No-go theorems for logical gates on product quantum codes

TL;DR

This work addresses the question of which fault-tolerant logical gates can be implemented on quantum LDPC hypergraph product (HGP) codes. It introduces a generalized Bravyi–König framework for algebraic, non-geometrically-local codes and leverages canonical/logical-representative techniques, punctures, and hypertube geometry to derive no-go theorems that exclude transversal non-Clifford gates for HGP codes of any product-dimension and constrain constant-depth circuits by the Clifford hierarchy level. The paper also constructs yes-go examples in both topological (local) and non-local code families that saturate these bounds, illustrating the algebraic origin of gate restrictions beyond geometry. Collectively, the results sharpen our understanding of the trade-off between information protection and logical computation in qLDPC codes and guide future designs of fault-tolerant schemes, including higher-dimensional and non-local code constructions. The work thus provides a unifying, algebraic perspective on logical gates that extends beyond geometrically local, finite-qubit, or 2D product cases and points to practical directions for achieving robust quantum computation with qLDPC codes.

Abstract

Quantum error-correcting codes are essential to the implementation of fault-tolerant quantum computation. Homological products of classical codes offer a versatile framework for constructing quantum error-correcting codes with desirable properties, especially quantum low-density parity check (qLDPC) codes. Based on extensions of the Bravyi--König theorem that encompass codes without geometric locality, we establish a series of general no-go theorems for fault-tolerant logical gates supported by hypergraph product codes. Specifically, we show that non-Clifford logical gates cannot be implemented transversally on hypergraph product codes of all product dimensions, and that the dimensions impose various limitations on the accessible level of the Clifford hierarchy gates by constant-depth local circuits. We also discuss examples both with and without geometric locality which attain the Clifford hierarchy bounds. Our results reveal fundamental restrictions on logical gates originating from highly general algebraic structures, extending beyond existing knowledge only in geometrically local, finite logical qubits, transversal, or 2-dimensional product cases, and may guide the vital study of fault-tolerant quantum computation with qLDPC codes.

Figures (3)

• Figure 1: An illustration of $X$ and $Z$ canonical logical representatives supported on either sectors of a 2-dimensional hypergraph product code. An $X$ canonical logical representative, indicated by a red line, is supported on a single column (row) given by the puncture $e_b^{(j)}$$(e_a^{(i)})$ on the first (second) sector. A $Z$ canonical logical representative, indicated by a blue line, is supported on a single row (column) given by the puncture $f_a^{(i)}$$(f_b^{(j)})$ on the first (second) sector.
• Figure 2: Support of logical operators in two sectors of a hypergraph product code under a constant-depth circuit. This diagram illustrates the case where both $L_1$ and $L_2$ are $X$-type logical operators, with the canonical representatives for $L_1$ and $L_2$ found on the left and right sector respectively, but the argument can be easily extended from here. We begin with a pink hyperplane representing the support of a canonical $X$-type logical operator $L_1$. Under conjugation by a constant-depth circuit $U$, the resulting operator $K_1 = U L_1 U^\dagger$ has support on a bounded number of red hyperplanes in each sector. In the right sector, a new logical basis $\mathcal{B}_2$ is chosen such that each logical operator $L_2 \in \mathcal{B}_2$ is supported on a single green hyperplane that is disjoint from the red hyperplanes in that sector. As a result, the intersection $\mathrm{supp}(L_2) \cap \mathrm{supp}(K_1)$ is empty, so the commutator $K_2 = K_1 L_2 K_1^\dagger L_2^\dagger$ acts trivially on the code space.
• Figure 3: Intersection of logical $X$ operators for three copies of $3$-dimensional toric code from red, green, and blue, respectively. The intersection of the red and green planes is drawn with a black dashed line, which intersects perpendicularly to the blue plane. Such an intersection in logical operators indicates a nontrivial action of logical $\mathrm{C}\mathrm{C}{Z}$ (which is realized through physical $\mathrm{C}\mathrm{C}{Z}$s acting on six paths of each cube).

Theorems & Definitions (88)

• Definition 1: Classical code
• Definition 2: Information set
• Definition 3: Stabilizer code
• Remark
• Definition 4: Code distance
• Definition 5: Quantum LDPC code
• Definition 6: CSS code and chain complex representation
• Definition 7: Support
• Definition 8: Code projector
• Definition 9: Logical