Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.
We present a quantum error correcting code with dynamically generated logical qubits. When viewed as a subsystem code, the code has no logical qubits. Nevertheless, our measurement patterns generate logical qubits, allowing the code to act as a fault-tolerant quantum memory. Our particular code gives a model very similar to the two-dimensional toric code, but each measurement is a two-qubit Pauli measurement.
A protocol is proposed to generate singlet states of three logical qubits constructed by pairs of spins. Single and multiple operations of logical qubits are studied for the construction of an effective Hamiltonian, with which robust control fields are derived with invariant-based reverse engineering and optimal control. Moreover, systematic errors are further compensated by periodic modulation for better robustness. Furthermore, resistance to decoherence of the protocol is also shown with numerical simulations. Therefore, the protocol may provide useful perspectives for generations of logical…
Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and …
Future quantum computers will require quantum error correction for faithful operation. The correction capabilities come with an overhead for performing fault-tolerant logical operations on the encoded qubits. One of the most resource efficient ways to implement logical operations is lattice surgery, where groups of physical qubits, arranged on lattices, can be merged and split to realize entangling gates and teleport logical information. Here, we report on the experimental realization of lattice surgery between two topologically encoded qubits in a 10-qubit ion trap quantum information process…
Fault-tolerant quantum computing demands many qubits with long lifetimes to conduct accurate quantum gate operations. However, external noise limits the computing time of physical qubits. Quantum error correction codes may extend such limits, but imperfect gate operations introduce errors to the correction procedure as well. The additional gate operations required due to the physical layout of qubits exacerbate the situation. Here, we use density-matrix simulations to investigate the performance change of logical qubits according to quantum error correction codes and qubit layouts and the expe…
Large-scale universal quantum computing requires the implementation of quantum error correction (QEC). While the implementation of QEC has already been demonstrated for quantum memories, reliable quantum computing requires also the application of nontrivial logical gate operations to the encoded qubits. Here, we present examples of such operations by implementing, in addition to the identity operation, the NOT and the Hadamard gate to a logical qubit encoded in a five qubit system that allows correction of arbitrary single qubit errors. We perform quantum process tomography of the encoded gate…
Coherence times for superconducting qubits have greatly improved over time. Moreover, small logical qubit architectures using engineered dissipation have shown great promise for further improvements in the coherence of a logical qubit manifold comprised of few physical qubits. Nevertheless, optimal working parameters for small logical qubits are generally not well understood. This work presents several approaches to finding preferential parameter configurations by looking at three different cases of increasing complexity. We begin by looking at state stabilization of a single qubit using dissi…
We demonstrate coherent control of two logical qubits encoded in a decoherence free subspace (DFS) of four dipolar-coupled protons in an NMR quantum information processor. A pseudo-pure fiducial state is created in the DFS, and a unitary logical qubit entangling operator evolves the system to a logical Bell state. The four-spin molecule is partially aligned by a liquid crystal solvent, which introduces strong dipolar couplings among the spins. Although the system Hamiltonian is never fully specified, we demonstrate high fidelity control over the logical degrees of freedom. In fact, the DFS enc…
Quantum information stored in a qubit is rapidly lost to the environment. The realization of robust qubits is one of the most important challenges in quantum computing. Herein we propose to embed a logical qubit within the manifold of a qudit as a scheme to preserve quantum information over extended periods of time. Under identical conditions (e.g. decoherence channels) the submanifold of the logical qubit exhibits extended lifetimes compared to a pure two-level system (qubit). The retention of quantum information further improves with separation between the sublevels of the logical qubit. Lif…
We consider tensor-network stabilizer codes and show that their tensor-network decoder has the property that independent logical qubits can be decoded in parallel. As long as the error rate is below threshold, we show that this parallel decoder is essentially optimal. As an application, we verify this for the max-rate holographic Steane (heptagon) code. For holographic codes this tensor-network decoder was shown to be efficient with complexity polynomial in n, the number of physical qubits. Here we show that, by using the parallel decoding scheme, the complexity is also linear in k, the number…
Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherently error tolerance. However, the rigorous geometric conditions lead to complex and/or error-disturbed quantum controls, especially for logical qubits that involve more physical qubits, whose error tolerance is effective in principle though, their experimental demonstration is still demanding. Thus, how to best simplify the needed control and manifest its full advantage has become the key to widespread applications of geometric quantum computation. Here we propose a new fast and robust…
We propose a proof-of-principle experiment to encode one logical qubit in noise protected subspace of three identical spins in a methyl group. The symmetry analysis of the wavefunction shows that this fermionic system exhibits a symmetry correlation between the spatial degree of freedom and the spin degree of freedom. We show that one can use this correlation to populate the noiseless subsystem by relying on the interaction between the electric dipole moment of the methyl group with a circularly polarized microwave field. Logical gates are implemented by controlling both the intensity and phas…
