Logical Entangling with Phantom Codes in Hypergraph Products

Abstract: Logical entangling gates are a major source of physical spacetime overhead in fault-tolerant quantum computation. Phantom codes reduce this cost by implementing every ordered in-block logical CNOT through physical qubit permutations and Pauli-frame updates. Whether this mechanism can coexist with the low-weight stabilizer structure of qLDPC codes is a central question for low-overhead fault-tolerant architectures. We give a deterministic answer within binary CSS hypergraph product (HGP) codes. Up to natural equivalences, the simplex-repetition family is the unique HGP family satisfying the phantom condition. We then evaluate this family under circuit-level noise in logical GHZ-state preparation and Trotterized many-body quantum simulation. The codes retain low-weight stabilizer checks and yield concrete advantages over rotated surface-code baselines in both benchmarks. Reconfigurable neutral-atom arrays offer a natural setting for this approach, supporting nonlocal qLDPC operations while enabling in-block logical CNOTs without additional physical operations. Together, these results make precise how permutation-based logical entangling constrains code design within the HGP framework, demonstrate the circuit-level benefits of the unique family, and guide the search for phantom qLDPC families with better asymptotic parameters for low-overhead fault tolerance on neutral-atom hardware.

• The paper establishes that only simplex–repetition HGP codes permit permutation-based in-block logical CNOTs, enabling zero-overhead entangling.
• Circuit benchmarks demonstrate lower logical infidelities compared to surface codes in tasks like GHZ state preparation and many-body simulations.
• This classification restricts binary CSS HGP designs, emphasizing hardware-specific advantages for architectures like neutral-atom quantum processors.

Logical Entangling with Phantom Codes in Hypergraph Products: An Expert Summary

Introduction and Motivation

The implementation of logical entangling gates, especially logical CNOTs, is a significant source of spacetime overhead in fault-tolerant quantum computation. Phantom codes have emerged as a promising class wherein every ordered in-block logical CNOT can be implemented solely via physical qubit permutations and Pauli-frame updates. This mechanism offers strong potential for reducing hardware overhead, particularly for architectures supporting nonlocal connectivity, such as neutral-atom arrays. However, a central question pertains to whether phantom code constructions are compatible with low-weight stabilizer structure and scalability, such as that provided by quantum low-density parity-check (qLDPC) codes.

This work establishes that, within the framework of binary CSS codes constructed via hypergraph product (HGP), the simplex--repetition family is, up to natural equivalence, the unique HGP construction satisfying the phantom gate implementation condition. The paper provides a structural classification, validates this family through circuit-level benchmarks, and discusses implications for future fault-tolerant quantum computing architectures (2607.12948).

Theoretical Classification: Uniqueness of HGP Phantom Codes

The hypergraph product construction provides a method for synthesizing CSS quantum codes from classical seed codes, generating codes with desirable sparsity properties. The main theoretical result demonstrates that, for binary CSS HGP codes, requiring every ordered in-block logical CNOT to be executable by a physical qubit permutation restricts one, up to relabelings and trivial code equivalences, to HGPs of a simplex code and a repetition code.

The technical core involves characterizing when the group of physical qubit permutations can act transitively on the set of nontrivial logical operators. By a detailed algebraic analysis leveraging the Künneth decomposition of the logical operator space, it is established that only the simplex code (as the unique binary one-weight code of given dimension) paired with a repetition code satisfies the aggressive constraint imposed by the phantom property. All other HGP constructions necessarily violate transitivity or create logical operator classes with differing minimum weight, precluding a canonical permutation-based implementation of logical CNOTs.

This result provides sharp restrictions on code design. It not only complements general no-go results for high-rate codes with permutation-based logical gates [Koh2026Phantom, MorrisMalz2026PhantomBounds, guyot2026addressability], but gives a complete constructive classification in the HGP setting.

The Simplex--Repetition HGP Family: Structure and Properties

The simplex--repetition HGP family is thus isolated as the sole binary CSS HGP family exhibiting the phantom property. The construction employs $H_1$ as the parity-check of the binary simplex code, whose kernel consists of all nontrivial linear functionals on $\mathbb{F}_2^k$, and $H_2$ as the parity-check matrix for the length-$d$ repetition code. The resulting code encodes $k$ logical qubits, with parameters

$[[n_1 n_2 + m_1 m_2,\; k,\; \min\{2^{k-1}, n_2\}]]$

where $n_1=2^k-1$ and $m_1 = \frac{(2^k-1)(2^{k-1}-1)}{3}$.

Stabilizer checks inherit low weight: $X$-checks are at most weight $5$ (for $\mathbb{F}_2^k$0), and $\mathbb{F}_2^k$1-checks are weight $\mathbb{F}_2^k$2. The structure allows all ordered in-block logical CNOTs to be mapped to physical qubit permutations associated to $\mathbb{F}_2^k$3 actions on the simplex indices, followed by Pauli-frame updates, and without active quantum operations—yielding "zero-overhead" in-block logical entangling.

Circuit-Level Benchmarks

Two types of benchmark quantum circuits are examined to ascertain the operational impact of the simplex--repetition HGP family relative to standard rotated surface-code baselines of matching distance:

• Logical GHZ-State Preparation: For $\mathbb{F}_2^k$4 logical qubits, the HGP code enables in-block fanout via phantom CNOTs and expands the GHZ state across blocks using only transversal inter-block CNOTs. The simplex--repetition HGP code (e.g., [[365,4,8]]) exhibits a lower logical infidelity compared to surface code implementations, e.g., $\mathbb{F}_2^k$5 vs. $\mathbb{F}_2^k$6 for $\mathbb{F}_2^k$7 at $\mathbb{F}_2^k$8.
• Trotterized Many-Body Quantum Simulation: Simulating Clifford circuits for eight-body Ising interactions interleaved with transverse-field terms, where in-block CNOT ladders are absorbed by phantom relabellings. For $\mathbb{F}_2^k$9 at $H_2$0, the HGP code achieves a logical infidelity of $H_2$1 versus $H_2$2 for the matched surface code.

These results persist over multiple circuit sizes and physical error rates, and the advantage is more pronounced when the quantum workload allows a larger fraction of entangling layers to be implemented by permutations, as in GHZ state preparation.

Implications and Outlook

From a theoretical perspective, the uniqueness proof elucidates how stringent the requirement is for implementing all in-block logical CNOTs via permutations within binary CSS HGP constructions—admitting only the simplex--repetition code family. Practically, these codes deliver lower logical infidelity in realistic circuit-level noise models while retaining low-weight stabilizers, and they are naturally suited to architectures with reconfigurable connectivity such as neutral-atom arrays, where nonlocal operations and classical relabelling are feasible.

The results establish that, for the aggressive form of the phantom property, there is little room for scalable high-rate qLDPC codes within the HGP paradigm. This classification highlights an important research direction: the search for qLDPC code families with relaxed forms of phantomness, e.g., supporting permutation-based CNOTs on a selected subset of logical qubits or within alternative LDPC frameworks such as balanced products or lifted products, possibly combined with hardware-specific modifications for neutral-atom quantum processors.

Conclusion

The paper provides an exact structural classification: simplex--repetition HGP codes uniquely satisfy the phantom CNOT condition among binary CSS HGP codes. Circuit benchmarks demonstrate tangible reductions in logical error rates compared to established surface code baselines, substantiating the practical value of this construction in logical state preparation and many-body simulation tasks. The findings map the boundaries of permutation-based logical entangling in standard qLDPC constructions and motivate the exploration of broader code families for realizing scalable, low-overhead, and hardware-adapted fault-tolerant quantum computation (2607.12948).