S action suffices for the full Clifford group.The method follows, producing fold-type gates; broader embedded-code permutations from are not used here.Source: c. Logical diagonal fold gates · p. 63
  • Exhaustive fold-gate identification is generally intractable because valid embedded-code kernel elements may translate to non-depth-one circuits; larger-code results are therefore partial, algebraic, or unknown.Partial searches are reported for phantom codes with n ≲ 20, while larger-code fold gates are algebraically constructed or remain unknown.Source: c. Logical diagonal fold gates · p. 63

Appendix I: Numerical benchmarking of logical performance … 7. Benchmarking results at p = 3 × 10−3 physical error rate

This section benchmarks phantom-code logical performance using neutral-atom-calibrated circuit-level noise, fault-tolerant state preparation, correlation-aware decoding, and simulations of repeated in-block CNOTs, GHZ preparation, and Trotterized dynamics. It examines logical-failure scaling and compares phantom codes with matched surface-code baselines while accounting for preselection rates and decoding or circuit-design conditions.

1. Noise model

  • Benchmarking adopts a circuit-level noise model calibrated to recent neutral-atom experiments, with p = 3 × 10−3 representing current-generation hardware.The simulations also consider p = 1×10−3 for near-term and p = 5×10−4 for future error rates.Source: §1 · p. 63, Table VIII · p. 63

a. Circuit structure and matching-based correlated decoding

  • Surface-code logical circuits use correlated MWPM decoding that decomposes hypergraph decoding into lightcone-based graphs and reconstructs logical-observable flips.The method uses one graph per initial logical stabilizer and standard MWPM on the resulting graphs.Source: a. Circuit structure and… · p. 64

a. Fault-tolerant state preparation protocol

  • Fault-tolerant phantom-code state preparation uses four-to-one logical state certification and exploits state automorphisms beyond the code automorphism group.Qubit permutations are compiled away through relabelling, and larger state-automorphism spaces are available for k > 1 codes.Source: a. Fault-tolerant state preparation… · p. 65
  • The state-preparation protocols exhibit p^4 logical-failure scaling from p = 5 × 10−4 to p = 3 × 10−3 despite residual malignant faults.The |0000⟩ protocol has no order-two and fewer than 100 order-three malignant faults, while |++++⟩ has one order-two and approximately 200 order-three faults.Source: a. Fault-tolerant state preparation… · p. 65

b. Interfacing fault-tolerant state-preparation with logical circuit simulations

  • Prepared-codeblock residual errors are sampled from Monte Carlo state-preparation simulations and injected into subsequent logical-circuit simulations under strict or relaxed preselection.This interfaces noisy state preparation with otherwise initialized ancillary codeblocks during circuit benchmarking.Source: b. Interfacing fault-tolerant… · pp. 65–66

c. Spatiotemporal sliding-window list and most-likely-error decoding

  • List decoding is >100× faster but approximately half as accurate per window as MLE decoding because it uses fewer error correlations.MLE jointly uses bit- and phase-flip syndromes across control and target codeblocks, whereas list decoding separates X and Z errors.Source: c. Spatiotemporal sliding-window list… · p. 67

4. CNOT circuit benchmark

  • CNOT benchmarks compare transversal surface-code gates with in-block phantom-code permutations, reporting logical failure rate per logical qubit across Pauli bases.The circuits vary depth on four logical qubits and use |0000⟩ or |++++⟩ initialization with deterministic logical measurements.Source: §4 · p. 67

5. Logical GHZ state preparation benchmark

  • GHZ benchmarks prepare up to K = 64 logical qubits using a log-depth CNOT circuit, with phantom-code mixed-basis initialization enabled by teleportation between prepared codeblocks.The |+000⟩ preselection check has a rejection rate below 0.05%, and logical fidelity is estimated from a complete stabilizer-generator basis.Source: §5 · pp. 67–68
  • For GHZ benchmarking, sliding-window decoding excludes the noiseless SE round and preceding round for phantom codes, while surface-code decoding retains a relative time-boundary advantage.This boundary treatment prevents phantom-code logical infidelity from being drastically underestimated, but gives the surface code an unfair decoding advantage.Source: §5 · p. 68

6. Trotterized quantum simulation circuit benchmark

  • The Trotterized benchmark uses θ = ϕ = π/2 for tractable simulation, while small-angle rotations can use resource-efficient STAR injection on both code families.The circuit comprises forward and inverse CNOT ladders around a Z rotation; surface-code CNOTs are transversal, whereas only one phantom-code CNOT is transversal and the rest are in-block.Source: §6 · p. 68

7. Benchmarking results at p = 3 × 10−3 physical error rate

  • The benchmark compares phantom codes with surface codes matched by spatial footprint or code distance, despite phantom-code preselection rates of 1.3% strict and 5.1% relaxed.The phantom code continues outperforming d = 8 under relaxed preselection for Trotterized dynamics, and the advantage persists across GHZ sizes K.Source: §7 · pp. 68–69
  • At p = 3 × 10−3, phantom codes improve logical performance by up to ∼430× over d = 6 and ∼160× over d = 8 surface codes for depth-10 in-block CNOT circuits, while retaining ∼26× lower GHZ infidelity at K = 64 and ∼17× lower Trotterized-simulation infidelity over d = 6.The GHZ advantage over d = 8 is ∼7× with approximately 1.8× fewer physical qubits; Trotterized-simulation improvements over d = 8 are ∼13×.Source: Figure 18 · p. 69, §7 · p. 69

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