High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction

To implement fault-tolerant quantum computation with continuous variables, the Gottesman-Kitaev-Preskill (GKP) qubit has been recognized as an important technological element. However, it is still challenging to experimentally generate the GKP qubit with the required squeezing level, 14.8 dB, of the existing fault-tolerant quantum computation. To reduce this requirement, we propose a high-threshold fault-tolerant quantum computation with GKP qubits using topologically protected measurement-based quantum computation with the surface code. By harnessing analog information contained in the GKP qubits, we apply analog quantum error correction to the surface code. Furthermore, we develop a method to prevent the squeezing level from decreasing during the construction of the large-scale cluster states for the topologically protected, measurement-based, quantum computation. We numerically show that the required squeezing level can be relaxed to less than 10 dB, which is within the reach of the current experimental technology. Hence, this work can considerably alleviate this experimental requirement and take a step closer to the realization of large-scale quantum computation.

Popular Summary

Quantum computation has the potential to solve problems that are intractable with traditional computers. However, large-scale quantum computation is challenging because current experimental setups cannot sufficiently suppress errors. One promising approach to fault-tolerant computing is the GKP qubit, a quantum bit that encodes information in both the position and momentum of an oscillator. Near-term setups, however, are unable to achieve the technical requirements needed to set up a GKP qubit for fault-tolerant operation. Here, we propose a way to reduce these requirements.

A GKP qubit relies on squeezed states, in which the quantum uncertainty of one parameter (e.g., position) is reduced at a cost of increased uncertainty in another (e.g., momentum). Existing approaches to quantum computing require a squeezing level of at least 14.8 dB. Unfortunately, current technologies can achieve only 10 dB. Using theoretical calculations, we found a way to reduce the requirement to below 10 dB.

Our method has two parts. First, we use a maximum likelihood method to apply an analog quantum error correction to a surface code, a simple qubit architecture that demonstrates high resiliency to errors. Second, we construct the high-purity resource state for computation with a surface code by using a threshold-decision method.

Our method provides versatile error correction because it can be applied to a variety of codes used to digitize continuous variables. We believe our approach to fault-tolerant computation will open up a new approach to practical quantum computers.

Figure 1

Introduction of a likelihood function. (a) Measurement outcome and deviation from the peak value in $q$ quadrature. The dotted line shows the measurement outcome $q_{\mathrm{m}}$ equal to $(2t+k)\sqrt{\pi }+{\mathrm{\Delta }}_{\mathrm{m}}$ $(t=0,\pm 1,\pm 2,\dots ,k=0,1)$, where $k$ is defined as the bit value that minimizes the deviation ${\mathrm{\Delta }}_{\mathrm{m}}$. The red areas indicate the area that yields the code word ($k+1$) mod 2, whereas the white area denotes the area that yields the code word $k$. (b,c) Gaussian distribution functions as likelihood functions of the true deviation $\overline{\mathrm{\Delta }}$ represented by the arrows. (b) The case of the correct decision, where the amplitude of the true deviation is $|\overline{\mathrm{\Delta }}|<\sqrt{\pi }/2$. (c) The case of the incorrect decision $\sqrt{\pi }/2<|\overline{\mathrm{\Delta }}|<\sqrt{\pi }$.

Figure 2

Simulation results for the logical error probabilities of the surface code with ideal syndrome measurements using (a) the digital QEC and (b) the analog QEC for several distances $d$, which are the size of the 3D cluster state. The simulation results for the digital QEC are obtained from 50 000 samples. The simulation results for the analog QEC are obtained from 50 000 samples (for $d=5–15$) and 10 000 samples (for $d=17–25$).

Figure 3

Simulation results for the logical error probabilities of the surface code with noisy syndrome measurements using (a) the digital QEC and (b) the analog QEC. The simulation results for the digital QEC are obtained from 50 000 samples. The simulation results for the analog QEC are obtained from 10 000 samples.

Figure 4

Introduction of the postselected measurement. (a) The conventional measurement of the GKP qubit, with the Gaussian distribution followed by the deviation of the GKP qubit that has variance ${\sigma }^2$. The plain (blue) region and the region with the vertical (red) line represent the different code words ($k-1$) mod 2 and ($k+1$) mod 2, respectively. The vertical line corresponds to the probability of an incorrect decision of the bit value. (b) The postselected measurement. The dotted line represents an upper limit $v_{\mathrm{up}}$. The horizontal line shows the probability that the results of the measurement are discarded by introducing $v_{\mathrm{up}}$. The vertical line shows the probability that our method fails.

Figure 5

The error probabilities of the postselected measurement $E_{\text{post}}$ and the success probabilities of the postselected measurement $P_{\mathrm{suc}}$ on the qubit of variance $3{\sigma }^2$. (a) The error probabilities with the method using only the CZ gate, and our method using the postselected measurement for the upper limit $v_{\mathrm{up}}=0$ (red solid line), $v_{\mathrm{up}}=\sqrt{\pi }/10$ (red dashed line), $v_{\mathrm{up}}=\sqrt{\pi }/6$ (blue solid line), and $v_{\mathrm{up}}=\sqrt{\pi }/4$ (blue dashed line), respectively. (b) The success probability for our method. The squeezing level is equal to $-10{\log }_{10}6{\sigma }^2$.

Figure 6

The 3D cluster-state construction. (a) The preparation of the 3-tree cluster state by using the CZ gate. (b) The single-qubit-level QEC using the additional ancilla qubit with the postselected measurement. (c)–(e) The construction of the hexagonal cluster state from the 3-tree qubit with the postselected measurement. (f) The construction of the 3D cluster state from the hexagonal cluster states, where the entanglement is generated between the neighboring hexagonal cluster states without the postselected measurement.

Figure 7

Simulation results for the logical error probabilities of the surface code by using the 3D cluster state prepared by the proposed method for $v_{\mathrm{up}}=2\sqrt{\pi }/5$ with noisy syndrome measurements using (a) the digital QEC and (b) the analog QEC, respectively. The simulation results for the digital QEC are obtained from 50 000 samples. The simulation results for the analog QEC are obtained from 10 000 samples.