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arXiv:1907.04379 · Physical review. B./Physical review. B

Set of holonomic and protected gates on topological qubits for a realistic quantum computer

Andrey R. Klots, L. B. Ioffe

Published Oct 6, 2021• 14 citations
arXivHTMLPDF

Abstract

In recent years qubit designs such as transmons have approached fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits are arguably the most prospective for building a realistic quantum computer as they are intrinsically protected from noise and leakage errors that occur in transmons. We propose a topologically protected qubit design based on a $\ensuremath{\pi}$-periodic Josephson element and a universal set of gates: A protected Clifford group and highly robust (with infidelity $\ensuremath{\lesssim}{10}^{\ensuremath{-}4}$) nondiscrete holonomic phase gate. The qubit is controlled via charge $Q$ and flux $\mathrm{\ensuremath{\Phi}}$ biases. The holonomic gate is realized by quickly, but adiabatically, going along a particular closed path in the two-dimensional ${\mathrm{\ensuremath{\Phi}},Q}$ space---a path where computational states are always degenerate but Berry curvature is localized inside the path. This gate is robust against currently achievable noise levels. This qubit architecture allows building a realistic scalable superconducting quantum computer with leakage and noise-induced errors below ${10}^{\ensuremath{-}4}$, which allows performing realistic error correction codes with currently available fabrication techniques.

