Measurement-free implementations of small-scale surface codes for quantum-dot qubits

The performance of quantum-error-correction schemes depends sensitively on the physical realizations of the qubits and the implementations of various operations. For example, in quantum-dot spin qubits, readout is typically much slower than gate operations, and conventional surface-code implementations that rely heavily on syndrome measurements could therefore be challenging. However, fast and accurate reset of quantum-dot qubits, without readout, can be achieved via tunneling to a reservoir. Here we propose small-scale surface-code implementations for which syndrome measurements are replaced by a combination of Toffoli gates and qubit reset. For quantum-dot qubits, this enables much faster error correction than measurement-based schemes, but requires additional ancilla qubits and non-nearest-neighbor interactions. We have performed numerical simulations of two different coding schemes, obtaining error thresholds on the orders of ${10}^{-2}$ for a one-dimensional architecture that only corrects bit-flip errors and ${10}^{-4}$ for a two-dimensional architecture that corrects bit- and phase-flip errors.

Figure 1

A unitary operation determined by the result of a measurement of an ancilla followed by reinitialization of the ancilla (left circuit) is equivalent to a unitary controlled operation followed by the reinitialization of the control qubit (right circuit). Double lines in the left circuit represent a single classical bit. In both circuits, $|0\rangle$ indicates a rapid reinitialization of the qubit to its 0 state. For quantum dots, this involves tunneling to and from a reservoir.

Figure 2

Error-extraction and -correction circuit for the bit-flip code, a one-dimensional surface code that only corrects bit-flip errors. Here $d_1$, $d_2$, and $d_3$ are data qubits and the rest are ancillas. The error information is extracted to ancilla pairs $\{a_1,a_2\}$ and $\{b_1,b_2\}$ in alternate time cycles, as shown in the figure. (A given cycle corresponds to time steps 1–11, where a particular step may be comprised of 1–2 gates that can be implemented simultaneously.) This two-cycle sequence ensures that single-data-qubit errors are always signaled by exactly two syndromes. A data error on $d_2$ flips both ancilla qubits that interact with $d_2$ in that cycle. A data error on $d_1$ or $d_3$ flips the pair $\{a_1,b_1\}$ or $\{a_2,b_2\}$, respectively. Using these ancilla pairs as the controls and the relevant data qubits as the targets of the Toffoli gates, all single bit-flip errors are corrected. Whenever a Toffoli gate corrects an error, the control qubits must return to state $|0\rangle$ so that the matching is perfect, i.e., any syndrome is associated with at most one error. To do that, for each error-correcting Toffoli gate we employ another one targeting either $c_1$ or $c_2$ ancilla qubits and use these ancillas as the control qubits for two controlled-not (cnot) gates whose targets are the two control qubits of the error-correcting Toffoli gate. The portion of the circuit responsible for perfect matching is enclosed in the blue box in the figure. Excluding this subcircuit simplifies the QEC procedure considerably without significantly affecting its ability to correct single errors, as explained in the Appendix.

Figure 3

(a) Architecture of a 17-qubit, distance-3 surface code, introduced in Ref. [21]. Data qubits are denoted by the open circles whereas the $X$- and $Z$-syndrome qubits are denoted by orange (light) and green (dark) closed circles, respectively. (b) Standard syndrome-extraction circuits for $X$ and $Z$ syndromes, for the qubit orientations shown on the right. The circuits for the exterior ancillas that have only two neighboring data qubits involve fewer gates, with the same relative ordering.

