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Realization of Three-Qubit Quantum Error Correction with Superconducting Circuits
M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, R. J. Schoelkopf
TL;DR
The paper develops superconducting-circuit procedures for encoding states, correcting errors with ancilla-resolved syndromes, and implementing a CCNot gate. The gate achieves over 99% quantum phase fidelity for six relevant phases, while bit-flip correction reaches 75.7% state fidelity and resolves error syndromes with finite fidelity.
Problem
The experiment addresses how to encode a quantum state into a protected three-qubit subspace that can recover from single-qubit errors.
Method
The procedure uses entangled-state encoding, ancilla rotations, phase-gate-based CCNot construction, and syndrome extraction to correct bit-flip errors.
Results
75.7% state fidelity is obtained after correcting errors on all three qubits, while measured syndrome fidelities range from 61.2% to 81.3%.
Takeaways & Limitations
The protected subspace recovers from any single spurious y rotation, and the measured ancilla states encode the four error syndromes as expected.
Takeaways & Limitations
The three-qubit phase-gate procedure leaves the non-nearest-neighbor phase φ101 uncorrected because the ancillas are reset and their final state does not matter.
Abstract
from arXiv · showhide
Quantum computers promise to solve certain problems exponentially faster than possible classically but are challenging to build because of their increased susceptibility to errors. Remarkably, however, it is possible to detect and correct errors without destroying coherence by using quantum error correcting codes [1]. The simplest of these are the three-qubit codes, which map a one-qubit state to an entangled three-qubit state and can correct any single phase-flip or bit-flip error of one of the three qubits, depending on the code used [2]. Here we demonstrate both codes in a superconducting circuit by encoding a quantum state as previously shown [3,4], inducing errors on all three qubits with some probability, and decoding the error syndrome by reversing the encoding process. This syndrome is then used as the input to a three-qubit gate which corrects the primary qubit if it was flipped. As the code can recover from a single error on any qubit, the fidelity of this process should decrease only quadratically with error probability. We implement the correcting three-qubit gate, known as a conditional-conditional NOT (CCNot) or Toffoli gate, using an interaction with the third excited state of a single qubit, in 63 ns. We find 85\pm1% fidelity to the expected classical action of this gate and 78\pm1% fidelity to the ideal quantum process matrix. Using it, we perform a single pass of both quantum bit- and phase-flip error correction with 76\pm0.5% process fidelity and demonstrate the predicted first-order insensitivity to errors. Concatenating these two codes and performing them on a nine-qubit device would correct arbitrary single-qubit errors. When combined with recent advances in superconducting qubit coherence times [5,6], this may lead to scalable quantum technology.
Hamiltonian parameters
The system is modeled as four transmon qubits coupled to a microwave cavity through a Tavis–Cummings Hamiltonian. Its parameters include cavity and qubit frequencies, coupling strengths, and flux-dependent Josephson energies.
- Hamiltonian parameters: The Tavis–Cummings Hamiltonian describes four transmon qubits coupled to a single microwave cavity mode.The model contains cavity energy, qubit transition energies, and qubit–cavity coupling terms.
- Hamiltonian parameters: The Hamiltonian uses qubit transition frequencies ω⁽q⁾₀ⱼ and coupling elements g⁽q⁾ⱼₖ, with couplings determined by bare qubit–cavity strengths and matrix elements.The transition frequencies and coupling matrix elements depend on each transmon’s charging and Josephson energies.
- Hamiltonian parameters: The Josephson energy varies with applied flux as E_Jq = E^max_Jq |cos(πΦq/Φ0)|, while flux-voltage relations include crosstalk and offsets.Here Φq is the flux through the transmon SQUID loop and Φ0 is the flux quantum.
- Hamiltonian parameters: Spectroscopy and transmission measurements give ωc/2π = 9.070 GHz, E^max_Jq/h = {33, 35, 26, 57} GHz, and gq/2π ≈220 MHz.The Josephson-energy values are listed for Q1–Q4.
- Hamiltonian parameters: The measured qubit lifetimes for Q1–Q3 are T1 = (1.3, 0.9, 0.7) µs, with coherence times T*2 = (0.5, 0.6, 1.3) µs.These values characterize the device used for the three-qubit operations.
Qubit rotations and gate calibration
The experiment calibrates arbitrary single-qubit rotations and multi-qubit phases using microwave pulses, Ramsey measurements, and compensating z rotations. These calibrated operations support the phase-flip correction protocol and its implementation context.
