This paper introduces a framework for constructing Calderbank-Shor-Steane (CSS) codes that support fault-tolerant logical transversal
Z-rotations. Using this framework, we obtain asymptotically good CSS codes that fault-tolerantly realize the logical transversal Clifford group (i.e., transversal single-qubit Clifford gates are realized within a single code block, while transversal two-qubit Clifford gates are realized across two identical code blocks). Furthermore, investigating CSS-T codes, we: (a) demonstrate asymptotically good CSS-T codes wherein the transversal
T realizes the logical transversal
S†; (b) show that the condition
C2∗C1⊆C1⊥ is necessary but not sufficient for CSS-T codes; and (c) revise the characterizations of CSS-T codes wherein the transversal
T implements the logical identity and the logical transversal
T, respectively.
Quantum circuits for multiplication are necessary for scientific computing on quantum computers. Clifford + T circuits are widely used in fault-tolerant implementations. The costs of implementing T and double-qubit gates are higher than those of other one-qubit gates in the Clifford + T group. In addition, the small number of qubits available in existing quantum devices is another constraint on quantum circuits. Therefore, we reduce T and CNOT gates and circuit width as the primary optimization goal in this paper. We propose an algorithm for multiplication with two unchanged operands. Preserving both operands of the multiplication is important for realizing some quantum algorithms, such as quantum bilinear interpolation. Using this algorithm, we design the circuit of the integer multiplier with two unchanged operands reducing CNOT gates. Next, we develop a Clifford + T circuit for the multiplier and introduce new optimization rules to reduce T gates. Comparative analysis shows that the proposed multiplier achieves the best width among existing multipliers. Compared to multipliers with two unchanged operands that use at most one ancillary qubit, our proposed multiplier has the best T-count, T-depth, and CNOT-count.
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.