Also known as NMR · ESR · EPR · MRI · NV center · nitrogen-vacancy · spin qubits · quantum dots
Spin systems are where quantum control became an engineering discipline. In nuclear magnetic resonance (NMR),
electron spin resonance (ESR/EPR) and MRI, radio-frequency and microwave pulses have manipulated spins for decades.
Echoes, composite pulses and adiabatic sweeps all come from this tradition [1, 2, 3]. Today the family also includes solid-state spins: colour centres such as the nitrogen-vacancy
(NV) centre in diamond, and electron spins in semiconductor quantum dots and donors.
Magnetic resonance
The first quantum processors. NMR provided the earliest testbed for quantum algorithms, with up to seven
qubits, and developed many of the pulse and tomography techniques other platforms now use [4].
Bounds and optimal sequences. Unitary dynamics bounds how much signal any pulse sequence can transfer between
spin states, and optimal sequences reach those bounds [5]. Time-optimal control of coupled spins
was solved with geometric methods [6]. See Geometric & time-optimal control.
GRAPE and broadband pulses. GRAPE was introduced to design NMR pulse sequences [7]. Optimal
control gives excitation and inversion pulses that work over wide ranges of frequency offsets and field strengths
[8, 9]. Shaped optimal-control pulses also increased the excitation
bandwidth in pulsed EPR [10]. See GRAPE and Robust & ensemble optimal control.
Imaging. Pulse design for MRI was among the first engineering applications of optimal control
[11].
NV centres and other colour centres
The NV centre has a spin that can be initialized and read out optically at room temperature
[12]. It is a leading platform for nanoscale sensing [13, 14].
Control tasks include:
Gates and entanglement. Optimal control of two nearby NV centres gave single-spin gates with fidelity of about
0.99 and, combined with decoupling, entanglement between them [15].
Nuclear spins. Decoupling sequences on the electron spin select and control individual weakly coupled
\(^{13}\)C nuclear spins [16].
Robust sensing. Smooth optimal control made wide-field magnetometry robust to inhomogeneous control amplitudes
and detunings [17]. A cost function built for sensing improved sensitivity to time-varying
fields [18]. See Control for sensing & metrology and the review [19].
Semiconductor spin qubits
Electron spins in quantum dots are controlled electrically, through gate voltages that tune tunnelling and
exchange, or with microwaves [20, 21]. Coherent manipulation of two coupled spins
established singlet–triplet qubits [22]. Exchange-based gates require pulse optimization against
charge noise, nuclear-spin noise and leakage. With careful pulse optimization and closed-loop tuning, single-qubit
gates of singlet–triplet qubits in GaAs reached 99.5% fidelity with 0.13% leakage [23].
Silicon processors have crossed the surface-code threshold, with all gate fidelities above 99.5%
[24]. See Quantum gate synthesis and Closed-loop & in-situ optimization.
Limitations
In ensembles, inhomogeneity of fields and couplings is unavoidable, so robustness costs pulse length.
Electrical control of quantum dots relies on accurate, device-specific models of gate voltages and couplings.
Key papers
Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen and Steffen J. GlaserOptimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithmsJournal of Magnetic Resonance172, 296-305 (2005)doi
GRAPE, introduced for designing NMR pulse sequences in coupled spin systems.
L. M. K. Vandersypen and I. L. ChuangNMR techniques for quantum control and computationRev. Mod. Phys.76, 1037-1069 (2005)doi
Review of the NMR pulse and tomography techniques that served as the first quantum-computing testbed.
S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, et al.Unitary Control in Quantum Ensembles: Maximizing Signal Intensity in Coherent SpectroscopyScience280, 421-424 (1998)doi
Unitary bounds on spin transformations and pulse sequences that maximize signal in coherent spectroscopy.
Florian Dolde, Ville Bergholm, Ya Wang, Ingmar Jakobi, Boris Naydenov, et al.High-fidelity spin entanglement using optimal controlNat Commun5, 3371 (2014)doi
Optimal control of two NV centres, with high-fidelity single-spin gates and entanglement between them.
Pascal Cerfontaine, Tim Botzem, Julian Ritzmann, Simon Sebastian Humpohl, Arne Ludwig, et al.Closed-loop control of a GaAs-based singlet-triplet spin qubit with 99.5% gate fidelity and low leakageNat Commun11, 4144 (2020)doi
Optimized, closed-loop-tuned exchange gates for singlet–triplet qubits in GaAs at 99.5% fidelity.
Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol and Jason R. PettaSemiconductor spin qubitsRev. Mod. Phys.95, 025003 (2023)doi
Review of semiconductor spin qubits — single spins, donors, singlet–triplet and exchange-only qubits.
