The main aim of this thesis is to look for a logical deductive calculus (we will adopt sequent calculus, originally introduced in Gentzen, 1935), which could describe quantum information and its properties. More precisely, we intended to describe in logical terms the formation of the qubit (the unit of quantum information) which is a particular linear superposition of the two classical bits 0 and 1. To do so, we had to introduce the new connective "quantum superposition", in the logic of one qubit, Lq, as the classical conjunction cannot describe this quantum link.
The existence of a non-algorithmic side of the mind, conjectured by Penrose on the basis of Gödel's first incompleteness theorem, is investigated here in terms of a quantum metalanguage. We suggest that, besides human ordinary thought, which can be formalized in a computable, logical language, there is another important kind of human thought, which is Turing-non-computable. This is methatought, the process of thinking about ordinary thought. Metathought can be formalized as a metalanguage, which speaks about and controls the logical language of ordinary thought. Ordinary thought has two comput…
Dipartimento di Matematica Pura ed Applicata（纯数学与应用数学系）
The logic which describes quantum robots is not orthodox quantum logic, but a deductive calculus which reproduces the quantum tasks (computational processes, and actions) taking into account quantum superposition and quantum entanglement. A way toward the realization of intelligent quantum robots is to adopt a quantum metalanguage to control quantum robots. A physical implementation of a quantum metalanguage might be the use of coherent states in brain signals.
Logical propositions with the fuzzy modality "Probably" are shown to obey an uncertainty principle very similar to that of Quantum Optics. In the case of such propositions, the partial truth values are in fact probabilities. The corresponding assertions in the metalanguage, have complex assertion degrees which can be interpreted as probability amplitudes. In the logical case, the uncertainty relation is about the assertion degree, which plays the role of the phase, and the total number of atomic propositions, which plays the role of the number of modes. In analogy with coherent states in quant…
We show that self-reference can be formalized in Basic logic by means of the new connective @, called "entanglement". In fact, the property of non-idempotence of the connective @ is a metatheorem, which states that a self-entangled sentence loses its own identity. This prevents having self-referential paradoxes in the corresponding metalanguage. In this context, we introduce a generalized definition of self-reference, which is needed to deal with the multiplicative connectives of substructural logics.
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…
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 fai…
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 …
Logic-qubit entanglement has attracted much attention in both quantum communication and quantum computation. Here, we present an efficient protocol to distill the logic-qubit entanglement with the help of cross-Kerr nonlinearity. This protocol not only can purify the logic bit-flip error and logic phase-flip error, but also can correct the physical bit-flip error completely. We use cross-Kerr nonlinearity to construct quantum nondemolition detectors. Our distillation protocol for logic-qubit entanglement may be useful for the practical applications in quantum information, especially in long-di…
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…
We establish theoretical bounds on qubit detuning for high fidelity controlled-NOT logic gate implementations with weakly coupled Josephson phase qubits. It is found that the value of qubit detuning during the entangling pulses must not exceed 2g for two-step, and g for single-step control sequences, where g is the relevant coupling constant.
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…
Decoherence is one of the main obstacles in long-distance quantum communication. Recently, the theoretical work of Fröwis and W. Dür (Phys. Rev. Lett. \textbf{106}, 110402 (2011)) and the experiment of Lu \emph{et al.} (Nat. Photon. \textbf{8}, 364 (2014)) both showed that the logic qubits entanglement say the concatenated Greenberger-Horne-Zeilinger (C-GHZ) state is more robust under decoherence. In this paper, we describe a protocol for Bell-state analysis for this logic qubits entanglement. This protocol can also be extended to the multipartite C-GHZ state analysis. Also, we discuss its app…
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…
Recently, the theoretical work of Fröwis and W. Dür (Phys. Rev. Lett. \textbf{106}, 110402 (2011)) and the experiment of Lu \emph{et al.} (Nat. Photon. \textbf{8}, 364 (2014)) both showed that the logic-qubit entanglement has its potential application in future quantum communication and quantum network. However, the entanglement will suffer from the noise and decoherence. In this paper, we will investigate the entanglement purification for logic-qubit entanglement. We show that both the bit-flip error and phase-flip error in logic-qubit entanglement can be well purified. Moreover, the bit-flip…
In this paper, we propose a scheme to implement two-qubit logic gates with a controllable and selective interaction in a scalable superconducting circuit of charge qubits. A nanomechanical resonator is used as a data bus to connect qubits. It is indicated that a controllable interaction between qubits can be obtained by making use of the data bus. It is shown that a selective interaction between qubits can be realized when many qubits are involved in the system under our consideration.
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 discuss the implementation of quantum logic in a system of strongly interacting particles. The implementation is qubitless since ``logical qubits'' don't correspond to any physical two-state subsystems. As an illustration, we present the results of simulations of the quantum controlled-NOT gate and Shor's algorithm for a chain of spin-1/2 particles with Heisenberg coupling. Our proposal extends the current theory of quantum information processing to include systems with permanent strong coupling between the two-state subsystems.
