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arXiv:quant-ph/0211167

Maximum speed of quantum gate operation

Lev B. Levitin, Tommaso Toffoli, Zachary Walton

Published Nov 26, 2002• 0 citations
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Abstract

We consider a quantum gate, driven by a general time-dependent Hamiltonian, that complements the state of a qubit and then adds to it an arbitrary phase shift. It is shown that the minimum operation time of the gate is tau = (h/4E)(1+2 theta/pi), where h is Planck's constant, E is the average over time of the quantum-mechanical average energy, and theta is the phase shift modulo pi.

DOI
10.48550/arxiv.quant-ph/0211167

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We consider a quantum gate, driven by a general time-dependent Hamiltonian, that complements the state of a qubit and then adds to it an arbitrary phase shift. It is shown that the minimum operation time of the gate is tau = (h/4E)(1+2 theta/pi), where h is Planck's constant, E is the average over time of the quantum-mechanical average energy, and theta is the phase shift modulo pi.
Abstract The T gate (T operator) is a single-qubit quantum gate that applies a phase shift of π/4\pi/4π/4 around the ZZZ-axis of the Bloch sphere. It modifies the phase of the qubit’s state without changing its amplitude. This phase rotation is crucial for implementing complex quantum algorithms and precise control of quantum states in computation. The T gate plays an essential role in universal quantum computation, enabling the construction of arbitrary single-qubit rotations when combined with other gates. It is frequently used in error correction schemes, phase manipulation, and as a building block in quantum circuits that require fine-grained phase control.
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We consider the implementation of two-qubit gates when the physical systems used to realize the qubits are weakly anharmonic and therefore possess additional quantum states in the accessible energy range. We analyze the effect of the additional quantum states on the maximum achievable speed for quantum gates in the qubit state space. By calculating the minimum gate time using optimal control theory, we find that higher energy levels can help make two-qubit gates significantly faster than the reference value based on simple qubits. This speedup is a result of the higher coupling strength between higher energy levels. We then analyze the situation where the pulse optimization algorithm avoids pulses that excite the higher levels. We find that in this case the presence of the additional states can lead to a significant reduction in the maximum achievable gate speed. We also compare the optimal control gate times with those obtained using the cross-resonance/selective-darkening gate protocol. We find that the latter, with some parameter optimization, can be used to achieve a relatively fast implementation of the CNOT gate. These results can help the search for optimized gate implementations in realistic quantum computing architectures, such as those based on superconducting qubits. They also provide guidelines for desirable conditions on anharmonicity that would allow optimal utilization of the higher levels to achieve fast quantum gates.
A relationship between the phase factor of a quantum gate, the layout of energy levels of its effective Hamiltonian, and the implementation time of the gate is demonstrated. By an example of the direct and inverse quantum Fourier transforms (QFT) gates for a qutrit represented by a quadrupole nucleus with spin I = 1, as well as for a system of two qubits ( I = 1/2), effective Hamiltonians and minimum implementation times corresponding to different global phases are obtained. Implementation schemes are proposed for these Hamiltonians by the nuclear magnetic resonance (NMR) technique with the use of sequences of radio-frequency (RF) pulses separated by intervals of free evolution. Analytic results for the minimum times of gates are in agreement with the results obtained by numerical optimization methods. The phase considered is divided into dynamic and geometric parts.
We present a detailed analysis of the time evolution of a charge qubit realized in the lowest bound states of a double well potential. A mechanism for quantum gate operations on this qubit is modeled by directly solving the corresponding time-dependent Schrodinger equation. We demonstrate that the mechanism induces complicated non-adiabatic time evolution of the qubit system. With ideal potentials and instantaneous switching, this mechanism is shown to be capable of carrying out a NOT gate operation, and rotations by pi/4 and 3 pi/4. Effects of non-ideal potentials and smooth switching are also examined.