Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.
Low-space-overhead fault-tolerant quantum computation requires not only high-rate quantum error-correcting codes but also space-efficient implementations of logical operations. Transversal and fold-transversal gates are promising since they limit error propagation and require no additional qubits. However, the logical operations they enable are typically restricted, and a central challenge is to construct codes that combine a complete set of logical Clifford gates with favorable code parameters. In this work, we introduce a family of quantum codes with an asymptotically constant encoding rate and sublogarithmically growing stabilizer weight, while supporting the entire logical Clifford group using only transversal and fold-transversal gates. Our construction is based on classical codes whose code spaces are absolutely irreducible Steinberg modules of their Tanner-graph automorphism groups. Taking hypergraph products of these classical codes yields quantum codes for which all logical Clifford operations can be synthesized from a fixed set of transversal and fold-transversal gates. Moreover, the slow growth of the stabilizer weight enables a high error-correcting performance in small instances. These results provide a path toward fault-tolerant quantum computation with low space overhead using transversal and fold-transversal gates.
To build large-scale quantum computers while minimizing resource requirements, one may want to use high-rate quantum error-correcting codes that can efficiently encode information. However, realizing an addressable gate$\unicode{x2014}$a logical gate on a subset of logical qubits within a high-rate code$\unicode{x2014}$in a fault-tolerant manner can be challenging and may require ancilla qubits. Transversal and fold-transversal gates could provide a means to fault-tolerantly implement logical gates using a constant-depth circuit without ancilla qubits, but available gates of these types could be limited depending on the code and might not be addressable. In this work, we study a family of [[n=2m,k=(m/2m)≈n/πlog2(n)/2,d=2m/2=n]] self-dual quantum Reed$\unicode{x2013}$Muller codes, where m is a positive even number. For any code in this family, we construct a generating set of the full logical Clifford group comprising only transversal and fold-transversal gates, thus enabling the implementation of any addressable Clifford gate. To our knowledge, this is the first known construction of the full logical Clifford group using only transversal and fold-transversal gates without requiring ancilla qubits for a family of codes in which k grows near-linearly in n.
It is a major challenge in quantum fault-tolerance to obtain low-overhead protocols for performing non-Clifford gates. In this vein, we construct quantum codes with low-weight stabilizers that support transversal (i.e. low-depth) implementations of the non-Clifford Cr−1Z gate, for every constant r≥3. In particular, we obtain length-n quantum LDPC codes (with constant-weight stabilizers) of polynomial distance d≥n(1−ε)/r supporting transversal Cr−1Z gates on a close-to-linear number k≥n1−ε of disjoint tuples of logical qubits, for arbitrarily small ε>0. Our construction is the first with constant-weight stabilizers that obtains dk≫n, and as a consequence achieves arbitrarily small magic state overhead exponent γ=log(n/k)/log(d)>0. Comparable prior constructions instead required at least polylogarithmic stabilizer weight. We also show how to obtain linearly many k=Ω(n) logical Cr−1Z gates, though with stabilizer weight and physical circuit depth nε. We show that our transversal gates also support addressing (i.e. targeting) of specific logical qubits.
To obtain our codes, we develop a general transformation based on cup products that maps classical codes satisfying a multiplication property to quantum codes with transversal Cr−1Z. We apply this transformation to a new family of classical Tanner codes that we construct from punctured tensor products of algebraic codes.
Quantum LDPC codes promise to reduce the overhead of fault-tolerant quantum computation, but their practical use requires efficient logical gates, measurements, and access to non-Clifford resources. Here, we develop a complete set of computational primitives for La-cross hypergraph product codes. For a subclass of these codes, we show that seed automorphisms together with fold-transversal gates generate the full logical Clifford group up to Pauli corrections. We further prove that every rank one logical operator admits D pairwise disjoint physical representatives, enabling parallel gauging and a space--time tradeoff for logical measurements. Finally, we construct an addressable surface code interface for injecting externally prepared resource states into selected logical qubits. Together, these results establish La-cross codes as a simple, low-connectivity generalisation of surface code computation, retaining analogous computational primitives while providing a complete Clifford+T instruction set across multiple encoded qubits.
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 study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S, SHS=X, and CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
Designing quantum codes with both transversal non-Clifford gates and good error-correcting parameters is an important goal in fault-tolerant quantum computation. Here, we construct an explicit family of binary CSS codes with constant rate and linear distance that admit fully addressable transversal T gates. Specifically, any prescribed tensor product of logical powers of T is implemented by a tensor product of physical powers of T, without any subsequent correction. Our construction combines algebraic-geometry codes with suitable binary embedding to obtain generalized divisibility of binary codes. This divisibility then enables fully addressable transversal T gates. Furthermore, we formulate the minimum binary embedding length problem as a minimum-weight problem over an affine space and numerically improve the constants in the asymptotic rate and relative-distance bounds.