We find the optimal universal way of manipulating a single qubit, $|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,$ such that $(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).$ Such optimal transformations fall into two classes. For $0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\ensuremath{\alpha}=0)$ to $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to $\frac{2}{3}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity $\frac{2}{3}$ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ the fidelity is lower than $\frac{2}{3}.$ For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.We find the optimal universal way of manipulating a single qubit, $|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,$ such that $(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).$ Such optimal transformations fall into two classes. For $0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\ensuremath{\alpha}=0)$ to $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to $\frac{2}{3}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity $\frac{2}{3}$ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ the fidelity is lower than $\frac{2}{3}.$ For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.

arXiv:quant-ph/0008011Physical Review AFeb 8, 200115 citations

Average fidelity in n-qubit systems

Renán Cabrera, W. E. Baylis

arXiv:quant-ph/0701084Physics Letters AApr 2, 200727 citations

Leakage and fidelity of real qubits

Rosario Fazio, G. Massimo Palma, Emilio Sciacca +1 more

Physica B Condensed MatterJul 1, 20000 citations

QuCloud+: A Holistic Qubit Mapping Scheme for Single/Multi-programming on 2D/3D NISQ Quantum Computers

Lei Liu, Xinglei Dou

Qubit mapping for NISQ superconducting quantum computers is essential to fidelity and resource utilization. The existing qubit mapping schemes meet challenges, e.g., crosstalk, SWAP overheads, diverse device topologies, etc., leading to qubit resource underutilization and low fidelity in computing results. This article introduces QuCloud+, a new qubit mapping scheme that tackles these challenges. QuCloud+ has several new designs. (1) QuCloud+ supports single/multi-programming quantum computing on quantum chips with 2D/3D topology. (2) QuCloud+ partitions physical qubits for concurrent quantum programs with the crosstalk-aware community detection technique and further allocates qubits according to qubit degree, improving fidelity, and resource utilization. (3) QuCloud+ includes an X-SWAP mechanism that avoids SWAPs with high crosstalk errors and enables inter-program SWAPs to reduce the SWAP overheads. (4) QuCloud+ schedules concurrent quantum programs to be mapped and executed based on estimated fidelity for the best practice. Experimental results show that, compared with the existing typical multi-programming study [ 12 ], QuCloud+ achieves up to 9.03% higher fidelity and saves on the required SWAPs during mapping, reducing the number of CNOT gates inserted by 40.92%. Compared with a recent study [ 30 ] that enables post-mapping gate optimizations to further reduce gates, QuCloud+ reduces the post-mapping circuit depth by 21.91% while using a similar number of gates.Qubit mapping for NISQ superconducting quantum computers is essential to fidelity and resource utilization. The existing qubit mapping schemes meet challenges, e.g., crosstalk, SWAP overheads, diverse device topologies, etc., leading to qubit resource underutilization and low fidelity in computing results. This article introduces QuCloud+, a new qubit mapping scheme that tackles these challenges. QuCloud+ has several new designs. (1) QuCloud+ supports single/multi-programming quantum computing on quantum chips with 2D/3D topology. (2) QuCloud+ partitions physical qubits for concurrent quantum programs with the crosstalk-aware community detection technique and further allocates qubits according to qubit degree, improving fidelity, and resource utilization. (3) QuCloud+ includes an X-SWAP mechanism that avoids SWAPs with high crosstalk errors and enables inter-program SWAPs to reduce the SWAP overheads. (4) QuCloud+ schedules concurrent quantum programs to be mapped and executed based on estimated fidelity for the best practice. Experimental results show that, compared with the existing typical multi-programming study [ 12 ], QuCloud+ achieves up to 9.03% higher fidelity and saves on the required SWAPs during mapping, reducing the number of CNOT gates inserted by 40.92%. Compared with a recent study [ 30 ] that enables post-mapping gate optimizations to further reduce gates, QuCloud+ reduces the post-mapping circuit depth by 21.91% while using a similar number of gates.

arXiv:2207.14483ACM Transactions on Architecture and Code OptimizationNov 7, 202337 citations

High-fidelity entanglement of Ca+43 hyperfine clock states

Gerhard Kirchmair, Jan Benhelm, F. Zähringer +3 more

In an experiment using the odd calcium isotope $^{43}\mathrm{Ca}^{+}$ , we combine the merits of a high-fidelity entangling operation on an optical transition (optical qubit) with the long coherence times of two ``clock'' states in the hyperfine ground state (hyperfine qubit) by mapping between these two qubits. For state initialization, state detection, global qubit rotations, and mapping operations, errors smaller than 1% are achieved, whereas the entangling gate adds errors of 2.3%. Based on these operations, we create Bell states with a fidelity of 96.9(3)% in the optical qubit and a fidelity of 96.7(3)% when mapped to the hyperfine states. In the latter case, the entanglement is preserved for $96(3)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$ , exceeding the duration of a single gate operation by three orders of magnitude.In an experiment using the odd calcium isotope $^{43}\mathrm{Ca}^{+}$, we combine the merits of a high-fidelity entangling operation on an optical transition (optical qubit) with the long coherence times of two ``clock'' states in the hyperfine ground state (hyperfine qubit) by mapping between these two qubits. For state initialization, state detection, global qubit rotations, and mapping operations, errors smaller than 1% are achieved, whereas the entangling gate adds errors of 2.3%. Based on these operations, we create Bell states with a fidelity of 96.9(3)% in the optical qubit and a fidelity of 96.7(3)% when mapped to the hyperfine states. In the latter case, the entanglement is preserved for $96(3)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}$, exceeding the duration of a single gate operation by three orders of magnitude.

arXiv:0812.1976Physical Review AFeb 17, 200927 citations

\frac{1}{2}

(for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from

\frac{1}{2}

(for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to

\frac{2}{3}

(for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity

\frac{2}{3}

is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ the fidelity is lower than

\frac{2}{3}.

For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.

In an experiment using the odd calcium isotope

^{43}\mathrm{Ca}^{+}

, we combine the merits of a high-fidelity entangling operation on an optical transition (optical qubit) with the long coherence times of two ``clock'' states in the hyperfine ground state (hyperfine qubit) by mapping between these two qubits. For state initialization, state detection, global qubit rotations, and mapping operations, errors smaller than 1% are achieved, whereas the entangling gate adds errors of 2.3%. Based on these operations, we create Bell states with a fidelity of 96.9(3)% in the optical qubit and a fidelity of 96.7(3)% when mapped to the hyperfine states. In the latter case, the entanglement is preserved for

96(3)\phantom{\rule{0.3em}{0ex}}\mathrm{ms}

, exceeding the duration of a single gate operation by three orders of magnitude.