Integrated photonics provides a versatile platform for encoding and processing quantum information.However, the encoded quantum states are sensitive to noise, which limits their capability to perform complicated quantum computations.Here, we use a five-qubit linear cluster state on a silicon photonic chip to implement a quantum error-correction code and demonstrate its capability of identifying and correcting a single-qubit error.The encoded quantum information is reconstructed from a single-qubit error and an average state fidelity of 0.863 ± 0.032 is achieved for different input states.We further extend the scheme to demonstrate a fault-tolerant measurement-based quantum computation (MBQC) on stabilizer formalism that allows us to redo the qubit operation against the failure of the teleportation process.Our work provides a proof-of-concept working prototype of error correction and MBQC in an integrated photonic chip.
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.
Quantum error correction protects fragile quantum information by encoding it into a larger quantum system. These extra degrees of freedom enable the detection and correction of errors, but also increase the operational complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while operating the logical qubit, and are essential for realizing error suppression in practice. While fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. In this work, we experimentally demonstrate fault-tolerant preparation, measurement, rotation, and stabilizer measurement of a Bacon-Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6% and a Clifford gate error of 0.3% after error correction. Additionally, we prepare magic states with fidelities exceeding the distillation threshold, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant operation. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved.
A critical milestone for quantum computers is to demonstrate fault-tolerant computation that outperforms computation on physical qubits. The tesseract subsystem color code protects four logical qubits in 16 physical qubits, to distance four. Using the tesseract code on Quantinuum's trapped-ion quantum computers, we prepare high-fidelity encoded graph states on up to 12 logical qubits, beneficially combining for the first time fault-tolerant error correction and computation. We also protect encoded states through up to five rounds of error correction. Using performant quantum software and hardware together allows moderate-depth logical quantum circuits to have an order of magnitude less error than the equivalent unencoded circuits.
Correcting errors in real-time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD (quantum charge-coupled device) trapped-ion quantum computer to encode a single logical qubit using the color code. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of 1.7(2) 10^{-3}$, compared to the average physical SPAM error 2.4(4) 10^{-3} of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections, which are tracked software or applied physically.
Quantum error correction codes are essential for fault-tolerant quantum computing. They are a method of detecting and correcting errors in a quantum computer by encoding logical qubits across multiple physical qubits. Several prominent error correction codes have been developed and larger and more complex codes are being developed continuously. However, evaluating the fault-tolerance of these larger codes is becoming exponentially more complicated. While conventional methods of proving and evaluating the fault-tolerance of these circuits have worked thus far, they are soon to be obsolete. The aim of this thesis is to develop a framework that automatically evaluates the fault-tolerance of a quantum error correction circuit across different hardware architectures. The framework works as follows: it takes a noiseless quantum error correction code; injects hardware-realistic noise into it; and evaluates the fault-tolerance of the resulting noisy circuit by estimating its logical error rate through Monte Carlo sampling. For the injection of hardware-realistic noise, this thesis considers noise stemming from five different quantum computing hardware architectures: superconducting qubits, neutral atoms, trapped ions, photonic qubits, and spin qubits. The noise model was constructed by extracting single-qubit gate, two-qubit gate and measurement error fidelities from current experimental literature. The noise model and circuit generation is implemented in Python, circuit representation and noise injection is executed in Stim and decoding and estimating the logical error rate is achieved through PyMatching. The framework is evaluated on the repetition code, the surface code, and the color code; the logical error rate is estimated across all five hardware architectures under the noise model; and the results are discussed in the context of hardware compatibility for the given codes.