Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime

Abstract

The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group $G$ and two subsets of $G$ whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find $[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]$, and $[[ 85, 8, 9 ]]$ codes, all of which are not CSS; we also find several weight-7 codes with $kd > n$. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of $0.2\%$, approximately matching that of the $[[ 144, 12, 12 ]]$ BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.

Figures (5)

• Figure 1: Visualization of two mirror codes. (a) An abelian mirror code with $G = \mathbb{Z}_6 \times Z_6$, i.e. a $6 \times 6$ square lattice with periodic boundary conditions. Each black point is a data qubit. Qubits highlighted in solid green (red) comprise $A=\{(1,2),(4,3),(4,4)\}$ ($B=\{(2,4),(3,1),(4,1)\}$), and a stabilizer is given by $\mathbf{S}((0,0)) = \mathbf{Z}(A) \mathbf{X}(B)$. Two other stabilizers are shown, one in light green/red corresponding to $\mathbf{S}((1,0)) = \mathbf{Z}(A + (1,0)) \mathbf{X}(B -(1,0))$, and one in striped green/red corresponding to $\mathbf{S}((0,1)) = \mathbf{Z}(A + (0,1)) \mathbf{X}(B - (0,1))$. In general, there are $n$ stabilizers, one for each element in $G$, and $\mathbf{Z}$ and $\mathbf{X}$ components translate in opposite directions. This code has parameters $\llbracket 36, 6, 6 \rrbracket$ with check weight 6. (b) An example of a non-abelian construction of a mirror code, with $G = S_4$ the group of permutations on $4$ elements, visualized on its Cayley graph which forms a permutohedron. Once again, $A, B$ are highlighted in solid green/red, forming a stabilizer $\mathbf{S}(e) = \mathbf{Z}(A) \mathbf{X}(B)$. Another stabilizer, shown in light green/red, is given by $\mathbf{S}((12)) = \mathbf{Z}(A (12)) \mathbf{X}(B(12))$. This purpose of this non-abelian construction is largely illustrative, as the code shown, while well-defined, has logical dimension 0.
• Figure 2: Comparison of mirror codes with its closest relatives, including two-block group algebra (2BGA), bivariate bicycle (BB), and generalized bicycle (GB) codes. "Normal" denotes a construction with group $G$ and subsets $A, B \subseteq G$ which satisfy $g A g^{-1} = A,\; gBg^{-1} = B$ for all $g \in G$; "abelian" denotes a construction with an abelian group $G$.
• Figure 3: Five syndrome extraction circuits offering various levels of fault tolerance. Colours indicate the basis of the measurement. Measurements in the $\mathbf{X}$ basis are used to compute syndromes, while $\mathbf{Z}$ basis measurements are flags, which should return the $\bra{0}$ outcome if no faults occurred. These are also denoted with flags. (a) The bare ancilla syndrome extraction circuit is the most efficient but least fault tolerant circuit. (b) The loop circuit uses a single flag qubit to detect if any errors occur on the ancilla qubit that might propagate to multiple data qubits. In this circuit and later ones, the dots with dotted lines indicate controlled Pauli operations just like in the bare ancilla circuit. This is more fault tolerant than the bare ancilla circuit. (c) The superdense circuit pairs up stabilizers, using them to flag each other. If an error happens that might propagate to the qubits of one stabilizer, this will flip the other stabilizer's syndrome. The top (bottom) measurement reports the syndrome outcome for $\mathbf{S}_1$ ($\mathbf{S}_2$). (d) The $\text{CSS-FT}_6$ circuit uses two flag qubits and distributes the controlled operations among its three ancilla qubits. This is fault tolerant when specifically measuring an all-$\mathbf{X}$ stabilizer of a CSS code with weight $\leq6$. To measure an all-$\mathbf{Z}$ stabilizer of such a code, everything must be done in the other basis, which means swapping the controls and targets of the $\leq6$ external and 4 internal CNOT gates, and changing the state preparation and measurement bases from $\mathbf{X}$ to $\mathbf{Z}$ and vice versa. In circuits (a)-(d), the $\mathbf{X}$ measurements compute the stabilizer's syndrome. (e) This circuit is fully fault-tolerant for any stabilizer code of weight $\leq6$. The circuit has been laid out so each operation that involves a data qubit is at a different timestep. If it is desirable, the three flags could be measured by a single reused qubit to bring the ancilla count down from 6 to 4. The stabilizer's syndrome is computed by adding the parities of the three $\mathbf{X}$ measurements.
• Figure 4: A comparison of a several mirror codes under various syndrome extraction circuits using the SI1000 circuit-level noise model.
• Figure 5: A comparison of several syndrome extraction circuits for various mirror codes using the SI1000 circuit-level noise model.

Theorems & Definitions (45)

• Theorem 2.1: Fundamental theorem of finite abelian groups
• Proposition 3.1: Mirror codes are LDPC
• proof
• Proposition 3.2: Well-defined mirror code characterization
• proof
• Lemma 3.3: Equivalence of alternative constructions
• proof
• Lemma 3.4: Center-based sufficient conditions for valid mirror codes
• proof
• Lemma 3.5: Normality-based sufficient conditions for well-defined mirror codes