We study the generation of two-qudit entangling quantum logic gates using two techniques in quantum optimal control.We take advantage of both continuous, Lie algebraic control and digital, Lie group control.In both cases, the key is access to a time-dependent Hamiltonian, which can generate an arbitrary unitary matrix in the group SU(d 2 ).We find efficient protocols for creating high-fidelity entangling gates.As a test of our theory, we study the case of qudits robustly encoded in nuclear spins of alkaline earth atoms and manipulated with magnetic and optical fields, with entangling interactions arising from the wellknown Rydberg blockade.We applied this in a case study based on a d = 10 dimensional qudit encoded in the I = 9/2 nuclear spin in 87 Sr, controlled through a combination of nuclear spin resonance, a tensor ac-Stark shift, and Rydberg dressing, which allows us to generate an arbitrary symmetric entangling twoqudit gate, such as CPhase.Our techniques can be used to implement qudit entangling gates for any 2 ≤ d ≤ 10 encoded in the nuclear spin.We also studied how decoherence due to the finite lifetime of the Rydberg states affects the creation of the CPhase gate and found, through numerical optimization, a fidelity of 0.9985, 0.9980, 0.9942, and 0.9800 for d = 2, d = 3, d = 5, and d = 7, respectively.This provides a powerful platform to explore the various applications of quantum information processing of qudits, including metrological enhancement with qudits, quantum simulation, universal quantum computation, and quantum error correction.We study the generation of two-qudit entangling quantum logic gates using two techniques in quantum optimal control.We take advantage of both continuous, Lie algebraic control and digital, Lie group control.In both cases, the key is access to a time-dependent Hamiltonian, which can generate an arbitrary unitary matrix in the group SU(d 2 ).We find efficient protocols for creating high-fidelity entangling gates.As a test of our theory, we study the case of qudits robustly encoded in nuclear spins of alkaline earth atoms and manipulated with magnetic and optical fields, with entangling interactions arising from the wellknown Rydberg blockade.We applied this in a case study based on a d = 10 dimensional qudit encoded in the I = 9/2 nuclear spin in 87 Sr, controlled through a combination of nuclear spin resonance, a tensor ac-Stark shift, and Rydberg dressing, which allows us to generate an arbitrary symmetric entangling twoqudit gate, such as CPhase.Our techniques can be used to implement qudit entangling gates for any 2 ≤ d ≤ 10 encoded in the nuclear spin.We also studied how decoherence due to the finite lifetime of the Rydberg states affects the creation of the CPhase gate and found, through numerical optimization, a fidelity of 0.9985, 0.9980, 0.9942, and 0.9800 for d = 2, d = 3, d = 5, and d = 7, respectively.This provides a powerful platform to explore the various applications of quantum information processing of qudits, including metrological enhancement with qudits, quantum simulation, universal quantum computation, and quantum error correction.

arXiv:2212.08799PRX QuantumNov 29, 202330 citations

Demonstrating Experimentally the Encoding and Dynamics of an Error-Correctable Logical Qubit on a Hyperfine-Coupled Nuclear Spin Qudit

Sumin Lim, Mikhail V. Vaganov, Junjie Liu +1 more

The realization of effective quantum error correction protocols remains a central challenge in the development of scalable quantum computers. Employing high-dimensional quantum systems (qudits) can offer more hardware-efficient protocols than qubit-based approaches. Using electron-nuclear double resonance, we implement a logical qubit encoded on the four states of a I=3/2 nuclear spin hyperfine-coupled to an S=1/2 electron spin qubit; the encoding protects against the dominant decoherence mechanism in such systems, i.e., fluctuations of the quantizing magnetic field. We explore the dynamics of the encoded state both under a controlled application of the fluctuation and under natural decoherence processes. Our results confirm the potential of these proposals for practical, implementable, fault-tolerant quantum memories.The realization of effective quantum error correction protocols remains a central challenge in the development of scalable quantum computers. Employing high-dimensional quantum systems (qudits) can offer more hardware-efficient protocols than qubit-based approaches. Using electron-nuclear double resonance, we implement a logical qubit encoded on the four states of a I=3/2 nuclear spin hyperfine-coupled to an S=1/2 electron spin qubit; the encoding protects against the dominant decoherence mechanism in such systems, i.e., fluctuations of the quantizing magnetic field. We explore the dynamics of the encoded state both under a controlled application of the fluctuation and under natural decoherence processes. Our results confirm the potential of these proposals for practical, implementable, fault-tolerant quantum memories.

arXiv:2405.20827Physical Review LettersFeb 20, 202513 citations

Optimal Control and Circuit Synthesis of Quantum Gates

Hyeong-Jin Kim

Jan 1, 20210 citations

Quantum optimal control of ten-level nuclear spin qudits in Sr 87

Sivaprasad Omanakuttan, Anupam Mitra, Michael J. Martin +1 more

We study the ability to implement unitary maps on states of the $I=9/2$ nuclear spin in $^{87}\mathrm{Sr}$ , a $d=10$ dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin resonance and a tensor ac Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline-earth-metal atoms, such as $^{87}\mathrm{Sr}$ , have a very favorable figure of merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf Rabi frequency of ${\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{\ensuremath{\psi}}\ensuremath{\rangle}=0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.We study the ability to implement unitary maps on states of the $I=9/2$ nuclear spin in $^{87}\mathrm{Sr}$, a $d=10$ dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin resonance and a tensor ac Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline-earth-metal atoms, such as $^{87}\mathrm{Sr}$, have a very favorable figure of merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf Rabi frequency of ${\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{\ensuremath{\psi}}\ensuremath{\rangle}=0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.

arXiv:2106.13705Physical Review ADec 16, 202137 citations

We study the ability to implement unitary maps on states of the

I=9/2

nuclear spin in

^{87}\mathrm{Sr}

, a

d=10

dimensional (qudecimal) Hilbert space, using quantum optimal control. Through a combination of nuclear spin resonance and a tensor ac Stark shift, by solely modulating the phase of a radio-frequency magnetic field, the system is quantum controllable. Alkaline-earth-metal atoms, such as

^{87}\mathrm{Sr}

, have a very favorable figure of merit for such control due to narrow intercombination lines and the large hyperfine splitting in the excited states. We numerically study the quantum speed limit, optimal parameters, and the fidelity of arbitrary state preparation and full SU(10) maps, including the presence of decoherence due to optical pumping induced by the light-shifting laser. We also study the use of robust control to mitigate some dephasing due to inhomogeneities in the light shift. We find that with an rf Rabi frequency of ${\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ and 0.5% inhomogeneity in the the light shift we can prepare an arbitrary Haar-random state in a time $T=4.5\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{\ensuremath{\psi}}\ensuremath{\rangle}=0.9992$, and an arbitrary Haar-random SU(10) map in a time $T=24\ensuremath{\pi}/{\mathrm{\ensuremath{\Omega}}}_{\text{rf}}$ with average fidelity $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.