The color code, the surface code, and the transversal CNOT: NP-hardness of minimum-weight decoding

Abstract

The decoding problem is a ubiquitous algorithmic task in fault-tolerant quantum computing, and solving it efficiently is essential for scalable quantum computing. Here, we prove that minimum-weight decoding is NP-hard in three quintessential settings: (i) the color code with Pauli $Z$ errors, (ii) the surface code with Pauli $X$, $Y$ and $Z$ errors, and (iii) the surface code with a transversal CNOT gate, Pauli $Z$ and measurement bit-flip errors. Our results show that computational intractability already arises in basic and practically relevant decoding problems central to both quantum memories and logical circuit implementations, highlighting a sharp computational complexity separation between minimum-weight decoding and its approximate realizations.

Figures (12)

• Figure 1: Overview of the proof of Theorem \ref{['thm:main']}. (a) A 3DM instance, where the problem is to decide whether there exists a disjoint subset of hyperedges $t_i$ that covers all elements in $A\cup B\cup C$. (b) Symbols used to represent the wire, crossing, element, and splitting gadgets (from left to right). (c) The implementation of the splitting gadget in the color code (top) and the surface code (bottom). Dots represent defects, with the nodes circled in yellow and the partner nodes (which are defects in another gadget) circled in black. The neighborhood of the gadget is highlighted in light gray. A minimal cover in the color code with all nodes TRUE and one in the surface code with all nodes FALSE are shown. (d) An example of two gadgets linked together. The nodes from both gadgets have the same truth value, FALSE in this case. (e) Schematic for the placement of gadgets in the syndrome to be decoded. The same coarse-grained layout is used when reducing 3DM to ColorCodeZ or SurfaceCodeXYZ, with differences in the implementation of the gadgets. (f) The placement of gadgets defining the syndrome when reducing 3DM to tCNOTZ.
• Figure 2: (a)(b) The wire gadget, (c)(d) splitting gadget, and (e)-(h) crossing gadget between wires of different defect types in the ColorCodeZ decoding problem, as introduced in Ref. walters2026CCNPhard. The different minimal covers are shown. In all figures, dots represent defects (unsatisfied $X$ stabilizers in the color code), with the nodes circled in yellow and the partner nodes (which are defects in another gadget) circled in black. The neighborhood of the gadget is highlighted in light gray.
• Figure 3: The crossing gadget for wires of the same defect type is constructed from two splitting gadgets and two crossing gadgets between wires of different defect types.
• Figure 4: The element gadget with $R$ defects. Here, we illustrate the minimal cover with the second node set to TRUE.
• Figure 5: Two types of wire gadgets with $Z$ defects in the surface code. Minimal covers with both nodes TRUE and both nodes FALSE are shown in (a)(c) and (b)(d), respectively. In our surface code figures, blue strings denote $Z$ errors and red strings denote $X$ errors (with $Y$ errors on any overlap). Having access to both gadgets in (a)(b) and (c)(d) allows us to place the nodes at any vertex on the lattice.

Theorems & Definitions (8)

• Theorem 1
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 6
• proof
• Theorem 7
• proof : Proof of Theorem \ref{['thm:logical']}