Dynamical decoupling can enforce a symmetry on the dynamics of an open quantum system. Here we develop an efficient dynamical-decoupling-based strategy to create the decoherence-free subspaces (DFSs) for a set of qubits by optimally transferring the states of these qubits. We design to transfer the state of each qubit to all of the other qubits in an optimal and efficient manner, so as to produce an effective collective-type noise for all qubits. This collective pseudo-noise is essentially derived from arbitrarily independent real baths and needs only nearest-neighbor state transfers for its i…
We show how to carry out quantum logical operations (controlled-not and Toffoli gates) on encoded qubits for several encodings which protect against various 1-bit errors. This improves the reliability of these operations by allowing one to correct for one bit errors which either preexisted or occurred in course of operation. The logical operations we consider allow one to cary out the vast majority of the steps in the quantum factoring algorithm. Thus, our results help bring quantum factoring and other quantum computations closer to reality
Stabilization of encoded logical qubits using quantum error correction is key to the realization of reliable quantum computers. While qubit codes require many physical systems to be controlled, oscillator codes offer the possibility to perform error correction on a single physical entity. One powerful encoding for oscillators is the grid state or GKP encoding, which allows small displacement errors to be corrected. Here we introduce and implement a dissipative map designed for physically realistic finite GKP codes which performs quantum error correction of a logical qubit implemented in the mo…
We propose a scheme of a programmable quantum motherboard based on the system of three interacting high-Q resonators coupled with two-level atoms. By using the algebraic methods, we found that the investigated atomic-resonator platform can possess an equidistant spectrum of the eigenfrequencies at which simple reversible dynamics of the single photon excitation becomes possible. It was shown that such multiresonator quantum motherboard scheme allows to achieve efficient and programmable quantum state transfer between distributed atoms and generate logical qubits and qutrits. The use of the stu…
In a digital quantum simulator, basic two-qubit interactions are manipulated by means of fast local control operations to establish a desired target Hamiltonian. Here we consider a quantum simulator based on logical systems, i.e. where several physical qubits are used to represent a single logical two-level system to obtain enhanced and simple control over effective interactions between logical systems. Fixed, distance-dependent pairwise interactions between the physical qubits lead to effective interactions between the logical systems, which can be fully controlled solely by the choice of the…
Genuine Multipartite Entanglement between Logical Qubits via Cross-Code Lattice Surgery
Universität Innsbruck（因斯布鲁克大学）；Technische Universität Wien（维也纳工业大学）；Institute for Theoretical Physics, University of Innsbruck（因斯布鲁克大学理论物理研究所）；neQxt GmbH（neQxt有限公司）；Alpine Quantum Technologies GmbH（阿尔卑斯量子技术有限公司）
We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS…
A logical qubit is a two-dimensional subspace of a higher dimensional system, chosen such that it is possible to detect and correct the occurrence of certain errors. Manipulation of the encoded information generally requires arbitrary and precise control over the entire system. Whether based on multiple physical qubits or larger dimensional modes such as oscillators, the individual elements in realistic devices will always have residual interactions which must be accounted for when designing logical operations. Here we demonstrate a holistic control strategy which exploits accurate knowledge o…
Lattice surgery for near-term experimental logical qubit entanglement creation in planar architectures
Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich（于利希研究中心理论纳米电子学研究所）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Department of Physics, ETH Zurich（苏黎世联邦理工学院物理系）；ETH Zurich - PSI Quantum Computing Hub, Paul Scherrer Institute（保罗谢勒研究所苏黎世联邦理工学院-PSI量子计算中心）；Quantum Center, ETH Zurich（苏黎世联邦理工学院量子中心）
Liquid phase NMR is a general purpose test-bed for developing methods of coherent control relevant to quantum information processing. Here we extend these studies to the coherent control of logical qubits and in particular to the unitary gates necessary to create entanglement between logical qubits. We report an experimental implementation of a conditional logical gate between two logical qubits that are each in decoherence free subspaces that protect the quantum information from fully correlated dephasing.
We study variants of Shor's code that are adept at handling single-axis correlated idling errors, which are commonly observed in many quantum systems. By using the repetition code structure of the Shor's code basis states, we calculate the logical channel applied to the encoded information when subjected to coherent and correlated single qubit idling errors, followed by stabilizer measurement. Changing the signs of the stabilizer generators allows us to change how the coherent errors interfere, leading to a quantum error correcting code which performs as well as a classical repetition code of …
We derive a set of composite pulse sequences that generates CNOT gates and correct all systematic errors within the logical subspace to arbitrary order. These sequences are applicable for any two-qubit interaction Hamiltonian, and make no assumptions about the underlying noise mechanism except that it is constant on the timescale of the operation. We do assume access to error-free single-qubit gates, so single-qubit gate imperfections eventually limit the achievable fidelity. However, since single-qubit gates generally have much higher fidelities than two-qubit gates in practice, these pulse s…
We simulate four quantum error correcting codes under error models inspired by realistic noise sources in near-term ion trap quantum computers: $T_2$ dephasing, gate overrotation, and crosstalk. We use this data to find preferred codes for given error parameters along with logical error biases and a pseudothreshold which compares the physical and logical gate failure rates for a CNOT gate. Using these results we conclude that Bacon-Shor-13 is the most promising near term candidate as long as the impact of crosstalk can be mitigated through other means.
