It is hard to achieve a theoretical quantum advantage on NISQ devices. Besides the attempts to reduce error using error mitigation and dynamical decoupling, small quantum error correction and fault-tolerant schemes that reduce the high overhead of traditional schemes have also been proposed. According to the recent advancements in fault tolerance, it is possible to minimize the number of ancillary qubits using flags. While implementing those schemes is still impossible, it is worthwhile to bridge the gap between the NISQ era and the FTQC era. Here, we introduce a benchmarking method to test fault-tolerant quantum error correction with flags for the [[5,1,3]] code on NISQ devices. Based on results obtained using IBM's qasm simulator and its 15-qubit Melbourne processor, we show that this flagged scheme is testable on NISQ devices by checking how much the subspace of intermediate state overlaps with the expected state in the presence of noise.
Quantum computers will require encoding of quantum information to protect them from noise. Fault-tolerant quantum computing architectures illustrate how this might be done but have not yet shown a conclusive practical advantage. Here we demonstrate that a small but useful error detecting code improves the fidelity of the fault-tolerant gates implemented in the code space as compared to the fidelity of physically equivalent gates implemented on physical qubits. By running a randomized benchmarking protocol in the logical code space of the [4,2,2] code, we observe an order of magnitude improveme…
We extensively test a recent protocol to demonstrate quantum fault tolerance on three systems: (1) a real-time simulation of five spin qubits coupled to an environment with two-level defects, (2) a real-time simulation of transmon quantum computers, and (3) the 16-qubit processor of the IBM Q Experience. In the simulations, the dynamics of the full system is obtained by numerically solving the time-dependent Schrödinger equation. We find that the fault-tolerant scheme provides a systematic way to improve the results when the errors are dominated by the inherent control and measurement errors p…
Fault-tolerant (FT) computation by using quantum error correction (QEC) is essential for realizing large-scale quantum algorithms. Devices are expected to have enough qubits to demonstrate aspects of fault tolerance in the near future. However, these near-term quantum processors will only contain a small amount of noisy qubits and allow limited qubit connectivity. Fault-tolerant schemes that not only have low qubit overhead but also comply with geometrical interaction constraints are therefore necessary. In this work, we combine flag fault tolerance with quantum circuit mapping, to enable an e…
Experiments conducted on open-access cloud-based IBM Quantum devices are presented for characterizing their fault tolerance using $[4,2,2]$-encoded gate sequences. Up to 100 logical gates are activated in the IBMQ Bogota and IBMQ Santiago devices and we found that a $[4,2,2]$ code's logical gate set may be deemed fault-tolerant for gate sequences larger than 10 gates. However, certain circuits did not satisfy the fault tolerance criterion. In some cases, the encoded-gate sequences show a high error rate that is lower bounded at $\approx 0.1$, whereby the error inherent in these circuits cannot…
Fault tolerance of quantum circuits with tensor networks and symplectic geometry
Technical University of Munich（慕尼黑工业大学）；BMFTR Research Hub 6G-Life, Germany（德国 BMFTR 研究枢纽 6G-Life）；Munich Centre for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Munich Quantum Valley (MQV)（慕尼黑量子谷）；Cluster of Excellence CeTI, Technical University of Dresden（德累斯顿工业大学卓越集群 CeTI）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Nanyang Quantum Hub, Nanyang Technological University（南洋理工大学南洋量子枢纽）
Attaining fault tolerance while maintaining low overhead is one of the main challenges in a practical implementation of quantum circuits. One major technique that can overcome this problem is the flag technique, in which high-weight errors arising from a few faults can be detected by a few ancillas and distinguished using subsequent syndrome measurements. The technique can be further improved using the fact that for some families of codes, errors of any weight are logically equivalent if they have the same syndrome and weight parity, as previously shown in [Phys. Rev. A 104, 042410 (2021)]. In…
We present a [[7, 1, 3]] quantum error-correcting code that is able to achieve fault-tolerant syndrome measurement using one ancillary qubit per stabilizer for an error model of independent single-qubit Pauli errors. All single-qubit Pauli errors on the ancillary qubits propagate to form exclusively correctable errors on the data qubits. The situation changes for error models with two-qubit Pauli errors. We compare the level-1 logical error rates under two noise models: the standard Pauli symmetric depolarizing error model and an anisotropic error model. The anisotropic model is motivated by c…
The advent of noisy intermediate-scale quantum (NISQ) devices offers crucial opportunities for the development of quantum algorithms. Here we evaluate the noise tolerance of two quantum neural network (QNN) architectures on IBM's NISQ devices, namely, dissipative QNN (DQNN) whose building-block perceptron is a completely positive map, and the quantum approximate optimization algorithm (QAOA). We compare these two approaches to learning an unknown unitary. While both networks succeed in this learning task, we find that a DQNN learns an unknown unitary more reliably than QAOA and is less suscept…
Steane's seven-qubit quantum code is a natural choice for fault-tolerance experiments because it is small and just two extra qubits are enough to correct errors. However, the two-qubit error-correction technique, known as "flagged" syndrome extraction, works slowly, measuring only one syndrome at a time. This is a disadvantage in experiments with high qubit rest error rates. We extend the technique to extract multiple syndromes at once, without needing more qubits. Qubits for different syndromes can flag errors in each other. This gives equally fast and more qubit-efficient alternatives to Ste…
