Hardness results for decoding the surface code with Pauli noise

TL;DR

This work shows that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively, and reduces directly from SAT for QMLD and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula.

Abstract

Real quantum computers will be subject to complicated, qubit-dependent noise, instead of simple noise such as depolarizing noise with the same strength for all qubits. We can do quantum error correction more effectively if our decoding algorithms take into account this prior information about the specific noise present. This motivates us to consider the complexity of surface code decoding where the input to the decoding problem is not only the syndrome-measurement results, but also a noise model in the form of probabilities of single-qubit Pauli errors for every qubit. In this setting, we show that quantum maximum likelihood decoding (QMLD) and degenerate quantum maximum likelihood decoding (DQMLD) for the surface code are NP-hard and #P-hard, respectively. We reduce directly from SAT for QMLD, and from #SAT for DQMLD, by showing how to transform a boolean formula into a qubit-dependent Pauli noise model and set of syndromes that encode the satisfiability properties of the formula. We also give hardness of approximation results for QMLD and DQMLD. These are worst-case hardness results that do not contradict the empirical fact that many efficient surface code decoders are correct in the average case (i.e., for most sets of syndromes and for most reasonable noise models). These hardness results are nicely analogous with the known hardness results for QMLD and DQMLD for arbitrary stabilizer codes with independent $X$ and $Z$ noise.

Figures (24)

• Figure 1: \ref{['surfaceCodeDef']} The surface code with the boundary conditions we use. Qubits are at the intersections of lines. The blue (darker) faces are $Z$ stabilizers and the green (lighter) faces are $X$ stabilizers. The boundary consists of 2-qubit stabilizers: $X$ stabilizers on the bottom and top boundaries, and $Z$ stabilizers on the left and right boundaries. \ref{['boundaryConditionStrings']} This choice of boundary conditions has an important consequence. Strings of $X$ errors can run to the top or bottom boundary without having a $-1$ syndrome at the end of the string. Likewise, strings of $Z$ errors can run to the left or right boundary without having a $-1$ syndrome at the end of the string. Red labels are a possible error consistent with these syndromes.
• Figure 2: Any boolean formula can be viewed as a boolean circuit with a layer of FAN-OUT gates on the input wires, then some wire crossings, then AND, OR, and NOT gates without any wire crossings. This is because the part of the circuit graph above the FAN-OUT gates is a tree, and all trees are planar.
• Figure 3: \ref{['exampleNoiseModel']} An example of our graphical notation for writing down noise models, which in this paper are just probability distributions of Pauli errors. For every operator drawn on a qubit, that error occurs on that qubit with the probability $p$, which we can choose to be any fixed constant in $(0,0.25]$. If no operators are drawn on a qubit, then errors have 0 probability for that qubit. The errors for different qubits are independent. \ref{['exampleErrorNoiseModel']} An example of an error that could occur with this noise model. We denote errors with bold red letters, and often draw them on top of the noise model, as done here. This error occurs with probability $p^3(1-p)(1-2p)(1-3p)$. Here we do not draw the syndromes that result from this error.
• Figure 4: Diagram of our Turing reduction from SAT to QMLD. In Section \ref{['qmldHardSection']} we give a polynomial-time algorithm that converts a boolean formula to a QMLD problem instance (surface code size, list of error probabilities, and syndromes); this algorithm is (a). A hypothetical algorithm that exactly solves QMLD would produce a Pauli error $E$ when given that QMLD problem as its input; this hypothetical algorithm is (b), the QMLD oracle. Given that error $E$, one can determine in polynomial time whether the original boolean formula is satisfiable; the algorithm that does this is (c) and is also described in Section 3. This means that the existence of an algorithm that exactly solves QMLD in polynomial time implies the existence of an algorithm that solves SAT in polynomial time; this algorithm is (d). This establishes QMLD as NP-hard, via a Turing reduction.
• Figure 5: The variable gadget. The noise model and lack of $-1$ syndromes given means that either there is one string of $X$ errors starting at the bottom boundary and continuing up (which corresponds to the variable being true), or there are no errors (which corresponds to the variable being false).

Theorems & Definitions (17)

• Theorem 1
• Theorem 1
• Corollary 1
• Corollary 1
• definition 1
• definition 2
• Theorem 1
• definition 2
• Corollary 1
• proof