Universal fault-tolerant quantum supercomputing relies on the implementation of quantum error correction. An essential milestone is the achievement of error-corrected logical qubits that genuinely benefit from error correction, outperforming simple physical qubits. Although tremendous efforts have been devoted to demonstrating quantum error-correcting codes with different quantum hardware, previous realizations are limited to be against certain types of errors or to prepare special logical states. It remains one of the greatest and also notoriously difficult challenges to realize a universal quantum error-correcting code for more than a decade.

In a new research article published in the Beijing-based National Science Review, scientists at the University of Science and Technology of China, the Tsinghua University, and at the University of Oxford, present their latest work on the experimental exploration of five-qubit quantum error-correcting code with superconducting qubits. The authors realized the [[5,1,3]] code on a superconducting quantum processor, verified the viability of experimental realization of quantum error-correcting codes with superconducting qubits.

These scientists completed the important step towards the implementation of quantum error correction. This is achieved first by dedicated experimental optimization of superconducting quantum qubits, enabling the realization of more than a hundred quantum gates. Focusing on the five-qubit quantum error-correcting code, the so-called 'perfect code' that corrects single generic qubit errors, they theoretically compiled and optimized its encoding process to have the minimal possible number (eight) of nearest-neighbor controlled-phase gates. These experimental and theoretical advances finally enabled the realization of the basic ingredients of a fully functional five-qubit error-correcting code, involving the encoding of a general logical qubit into an error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the power for transversal manipulation of the logical state, and state decoding.

"The device for the implementation of the five-qubit error-correcting code is a 12-qubit superconducting quantum processor. Among these 12 qubits, we chose five adjacent qubits to perform the experiment. The qubits are capacitively coupled to their nearest neighbors. The capacitively coupled XY control lines enable the application of single-qubit rotation gates by applying microwave pulses, and the inductively coupled Z control lines enable the double-qubit controlled-phase gates by adiabatically tune the two-qubit state |11> close to the avoid level crossing of |11> and |02>. After careful calibrations and gate optimizations, we have the average gate fidelities as high as 0.9993 for single-qubit gates and 0.986 for two-qubit gates. With the implementation of only single-qubit rotation gates and double-qubit controlled-phase gates, we realized the circuit for encoding and decoding of the logical state." they state in an article titled "Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits."

"On a superconducting quantum processor, we experimentally realized the logical states |0>_L, |1>_L, |±>_L, and |±i>_L that are eigenstates of the logical Pauli operators X_L, Y_L, and Z_L, and the magic state |T>_L= (|0>_L+e^{i\pi/4}|1>_L)/\sqrt{2} that cannot be realized by applying Clifford operations on any eigenstate of the logical Pauli operators," they add. "Finally, the state fidelity of |T>_L reaches 54.5(4)%."

"The quality of the prepared logical states can also be divided into its overlap with the logical code space and its agreement with the target logical state after projecting it into the code space," they stated. After projecting to the code space, the average value is as high as 98.6(1)%. "Since projecting to the code space is equivalent to post-selecting all +1 stabilizer measurements, our result also indicates the possibility of high fidelity logical state preparation with future non-destructive stabilizer measurements."

After the realization of the logical state, the scientists proceed with the verification of error correction/detection ability of the five qubit code. "As shown in Fig.2(a) we do indeed find, for each case, the corresponding syndrome pattern that identifies the location of the single-qubit error," they added.

Then, the scientists implemented and verified the transversal logical operations, and performed the quantum process tomography within the code space to characterize these logical operations. "We determine gate fidelities of the logical X_L, Y_L, and Z_L operations to be 97.2(2)%, 97.8(2)%, and 97.3(2)%, respectively," they stated.

"Finally, after encoding the single-qubit input state into the logical state, we apply the decoding circuit, see Fig. 4(a), to map it back to the input state," they added. "After quantum process tomography from the four output states, the process fidelity is determined as 74.5(6)% as shown in Fig. 4(b)."

"An essential milestone on the road to fault-tolerant quantum computing is the achievement of error-corrected logical qubits that genuinely benefit from error correction, outperforming simple physical qubits," they add. "Direction for future works include the realization of non-destructive error detection and error correction, and the implementation of logical operations on multiple logical qubits for the five-qubit code. Our work also has applications in error mitigation for near-term quantum supercomputing."