Tunable Coupling Scheme for Implementing High-Fidelity Two-Qubit Gates

Fei Yan, Philip Krantz, Youngkyu Sung, Morten Kjaergaard, Daniel L. Campbell, Terry P. Orlando, Simon Gustavsson, and William D. Oliver

Phys. Rev. Applied 10, 054062 – Published 28 November 2018

Abstract

The prospect of computational hardware with quantum advantage relies critically on the quality of quantum-gate operations. Imperfect two-qubit gates are a major bottleneck for achieving scalable quantum-information processors. Here, we propose a generalizable and extensible scheme for a two-qubit tunable coupler that controls the qubit-qubit coupling by modulating the coupler frequency. Two-qubit gate operations can be implemented by operating the coupler in the dispersive regime, which is noninvasive to the qubit states. We investigate the performance of the scheme by simulating a universal two-qubit gate on a superconducting quantum circuit, and find that errors from known parasitic effects are strongly suppressed. The scheme is compatible with existing high-coherence hardware, thereby promising a higher gate fidelity with current technologies.

Received 26 March 2018
Revised 17 July 2018

DOI:https://doi.org/10.1103/PhysRevApplied.10.054062

Physics Subject Headings (PhySH)

Quantum computation Quantum control Quantum gates Quantum information architectures & platforms Quantum information with solid state qubits

Coupled oscillators Superconducting qubits

Quantum Information, Science & Technology

Authors & Affiliations

Fei Yan¹, Philip Krantz¹, Youngkyu Sung¹, Morten Kjaergaard¹, Daniel L. Campbell¹, Terry P. Orlando¹, Simon Gustavsson^1,*, and William D. Oliver^1,2,3

¹Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA
³Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

^*simongus@mit.edu

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Vol. 10, Iss. 5 — November 2018

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Figure 1
(a) Sketch of a generic three-body system in a chain geometry, where the center mode is a tunable coupler. (b) Level diagram of the ground and one-excitation states of the system. The ket symbol follows the chain order $| ω_{1}, ω_{c}, ω_{2} ⟩$ . The round-trip arrows indicate NN (orange) and NNN (green) coupling. (c) Level diagrams of the reduced two-qubit system after Schrieffer-Wolff transformation of the level diagram in (b). Each figure corresponds to the case of an effective negative (left), zero (center), and positive (right) net coupling, $\tilde{g}$ . The double-headed arrows indicate the sign and magnitude of coupling. In this example, the NNN coupling (green) is positive and fixed. The NN coupling (orange) is negative and tunable with the coupler energy (solid black line).
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Figure 2
(a) Circuit diagram of a superconducting circuit implementing a tunable coupler. Each mode is constructed by a tunable transmon qubit. (b) The $ω_{c}$ dependence of $2 \tilde{g}$ . The crossing at $2 \tilde{g} = 0$ (black dot) indicates the switch-off bias $ω_{c}^{off}$ . (c) Calculated eigenenergies of the one-excitation manifold as a function of the coupler frequency. The parameters used are $C_{1} = 70$ fF, $C_{2} = 72$ fF, $C_{c} = 200$ fF, $C_{1 c} = 4$ fF, $C_{2 c} = 4.2$ fF, $C_{12} = 0.1$ fF, $ω_{1} = ω_{2} = 4$ GHz. We intentionally create a 5–10% variation between $C_{1}$ and $C_{2}$ as well as between $C_{1 c}$ and $C_{2 c}$ to emulate fabrication variation (not a requirement for proof of concept). Since the two-qubit modes are degenerate, the eigenstates are symmetric (solid line) and antisymmetric (dashed line) combination of the their wavefunctions, and the energy gap (shaded) corresponds to the effective coupling $2 \tilde{g}$ shown in (b). The inset illustrates the pulse that turns the coupling on and off, executing an iSWAP gate.
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Figure 3
Relation between error per gate $ϵ$ and gate length $τ$ due to energy relaxation and other parasitic effects, respectively. Purple circles are simulation results in the absence of decoherence (bare dynamics) and using a negative coupler anharmonicity $α_{c} = - 100$ MHz, consistent with a transmon design. The solid purple line is a power-law fit, $ϵ = 1.5 \times 10^{6} {(τ / 1 ns)}^{- 5.7}$ . Pink squares are simulation results in the absence of decoherence and using a positive coupler anharmonicity, $α_{c} = + 100$ MHz, achievable using a capacitively shunted flux qubit. The solid pink line is an exponential fit, $ϵ = 73 \exp (- 0.4 τ / 1 ns)$ . Dashed lines are calculated errors from $T_{1}$ process only (different $T_{1}$ values assumed). The crossings between the curves in the noise-free case and the $T_{1}$ case indicate approximately the break-even point for the optimal gate time and gate fidelity.
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Figure 4
Circuit diagram of the implemented superconducting circuits, consisting of qubit mode 1 (red), qubit mode 2 (blue), and coupler mode c (black). Each mode is a tunable transmon qubit. $E_{J_{λ, L (R)}}$ is the Josepshon energy of the left (right) junction in mode $λ$ . $C_{λ}$ is the dominant mode capacitance. $C_{j c}$ ( $j = 1, 2$ ) is the coupling capacitance between qubit $j$ and coupler. $C_{12}$ is the direct coupling capacitance between the two qubits. $Φ_{e, λ}$ is the external magnetic flux threading each loop. $ϕ_{λ}$ is the reduced node flux.
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