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arXiv:2212.08799 · PRX Quantum

Qudit Entanglers Using Quantum Optimal Control

Sivaprasad Omanakuttan, Anupam Mitra, Η. Meier, Michael J. Martin, Ivan Deutsch

Published Nov 29, 2023• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.4.040333

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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.
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We study the ability to implement unitary maps on states of the I=9/2I=9/2I=9/2 nuclear spin in 87Sr^{87}\mathrm{Sr}87Sr, a d=10d=10d=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 87Sr^{87}\mathrm{Sr}87Sr, 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 $\ensuremath{\langle}{\mathcal{F}}_{U}\ensuremath{\rangle}=0.9923$.
Gate-based quantum computers typically encode and process information in two-dimensional units called qubits.Using d-dimensional qudits instead may offer intrinsic advantages, including more efficient circuit synthesis, problem-tailored encodings and embedded error correction.In this work, we design a superconducting qudit-based quantum processor wherein the logical space of transmon qubits is extended to higher-excited levels.We propose a universal gate set featuring a two-qudit cross-resonance entangling gate, for which we predict fidelities beyond 99% in the d = 4 case of ququarts with realistic experimental parameters.Furthermore, we present a decomposition routine that compiles general qudit unitaries into these elementary gates, requiring fewer entangling gates than qubit alternatives.As proof-of-concept applications, we numerically demonstrate the synthesis of SU(16) gates for noisy quantum hardware and an embedded error-correction sequence that encodes a qubit memory in a transmon ququart to protect against pure dephasing noise.We conclude that universal qudit control-a valuable extension to the operational toolbox of superconducting quantum information processing-is within reach of current transmon-based architectures and has applications to near-term and long-term hardware.
Robust quantum computation with ddd-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high-fidelity two-qudit gates. We first describe how to implement parallel single-qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth (time) of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the alkali-metal atoms 87Rb^{87}\mathrm{Rb}87Rb and 133Cs^{133}\mathrm{Cs}133Cs. Second, we provide a protocol for implementing parallelized nonlocal two-qudit gates using the assistance of entangled qubit pairs. Using known protocols for qubit entanglement purification, this offers the possibility of high-fidelity two-qudit gates.
The generation of a high-dimensional quantum entangled state provides a crucial resource for achieving more complex and diverse quantum information processing tasks. In this paper, we present two deterministic protocols designed for the generation of a four-dimensional two-qudit Bell state and three-qudit Greenberger-Horne-Zeilinger (GHZ) state assisted by cross-Kerr nonlinearity, where the four-dimensional qudit is encoded in the hybrid polarization-spatial state of the single photon to conserve quantum resources and alleviate environmental noise. The proposed protocols for generating a four-dimensional two-qudit Bell state and three-qudit GHZ state are circuit concise, high success, robust fidelity, noise resistant, and resource saving. Moreover, the four-dimensional NNN-qudit (N>3N>3N>3) GHZ states can be generated with the help of the $N\ensuremath{-}1$ generating devices of the four-dimensional Bell state. Consequently, its efficiency and fidelity decay exponentially by a factor of $N\ensuremath{-}1$. Further, the arbitrary high-dimensional qudit-encoded entangled states can be achieved with linear-optics four-dimensional single-qudit XXX gates and ZZZ gates. These protocols offer a reference for further advancements in high-dimensional state generation technology and providing insights for the practical implementations of metrological enhancement, quantum simulation, high-dimensional quantum computation, and quantum error correction.
A key ingredient for fault-tolerant quantum computers are sufficiently accurate logic gates on single and multiple qubits in the presence of decohering noise. In this thesis, we theoretically develop and experimentally demonstrate such high-fidelity quantum gates for semiconductor-based quantum computing. Specifically, we consider a qubit encoded in the singlet and one triplet state of two electron spins in GaAs. While its potential for optical coupling makes GaAs an attractive material, highly auto-correlated magnetic field noise from fluctuating nuclear spins leads to considerably lower coherence times than for Si-based devices, where nuclear spins can be removed by isotopic purification. In addition to noise, the control methods used in earlier experiments are based on unrealistic approximations. For these reasons, fidelities well above 99 %, as required by current quantum error correction (QEC) schemes, have not been obtained before this thesis. To tackle these issues, we extend the filter function formalism to describe quantum gates and processes in the presence of experimentally relevant non-Markovian noise. Since this formalism can consider all relevant noise sources in a computationally efficient manner, it can be used to find optimal control pulses by numerical optimization, leading to predicted single-qubit gate fidelities of 99.9 %. Furthermore, we deal with the considerable control challenges associated with this qubit type by experimental calibration of the optimized control pulses. To this end, we extend our previous experimental gate set calibration (GSC) routine to remove systematic gate errors on an arbitrary number of qubits. We apply the numerically optimized single-qubit control pulses to our GaAs sample and experimentally calibrate them with GSC. This procedure yields an average gate fidelity of 99.50 ± 0.04 % and a low leakage rate of 0.13 ± 0.03 % out of the computational subspace, characterized by randomized benchmarking. We also optimize realistic two-qubit control pulses, considering current control hardware as well as interqubit Coulomb and exchange coupling that cannot be fully turned off. Using measured noise spectra, we show that two-qubit gate fidelities of 99.90 % can be reached in GaAs, while 99.99 % can be achieved in Si. These results demonstrate that high-fidelity gates can be realized even in the presence of nuclear spins as in GaAs.