Minimum Weight Decoding in the Colour Code is NP-hard

TL;DR

It is proved that exact decoding of the colour code is NP-hard -- that is, there does not exist a polynomial time algorithm unless P=NP, which highlights a notable contrast to some of the colour code's key competitors, and motivates continued work in the narrower space of heuristic and approximate algorithms for fast, accurate and scalable colour code decoding.

Abstract

All utility-scale quantum computers will require some form of Quantum Error Correction in which logical qubits are encoded in a larger number of physical qubits. One promising encoding is known as the colour code which has broad applicability across all qubit types and can decisively reduce the overhead of certain logical operations when compared to other two-dimensional topological codes such as the surface code. However, whereas the surface code decoding problem can be solved exactly in polynomial time by finding minimum weight matchings in a graph, prior to this work, it was not known whether exact and efficient colour code decoding was possible. Optimism in this area, stemming from the colour code's significant structure and well understood similarities to the surface code, fanned this uncertainty. In this paper we resolve this, proving that exact decoding of the colour code is NP-hard -- that is, there does not exist a polynomial time algorithm unless P=NP. This highlights a notable contrast to some of the colour code's key competitors, such as the surface code, and motivates continued work in the narrower space of heuristic and approximate algorithms for fast, accurate and scalable colour code decoding.

Figures (13)

• Figure 1: The colour code and its error model. (a) The $6.6.6$ colour code which can be defined with data qubits on the vertices of a hexagonal lattice and checks on the coloured faces. A Pauli error on a bulk data qubit (the yellow star in the centre of the picture) anti-commutes with all three stabiliser checks acting on that qubit, triggering the three checks ringed in black. However, if the data qubit lies on a boundary, then only the two checks it participates in are triggered (the yellow star on the middle right); and if it lies at a corner, it triggers the single check that it is contained in (the yellow star in the bottom right). We have not shown the other boundary (the red boundary) as our constructions will need rather large codes. The coloured nodes in the centres of the hexagons are the vertices of the dual lattice, which is a representation of the colour code's error model where the nodes correspond to checks and the data qubit errors correspond to faces. (b) and (c) show errors (the yellow triangles) on the dual lattice of the colour code. The check nodes fire (ringed in black) if they meet an odd number of errors so these errors partially cancel leaving the defect patterns shown.
• Figure 2: Outline of the syndrome construction (see Section \ref{['s:proof-outline']}). In order to prove the NP-hardness of decoding in the colour code, we construct a specific syndrome that represents a 3-SAT problem and is, therefore, hard to decode. This figure illustrates the construction for the formula $(X_1\vee \bar{X}_2\vee X_4)\wedge (X_2\vee X_3\vee X_5)\wedge (\bar{X}_2\vee\bar{X}_3\vee \bar{X}_4)$. The boxes represent the main gadgets (which are collections of defects) -- for simplicity we have used lines to represent the wires. If the constructed syndrome can be covered exactly (see Section \ref{['s:exact-cover']}), then each variable gadget must be in the TRUE or FALSE state, and each clause gadget must have at least one TRUE input, which means that the assignment is a satisfying assignment for the 3-SAT formula. The converse also holds: if the formula is satisfiable then putting the variable gadgets in the corresponding state allows us to exactly cover all the variable gadgets, wires and clause gadgets (since a clause gadget can be exactly covered providing at least one of its inputs is TRUE). Finally, we use the technical properties of the garbage collection gadget to complete the cover to an exact cover of the whole syndrome.
• Figure 3: An example of the wire gadget. (a) The nodes circled in black are defects; the cyan discs are the wire gadget's link nodes through which the gadget is connected to other gadgets; the grey discs are the partner link nodes in the neighbouring gadgets. The key point is that a wire has a 'path like' structure, and has an even number of nodes. We see that the state of the link node in the top left is copied to the state of the link node in the bottom right -- TRUE to TRUE and FALSE to FALSE in (b) and (c) respectively.
• Figure 4: The clause gadget. (a) To model 3-SAT, we want the clause gadget to have an exact cover if and only if at least one input is TRUE (the link node matched to its partner). The gadget has three input links and three links that are connected to the garbage collection gadget (GC). In (b) we see that the gadget cannot be covered exactly if all three inputs are FALSE -- the errors show what is forced by the three inputs being FALSE, and we see the central node is uncovered. In contrast, (c), (d) and (e) show that the gadget can be covered exactly if one, two or three inputs are TRUE. Since the gadget is rotationally symmetric, the other cases are just rotations of the cover shown in each figure.
• Figure 5: The variable gadget. In order to model 3-SAT we want the gadget to have two exact covers, corresponding to the variable being TRUE or FALSE, and for it to have several outputs in the same state as the variable, and several outputs in the opposite state. We construct the gadget out of smaller subgadgets. (a), (b) and (c) show the RGB-duplicator subgadget, where all 3 links have the same state. (d), (e) and (f) show these combined to form the duplicator subgadget which has several red links which must all be in the same state in any exact cover. Finally, (g) and (h) show the full variable gadget. Although we have shown four links nodes in $A$ and $B$, we can extend the duplicator subgadgets arbitrarily to ensure we have enough link nodes to enable connections to all the necessary clauses. Note the variable gadget in the FALSE state is just the mirror image of the variable gadget in the TRUE state shown in (h).

Theorems & Definitions (10)

• Theorem 1
• Lemma 2
• Theorem 3
• Definition
• Definition
• Lemma 4
• proof
• Lemma 5
• proof
• Definition