We describe a fault-tolerant memory for an error-corrected logical qubit based on silicon double quantum dot physical qubits. Our design accounts for constraints imposed by supporting classical electronics. A significant consequence of the constraints is to add error-prone idle steps for the physical qubits. Even using a schedule with provably minimum idle time, for our noise model and choice of error-correction code, we find that these additional idles negate any benefits of error correction. Using additional qubit operations, we can greatly suppress idle-induced errors, making error correcti…
Superconducting qubits are among the most promising platforms for building a quantum computer. However, individual qubit coherence times are not far past the scalability threshold for quantum error correction, meaning that millions of physical devices would be required to construct a useful quantum computer. Consequently, further increases in coherence time are very desirable. In this letter, we blueprint a simple circuit consisting of two transmon qubits and two additional lossy qubits or resonators, which is passively protected against all single qubit quantum error channels through a combin…
We present a scheme to achieve maximally entangled states, controlled phase-shift gate, and SWAP gate for two superconducting-quantum-interference-device (SQUID) qubits, by placing SQUIDs in a microwave cavity. We also show how to transfer quantum information from one SQUID qubit to another. In this scheme, no transfer of quantum information between the SQUIDs and the cavity is required, the cavity field is only virtually excited and thus the requirement on the quality factor of the cavity is greatly relaxed.
We investigate a scheme of fault-tolerant quantum computation based on the cluster model. Logical qubits are encoded by a suitable code such as the Steane's 7-qubit code. Cluster states of logical qubits are prepared by post-selection through verification at high fidelity level, where the unsuccessful ones are discarded without recovery operation. Then, gate operations are implemented by transversal measurements on the prepared logical cluster states. The noise threshold is improved significantly by making the high fidelity preparation and transversal measurement. It is estimated to be about 3…
In this paper, we describe and experimentally demonstrate an error detection scheme that does not employ ancilla qubits or mid-circuit measurements. This is achieved by expanding the Hilbert space where a single logical qubit is encoded using several physical qubits. For example, one possible two qubit encoding identifies $|0\rangle_L=|01\rangle$ and $|1\rangle_L=|10\rangle$. If during the final measurement a $|11\rangle$ or $|00\rangle$ is observed an error is declared and the run is not included in subsequent analysis. We provide codewords for a simple bit-flip encoding, a way to encode the …
To realize fault-tolerant quantum computing, it is necessary to store quantum information in logical qubits with error correction functions, realized by distributing a logical state among multiple physical qubits or by encoding it in the Hilbert space of a high-dimensional system. Quantum gate operations between these error-correctable logical qubits, which are essential for implementation of any practical quantum computational task, have not been experimentally demonstrated yet. Here we demonstrate a geometric method for realizing controlled-phase gates between two logical qubits encoded in p…
We design and analyze a logical qubit composed of a linear array of electron spins in semiconductor quantum dots. To avoid the difficulty of fully controlling a two-dimensional array of dots, we adapt spin control and error correction to a one-dimensional line of silicon quantum dots. Control speed and efficiency are maintained via a scheme in which electron spin states are controlled globally using broadband microwave pulses for magnetic resonance while two-qubit gates are provided by local electrical control of the exchange interaction between neighboring dots. Error correction with two-, th…
We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant app…
Large-scale quantum computers will inevitably need quantum error correction to protect information against decoherence. Traditional error correction typically requires many qubits, along with high-efficiency error syndrome measurement and real-time feedback. Autonomous quantum error correction (AQEC) instead uses steady-state bath engineering to perform the correction in a hardware-efficient manner. We realize an AQEC scheme, implemented with only two transmon qubits in a 2D scalable architecture, that actively corrects single-photon loss and passively suppresses low-frequency dephasing using …
The technological world is in the midst of a quantum computing and quantum information revolution. Since Richard Feynman's famous "plenty of room at the bottom" lecture, hinting at the notion of novel devices employing quantum mechanics, the quantum information community has taken gigantic strides in understanding the potential applications of a quantum computer and laid the foundational requirements for building one. We believe that the next significant step will be to demonstrate a quantum memory, in which a system of interacting qubits stores an encoded logical qubit state longer than the i…
