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arXiv:quant-ph/0305040 · Physical Review A

Fast nonadiabatic two-qubit gates for the Kane quantum computer

Charles D. Hill, Hsi‐Sheng Goan

Published Jul 22, 2003• 27 citations
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Abstract

In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.

DOI
10.1103/physreva.68.012321

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arXiv:2109.01469•Physical Review A•Apr 25, 2022•16 citations
In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.
We propose and find pulse sequences for non‐adiabatic two‐qubit gates (CNOT, swap and controlled Z gates) for the Kane silicon‐based electron mediated nuclear spin quantum computer architecture. These quantum gates are simpler, with higher fidelity and faster than existing proposals. Any two qubit gate may be implemented using similar pulse sequences. In this paper, we investigate the effect of decoherence on the operation of these quantum gates and some simple algorithms for a variety of dephasing rates. Our model shows that conservative estimates of the dephasing rates in the Kane architecture give gate fidelities comparable to that required for fault tolerant quantum computing.
The benefit of exchange-only qubits compared to other spin-qubit types is the universal control using only voltage-controlled exchange interactions between neighboring spins. As a compromise, qubit operations have to be constructed from nonorthogonal rotation axes of the Bloch sphere and result in rather long pulsing sequences. This paper aims to develop a faster implementation of single-qubit gates using simultaneous exchange pulses and manifests their potential for the construction of two-qubit gates. We introduce pulse sequences in which single-qubit gates could be executed faster and show that subsequences on three spins in two-qubit gates could be implemented in fewer steps. Our findings can particularly speed up gate sequences for realistic idle times between sequential pulses and we show that this advantage increases with more interconnectivity of the quantum dots. We further demonstrate how a phase operation can introduce a relative phase between the computational and some of the leakage states, which can be advantageous for the construction of two-qubit gates. In addition to our theoretical analysis, we experimentally demonstrate and characterize a simultaneous exchange implementation of X rotations in a Si Ge quantum dot device and compare to the state of the art with sequential exchange pulses.
We design efficient controlled-rotation gates with arbitrary angle acting on three-spin encoded qubits for exchange-only quantum computation. Two pulse sequence constructions are given. The first is motivated by an analytic derivation of the efficient Fong-Wandzura sequence for an exact cnot gate. This derivation, briefly reviewed here, is based on elevating short sequences of swap pulses to an entangling two-qubit gate. To go beyond cnot, we apply a similar elevation to a modified short sequence consisting of swaps and one pulse of arbitrary duration. This results in two-qubit sequences that carry out controlled-rotation gates of arbitrary angle. The second construction streamlines a class of arbitrary cphase gates established earlier. Both constructions are based on building two-qubit sequences out of subsequences with special properties that render each step of the construction analytically tractable.
The method of iterated resolvents is used to obtain an effective Hamiltonian for neighboring qubits in the Kane solid-state quantum computer. In contrast to the adiabatic gate processes inherent in the Kane proposal we show that free evolution of the qubit-qubit system, as generated by this effective Hamiltonian, combined with single-qubit operations, is sufficient to produce a controlled-NOT gate. Thus the usual set of universal gates can be obtained on the Kane quantum computer without the need for adiabatic switching of the controllable parameters as prescribed by Kane [Nature (London) 393, 133 (1998)]. Both the fidelity and the gate time of this nonadiabatic controlled-NOT gate are determined by numerical simulation.
Nonadiabatic holonomic gates provide a promising way to realize high-fidelity quantum gates because of their robustness and high-speed implementation. Compared with quantum computation that consists of many nonadiabatic holonomic one-qubit and two-qubit gates, the nonadiabatic holonomic multiqubit gate, which can reduce the number of quantum gates, is more efficient to construct simpler and faster complicated quantum computation. We propose the simplest scheme to realize nonadiabatic holonomic three-qubit controlled gates. On the one hand, the parameters we set to realize the three-qubit controlled gates are all time independent, which makes it easy to control and operate the quantum systems. On the other hand, our scheme avoids the combination of the elementary gates and realizes the three-qubit controlled gates only by one-shot implementation.
We consider the implementation of two-qubit gates when the physical systems used to realize the qubits possess additional quantum states in the accessible energy range. We use optimal control theory to determine the maximum achievable gate speed for two-qubit gates in the qubit subspace of the many-level Hilbert space, and we analyze the effect of the additional quantum states on the gate speed. We identify two competing mechanisms. On one hand, higher energy levels are generally more strongly coupled to each other. Under suitable conditions, this stronger coupling can be utilized to make two-qubit gates significantly faster than the reference value based on simple qubits. On the other hand, a weak anharmonicity constrains the speed at which the system can be adequately controlled: according to the intuitive picture, faster operations require stronger control fields, which are more likely to excite higher levels in a weakly anharmonic system, which in turn leads to faster decoherence and uncontrolled leakage outside the qubit space. To account for this constraint, we modify the pulse optimization algorithm to avoid pulses that lead to appreciable population of the higher levels. In this case, we find that the presence of the higher levels can lead to a significant reduction in the maximum achievable gate speed. We also compare the optimal-control gate speeds with those obtained using the cross-resonance or selective-darkening gate protocol. We find that this protocol, with some parameter optimization, can be used to achieve a relatively fast implementation of the controlled-NOT gate. These results can help the search for optimized gate implementations in realistic quantum computing architectures, such as those based on superconducting circuits. They also provide guidelines for desirable conditions on anharmonicity that would allow optimal utilization of the higher levels to achieve fast quantum gates.