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arXiv:1810.04690 · Physical Review Letters

Demonstration of Controlled-Phase Gates between Two Error-Correctable Photonic Qubits

Yuan Xu, Y. Ma, Weizhou Cai, X D Mu, W. Dai, Wei Wang, Lian Hu, X. Li, J. Han, Hanrui Wang, Yipu Song, Zhen‐Biao Yang, Shi‐Biao Zheng, Luyan Sun

Published Mar 24, 2020• 77 citations
arXivHTMLPDF

Abstract

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 photonic fields stored in cavities. The gates are realized by dispersively coupling an ancillary superconducting qubit to these cavities and driving it to make a cyclic evolution depending on the joint photonic state of the cavities, which produces a conditional geometric phase. We first realize phase gates for photonic qubits with the logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. Then we use this geometric method to implement a controlled-phase gate between two binomially encoded logical qubits, which have an error-correctable function.

DOI
10.1103/physrevlett.124.120501

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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 photonic fields stored in cavities. The gates are realized by dispersively coupling an ancillary superconducting qubit to these cavities and driving it to make a cyclic evolution depending on the joint photonic state of the cavities, which produces a conditional geometric phase. We first realize phase gates for photonic qubits with the logical basis states encoded in two quasiorthogonal coherent states, which have important implications for continuous-variable-based quantum computation. Then we use this geometric method to implement a controlled-phase gate between two binomially encoded logical qubits, which have an error-correctable function.
High-fidelity two-logical-qubit gates are essential for realizing fault-tolerant quantum computation with bosonic codes, yet experimentally reported fidelities have rarely exceeded 90\%. Here, we propose a geometric phase engineering approach for implementing controlled-phase gates for binomially encoded logical qubits. This method leverages the structural simplicity of geometric drives to reduce the numerical optimization dimensionality while fully incorporating system nonlinearities, enabling fast and high-fidelity logical operations. As an example, we experimentally demonstrate a process fidelity of 97.4$\pm$0.8\% for a controlled-Z gate between two binomial codes, surpassing all previously reported two-logical-qubit gates in bosonic codes. This work demonstrates that geometric phase engineering provides an effective and experimentally feasible route to fast, high-fidelity logical operations in bosonic quantum processors.
Schr\{o}dinger cat states are crucial for exploration of fundamental issues of quantum mechanics and have important applications in quantum information processing. Here, we propose and experimentally demonstrate a method for manipulating cat states in a cavity with the Aharonov-Anandan phase acquired by a superconducting qubit, which is dispersively coupled to the cavity. Based on this dispersive coupling, the qubit can be forced to trace out a circuit in the projective Hilbert space conditional on one coherent state. By preparing the cavity in a superposition of two coherent states, the geometric phase associated with this transport is encoded to the relative probability amplitude of these two coherent states. We demonstrate the photon-number parity of a cat state in a cavity can be controlled by adjusting this geometric phase, which offers the possibility for protecting its quantum coherence from single-photon loss. Based on this geometric effect, we realize phase gates for one and two photonic qubits whose logical basis states are encoded in two quasi-orthogonal coherent states. We further demonstrate two-cavity gates with symmetric and asymmetric Fock state encoding schemes. Our method can be directly extended to implementation of controlled-phase gates between error-correctable logical qubits.
Geometric quantum computation offers a practical strategy toward robust quantum computation due to its inherent 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, but 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 fast and robust geometric scheme, with decoherence-free subspace encoding, and present its physical implementation on superconducting quantum circuits, where we only utilize the experimentally demonstrated parametrically tunable coupling to achieve high-fidelity geometric control over logical qubits. Numerical simulation verifies that it can efficiently combine the error tolerance from both the geometric phase and logical-qubit encoding, displaying our gate-performance superiority over the conventional dynamical one without encoding, in terms of both gate fidelity and robustness. Therefore, our scheme can consolidate both error suppression methods for logical-qubit control, which sheds light on the future large-scale quantum computation.
Quantum phase gates are of fundamental interest in quantum physics and play an important role in quantum communication and quantum computing. Here, we propose a scheme to implement a hybrid discrete-variable–continuous-variable (DV–CV) phase gate on multiple photonic qubits by utilizing multiple control microwave cavities and a single target microwave cavity, all coupled to a common superconducting qutrit (a three-level quantum system) in circuit quantum electrodynamics (QED). In our scheme, two logic states of the target qubit are encoded via the vacuum state and the coherent state (CV state) of the microwave cavity, while two logic states of the control qubits are encoded through the vacuum and single-photon states (DV states) of the microwave cavity. This hybrid DV–CV phase gate enables target qubit to acquire a phase that depends on the total number of control qubits in non-vacuum states, when the target qubit is in the coherent state. Our protocol is extensible and operationally efficient, involving just one step and the measurement of only one target qubit. In addition, as a practical application, this hybrid DV–CV phase gate can be employed as a quantum voting machine. Our numerical simulation demonstrates that the high-fidelity construction of a hybrid three-qubit DV–CV phase gate is feasible with current circuit QED technology.
We propose a protocol to implement multi-qubit geometric gates (i.e., the Molmer-Sorensen gate) using photonic cat qubits. These cat qubits stored in high-QQQ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat qubits are effectively suppressed, leaving only a bit-flip error to be corrected. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against control imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multi-mode entangled cat states in a short time with high fidelities.
We propose a theoretical protocol to implement multiqubit geometric gates (i.e., the M\o{}lmer-S\o{}rensen gate) using photonic cat-state qubits. These cat-state qubits stored in high-QQQ resonators are promising for hardware-efficient universal quantum computing. Specifically, in the limit of strong two-photon drivings, phase-flip errors of the cat-state qubits are effectively suppressed, leaving only a bit-flip error to be corrected. Because this dominant error commutes with the evolution operator, our protocol preserves the error bias, and, thus, can lower the code-capacity threshold for error correction. A geometric evolution guarantees the robustness of the protocol against stochastic noise along the evolution path. Moreover, by changing detunings of the cavity-cavity couplings at a proper time, the protocol can be robust against parameter imperfections (e.g., the total evolution time) without introducing extra noises into the system. As a result, the gate can produce multimode entangled cat states in a short time with high fidelities.