Software
Spinach — spin dynamics simulation for magnetic resonance, with optimal control (GRAPE) built in
References
E. L. HahnSpin EchoesPhys. Rev.80, 580-594 (1950)doi
Malcolm H. LevittComposite pulsesProgress in Nuclear Magnetic Resonance Spectroscopy18, 61-122 (1986)doi
Michael Garwood and Lance DelaBarreThe Return of the Frequency Sweep: Designing Adiabatic Pulses for Contemporary NMRJournal of Magnetic Resonance153, 155-177 (2001)doi
L. M. K. Vandersypen and I. L. ChuangNMR techniques for quantum control and computationRev. Mod. Phys.76, 1037-1069 (2005)doi
S. J. Glaser, T. Schulte-Herbrüggen, M. Sieveking, O. Schedletzky, N. C. Nielsen, et al.Unitary Control in Quantum Ensembles: Maximizing Signal Intensity in Coherent SpectroscopyScience280, 421-424 (1998)doi
Navin Khaneja, Roger Brockett and Steffen J. GlaserTime optimal control in spin systemsPhys. Rev. A63, 032308 (2001)doi
Navin Khaneja, Timo Reiss, Cindie Kehlet, Thomas Schulte-Herbrüggen and Steffen J. GlaserOptimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithmsJournal of Magnetic Resonance172, 296-305 (2005)doi
Thomas E Skinner, Timo O Reiss, Burkhard Luy, Navin Khaneja and Steffen J GlaserApplication of optimal control theory to the design of broadband excitation pulses for high-resolution NMRJournal of Magnetic Resonance163, 8-15 (2003)doi
Kyryl Kobzar, Thomas E. Skinner, Navin Khaneja, Steffen J. Glaser and Burkhard LuyExploring the limits of broadband excitation and inversion pulsesJournal of Magnetic Resonance170, 236-243 (2004)doi
Philipp E. Spindler, Yun Zhang, Burkhard Endeward, Naum Gershernzon, Thomas E. Skinner, et al.Shaped optimal control pulses for increased excitation bandwidth in EPRJournal of Magnetic Resonance218, 49-58 (2012)doi
Steven Conolly, Dwight Nishimura and Albert MacovskiOptimal Control Solutions to the Magnetic Resonance Selective Excitation ProblemIEEE Trans. Med. Imaging5, 106-115 (1986)doi
Marcus W. Doherty, Neil B. Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup and Lloyd C.L. HollenbergThe nitrogen-vacancy colour centre in diamondPhysics Reports528, 1-45 (2013)doi
J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, et al.High-sensitivity diamond magnetometer with nanoscale resolutionNature Phys4, 810-816 (2008)doi
C. L. Degen, F. Reinhard and P. CappellaroQuantum sensingRev. Mod. Phys.89, 035002 (2017)doi
Florian Dolde, Ville Bergholm, Ya Wang, Ingmar Jakobi, Boris Naydenov, et al.High-fidelity spin entanglement using optimal controlNat Commun5, 3371 (2014)doi
T. H. Taminiau, J. J. T. Wagenaar, T. van der Sar, F. Jelezko, V. V. Dobrovitski and R. HansonDetection and Control of Individual Nuclear Spins Using a Weakly Coupled Electron SpinPhys. Rev. Lett.109, 137602 (2012)doi
Tobias Nöbauer, Andreas Angerer, Björn Bartels, Michael Trupke, Stefan Rotter, et al.Smooth Optimal Quantum Control for Robust Solid-State Spin MagnetometryPhys. Rev. Lett.115, 190801 (2015)doi
F. Poggiali, P. Cappellaro and N. FabbriOptimal Control for One-Qubit Quantum SensingPhys. Rev. X8, 021059 (2018)doi
Phila Rembold, Nimba Oshnik, Matthias M. Müller, Simone Montangero, Tommaso Calarco and Elke NeuIntroduction to quantum optimal control for quantum sensing with nitrogen-vacancy centers in diamondAVS Quantum Science2, 024701 (2020)doi
Daniel Loss and David P. DiVincenzoQuantum computation with quantum dotsPhys. Rev. A57, 120-126 (1998)doi
Guido Burkard, Thaddeus D. Ladd, Andrew Pan, John M. Nichol and Jason R. PettaSemiconductor spin qubitsRev. Mod. Phys.95, 025003 (2023)doi
J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird, A. Yacoby, et al.Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum DotsScience309, 2180-2184 (2005)doi
Pascal Cerfontaine, Tim Botzem, Julian Ritzmann, Simon Sebastian Humpohl, Arne Ludwig, et al.Closed-loop control of a GaAs-based singlet-triplet spin qubit with 99.5% gate fidelity and low leakageNat Commun11, 4144 (2020)doi
Xiao Xue, Maximilian Russ, Nodar Samkharadze, Brennan Undseth, Amir Sammak, et al.Quantum logic with spin qubits crossing the surface code thresholdNature601, 343-347 (2022)doi
Recent papers
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The use of reinforcement learning (RL) in the design of quantum control strategies for pulse level control of NISQ era quantum devices is attracting growing interest. Model-free RL, not requiring preexisting knowledge of the physical system, acts as a black-box optimization tool well suited for noisy devices. In this work, we provide a simulation-based analysis of the performance of model-free RL for state preparation and qubit gate design in a challenging setup: indirect control of NV nuclear spins in a spin-bath. In particular, the spin-bath is modeled via the generalized cluster correlation expansion accounting for magnetic impurities and intra-bath interactions. We analyze the performance of the RL agents by performing simulations with minimally observable environments and noisy, sparse rewards as introduced in (Phys. Rev. X 12, 011059), and compare them to other state-of-the-art black-box optimization tools.