We demonstrate a robust implementation of a deterministic linear-optical Controlled-NOT (CNOT) gate for single-photon two-qubit quantum logic. A polarization Sagnac interferometer with an embedded 45$^{\circ}$-oriented dove prism is used to enable the polarization control qubit to act on the momentum (spatial) target qubit of the same photon. The CNOT gate requires no active stabilization because the two spatial modes share a common path, and it is used to entangle the polarization and momentum qubits.
We report the first simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain in which selective excitations of spins is provided by the gradient of the external magnetic field is considered. The spins interact with their nearest neighbors. We simulate the quantum control-not (CN) gate implementation for remote qubits which provides the long-distance entanglement. Our approach can be applied to any implementation of quantum logic gates involving a large number of qubits.
A controlled-NOT logic gate based on conditional spectroscopy has been demonstrated recently for a pair of superconducting flux qubits [Plantenberg et al., Nature 447, 836 (2007)]. Here we study the fidelity of this type of gate applied to a phase qubit coupled to a resonator (or a pair of capacitively coupled phase qubits). Our results show that an intrinsic fidelity of more than 99% is achievable in 45ns.
Logical multi-qubit entanglement with dual-rail superconducting qubits
International Quantum Academy, Shenzhen（国际量子研究院）；Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学深圳量子科学与工程研究所）；Guangdong Provincial Key Laboratory of Quantum Science and Engineering, Southern University of Science and Technology（南方科技大学广东省量子科学与工程重点实验室）
Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schroedinger equation demonstrate that these operations can be performed with good fidelity.
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 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…
There are well-known protocols for performing CNOT quantum logic with qubits coupled by particular high-symmetry (Ising or Heisenberg) interactions. However, many architectures being considered for quantum computation involve qubits or qubits and resonators coupled by more complicated and less symmetric interactions. Here we consider a widely applicable model of weakly but otherwise arbitrarily coupled two-level systems, and use quantum gate design techniques to derive a simple and intuitive CNOT construction. Useful variations and extensions of the solution are given for common special cases.
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…
In this paper we generalize the usual model of quantum computer to a model in which the state is an operator of density matrix and the gates are general superoperators (quantum operations), not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum system (open or closed n-qubit system) is considered as an element of $4^{n}$-dimensional operator Hilbert space (Liouville space). It allows to use quantum computer (circuit) model with 4-valued logic. The gates of this model are general superoperators which act on n-ququats state. Ququat is quantum state in a 4-dime…
In the paper an approach is presented allowing to model quantum logic circuits by electronic gates for discrete spatially modulated electromagnetic signals. The designed circuitry is for modeling low scale quantum nets of general design and quantum devices based only on superposition principle of their work.
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…
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…
Unlike the standard Quantum Computational Logic (QCL), where the carrier of information (target) is conventionally assumed to be only the last qubit over a sequence of many qubits, here we propose an extended version of the QCL (we call Multi Target Quantum Computational Logic, briefly MTQCL) where the number and the position of the target qubits are arbitrary.
The big challenge in quantum computing is to realize scalable multi-qubit systems with cross-talk free addressability and efficient coupling of arbitrarily selected qubits. Quantum networks promise a solution by integrating smaller qubit modules to a larger computing cluster. Such a distributed architecture, however, requires the capability to execute quantum-logic gates between distant qubits. Here we experimentally realize such a gate over a distance of 60m. We employ an ancillary photon that we successively reflect from two remote qubit modules, followed by a heralding photon detection whic…
Logic qubit entanglement, which is also called the concatenated Greenberger-Horne-Zeilinger (C-GHZ) state, is robust in practical noisy environment. In this paper, we will describe an efficient approach to realize the complete polarization Bell-state analysis which is encoded in the logic qubit. We showed that the logic Bell-state can be distinguished in two steps with the help of the parity-check measurement (PCM), which is constructed by the cross-Kerr nonlinearity. We also explain that this approach can be used to distinguish arbitrary C-GHZ state with $N$ logic qubits. This protocol is use…
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…
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 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…
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）
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
We demonstrate laser-driven two-qubit and single-qubit logic gates with fidelities 99.9(1)% and 99.9934(3)% respectively, significantly above the approximately 99% minimum threshold level required for fault-tolerant quantum computation, using qubits stored in hyperfine ground states of calcium-43 ions held in a room-temperature trap. We study the speed/fidelity trade-off for the two-qubit gate, for gate times between 3.8$μ$s and 520$μ$s, and develop a theoretical error model which is consistent with the data and which allows us to identify the principal technical sources of infidelity.
Demonstration of High-Fidelity Entangled Logical Qubits using Transmons
University of Southern California（南加州大学）；Quantum Elements, Inc.（量子元素公司）；IBM Quantum（IBM量子）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）
