Quantum Metrology Enhanced by Repetitive Quantum Error Correction

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Quantum Metrology Enhanced by Repetitive Quantum Error Correction

Thomas Unden, Priya Balasubramanian, Daniel Louzon, Yuval Vinkler, Martin B. Plenio, Matthew Markham, Daniel Twitchen, Alastair Stacey, Igor Lovchinsky, Alexander O. Sushkov, Mikhail D. Lukin, Alex Retzker, Boris Naydenov, Liam P. McGuinness, and Fedor Jelezko

Phys. Rev. Lett. 116, 230502 – Published 9 June 2016

Abstract

We experimentally demonstrate the protection of a room-temperature hybrid spin register against environmental decoherence by performing repeated quantum error correction whilst maintaining sensitivity to signal fields. We use a long-lived nuclear spin to correct multiple phase errors on a sensitive electron spin in diamond and realize magnetic field sensing beyond the time scales set by natural decoherence. The universal extension of sensing time, robust to noise at any frequency, demonstrates the definitive advantage entangled multiqubit systems provide for quantum sensing and offers an important complement to quantum control techniques.

Received 10 February 2016

DOI:https://doi.org/10.1103/PhysRevLett.116.230502

Physics Subject Headings (PhySH)

Quantum error correction Quantum information architectures & platforms Quantum metrology

Nitrogen vacancy centers in diamond

Quantum Information, Science & Technology

Authors & Affiliations

Thomas Unden¹, Priya Balasubramanian¹, Daniel Louzon^1,3, Yuval Vinkler³, Martin B. Plenio^2,4, Matthew Markham⁵, Daniel Twitchen⁵, Alastair Stacey^5,†, Igor Lovchinsky⁶, Alexander O. Sushkov^6,‡, Mikhail D. Lukin⁶, Alex Retzker³, Boris Naydenov^1,2, Liam P. McGuinness¹, and Fedor Jelezko^1,2,*

¹Institute of Quantum Optics, Ulm University, 89081 Ulm, Germany
²Center for Integrated Quantum Science and Technology (IQST), Ulm University, 89081 Ulm, Germany
³Racah Institute of Physics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel
⁴Institute for Theoretical Physics, Ulm University, 89081 Ulm, Germany
⁵Element Six, Harwell Campus, Fermi Avenue, Didcot OX11 0QR, United Kingdom
⁶Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*fedor.jelezko@uni-ulm.de
^†Present address: Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Parkville, Melbourne VIC 3010, Australia.
^‡Present address: Department of Physics, Boston University, Boston, Massachusetts 02215, USA.