Figure 4

(a) The 29-qubit system for implementing measurement-free error correction of a distance-3 surface code for nine data qubits. The data qubits are denoted by open circles labeled 1–9. These qubits and their error-extraction syndromes $X_1,...,X_4,Z_1,...,Z_4$ are present in the measurement-based architecture shown in Fig. 3; twelve additional ancilla qubits (four $c$ qubits and $\widetilde{X_1},...,\widetilde{X_4},\widetilde{Z_1},...,\widetilde{Z_4}$) are present in the measurement-free architecture. Syndrome information is extracted into two sets of eight ancilla qubits, $S_1=\{X_1,...,X_4,Z_1,...,Z_4\}$ and $S_2=\{\widetilde{X_1},...,\widetilde{X_4},\widetilde{Z_1},...,\widetilde{Z_4}\}$, in alternate time cycles. Four ancilla qubits, denoted by blue closed circles labeled $c$, are used to remove used syndrome information from the system. The positions of the qubits in the figure do not necessarily represent the actual physical geometry, particularly since we have assumed long-distance couplings. (b) Portion of the circuit used to implement measurement-free $X$-error correction. Here the syndrome has already been extracted and stored on the $Z$ ancilla qubits. Two different cases are illustrated in the figure. In case 1, if there is an $X$ error on qubit 5, a Toffoli gate controlled by two $Z$-syndrome qubits from $S_1$ corrects this error. All data qubits that have two neighboring ancilla qubits, of the same type (i.e., $XX$ or $ZZ$), in the 17-qubit architecture are corrected in this way as well. In case 2 (in parentheses), if there is an $X$ error on qubit 1, the error-correcting Toffoli gate is controlled by one $Z$-syndrome qubit from $S_1$ and one from $S_2$, corresponding to the syndromes extracted in two consecutive cycles. Other data qubits with only one neighboring ancilla qubit, of a given type, in the 17-qubit architecture are corrected in this way as well. The remaining gates remove the used syndrome information from the ancilla qubits with the help of the $c$ qubits. The $Z$-error-correction circuit is analogous, using $X$-syndrome qubits instead of $Z$-syndrome qubits and ccz gates instead of Toffoli gates. As in Fig. 2, the portion of the circuit in the box is responsible for perfect matching.

Figure 5

(a) Numerical results for the logical-$X$-error rates $p_{\text{log}}$ for the one-dimensional BF code (blue and green lines) and the two-dimensional surface-17 code (red and orange lines) with measurement-free QEC and without error correction (black dashed line) as a function of the physical error rate $p$. The error threshold value is estimated as the value where the physical error rate is equal to the logical-error rate. For the BF code, $p_{\text{th,1D}}\approx 3.2\times {10}^{-3}$, and for the two-dimensional surface-17 code, $p_{\text{th,2D}}\approx 4.2\times {10}^{-5}$. The simplified circuits for the BF and surface-17 codes, not including perfect matching (green and orange lines), produce higher thresholds of $p_{\text{th,1D}}\approx 2.0\times {10}^{-2}$ and $p_{\text{th,2D}}\approx 1.3\times {10}^{-4}$. (b) Measurement-based-threshold values, which were simulated here by using the circuit and lookup-table decoder in Ref. [21], for the surface-17 code at different measurement time to gate time ($t_m/t_g$) ratios (blue triangles) in comparison to the measurement-free threshold (red dashed line). Increasing measurement time reduces the error threshold as during the measurement of the ancilla qubits, the data qubits sit idle without protection against errors. When the measurement time vs gate time ratio is around 10, the measurement-free method produces a better threshold than the measurement-based method.

Figure 6

Examples showing the effect of removing used syndrome information in the BF code. The three black open circles in the top row represent data qubits. The four red circles below the data qubits represent the ancillas, where the middle row corresponds to the first time cycle while the bottom row corresponds to the second time cycle. Flipped ancillas indicating a syndrome signal are closed and errors on data qubits are denoted by an X. (a) Here we show three different syndrome patterns that trigger different Toffoli gates to correct corresponding data errors. (b) In the case of a single data error, not removing used syndrome information is not harmful. For example, if there is an X error in the middle data qubit there will be two signals in the top row of ancillas, triggering a Toffoli gate that corrects the error. Even if these signals are not removed from the system, they will not trigger other Toffoli gates in the next cycle, and in the following cycle they will disappear completely. (c) We consider a case where errors appear in the middle data qubit in two consecutive time cycles. The first row shows that if the used information is removed both of these errors get corrected. In the second row, however, the information is not removed. This results in signals occurring in all the ancillas in the second cycle, which causes the Toffoli gates to correct errors in all three data qubits. One of these bit flips corrects the actual error, while the other two introduce new errors (shown in purple) resulting in signals in all ancillas in subsequent time cycles. The system then oscillates between having one and two errors indefinitely, making it vulnerable to errors that may occur in the future.