DOI
10.1103/physrevb.104.144502

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In recent years qubit designs such as transmons have approached fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits are arguably the most prospective for building a realistic quantum computer as they are intrinsically protected from noise and leakage errors that occur in transmons. We propose a topologically protected qubit design based on a \ensuremathπ\ensuremath{\pi}\ensuremathπ-periodic Josephson element and a universal set of gates: A protected Clifford group and highly robust (with infidelity \ensuremath≲10\ensuremath−4\ensuremath{\lesssim}{10}^{\ensuremath{-}4}\ensuremath≲10\ensuremath−4) nondiscrete holonomic phase gate. The qubit is controlled via charge QQQ and flux \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ biases. The holonomic gate is realized by quickly, but adiabatically, going along a particular closed path in the two-dimensional \ensuremathΦ,Q{\mathrm{\ensuremath{\Phi}},Q}\ensuremathΦ,Q space---a path where computational states are always degenerate but Berry curvature is localized inside the path. This gate is robust against currently achievable noise levels. This qubit architecture allows building a realistic scalable superconducting quantum computer with leakage and noise-induced errors below 10\ensuremath−4{10}^{\ensuremath{-}4}10\ensuremath−4, which allows performing realistic error correction codes with currently available fabrication techniques.
High-fidelity and robust quantum manipulation is the key for scalable quantum computation. Therefore, due to its intrinsic operational robustness, quantum manipulation induced by geometric phases is one of the promising strategies. However, the longer gate time for geometric operations and more physical difficulties with regard to implementation hinder its practical and wide application. Here, we propose a simplified implementation of universal holonomic quantum gates on superconducting circuits with experimentally demonstrated techniques, which can remove these two main challenges by introducing time-optimal control into the construction of quantum gates. Notably, our scheme is also based on a decoherence-free subspace encoding and requires minimal physical-qubit resources, which can be partially immune to error caused by qubit-frequency drift, one of the main sources of error for large-scale superconducting circuits. Meanwhile, gate error caused by unwanted leakage can also be eliminated by our deliberate design of quantum evolution paths. Finally, our scheme is numerically shown to be more robust than the conventional ones and thus provides a promising strategy for scalable solid-state fault-tolerant quantum computation.
Geometric phase is an indispensable element for achieving robust and high-fidelity quantum gates due to its built-in noise-resilience feature. However, due to the complexity of manipulation and the intrinsic leakage of the encoded quantum information to a non-logical-qubit basis, the experimental realization of universal nonadiabatic holonomic quantum computation is very difficult. Here we propose to implement scalable nonadiabatic holonomic quantum computation with a decoherence-free subspace encoding on a two-dimensional square superconducting transmon-qubit lattice, where only the two-body interaction of neighboring qubits, from the simplest capacitive coupling, is needed. Meanwhile, we introduce qubit-frequency driving to achieve tunable resonant coupling for the neighboring transmon qubits, and thus avoid the leakage problem. In addition, our presented numerical simulation shows that high-fidelity quantum gates can be obtained, verifying the advantages of the robustness and scalability of our scheme. Therefore, our scheme provides a promising way towards the physical implementation of robust and scalable quantum computation.
Topological quantum codes are intrinsically fault-tolerant to local noise, and underlie the theory of topological phases of matter. We explore geometry to enhance the performance of topological quantum codes by rotating the four dimensional self-correcting quantum memory, and present codes targeted to both near-term and utility-scale quantum computers. We identify a full set of logical Clifford operations and with it design a universal fault-tolerant quantum architecture. Our design achieves single-shot error correction, significant reductions in required qubits, and low-depth logical operations. In turn, our proposed architecture relaxes the requirements for achieving fault tolerance and offers an efficient path for realization in several near-term quantum hardware implementations. Our [[96,6,8]] 4D Hadamard lattice code has low weight-6 stabilizers and depth-8 syndrome extraction circuits, a high pseudo-threshold of ∼0.01\sim 0.01∼0.01, and a logical error rate of ∼10−6\sim 10^{-6}∼10−6 per logical qubit per round of error correction at 10−310^{-3}10−3 physical error rate under a standard circuit-level noise model. A Clifford-complete logical gate set is presented, including a constructive and efficient method for Clifford gate synthesis.
Obtaining high-fidelity and robust quantum gates is the key for scalable quantum computation, and one of the promising ways is to implement quantum gates using geometric phases, where the influence of local noises can be greatly reduced. To obtain robust quantum gates, we here propose a scheme for quantum manipulation by combining the geometric phase approach with the dynamical correction technique, where the imperfection control induced X-error can be greatly suppressed. Moreover, to be robust against the decoherence effect and the randomized qubit-frequency shift Z-error, our scheme is also proposed based on the polariton qubit, the eigenstates of the light–matter interaction, which is immune to both errors up to the second order, due to its near symmetric energy spectrum. Finally, our scheme is implemented on the superconducting circuits, which also simplifies previous implementations. Since the main errors can be greatly reduced in our proposal, it provides a promising strategy for scalable solid-state fault-tolerant quantum computation.
Quantum holonomic gates hold built-in resilience to local noises and provide a promising approach for implementing fault-tolerant quantum computation. We propose to realize high-fidelity holonomic (N+1)(N+1)(N+1)-qubit controlled gates using Rydberg atoms confined in optical arrays or superconducting circuits. We identify the scheme, deduce the effective multibody Hamiltonian, and determine the working condition of the multiqubit gate. Uniquely, the multiqubit gate is immune to systematic errors, i.e., laser parameter fluctuations and motional dephasing, as the NNN control atoms largely remain in the very stable qubit space during the operation. We show that CN{C}_{N}CN​-not gates can reach the same level of fidelity at a given gate time for $N\ensuremath{\le}5$ under a suitable choice of parameters, and the gate tolerance against errors in systematic parameters can be further enhanced through optimal pulse engineering. In the case of Rydberg atoms, the proposed protocol is intrinsically different from typical schemes based on Rydberg blockade or antiblockade. Our study paves an alternative way to build robust multiqubit gates with Rydberg atoms trapped in optical arrays or with superconducting circuits. It contributes to current efforts to develop scalable quantum computation with trapped atoms and fabricable superconducting devices.
Geometric phases are well known to be noise resilient in quantum evolutions and operations. Holonomic quantum gates provide us with a robust way towards universal quantum computation, as these quantum gates are actually induced by non-Abelian geometric phases. Here we propose and elaborate how to efficiently implement universal nonadiabatic holonomic quantum gates on simpler superconducting circuits, with a single transmon serving as a qubit. In our proposal, an arbitrary single-qubit holonomic gate can be realized in a single-loop scenario by varying the amplitudes and phase difference of two microwave fields resonantly coupled to a transmon, while nontrivial two-qubit holonomic gates may be generated with a transmission-line resonator being simultaneously coupled to the two target transmons in an effective resonant way. Moreover, our scenario may readily be scaled up to a two-dimensional lattice configuration, which is able to support large scalable quantum computation, paving the way for practically implementing universal nonadiabatic holonomic quantum computation with superconducting circuits.