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Issue

Vol. 116, Iss. 23 — 10 June 2016

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Figure 1
The quantum system of interest and its fundamental properties. (a) Quantum register formed by a NV and a near-axial ${}^{13}C$ nucleus (auxiliary qubit) in the diamond lattice. (b) Energy level structure of the quantum register, showing the ground state spin manifold $| 0 ⟩$ , $| - 1 ⟩$ of the NV center and the spin manifold $| ↑ ⟩$ , $| ↓ ⟩$ of the ${}^{13}C$ spin. The NV-spin transitions are driven by a resonant microwave signal $Ω_{MW}$ , and the nuclear spin is driven by a radio frequency signal $Ω_{rf}$ . To resolve the energy degeneracy of the NV spin states $| \pm 1 ⟩$ and to define the NV qubit corresponding to the states $| 0 ⟩$ and $| - 1 ⟩$ , an external magnetic field of about 340 G was applied. The magnetic field is aligned to the symmetry axis of the NV center. The coupling between the NV center and nuclear spin is given by the hyperfine interaction, which in our case shows only a parallel component of 50 kHz (note that other nonaxial nuclear spins can also be employed as robust qubits). The depicted broadening of the $| - 1 ⟩$ energy level corresponds to fluctuations of the NV Larmor frequency induced by the environment. (c) Quantum state tomography performed in the subspace defined by the basis states $| 0 ⟩ | ↑ ⟩$ and $| - 1 ⟩ | ↓ ⟩$ . We estimate a fidelity of 80% to initialize the system to the state $| 0 ⟩ | ↑ ⟩ + i | - 1 ⟩ | ↓ ⟩$ , when the remaining entries of the full $6 \times 6$ density matrix are neglected. (d) Nuclear coherence under optical excitation of the NV center as measured by free induction decay of the nuclear spin. During the nuclear free evolution time, a continuous optical excitation of NV is performed. The NV was initially prepared in either the $| 0 ⟩$ or the $| - 1 ⟩$ state. After free evolution of the nuclear spin, a $π / 2$ pulse conditional on the electronic spin state $| 0 ⟩$ was applied to the nuclear spin to convert the nuclear coherence to population difference, and a projective measurement was subsequently performed.
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Figure 2
The temporal evolution of the hybrid register and the implemented QSEC scheme. (a) The QSEC protocol in a Bloch sphere representation, showing decoding (disentanglement), reset of the NV spin, and encoding. The error in this scheme is a single phase flip of the NV. No time evolution between the error and correction steps is assumed. The presented scheme is the basic correction step, which is applied repetitively (up to two times). The phase of the initial state corresponds to the signal. (b) The measurement sequence used in the experiment when QSEC is used. First, the electron and the nuclear spins were initialized via a laser pulse and single shot measurement. To encode the system in the state $| + ↓ ⟩ + | - ↑ ⟩$ , local operations are applied with microwave and radio frequency pulses. For entanglement creation with a controlled-NOT gate ( ${CNOT}_{z}$ ), the parallel hyperfine interaction between NV and ${}^{13}C$ is used. All control pulses are performed on resonance to a single NV transition corresponding to a certain nuclear spin state. After encoding, a microwave signal with a variable time duration is applied, resonant with the electronic part of the register, and coherent signal accumulation occurs. During the first error correction step, entanglement between the NV and nuclear spins is removed by a CNOT gate, then a reset of the NV is performed, and finally the entangled state is prepared for further sensing. After the second period of coherent interaction with the microwave signal, a further error correction step is applied. To read out the phase of the state $| + ↓ ⟩ + e^{i ϕ} | - ↑ ⟩$ , it is mapped onto the local phase of the nuclear spin and read out after a conditional $π / 2$ pulse by performing a nondemolition measurement. (c) The measurement sequence used in the experiment when no error correction is used. The measurement sequence follows that shown in (b), but without the intermediate and last error correction steps.
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Figure 3
The performance of QSEC against natural noise and sensitivity analysis. (a) NV Rabi oscillations based on the sequences in Figs. 2 and 2 comparing the signal when no EC is used and when two EC steps are performed. In the case where an EC was used, a slower oscillation appears. This oscillation corresponds to half of the hyperfine interaction strength (50 kHz) and is connected to the unitary evolution mediated by the hyperfine coupling within the register itself. For both experiments, the same overall measurement time was used. The solid lines correspond to fits. In the case of no EC, we used $f (x) = A e^{- γ x} \cos (ω x + ϕ) + c$ and in the case with an EC $f (x) = A_{1} e^{- Γ x} \cos (ω_{1} x + ϕ_{1}) + A_{2} e^{- γ x} \cos (ω_{2} x + ϕ_{2}) + c$ to fit the measurement data. (b) The probability distribution function (PDF) of the fit parameter $γ$ based on a Bayesian estimation. The fit parameter $γ$ corresponds to the rate of the observed decoherence with (blue line) and without (red line) QSEC. (c) A simulation of the measurement data, based on a master equation which takes the slow fluctuation of the NV Larmor frequency into account. More information about the simulation model can be found in Ref. [18]. (d) The magnetic sensitivity of the sensor as a function of the sensing time with and without QSEC. The shown sensitivities are calculated based on the estimated parameter describing the measurement data shown in (a). The dashed lines indicate the sensing duration which gives the best achievable sensitivity. The sensing duration is the time of sensing, neglecting the time needed for initialization, readout, and error correction.
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