Work sharing of qubits in topological error corrections

Topological error-correcting codes, such as surface codes and color codes, are promising because quantum operations are realized by two-dimensionally (2D) arrayed quantum bits (qubits). However, physical wiring of electrodes to qubits is complicated, and 3D integration for the wiring requires further development of fabrication technologies. Here, we propose a method to reduce the congestion of wiring to qubits by just adding a swap gate after each controlled-not (cnot) gate. swap gates exchange roles of qubits. Then, the roles of qubits are shared between different qubits. We found that our method transforms the qubit layout and reduces the number of qubits that cannot be two-dimensionally accessed. We show that fully 2D layouts including both qubits and control electrodes can be achieved for surface and color codes of minimum sizes. This method will be beneficial to simplifications of the fabrication process of quantum circuits in addition to improvements in the reliability of qubit systems.

Figure 1

Conventional layouts of quantum error-correcting codes. (a) A general circuit of $Z$-check for four qubit states (left) and its graphical expression (right) [24]. Data qubits are illustrated by open circles and syndrome qubits are illustrated by solid circles. The qubit-qubit operations in the conventional stabilizer measurement circuits consist of cnot gates. H indicates a Hadamard gate and M indicates a syndrome measurement in the computational basis. (b) A general circuit of $X$-check (left) and its graphical expression (right). Here, $|+\rangle \equiv [|0\rangle +|1\rangle ]/2$. (c) $Z$-checks of the seven-qubit color code [19]. A, B, and C indicate syndrome qubits. (d) $X$-checks of the seven-qubit color code. (e) Standard stabilizer measurement procedure of a distance-3 surface code based on cnot gates [14, 15]. Open circles indicate data qubits. Green (gray) and yellow (light gray) circles indicate a $Z$-check syndrome qubit and an $X$-check syndrome qubit, respectively. In (c)–(e), the order of cnot operations is described by black, dashed red, dotted blue, and dash-dotted green lines.

Figure 2

Qubit-layout transformations by cnot+swap gates instead of cnot gates. (a) A cnot+swap gate is equivalent to an iswap gate [28] with single-qubit rotations [29]. ${[\pi /2]}^Z$ indicates $\pi /2$ rotations around the $z$ axis. (b), (c) Geometrical description of (b) $Z$-check and (c) $X$-check after the replacement of cnot gates by cnot+swap gates in Figs. 1 and 1, respectively. The order of operations is described by black, dashed red, dotted blue, and dash-dotted green lines. See Eqs. (1) and (2). (d) Application of cnot+swap gates instead of cnot gates to the $Z$-check of the seven-qubit color code (left) and its rearrangement of qubits such that connected qubits become close (right). Qubits A, B, and C are initialized to $|0\rangle$ states. New syndrome qubits are qubits “a,” “b,” and “C.” (e) The $X$-check process after the $Z$-check of (d). The syndrome qubits a, b, and C of the previous $Z$-checks are initialized to $|+\rangle$ states. After the four steps of cnot+swap gates from black (solid) line to green (dash-dotted) line, qubits A, B, and C become the syndrome qubits. The quantum state shows after the $Z$-check with the initialization for the $X$-check. (f) New seven-qubit color-code layout with electrodes in the same physical plane as the qubits. Each electrode represents a set of wiring lines to a qubit such as $XY$, $Z$ and readout lines in Refs. [5, 6]. Now the role of syndrome qubits (originally qubits A, B, C) is shared by the five qubits (qubits A, B, C, a, and b). Because the readout process of quantum states requires additional physical loads to the syndrome qubits, the sharing of the role of syndrome qubits reduces the degradation of qubits.

Figure 3

The transformation of the next smallest 4.8.8 triangular color code. (a) Conventional 4.8.8 triangular color code of code distance 5 [19]. (b) Replacement of cnot gate by cnot+swap gate to the 4.8.8 triangular distance-5 color code. The transformation changes are dependent on the order of syndrome measurements. The orange (gray) circle indicates qubits that cannot be accessed two dimensionally. By this transformation, the number of orange (gray) circles can be reduced.

Figure 4

Detailed replacement process by cnot+swap gate in distance-3 standard surface code. Four directions of cnot gates are replaced by cnot+swap gates. In each step, the cnot+swap gates are applied to two qubits such that after the replacement, the quantum state is the same as that of the original cnot gate type. (a) First step. (b) Second step. (c) Third step. (d) Fourth step. (e) A final quantum state of the replacement. (f) A conventional final quantum state of Fig. 1 by using cnot gates. The difference between (e) and (f) is the arrangement of qubits.

