We calculate the fidelity with which an arbitrary state can be encoded into a [7,1,3] CSS quantum error correction code in a non-equiprobable Pauli operator error environment with the goal of determining whether this encoding can be used for practical implementations of quantum computation. This determination is accomplished by applying ideal error correction to the encoded state which demonstrates the correctability of errors that occurred during the encoding process. We then apply single-qubit Clifford gates to the encoded state and determine the accuracy with which these gates can be applied. Finally, fault tolerant noisy error correction is applied to the encoded states in the non-equiprobable Pauli operator error environment allowing us to compare noisy (realistic) and perfect error correction implementations. We note that this maintains the fidelity of the encoded state for certain error-probability values. These results have implications for when non-fault tolerant procedures may be used in practical quantum computation and whether quantum error correction should be applied at every step in a quantum protocol.
It has previously been shown by Rajput, Roggero, and Wiebe that
Z2 Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary
t-qubit errors, where
t is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for
t≤3 (distance
d≤7).
Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error correction code (QECC), with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state, and state decoding. To address this notoriously difficult challenge, we push the limits of the depth of superconducting quantum circuits and experimentally realise the universal five-qubit QECC, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the five-qubit QECC for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of
57.1(3)% while with a high fidelity of
98.6(1)% in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of
97.2(2)% within the code space. Finally, we realise the decoding circuit and recover the input state with a process fidelity of
57.4(7)%. After decoding, by identifying errors via the measurement of the four ancillae and then mathematically recovering the qubit state, we verify the power of error correction of this code. Thus, by demonstrating each key aspect of error correction with the five-qubit code, our work establishes the viability of experimental quantum error correction with superconducting qubits and paves the route to fault-tolerant quantum computing.