Scientists reach a new milestone in surface code error correction for quantum computing

Researchers at the Dutch quantum computing and quantum internet research center QuTech (a co-construction unit of Delft University of Technology and TNO) have reached a milestone in quantum error correction. They combined the high-fidelity operation of encoded quantum data with a scalable repetitive data stabilization scheme. The researchers reported their findings in the December 16th issue of "Nature · Physics."

Specifically, the researchers implemented a set of logical operations on a surface code qubit with a distance of 2. The qubit is composed of seven physical qubits and is stabilized by repeated error detection cycles. The logical operations include 1) initialization to an arbitrary state, 2) measurement on the basis of Bloch spheres, and 3) a set of universal single qubit gates. For each type of operation, the researchers observed that the fault-tolerant variant performed better than the non-fault-tolerant variant and quantified the difference . In particular, they used the concept of the logic Pauli transition matrix to demonstrate the process tomography of logic gates. The team said that this is a milestone on the way to use a longer-distance superconducting surface code for quantum error correction.

Principles of Quantum Error Correction

Physical qubits are prone to errors. These errors have different sources, including quantum decoherence, crosstalk, and incomplete alignment. Fortunately, the theory of quantum error correction guarantees the possibility of performing calculations while simultaneously protecting quantum data from such errors.
 
QuTech's Professor Leonardo DiCarlo said: “There are two capabilities that will distinguish error-correcting quantum computers from today's noisy medium-scale quantum (NISQ) processors. First, it will process logical qubits instead of physical qubits. Quantum information (each logical qubit is composed of many physical qubits). Secondly, it will use quantum parity interleaved with calculation steps to identify and correct errors that occur in physical qubits, thereby protecting the encoding when processing encoded information Information." According to theory, if the occurrence rate of physical errors is below a threshold, and the circuits used for logical operation and stabilization are fault-tolerant, the logical error rate can be suppressed exponentially.
 
Therefore, the basic idea of ​​error correction is that if you increase redundancy and use more and more qubits to encode data, the net error will decrease. Researchers from Delft University of Technology and colleagues from the Dutch National Institute of Applied Sciences (TNO) have now taken an important step towards this goal, realizing a seven physical qubit (transmon superconducting qubit) Composed of logical qubits.

Complete all operations required for the calculation

At present, the surface code is the most attractive quantum error correction code in solid-state implementation because it has actual nearest neighbor connection requirements and a high error threshold. Recent experiments have proved that repetitive stability can be achieved by selection after passing in the surface code. Due to the small size, quantum errors can be detected but not corrected. "This work shows that we can use encoded information to complete all the operations required for calculations," said Professor Barbara Terhal, also from QuTech.
 
The first author of the paper and PhD student Jorge Marques further explained: "So far, the researchers have coded and stabilized. We have now proved that we can also perform calculations. This is what a fault-tolerant computer must ultimately do: process and protect data at the same time. Protection from errors."
 
The surface code with a distance of 2 (Figure 1a) uses four data qubits (D1 to D4) to encode a logical qubit. Using three auxiliary qubits (A1, A2, and A3 in Figure 1a) the measurement stabilizer can detect all the errors of a single physical qubit. Surface codes cannot be used to correct such errors. Therefore, the state is stabilized after passing the selection.

  
Figure 1a ：A surface code with a distance of 2. Each circle represents a transmon qubit.

The layout of the qubits in the quantum chip is shown in Figure 1b.

  
Figure 1b Optical image of a quantum chip, with colors added to distinguish different circuit components

Researchers use stabilizer measurements to initialize logic states. They used full four-qubit state tomography to characterize the resulting state (Figure 1c-f). The fidelity F4Q of the ideal four-qubit target state is 90.0%, 92.9%, 77.80% and 77.09%, respectively. For each state, the logical fidelity FL can be extracted by further projecting the obtained four-qubit density matrix into the code space, which are respectively 99.83%, 99.97%, 97.02%, and 95.54%. FL>F4Q .

Figure 1c-f The estimated physical density matrix ρ after preparing the logical basic states |0L>(c), |1L>(d), |+L>(e) and |-L>(f).  

In the surface code, a series of data qubit results (each +1 or -1) can be obtained by measuring all the data qubits on the X(Z) base at the same time, so that XL (ZL) can be measured fault-tolerantly, despite the destruction Sexual. In addition, the result string is used to calculate the value of stable XD1XD2XD3XD4 (ZD1ZD3 and ZD2ZD4), which is the final step of error detection (Figure 2a). They also calculate the value of ZD2ZD4 by measuring D4 of the Z base, and detect the bit flip errors of D2 and D4, thereby reducing logic allocation errors.

Figure 2a is a set of data qubit measurements used to evaluate the logical operators ZL, XL, and YL with additional error detection.

The researchers demonstrated the measurement of logic states prepared on two orthogonal planes of the logic Bloch sphere. Figures 2c and 2e show the ZL, XL, and YL logic measurement results as a function of door angles φ and θ, respectively.

Figure 2c&2e ZL, XL and YL logic measurement results as a function of door angles φ and θ.

Finally, the researchers demonstrated a set of gates that can achieve universal logic qubit control (Figure 3). Full control of logic qubits requires a gate set consisting of Clifford and non-Clifford logic gates. Some Clifford gates such as ZL and XL can be implemented horizontally and are therefore fault-tolerant (Figure 3d).

Figure 3 The logic gate implemented in this article. (a&b) A general door-by-door measurement scheme that realizes arbitrary rotation around the Z-axis (a) and X-axis (b) of the Bloch ball. (c) Process tomography experiment of TL gate. Use the method shown in Figure 2 to initialize the basic logic state of the input. The output state is measured after the second round of stabilizer measurement. (d) Logic gates, ZL gates and XL gates are compiled using the hardware native gate set.

Next target

DiCarlo emphasized the multidisciplinary nature of this work: "This is a comprehensive result of experimental physics, theoretical physics of the Barbara Terhal team, and the joint development of electronic devices with TNO and external partners. The project is mainly sponsored by the US Intelligence Advanced Research Projects Agency. (IARPA) and Intel Corporation."
 
DiCarlo said: "Our ambitious goal is that when we increase coding redundancy, the net error rate actually decreases exponentially. Our current focus is on 17 physical qubits, and the next goal is 49. Our quantum computer architecture All of the layers are designed to achieve this kind of expansion."
 
 link: [1]https://www.nature.com/articles/s41567-021-01423-9#data-availability
         [2]https://phys.org/news/2021-12-team-important-quantum-error.html