Progress in quantum hardware requires a growing volume of experiments across diverse devices, straining the capacity of human experts. Automation remains challenging because device differences and drift can render workflows developed on one processor unsuitable for another. Here we present an agent-assisted experimental strategy for conducting experiments across three-qubit nuclear magnetic resonance processors. Developed on one processor and applied sequentially to four others, the strategy uses an agent to assess experimental evidence, select subsequent measurements and adapt procedures, with operators providing instrument preparation and physical intervention. Using three-qubit Grover state preparation as a common task, each deployment combined device-specific calibration, control-pulse optimization and measurement. The experimental strategy was refined through sequential deployment, and all four processors ultimately produced outputs dominated by their respective marked states. The marked-state populations of the reconstructed dominant eigenstates ranged from \(0.910\) to \(0.959\), with relative deviations below \(3.8\%\) from the ideal value of 0.945. These results demonstrate that an agent-assisted experimental strategy can be reused and refined across processors to realize a common quantum task, providing a basis for scaling experimental practice beyond individual devices.
Encoding quantum information in $d$-dimensional systems (qudits) with $d>2$ offers appealing new directions for resource-efficient quantum computing. However, protecting the coherence of nonclassical qudit resource states remains an open challenge. Dynamical decoupling (DD) is a well-established technique for extending coherence, but its application to qudits is hindered by the quadratic scaling of the decoupling sequence overhead with the dimension $d$. As a result, demonstrations of qudit DD have so far been limited to low-dimensional qudits. Here, we show that noise-biased qudits enable a nontrivial reduction of this overhead, for example, from quadratic to linear. We perform qudit DD on a model $d=8$ qudit, the nuclear spin of an antimony donor in silicon. We benchmark DD performance on highly nonclassical states such as uniform superpositions, cat states, and Haar random states and measure coherence improvements up to a factor $5$ to $10$. For a spin-cat error correction code, continuous DD extends the coherence time of the logical superposition $|+\rangle_L$ to $225$~ms, compared to $19$~ms under free evolution and $85$~ms with a Hahn echo. This improvement shows that DD sequences tailored to a system's noise bias can significantly enhance the lifetime of encoded logical qubits.
The nuclear spin of a phosphorus atom in silicon is considered as a promising quantum information carrier for quantum computing due to its long coherence time. Coherent control of the nuclear spin is typically realized using nuclear magnetic resonance (NMR). In this work, we theoretically propose that with the aid of hyperfine interactions and magnetic field gradients, fast electric-dipole-induced nuclear electric resonance (EDNER) can be achieved. The stronger EDNER (compared with NMR) arises from the combined effects of hyperfine interaction between the nuclear and electron spins and coupling between the electron spin and charge degrees of freedom mediated by magnetic field gradients in this scheme. By operating at specific working points, the nuclear-spin splitting becomes less sensitive to electrical noise, enabling long coherence times and high-fidelity quantum operations. All elctrical, addressable control of nuclear spins paves the way for scalable silicon-based quantum computing.
Nitrogen-vacancy (NV) centers in diamond are promising quantum sensors for AC magnetic fields. However, conventional dynamical-decoupling-based magnetometry becomes technically demanding at high frequencies because of the requirement for increasingly short and high-power microwave pulses. This paper describes an AC magnetometry technique based on continuous-wave optically detected magnetic resonance (CW-ODMR) measurements, which enables the detection of signals in the hundred-MHz range. By applying a strong static magnetic field transverse to the NV axis, the energy separation between the relevant electron-spin states was tuned and resonantly coupled with the target radiofrequency (RF) magnetic field, thereby forming RF-dressed states. Unlike pulsed dynamical-decoupling schemes, this approach does not require high-power microwave pulse sequences. The key feature of our method is that the detectable frequency can be tuned by controlling the bias magnetic field strength. Using this methodology, we have experimentally achieved the frequency-tunable detection of AC magnetic fields from 14.4-632 MHz. The sensitivity was 0.558 uT/Hz^(0.5) at 14.4 MHz and 24.4 uT/Hz^(0.5) at 632 MHz. We characterized the field-dependent ODMR contrast and linewidth, and found that their degradation at large transverse fields is a important factor limiting the high-frequency sensitivity. These results establish a simple CW-ODMR approach for widely tunable RF magnetic-field sensing without high-power pulsed spin control.