Proof of Finite Surface Code Threshold for Matching

The field of quantum computation currently lacks a formal proof of experimental feasibility. Qubits are fragile and sophisticated quantum error correction is required to achieve reliable quantum computation. The surface code is a promising quantum error correction code, requiring only a physically reasonable 2D lattice of qubits with nearest neighbor interactions. However, existing proofs that reliable quantum computation is possible using this code assume the ability to measure four-body operators and, despite making this difficult to realize assumption, require that the error rate of these operator measurements is less than ${10}^{-9}$, an unphysically low target. High error rates have been proved tolerable only when assuming tunable interactions of strength and error rate independent of distance, which is also unphysical. In this work, given a 2D lattice of qubits with only nearest neighbor two-qubit gates, and single-qubit measurement, initialization, and unitary gates, all of which have error rate $p$, we prove that arbitrarily reliable quantum computation is possible provided $p<7.4\times {10}^{-4}$, a target that many experiments have already achieved. This closes a long-standing open problem, formally proving the experimental feasibility of quantum computation under physically reasonable assumptions.

Figure 1

A small surface code. Larger codes can be constructed by expanding the pattern. Circles represent qubits. Each $M_i$ represents an operator (tensor product of Pauli $X$ or $Z$ operators) that is measured to detect errors. Note that all operators commute.

Figure 2

Quantum circuits measuring (a) $ZZZZ$ and (b) $XXXX$ operators. The $|0⟩$ represents initialization, the $H$ represents Hadamard, the $M_Z$ represents measurement of the operator $Z$, and the dot and target symbols connected by lines each represent a $C_X$ gate. The $C_X$ inverts the value of the target qubit if the control (dot) qubit is in the state $|1⟩$ and does nothing otherwise. For example, $C_X(\alpha |0⟩+\beta |1⟩)|0⟩=\alpha |00⟩+\beta |11⟩$. The interaction sequence is north, west, east, south, as shown in Fig. 3.

Figure 3

2D surface code (grey). Time runs vertically. Squares represent initialization to $|0⟩$, circles represent $Z$ basis measurement. Slashed squares represent initialization to $|+⟩$, slashed circles represent $X$ basis measurement (both achieved using Hadamard gates). (a) A single error leading to a pair of detection events (long vertical ellipses). A detection event is a sequential pair of measurements with differing value. Lines with arrows show the branching paths of error propagation. (b) An error leading to a single detection event due to proximity to a boundary of the lattice.

Figure 4

Distance 4 example of a lattice of dots and lines with stochastically generated detection events. Dots (small spheres) correspond to space-time locations where the endpoints of error chains could potentially be detected. Detection events (large spheres) correspond to space-time locations where error chain end points have been detected. Lines link pairs of dots where a pair of detection events could potentially be generated by a single error. Darker lines link spatial boundaries to a single dot where a single detection event could be generated by a single error.