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arXiv:2604.15111

Constraints on phantom codes from automorphism group bounds

Arthur S. Morris, Daniel Malz

quant-phSubmitted Apr 16, 2026• Announced Apr 17, 2026• 0 citations
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Abstract

Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical $\mathrm{CNOT}$ circuit can be implemented with a physical permutation, a property that has the potential to greatly reduce the depth of compiled circuits. Here we show that phantomness comes at the cost of low encoding rate. Specifically, we prove that any binary phantom code encoding $k$ logical qubits into $n$ physical qubits with distance $d\geq 2$ obeys the bound $k\leq \log_2(n+1)$ for all $k\neq 4$. For $k=4$ we explicitly construct a nonstabiliser $(\!(8, 2^4, 2)\!)$ phantom code that violates the bound and has a transversal non-Clifford gate. We further show that, within the class of nontrivial CSS phantom codes with $k\neq 4$, there is a unique family of codes saturating this bound. In addition, we prove that this logarithmic ceiling cannot be circumvented by permitting additional local unitary gates, or by making use of subsystem codes: any subspace or subsystem code admitting a $\mathrm{SWAP}$-transversal implementation of every logical $\mathrm{CNOT}$ circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group, a result which may find applications beyond phantom codes.

Comments
9 + 20 pages, 3 figures

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Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical CNOT\mathrm{CNOT}CNOT circuit can be implemented with a physical permutation, a property that has the potential to greatly reduce the depth of compiled circuits. Here we show that phantomness comes at the cost of low encoding rate. Specifically, we prove that any binary phantom code encoding kkk logical qubits into nnn physical qubits with distance d≥2d\geq 2d≥2 obeys the bound k≤log⁡2(n+1)k\leq \log_2(n+1)k≤log2​(n+1) for all k≠4k\neq 4k=4. For k=4k=4k=4 we explicitly construct a nonstabiliser ( ⁣(8,24,2) ⁣)(\!(8, 2^4, 2)\!)((8,24,2)) phantom code that violates the bound and has a transversal non-Clifford gate. We further show that, within the class of nontrivial CSS phantom codes with k≠4k\neq 4k=4, there is a unique family of codes saturating this bound. In addition, we prove that this logarithmic ceiling cannot be circumvented by permitting additional local unitary gates, or by making use of subsystem codes: any subspace or subsystem code admitting a SWAP\mathrm{SWAP}SWAP-transversal implementation of every logical CNOT\mathrm{CNOT}CNOT circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group, a result which may find applications beyond phantom codes.
An [[n,k,d]][[n,k,d]][[n,k,d]] stabilizer code is phantom if every in-block logical CNOT gate can be realized by a permutation of its physical qubits. This eliminates the large and complicated physical overhead normally required for logical entangling gates. Yet this symmetry is highly restrictive: phantom codes are rare, the number of logical qubits is limited to k=O(log⁡n)k=O(\log n)k=O(logn), and no phantom qLDPC family with growing logical dimension was previously known. We make three contributions. 1. We prove that recognizing phantomness of a given stabilizer code is at least as hard as Graph Isomorphism, even for CSS codes encoding only k=2k=2k=2 logical qubits. 2. We propose the first rate optimal phantom qLDPC families: for every fixed DDD, our CSS families achieve the maximal logical scaling k=Θ(log⁡n)k=Θ(\log n)k=Θ(logn) and distance d≥Dd\geq Dd≥D. 3. We prove a distance no-go theorem: every phantom code family with k=ω(log⁡n)k=ω(\sqrt{\log n})k=ω(logn​) and check weight w=O(1)w=O(1)w=O(1) satisfies d≤wd\leq wd≤w for sufficiently large nnn. Thus our fixed-distance qLDPC families are distance-scaling optimal at the maximal logical scaling k=Θ(log⁡n)k=Θ(\log n)k=Θ(logn).
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 2.71×10102.71 \times 10^{10}2.71×1010 inequivalent CSS codes up to n=14n=14n=14 and identify additional instances up to n=21n=21n=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.
Universal fault-tolerant quantum computing combines versatile but expensive operations with specialized but cheap ones. Its efficiency depends on how much computation can be pushed onto the cheap operations and on the size of the code needed to do so. Automorphism gates provide such cheap operations using only physical single-qubit Clifford gates and qubit permutations. Yet no general theory characterizes their maximum logical power or the minimum code size needed to attain it. We develop such a theory. For stabilizer codes encoding k≥3k\geq3k≥3 logical qubits, we show that the largest logical group attainable by automorphisms is generated by all addressable SSS and CX\mathrm{CX}CX gates, and we construct codes attaining it. While this group contains exponentially fewer gates than the full Clifford group, adding one suitable non-Clifford gate yields universality. We further classify the largest logical groups attainable using qubit permutations, physical single-qubit Cliffords, or both across general stabilizer and CSS codes, and derive refined bounds for self-dual CSS subclasses. Achieving the maximum-size logical group through automorphisms requires n=Θ(2k)n=Θ(2^k)n=Θ(2k) physical qubits. By contrast, all addressable diagonal Clifford gates, generated by SSS and CZ\mathrm{CZ}CZ, require only n=Θ(k2)n=Θ(k^2)n=Θ(k2) physical qubits when implemented using physical single-qubit Cliffords alone. Both bounds are tight. This polynomial qubit cost extends beyond Cliffords to all addressable diagonal gates at any fixed level of the Clifford hierarchy, using physical single-qubit diagonal gates. Thus, for full addressability, the sharpest physical-qubit cost divide lies between diagonal and CX\mathrm{CX}CX-type gates, not between Clifford and non-Clifford gates.
Identifying stabilizer codes that admit fault-tolerant implementations of the full logical Clifford group would significantly advance fault-tolerant quantum computation. Motivated by this goal, we study several classes of fault-tolerant gadget constructions consisting of Clifford gates acting on the physical qubits, including transversal gadgets, code automorphisms, and fold-transversal gadgets. While stabilizer codes encoding a single logical qubit, most notably the [[7,1,3]] Steane code, are known to admit transversal implementations of the full logical Clifford group, no analogous examples are known for codes encoding multiple logical qubits. In this work, we prove a no-go theorem establishing that no stabilizer code admits a fully transversal implementation of the Clifford group on more than one logical qubit. We further strengthen this result by showing that fold-transversal implementations of the full logical Clifford group are impossible for stabilizer codes encoding more than two logical qubits. More generally, we introduce the notion of k-fold transversal gadgets and prove that implementing the full Clifford group on k logical qubits requires at least k-fold transversal gadgets at the physical level. In addition, we analyze code-automorphism based constructions and demonstrate that they also fail to realize the full Clifford group on multiple logical qubits for any stabilizer code. Together, these results place fundamental constraints on fault-tolerant Clifford gadget design and show that stabilizer codes supporting the full logical Clifford group on multiple logical qubits via these architectures do not exist. Since the Clifford group is a core component of universal gate sets, our findings imply that quantum computing with codes encoding multiple logical qubits within a single code block necessarily entails more complex constructions for fault tolerance.
We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 231 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We tabulate 173 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, 10, 12, and 13, with respective best rates 2/5, 3/4, 34/77, 17/28, 14/25, 1/27, 3/56, and 1/33. We construct families of CSS codes from bipartite grids, cut-complements, and quadrics, and census the self-dual codes invariant under rank-3 permutation groups. Many of these codes realize the full logical Clifford group in this way.
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.