Hardware Implementation of Quantum Stabilizers in Superconducting Circuits

K. Dodge, Y. Liu, A. R. Klots, B. Cole, A. Shearrow, M. Senatore, S. Zhu, L. B. Ioffe, R. McDermott, and B. L. T. Plourde

Phys. Rev. Lett. 131, 150602 – Published 13 October 2023

Abstract

Stabilizer operations are at the heart of quantum error correction and are typically implemented in software-controlled entangling gates and measurements of groups of qubits. Alternatively, qubits can be designed so that the Hamiltonian corresponds directly to a stabilizer for protecting quantum information. We demonstrate such a hardware implementation of stabilizers in a superconducting circuit composed of chains of $π$ -periodic Josephson elements. With local on-chip flux and charge biasing, we observe a progressive softening of the energy band dispersion with respect to flux as the number of frustrated plaquette elements is increased, in close agreement with our numerical modeling.

Received 4 March 2023
Accepted 7 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.150602

Physics Subject Headings (PhySH)

Quantum computation Quantum error correction Quantum information with solid state qubits

Josephson junctions Superconducting qubits

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

Authors & Affiliations

K. Dodge ^1,*, Y. Liu ^1,*, A. R. Klots², B. Cole ¹, A. Shearrow³, M. Senatore ¹, S. Zhu ³, L. B. Ioffe², R. McDermott³, and B. L. T. Plourde ^1,†

¹Department of Physics, Syracuse University, Syracuse, New York 13244-1130, USA
²Google Quantum AI, Santa Barbara, California 93111, USA
³Department of Physics, University of Wisconsin–Madison, Madison, Wisconsin 53706, USA

^*These authors contributed equally to this work.
^†bplourde@syr.edu

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Vol. 131, Iss. 15 — 13 October 2023

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Images

Figure 1
Concatenation of $π$ -periodic plaquettes. (a) Schematic of single plaquette shunted by $C_{sh}$ . (b) $\cos 2 φ$ potential at frustration ( $Δ Φ = Φ - Φ_{0} / 2 = 0$ ) with localized wave functions in 0 and $π$ wells, drawn in the $0, π$ basis for vanishingly small tunneling. (c) Sketch of CW and CCW tunneling paths for $φ$ going between 0, $π$ wells indicated by blue and red dots, respectively. (d) Linear flux dispersion of 0 and $π$ levels for vanishing tunnel splitting. (e) Schematic of two plaquettes shunted by $C_{sh}$ with small capacitance $C_{isl}$ from intermediate island to ground. Potential with respect to phase across each plaquette displayed on (f) contour plot and (g) surface of torus; blue (red) lines correspond to hybridized even- (odd-) parity states; arrows indicate CW and CCW tunneling paths between wells of the same parity. (h) 1D cut of effective potential at double frustration. (i) Quadratic dispersion of even- (odd-) parity levels and flat dispersion of odd- (even-) parity levels near double frustration for simultaneous scan of plaquette fluxes along $Δ Φ_{1} = Δ Φ_{2}$ ( $Δ Φ_{1} = - Δ Φ_{2}$ ) on the left (right) (sketches do not include higher levels within a well).
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Figure 2
Multiplaquette flux biasing. 2D flux-modulation scans of readout cavity dispersive shift for (a) PB23 vs PB12 and (b) PB30 vs PB01. (c) Optical micrograph of device.
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Figure 3
Level transitions. Simulated level diagrams near (a) single and (b) double frustration; lines indicate example plasmons (red), heavy fluxons (blue), and light fluxons (magenta).
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Figure 4
Spectroscopy at different frustration points. Spectroscopy at (a) plaquette 2 single frustration, (b) plaquette (12) double frustration, and (c) triple frustration. Lines indicate modeled transitions with: $red = plasmons$ , $blue = heavy fluxons$ , $purple = light$ fluxons, $dotted = transitions$ out of 0 level, dash-dotted = transitions out of 1 level, $dashed = transitions$ out of 2 level, solid red $line = plasmon$ transition between antisymmetric levels in an even-parity well, and $orange = light$ fluxon plus cavity photon (Supplemental Material [14], Sec. X). (d) Comparison of dispersion of lowest heavy fluxon from modeled levels with linear, quadratic, and cubic fits for single (black circle), double (blue triangle), and triple (red square) frustration; frequency axis inverted for single and triple frustration for $Δ Φ < 0$ . (e) Repeated scans of cavity response vs offset charge bias to $C_{sh}$ island at plaquette 2 single frustration. 2D scan of spectroscopy at 0-1 transition frequency while scanning bias voltages to gate electrodes coupled to both intermediate islands for (f) plaquette (12) double frustration and (g) triple frustration. (h) Plot of $Δ_{SA}^{(i j)}$ and curvature of fluxon transition between even- and odd-parity ground states vs $E_{C}^{isl}$ showing measured values for plaquette (12), (23), and (13) double frustration (solid triangles) plus modeled values for a range of $C_{isl}$ (open circles).
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