The rapid development of quantum computing (QC) in the NISQ era urgently demands a low-level benchmark suite and insightful evaluation metrics for characterizing the properties of prototype NISQ devices, the efficiency of QC programming compilers, schedulers and assemblers, and the capability of quantum system simulators in a classical computer. In this work, we fill this gap by proposing a low-level, easy-to-use benchmark suite called QASMBench based on the OpenQASM assembly representation. It consolidates commonly used quantum routines and kernels from a variety of domains including chemistr…
One of the major objectives of the experimental programs at the LHC is the discovery of new physics. This requires the identification of rare signals in immense backgrounds. Using machine learning algorithms greatly enhances our ability to achieve this objective. With the progress of quantum technologies, quantum machine learning could become a powerful tool for data analysis in high energy physics. In this study, using IBM gate-model quantum computing systems, we employ the quantum variational classifier method in two recent LHC flagship physics analyses: $t\bar{t}H$ (Higgs boson production i…
Quantum computers can be protected from noise by encoding the logical quantum information redundantly into multiple qubits using error correcting codes. When manipulating the logical quantum states, it is imperative that errors caused by imperfect operations do not spread uncontrollably through the quantum register. This requires that all operations on the quantum register obey a fault-tolerant circuit design which, in general, increases the complexity of the implementation. Here, we demonstrate a fault-tolerant universal set of gates on two logical qubits in a trapped-ion quantum computer. In…
The overhead cost of performing universal fault-tolerant quantum computation for large scale quantum algorithms is very high. Despite several attempts at alternative schemes, magic state distillation remains one of the most efficient schemes for simulating non-Clifford gates in a fault-tolerant way. However, since magic state distillation circuits are not fault-tolerant, all Clifford operations must be encoded in a large distance code in order to have comparable failure rates with the magic states being distilled. In this work, we introduce a new concept which we call redundant ancilla encodin…
Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two ancilla qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the [[5,1,3]] code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the [[7,…
Quantum Computing has been evolving in the last years. Although nowadays quantum algorithms performance has shown superior to their classical counterparts, quantum decoherence and additional auxiliary qubits needed for error tolerance routines have been huge barriers for quantum algorithms efficient use. These restrictions lead us to search for ways to minimize algorithms costs, i.e the number of quantum logical gates and the depth of the circuit. For this, quantum circuit synthesis and quantum circuit optimization techniques are explored. We studied the viability of using Projective Simulatio…
As quantum computers approach the fault tolerance threshold, diagnosing and characterizing the noise on large scale quantum devices is increasingly important. One of the most important classes of noise channels is the class of Pauli channels, for reasons of both theoretical tractability and experimental relevance. Here we present a practical algorithm for estimating the $s$ nonzero Pauli error rates in an $s$-sparse, $n$-qubit Pauli noise channel, or more generally the $s$ largest Pauli error rates. The algorithm comes with rigorous recovery guarantees and uses only $O(n^2)$ measurements, $O(s…
In the era of quantum computing without full fault-tolerance, it is essential to suppress noise effects via the quantum error mitigation techniques to enhance the computational power of the quantum devices. One of the most effective noise-agnostic error mitigation schemes is the generalized quantum subspace expansion (GSE) method, which unifies various mitigation algorithms under the framework of the quantum subspace expansion. Specifically, the fault-subspace method, a subclass of GSE method, constructs an error-mitigated quantum state with copies of quantum states with different noise levels…
The presence of noise in quantum computers hinders their effective operation. Even though quantum error correction can theoretically remedy this problem, its practical realization is still a challenge. Testing and benchmarking noisy, intermediate-scale quantum (NISC) computers is therefore of high importance. Here, we suggest the application of the so-called quantum state matching protocol for testing purposes. This protocol was originally proposed to determine if an unknown quantum state falls in a prescribed neighborhood of a reference state. We decompose the unitary specific to the protocol…
Solid-state spin qubits are a promising platform for quantum computation and quantum networks. Recent experiments have demonstrated high-quality control over multi-qubit systems, elementary quantum algorithms and non-fault-tolerant error correction. Large-scale systems will require using error-corrected logical qubits that are operated fault-tolerantly, so that reliable computation is possible despite noisy operations. Overcoming imperfections in this way remains a major outstanding challenge for quantum science. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubi…
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…
Ensuring the correct functioning of quantum error correction (QEC) circuits is crucial to achieve fault tolerance in realistic quantum processors subjected to noise. The first checkpoint for a fully operational QEC circuit is to create genuine multipartite entanglement across all subsystems of physical qubits. We introduce a conditional witnessing technique to certify genuine multipartite entanglement (GME) that is efficient in the number of subsystems and, importantly, robust against experimental noise and imperfections. Specifically, we prove that the detection of entanglement in a linear nu…
Gottesman-Kitaev-Preskill (GKP) states appear to be amongst the leading candidates for correcting errors when encoding qubits into oscillators. However the preparation of GKP states remains a significant theoretical and experimental challenge. Until now, no clear definitions for fault-tolerantly preparing GKP states have been provided. Without careful consideration, a small number of faults can lead to large uncorrectable shift errors. After proposing a metric to compare approximate GKP states, we provide rigorous definitions of fault-tolerance and introduce a fault-tolerant phase estimation p…
