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arXiv:1710.04319 · Physical Review A

Optimal control of hybrid qubits: Implementing the quantum permutation algorithm

Carlos Mario Rivera Ruiz, Emanuel F. de Lima, F. F. Fanchini, Victor Lopez‐Richard, L. K. Castelano

Published Mar 21, 2018• 11 citations

Abstract

The optimal quantum control theory is employed to determine electric pulses capable of producing quantum gates with a fidelity higher than 0.9997, when noise is not taken into account. Particularly, these quantum gates were chosen to perform the permutation algorithm in hybrid qubits in double quantum dots (DQDs). The permutation algorithm is an oracle based quantum algorithm that solves the problem of the permutation parity faster than a classical algorithm without the necessity of entanglement between particles. The only requirement for achieving the speedup is the use of a one-particle quantum system with at least three levels. The high fidelity found in our results is closely related to the quantum speed limit, which is a measure of how fast a quantum state can be manipulated. Furthermore, we model charge noise by considering an average over the optimal field centered at different values of the reference detuning, which follows a Gaussian distribution. When the Gaussian spread is of the order of 5 $\ensuremath{\mu}\text{eV}$ (10% of the correct value), the fidelity is still higher than 0.95. Our scheme also can be used for the practical realization of different quantum algorithms in DQDs.

DOI: 10.1103/physreva.97.032332

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The optimal quantum control theory is employed to determine electric pulses capable of producing quantum gates with a fidelity higher than 0.9997, when noise is not taken into account. Particularly, these quantum gates were chosen to perform the permutation algorithm in hybrid qubits in double quantum dots (DQDs). The permutation algorithm is an oracle based quantum algorithm that solves the problem of the permutation parity faster than a classical algorithm without the necessity of entanglement between particles. The only requirement for achieving the speedup is the use of a one-particle quantum system with at least three levels. The high fidelity found in our results is closely related to the quantum speed limit, which is a measure of how fast a quantum state can be manipulated. Furthermore, we model charge noise by considering an average over the optimal field centered at different values of the reference detuning, which follows a Gaussian distribution. When the Gaussian spread is of the order of 5

\ensuremath{\mu}\text{eV}

(10% of the correct value), the fidelity is still higher than 0.95. Our scheme also can be used for the practical realization of different quantum algorithms in DQDs.

Semiconductor quantum dots in silicon are promising qubits because of long spin coherence times and their potential for scalability. However, such qubits with complete electrical control and fidelities above the threshold for quantum error correction have not yet been achieved. We show theoretically that the threshold fidelity can be achieved with ac gate operation of the quantum dot hybrid qubit. Formed by three electrons in a double dot, this qubit is electrically controlled, does not require magnetic fields, and runs at gigahertz (GHz) gate speeds. We analyze the decoherence caused by

1/f

charge noise in this qubit, find parameters that minimize the charge noise dependence in the qubit frequency, and determine the optimal working points for ac gate operations that drive the detuning and tunnel coupling.

We theoretically investigate electron spin operations driven by applied electric fields in a semiconductor double quantum dot (DQD) formed in a nanowire with longitudinal potential modulated by local gating. We develop a model that describes the process of loading and unloading the DQD taking into account the overlap between the electron wave function and the leads. Such a model considers the spatial occupation and the spin Pauli blockade in a time-dependent fashion due to the highly mixed states driven by the external electric field. Moreover, we present a road map based on the quantum optimal control theory (QOCT) to find a specific electric field that performs two-qubit quantum gates on a faster timescale and with higher possible fidelity. By employing the QOCT, we demonstrate the possibility of performing within high efficiency a universal set of quantum gates ${$cnot, H, and $\mathrm{T}}$, where cnot is the controlled-not gate, H is the Hadamard gate, and T is the $\ensuremath{\pi}/8$ gate, even in the presence of the loading/unloading process and charge noise effects. Furthermore, by varying the intensity of the applied magnetic field

B

, the optimized fidelity of the gates oscillates with a period inversely proportional to the gate operation time

{t}_{f}

. This behavior can be useful to attain higher fidelity for fast gate operations (

>1

GHz) by appropriately choosing

B

and

{t}_{f}

to produce a maximum of the oscillation.

The quantum permutation algorithm provides computational speed-up over classical algorithms for determining the parity of a given cyclic permutation. For its

n

-qubit implementations, the number of required quantum gates scales quadratically with

n

due to the quantum Fourier transforms included. We show here for the

n

-qubit case that the algorithm can be simplified so that it requires only

O(n)

quantum gates, which theoretically reduces the complexity of the implementation. To test our results experimentally, we utilize IBM's 5-qubit quantum processor to realize the algorithm by using the original and simplified recipes for the 2-qubit case. It turns out that the latter results in a significantly higher success probability which allows us to verify the algorithm more precisely than the previous experimental realizations. We also verify the algorithm for the first time for the 3-qubit case with a considerable success probability by taking the advantage of our simplified scheme.

A comparison of gate fidelities between different spin qubit types defined in quantum dots and a donor under different control errors is reported. We studied five qubit types, namely the quantum dot spin qubit, the double quantum dot singlet-triplet qubit, the double quantum dot hybrid qubit, the donor qubit and the quantum dot spin-donor qubit. For each one, we derived analytical time sequences that realize single qubit rotations along the principal axis of the Bloch sphere. We estimated the effects of control errors on the gate fidelity by using a Gaussian noise model. Then we compared the gate fidelities among qubit implementations due to pulse timing errors by using a realistic set of values for the error parameters of control amplitudes.

Quantum control for error correction is critical for the practical use of quantum computers. We address quantum optimal control for single-shot multi-qubit gates by framing it as a feasibility problem for the Hamiltonian model that is then solved with standard global optimization software. Our approach yields faster high-fidelity (>99.99%) single-shot three-qubit-gate control than obtained previously, and it has also enabled us to solve the quantum-control problem for a fast high-fidelity four-qubit gate.

Implementing high-fidelity two-qubit gates in single-electron spin qubits in silicon double quantum dots is still a major challenge. In this work we employ analytical methods to design control pulses that generate high-fidelity entangling gates for quantum computers based on this platform. Using realistic parameters and initially assuming a noise-free environment, we present simple control pulses that generate cnot, cphase, and cz gates with average fidelities greater than 99.99% and gate times as short as 45 ns. Moreover, using the local invariants of the system's evolution operator, we show that a simple square pulse generates a cnot gate in less than 27 ns and with a fidelity greater than 99.99%. Last, we use the same analytical methods to generate two-qubit gates locally equivalent to

\sqrt{\mathrm{CNOT}}

and

\sqrt{\mathrm{CZ}}

that are used to implement simple two-piece pulse sequences that produce high-fidelity cnot and cz gates in the presence of low-frequency noise.