Figure 5

Layout transformations of surface codes. (a) Replacement of cnot gates by cnot+swap gates in a distance-3 surface code. The replacement is carried out such that quantum states after the replacement are the same as those for cnot gates. (b) Rearrangement of (a) such that connected qubits become close. Twenty qubits share the work of syndrome qubits. (c) Distance-5 standard surface code. (d) Distance-5 surface code transformed by the replacement of cnot gates by cnot+swap gates. The order of operations is described by black, dashed red, dotted blue and dash-dotted green lines. Fifty-three qubits share the work of syndrome qubits. The alleviation of the concentration of the workload of the syndrome qubits contributes to reducing the degradation of qubits.

Figure 6

Layout transformation of a rotated distance-3 surface code. (a) Conventional layout of the rotated distance-3 surface code. (b) Replacement of cnot gates by cnot+swap gate in the rotated distance-3 surface code. There is one qubit (qubit 1) that cannot be accessed two dimensionally. In order to access all qubits, we need to cut one of the connections to around qubit 1. Here, we show a case of cutting the connection between qubit 0 and 14. (c)–(g) The cut technique to connect qubit 0 with 14 using cnot+swap gates. The initialization of two ancilla qubits can be carried out during the fourth process of (b). (h) Estimated logical error probabilities of the distance-3 rotated surface code of (b) as a function of physical qubit error. The symbols indicate the logical error probabilities estimated by numerical simulation, where the single-qubit error probability ($p_1$) is comparable to the two-qubit physical error probability ($p_2$) such as $p_1=p_2/2$. The initial states are encoded logical states of ${|0\rangle }_L$ and ${|+\rangle }_L$. (i) The estimated logical error probabilities when $p_1=0$.

Figure 7

Fault-tolerant rotated surface code with electrodes in the same physical plane as qubits. By cutting the connection between qubits 0 and 14 in Fig. 6, we can attach electrodes to all qubits in the same physical plane as the qubits. The distances between qubits will be optimized to avoid crosstalk between electrodes. Inset: Superconducting implementations of qubits and electrodes. Each electrode represents a set of wiring lines to a qubit such as $XY$, $Z$ and readout lines in Refs. [5, 6].

Figure 8

Estimated logical error probabilities of the distance-3 rotated surface code of Fig. 6 in the asymmetric depolarizing error model (see the text for details). The initial states are encoded logical states of ${|0\rangle }_L$ and ${|+\rangle }_L$. The fitting curves are defined as alpha $p_2^2$, where alpha is a fitting parameter.

Figure 9

Fault-tolerant seven-qubit color code in Z-check. Fault-tolerant seven-qubit color code is realized after forming three cat states as the syndrome qubits. The syndrome qubits A, B, and C in Figs. 1 and 1 are replaced by four qubits, A1–A4, B1–B4, and C1–C4, which are assumed to be in cat states, respectively. (a) When we can successfully obtain the cat state, an additional three steps (black, dashed red, and dotted blue arrows in the figure) of the cnot+swap gate are needed for the generation of a fault-tolerant seven-qubit color code. (b) Final state after the operations.

Figure 10

Generation process of a four-qubit cat state from five qubits. A four-qubit cat state is useful for the syndrome qubits of the fault-tolerant seven-qubit color code. (a) Circuit for generating a four-qubit cat state, given by $\frac{1}{\sqrt{2}}\left[\right|0\rangle |0\rangle |0\rangle |0\rangle +|1\rangle |1\rangle |1\rangle |1\rangle ]$, starting from $|00000\rangle$. (b) Generation process using cnot+swap.

Figure 11

Fault-tolerant seven-qubit color code, in which all qubits can be accessed two dimensionally. In order to access all qubits two dimensionally, we cut the connection between A4 and C4 of Fig. 9. The cnot+swap gate between the qubit A4 and C4 can be established by a series of cnot+swap gates in a A4-A3-B4-C1-C4 line, with additional ancilla qubits attached to the qubits A3, B4, and C1 (closed dark circles). The three ancilla qubits are initialized to the $|0\rangle$ state. (a) Operations before the connection between qubits A4 and C4. The cnot+swap gates expressed by the black arrows are carried out first, and those of the dashed red arrows are carried out next. (b) Quantum states after the processes of (a). (c) Apply cnot+swap gates from the three ancilla qubits to qubits A3, B4, and C1. Then, the three ancilla qubits store the quantum states of A3, B4, and C1. (d) Apply cnot+swap gates from the qubit of “e” to the two nearest qubits. Next, apply cnot+swap gates from the C4 qubit to the above two qubits. (e) Apply cnot+swap gates to put “$\mathrm{e}\oplus$ C4” and “e” states to their target qubits, respectively. (f) Return the qubit states in the three ancilla to the original qubits. (g) Apply cnot+swap gates to move the qubit state “g” to the appropriate position. (h) The final state, which has the same quantum states as Fig. 9.