We compare two different implementations of fault-tolerant entangling gates on logical qubits. In one instance, a twelve-qubit trapped-ion quantum computer is used to implement a non-transversal logical CNOT gate between two five qubit codes. The operation is evaluated with varying degrees of fault tolerance, which are provided by including quantum error correction circuit primitives known as flagging and pieceable fault tolerance. In the second instance, a twenty-qubit trapped-ion quantum computer is used to implement a transversal logical CNOT gate on two [[7,1,3]] color codes. The two codes were implemented on different but similar devices, and in both instances, all of the quantum error correction primitives, including the determination of corrections via decoding, are implemented during runtime using a classical compute environment that is tightly integrated with the quantum processor. For different combinations of the primitives, logical state fidelity measurements are made after applying the gate to different input states, providing bounds on the process fidelity. We find the highest fidelity operations with the color code, with the fault-tolerant SPAM operation achieving fidelities of 0.99939(15) and 0.99959(13) when preparing eigenstates of the logical X and Z operators, which is higher than the average physical qubit SPAM fidelities of 0.9968(2) and 0.9970(1) for the physical X and Z bases, respectively. When combined with a logical transversal CNOT gate, we find the color code to perform the sequence--state preparation, CNOT, measure out--with an average fidelity bounded by [0.9957,0.9963]. The logical fidelity bounds are higher than the analogous physical-level fidelity bounds, which we find to be [0.9850,0.9903], reflecting multiple physical noise sources such as SPAM errors for two qubits, several single-qubit gates, a two-qubit gate and some amount of memory error.
Quantum error correction is a crucial tool for mitigating hardware errors in quantum computers by encoding logical information into multiple physical qubits. However, no single error-correcting code allows for an intrinsically fault-tolerant implementation of all the gates needed for universal quantum computing [1-3]. One way to tackle this problem is to switch between two suitable error-correcting codes, while preserving the encoded logical information, which in combination give access to a fault-tolerant universal gate set [4-6]. In this work, we present the first experimental implementation of fault-tolerant code switching between two codes. One is the seven-qubit color code [7], which features fault-tolerant CNOT and
H quantum gates, while the other one, the 10-qubit code [8], allows for a fault-tolerant
T-gate implementation. Together they form a complementary universal gate set. Building on essential code switching building blocks, we construct logical circuits and prepare 12 different logical states which are not accessible natively in a fault-tolerant way within a single code. Finally, we use code switching to entangle two logical qubits employing the full universal gate set in a single logical quantum circuit. Our results experimentally open up a new route towards deterministic control over logical qubits with low auxiliary qubit overhead, not relying on the probabilistic preparation of resource states.
A scalable and programmable quantum computer holds the potential to solve computationally intensive tasks that classical computers cannot accomplish within a reasonable time frame, achieving quantum advantage. However, the vulnerability of the current generation of quantum processors to errors poses a significant challenge towards executing complex and deep quantum circuits required for practical problems. Quantum error-correction codes such as stabilizer codes offer a promising path forward for fault-tolerant quantum computing; however, their realization on quantum hardware is an on-going area of research. In particular, fault-tolerant quantum processing must employ logical gates on logical qubits with error suppression with realistically large size codes. This work has implemented a transversal controlled-not (cnot) gate between two logical qubits constructed using the repetition code with flag qubits, and demonstrated error suppression with increasing code size under multiple rounds of error detection and correction. Our results conducted on IBM quantum devices through cloud access show that despite the potential for error propagation among logical qubits during the transversal cnot gate operation, scaling the code distance as we increase the number of physical qubits from 21 to 39 and 57 can suppress errors, which persists over ten rounds of error detection and correction. Our work establishes the feasibility of employing logical cnot gates alongside error detection on a superconductor-based processor using current generation quantum hardware.
A technique for realizing a universal set of fault-tolerant quantum operations is the code switching method, which leverages two quantum codes with complementary sets of transversal gates. To date, the application of this technique has been largely limited to families of color codes supporting a logical
T gate. No analogous code switching protocols exist for many other prominent families, such as rotated surface codes, or for finer
Z-rotation gates. In this work, we first utilize the doubling technique as a unified framework to construct a class of quantum color codes encoding a single logical qubit with an arbitrarily large minimum distance, enabling the transversal realization of arbitrary small logical
Z-rotation gates. We investigate the structural properties of this code family, demonstrating that they improve upon the parameters of state-of-the-art triorthogonal codes, achieve lower qubit overhead compared to certain known color codes, and admit single-shot decoding of
Z-syndromes via meta-checks. Furthermore, we show that this framework extends beyond color codes; specifically, it enables the generation of
r-orthogonal quantum codes,
r≥2, that inherit the local geometry of rotated surface codes. We then provide an overhead optimization protocol alongside several candidate codes tailored for realizing logical
Z-rotation gates within rotated surface codes. Finally, we extend the fault-tolerant code switching protocol based on transversal CNOT gates to incorporate fault-tolerant realization of
Z-rotation gates at any level of the Clifford hierarchy for geometries compatible with rotated surface codes. We present the first demonstration of fault-tolerant magic state preparation by means of code switching within a distance-three rotated surface code using a total footprint of only 45 physical qubits, and evaluate its performance through a simulation.
We present a fault-tolerant universal quantum computing architecture based on a code concatenation of biased-noise qubits and the parity architecture. The parity architecture can be understood as a low-density parity-check code tailored specifically to obtain any desired logical connectivity from nearest-neighbor physical interactions. The code layout can be dynamically adjusted to algorithmic requirements on the fly. This allows for implementations with any desired code distance with a universal set of fault-tolerant gates. In addition to the previously explored toolsets for concatenated cat codes, our approach features parallelizable interactions between arbitrary sets of qubits by directly addressing the parity qubits in the code. The proposed scheme supports codes with lower physical qubit overhead than the repetition code at the same code distances, while requiring only weight-3 and weight-4 stabilizers and nearest-neighbor connectivity on a two-dimensional square lattice.
Quantum computing relies on quantum error correction for high-fidelity logical operations, but scaling to achieve near-term quantum utility is highly resource-intensive. High-rate quantum LDPC codes can reduce error correction overhead, yet realizing high-rate fault-tolerant computation with these codes remains a central challenge. Apart of the lattice surgery approach, standard schemes for realizing logical gates have so far been restricted to performing global operations on all logical qubits at the same time. Another approach relies on low-rate code switching methods. In this work, we introduce a gate-based protocol for addressable single- and multi-qubit Clifford operations on individual logical qubits encoded within one or more quantum LDPC codes. Our scheme leverages logical transversal operations via an auxiliary Bacon-Shor code to perform logical operations with constant time overhead enabled by teleportation. We demonstrate the implementation of an overcomplete logical Clifford gate set and perform numerical simulations to evaluate the error-correction performance of our protocol. Finally, we observe that our scheme can be integrated with magic state cultivation protocols to achieve universal, gate-based, and fully addressable quantum computation.