All-Microwave Leakage Reduction Units for Quantum Error Correction with Superconducting Transmon Qubits

J. F. Marques, H. Ali, B. M. Varbanov, M. Finkel, H. M. Veen, S. L. M. van der Meer, S. Valles-Sanclemente, N. Muthusubramanian, M. Beekman, N. Haider, B. M. Terhal, and L. DiCarlo

Phys. Rev. Lett. 130, 250602 – Published 22 June 2023

Abstract

Minimizing leakage from computational states is a challenge when using many-level systems like superconducting quantum circuits as qubits. We realize and extend the quantum-hardware-efficient, all-microwave leakage reduction unit (LRU) for transmons in a circuit QED architecture proposed by Battistel et al. This LRU effectively reduces leakage in the second- and third-excited transmon states with up to 99% efficacy in 220 ns, with minimum impact on the qubit subspace. As a first application in the context of quantum error correction, we show how multiple simultaneous LRUs can reduce the error detection rate and suppress leakage buildup within 1% in data and ancilla qubits over 50 cycles of a weight-2 stabilizer measurement.

Received 22 March 2023
Accepted 24 May 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.250602

Physics Subject Headings (PhySH)

Quantum control Quantum engineering Quantum error correction Quantum information processing

Superconducting qubits

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

J. F. Marques ^1,2, H. Ali^1,2, B. M. Varbanov¹, M. Finkel^1,2, H. M. Veen^1,2, S. L. M. van der Meer^1,2, S. Valles-Sanclemente ^1,2, N. Muthusubramanian^1,2, M. Beekman ^1,3, N. Haider^1,3, B. M. Terhal^1,4, and L. DiCarlo^1,2,*

¹QuTech, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
²Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands
³Netherlands Organisation for Applied Scientific Research (TNO), P.O. Box 96864, 2509 JG The Hague, Netherlands
⁴EEMCS Department, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, Netherlands

^*Corresponding author. l.dicarlo@tudelft.nl

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Issue

Vol. 130, Iss. 25 — 23 June 2023

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Figure 1
Leakage reduction unit scheme. (a) Schematic for the driven transmon-resonator system. A transmon ( $T$ , yellow) with three lowest-energy levels $| g ⟩$ , $| e ⟩$ , and $| f ⟩$ is coupled to a readout resonator ( $R$ ) with strength $g$ . The latter is coupled to a frequency-matched Purcell resonator ( $P$ ) with strength $J$ . The Purcell resonator also couples to a $50 Ω$ feedline through which its excitations quickly decay at rate $κ$ . The transmon is driven with a pulse of strength $Ω$ applied to its microwave drive line. (b) Energy level spectrum of the system. Levels are denoted as $| T, R, P ⟩$ , with numbers indicating photons in $R$ and $P$ . As the two resonators are frequency matched, the right-most degenerate states split by $2 J$ , and $g$ is shared equally among the two hybridized resonator modes $| 1^{-} ⟩$ and $| 1^{+} ⟩$ . An effective coupling $\tilde{g}$ arises between $| f 00 ⟩$ and the two hybridized states $| g 1^{\pm} ⟩$ via $| e 00 ⟩$ and $| e 1^{\pm} ⟩$ . (c) Spectroscopy of the $| f 00 ⟩ \leftrightarrow | g 1^{\pm} ⟩$ transition. Measured transmon population in $| f ⟩$ versus drive frequency, showing dips corresponding to the two transitions assisted by each of the hybridized resonator modes.
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Figure 2
Calibration of the leakage reduction unit pulse. (a) Pulse sequence used for LRU calibration. (b) Single-shot readout data obtained from the experiment. The blue, red, and green areas denote $m_{0}$ , $m_{1}$ , and $m_{2}$ assignment regions, respectively. The mean (white dots) and $3 σ$ standard deviation (white dashed circles) shown are obtained from Gaussian fits to the three input-state distributions. The blue data show the first $3 \times 10^{3}$ (from a total of $2^{15}$ ) shots of the experiment described in (a), indicating 99.(3)% $| f ⟩$ -state removal fraction. (c) Measured assignment fidelity matrix used for readout correction. (d),(e) Extracted $| f ⟩$ -state removal fraction versus pulse parameters. Added contours (white dashed curves) indicate 80%, 90%, and 97% removal fraction. The purple star indicates the pulse parameters used in (b).
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Figure 3
Process tomography of the leakage reduction unit. (a)–(f) Measured density matrices after the LRU gate for input states $| 0 ⟩$ , $| + ⟩$ , $| + i ⟩$ , $| 1 ⟩$ , $| - ⟩$ , and $| - i ⟩$ , respectively. (g) $Z$ -rotation angle induced on the qubit versus the LRU pulse duration. The linear best fit (black dashed line) indicates an ac-Stark shift of 71(9) kHz. (h),(i) Pauli transfer matrix of the LRU with (i) and without (h) virtual phase correction [ $t_{p} = 220 ns$ and $R = 84 . (7) %$ ].
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Figure 4
Repeated stabilizer measurement with leakage reduction. (a) Quantum circuit using ancilla $A$ to measure the $X$ -type parity of data qubits $D_{1}$ and $D_{2}$ . The dashed box shows the frequency arrangement for two-qubit CZ gates. A CZ gate is performed by fluxing the higher-frequency transmon down in frequency to the nearest avoided crossing (orange shaded trajectory). The duration of single-qubit gates, CZ gates, and measurement are 20, 60, and 340 ns, respectively, totalling 500 ns for the parity check (light-blue region). Performing the LRUs extends the circuit by 220 ns (blue-dashed region). Echo pulses on data qubits mitigate phase errors caused by residual $Z Z$ crosstalk and ac-Stark shift during the measurement and LRUs (light yellow slots). (b),(c) Measured error-detection probability (b) and leakage (c) versus the number of parity-check rounds in four settings. Here, leakage includes any population outside the computational subspace. The “No LRUs” setting (blue) does not apply any LRUs. LRU data (orange) and LRU ancilla (green) settings apply LRUs exclusively on the data qubits and the ancilla, respectively. The LRU both (red) setting applies LRUs on all qubits.
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