High-fidelity magic-state preparation with a biased-noise architecture

Magic state distillation is a resource intensive subroutine that consumes noisy input states to produce high-fidelity resource states that are used to perform logical operations in practical quantum-computing architectures. The resource cost of magic state distillation can be reduced by improving the fidelity of the raw input states. To this end, we propose an initialization protocol that offers a quadratic improvement in the error rate of the input magic states in architectures with biased noise. This is achieved by preparing an error-detecting code which detects the dominant errors that occur during state preparation. We obtain this advantage by exploiting the native gate operations of an underlying qubit architecture that experiences biases in its noise profile. We perform simulations to analyze the performance of our protocol with the XZZX surface code. Even at modest physical parameters with a two-qubit gate error rate of $0.7\%$ and total probability of dominant errors in the gate $O\left({10}^3\right)$ larger compared to that of nondominant errors, we find that our preparation scheme delivers magic states with logical error rate $O\left({10}^{-8}\right)$ after a single round of the standard 15-to-1 distillation protocol, two orders of magnitude lower than using conventional state preparation. Our approach therefore promises considerable savings in overheads with near-term technology.

Figure 1

(a) Illustration of the rectangular XZZX code with data qubits on the vertices of a rotated grid. The stabilizers are the product of two Pauli $X$ and two Pauli $Z$ operators on qubits arranged on the vertices around each face. The distance to $X$ and $Z$ errors is $d_x$ and $d_z$ respectively. The logical qubits Pauli $X_{\mathrm{L}}$ ($Z_{\mathrm{L}}$) are the product of Pauli $X$ ($Z$) on the qubits along the blue (left vertical) and red (upper horizontal) edges respectively. The order in which qubits are coupled to the ancilla at the center of each face (not shown) is indicated by the red arrow. (b) Circuit for stabilizer measurements. The ancilla is prepared in state $|+\rangle$, then coupled to data qubits with $\mathrm{CX}$ and $\mathrm{CZ}$ gates and finally read out in the $X$ basis

Figure 2

Illustration of the protocol for magic state preparation. In stage I the qubits in region I are initialized as shown, a $ZZ\left(\theta \right)$ gate is applied to the two grey qubits (top left), and the stabilizers are measured twice. The faces shaded in grey mark the fixed stabilizers for stage I. After stage I is successful and a $d_{x,1}\times d_{z,1}$ magic state is prepared, qubits in region II are initialized as shown. Stage II is then implemented and the $d_{x,1}\times d_{z,1}$ state is grown to a $d_{x,2}\times d_{z,2}$ state, where stabilizers are measured for $d_m=d_{z,2}$ rounds.

Figure 3

Logical error rate (${\varepsilon }_L^{\mathrm{raw}}$) and success rate after $d_m$ rounds of error correction in stage II with noise model A ($\mathrm{CX}$ slower than $\mathrm{CZ}$) so that $p_{\mathrm{CX}}=20p+120p/\eta$. The bias is $\eta ={10}^4$ in (a,c) and $\eta ={10}^3$ in (b,d). The code size in stage I is $d_{x,1}\times d_{z,1}=1\times 3$. Stage II code sizes $d_{x,2}\times d_{z,2}$ are shown in the legend, with $d_m=d_{z,2}$. The results for our scheme are shown using solid lines and those for the standard approach are shown using dotted lines. Error bars indicate standard error of the mean. Each data point is generated with ${10}^5$ Monte Carlo samples.

Figure 4

Logical error rate (${\varepsilon }_L^{\mathrm{raw}}$) and success rate after $d_m$ rounds of error correction in stage II with noise model B ($\mathrm{CX}$ as fast as $\mathrm{CZ}$) so that $p_{\mathrm{CX}}=2p+12p/\eta$. The bias is $\eta ={10}^4$ and the code size in stage I is $d_{x,1}\times d_{z,1}=1\times 3$. Stage II code sizes $d_{x,2}\times d_{z,2}$ are shown in the legend, with $d_m=d_{z,2}$. The results for our scheme are shown using solid lines and those for the standard approach are shown using dotted lines. Error bars indicate standard error of the mean. Each data point is generated with ${10}^5$ Monte Carlo samples.

Figure 5

Illustration of the protocol for preparing the magic state in the XZZX code with alternate stage II initialization pattern. The faces shaded in grey mark the fixed stabilizers for stage I.

Figure 6

Arrangement of qubits for preparing the magic state ${cos(\pi /8)|+i\rangle }_L-isin(\pi /8){|-i\rangle }_L$ in the tailored surface code. This code has two types of stabilizers: product of Pauli $Y$ on the qubits around the white squares and product of Pauli $X$ on the qubits around the grey squares. At the boundaries the stabilizers are $X\otimes X$ and $Y\otimes Y$ on two qubits. The fixed stabilizers for stage I are marked using black lines. The $ZZ\left(\theta \right)$ gate is applied to the two grey qubits on the top left.

Figure 7

Qubit arrangement in stage I of the standard scheme used for comparison in this paper. The faces shaded in grey mark the fixed stabilizers for stage I. Stage II is identical to Fig. 2

Figure 8

$X_L$ and $Z_L$ error rate in the magic state for $\eta ={10}^4$ (a,b) and $\eta ={10}^3$ (c,d) for noise model (A). The black (bold) dashed lines in (b,d) are found by fitting $Z_L$ error rate in the magic state prepared using our scheme, at low $p$ and large distances, to $A{p}^2$. In (b) we use the solid lines corresponding to $d_{x,2}\times d_{z,2}=3\times 15$ and $d_{x,2}\times d_{z,2}=5\times 25$ for the fit and find $A=(4.48\pm 0.07)\times {10}^3$. In (d) we use the solid lines corresponding to $d_{x,2}\times d_{z,2}=11\times 11$ and $d_{x,2}\times d_{z,2}=15\times 15$ for the fit and find $A=(4.34\pm 0.09)\times {10}^3$.

Figure 9

$X_L$ and $Z_L$ error rate in the magic state for $\eta ={10}^4$ for noise model (B). The black (bold) dashed lines in (b) are found by fitting $Z_L$ error rate in the magic state prepared using our scheme to $A{p}^2$. We use the solid lines corresponding to $d_{x,2}\times d_{z,2}=3\times 15$ and $d_{x,2}\times d_{z,2}=5\times 25$ for the fit and find $A=(1.78\pm 0.06)\times {10}^2$.