- Qubit rotations and gate calibration: Arbitrary x- and y-axis rotations use pulse-shaped resonant microwave tones, while z-axis rotations change the reference phase of later pulses.Flux-induced single-qubit dynamical phases are measured and canceled with z rotations.
- Qubit rotations and gate calibration: Two- and three-qubit phases are calibrated with Ramsey experiments that compare phase accumulation for control qubits in ground and excited states.For the Q2–Q3 interaction, Q3 is prepared along the y-axis and the phase difference is measured with Q2 in each state.
- Qubit rotations and gate calibration: The phase-flip correction protocol entangles two ancillas with the primary qubit, applies a π/2 primary-qubit rotation, induces z-axis errors, and reverses encoding to decode the syndrome.A CCNot operation then corrects the primary qubit when the syndrome indicates it was flipped.
- Qubit rotations and gate calibration: The protocol’s phase-flip probability is p = sin2(θ/2), and its protected-process fidelity is fit as f = (0.76 ± 0.005) −(1.46 ± 0.03)p2 + (0.72 ± 0.03)p3.The best-fit linear coefficient is 0.03±0.06 when a linear term is allowed.
DETAILS OF THREE-QUBIT PHASE GATE
The three-qubit phase gate is engineered through avoided crossings involving computational and non-computational states, then converted into a CCNot gate with single-qubit rotations. The design leaves one difficult two-qubit phase uncontrolled while calibrating the relevant phases and verifying computational-state action.
- DETAILS OF THREE-QUBIT PHASE GATE: The three-qubit phase arises from an effective interaction between |111⟩ and |003⟩ mediated by |102⟩.The direct transition is first-order prohibited because |111⟩ and |003⟩ differ by two excitations.
- DETAILS OF THREE-QUBIT PHASE GATE: The gate’s phases are organized into three one-qubit, three two-qubit, and one three-qubit phase, for seven unique phases overall.The intended correction scheme requires φ011 and φ110 to be set to zero, while φ101 remains uncorrected.
- DETAILS OF THREE-QUBIT PHASE GATE: The design leaves φ101 uncorrected because it is the most difficult non-nearest-neighbor interaction to control, making Q2 the target and Q1 and Q3 the ancillas.The ancillas’ final state does not matter because errors are transferred to them and they are subsequently reset.
- DETAILS OF THREE-QUBIT PHASE GATE: A CCNot gate is formed by placing π/2 and −π/2 pulses on Q2 before and after the phase gate, swapping |101⟩ and |111⟩ while leaving other computational states unchanged.The two pulses add for states with both ancillas excited and cancel otherwise.
- DETAILS OF THREE-QUBIT PHASE GATE: All relevant phases are controlled to one degree or better, implying a quantum phase fidelity above 99% for the six relevant phases.The control accuracy is limited by the arbitrary waveform generator’s voltage resolution.
BIT-FLIP ERROR CORRECTION
The bit-flip code encodes a qubit in an entangled three-qubit state, projects bit-flip syndromes into two ancillas, and uses a CCNot gate to reverse a primary-qubit flip. Single-error correction is demonstrated experimentally, with finite lifetimes limiting fidelity.
- BIT-FLIP ERROR CORRECTION: The bit-flip code encodes a quantum state in a three-qubit entangled state using CPhase gates and single-qubit rotations.The bit- and phase-flip codes differ only in the rotations applied after entanglement.
- BIT-FLIP ERROR CORRECTION: The protected subspace can recover from any single spurious y rotation on any of the three qubits.Partial rotations are interpreted as varying the probability of a full bit flip.
- BIT-FLIP ERROR CORRECTION: The two ancillas encode the error syndrome, becoming jointly excited if and only if the primary qubit has undergone a bit flip that the CCNot gate reverses.For finite rotations, the ancillas occupy a superposition of error and no-error syndromes, allowing coherent correction.
- BIT-FLIP ERROR CORRECTION: 75.7% centered state fidelity is measured after correction across errors on all three qubits.Qubit decay and differing excitation levels produce oscillations and displace the curves below ideal unit fidelity.
- BIT-FLIP ERROR CORRECTION: Ancilla syndrome fidelities are 81.3%, 69.7%, 73.1%, and 61.2% for no error and errors on Q3, Q1, and Q2, respectively.These measurements encode the four possible full bit-flip errors in the two-qubit ancilla subspace.