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arXiv:1303.0297

Single-qubit gates in two steps with rotation axes in a single plane

Yun-Pil Shim, Jianjia Fei, Sangchul Oh, Xuedong Hu, Mark G. Friesen

Published Mar 1, 2013• 7 citations
arXivHTMLPDF

Abstract

Any single-qubit unitary operation or quantum gate can be considered a rotation. Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes, and up to three rotation steps. Here we show that, if the rotation axes can be tuned arbitrarily in a fixed plane, then two rotation steps are sufficient for implementing a single-qubit gate, and one rotation step is sufficient for implementing a state transformation. The results are relevant for "exchange-only" logical qubits encoded in three-spin blocks, which are important for universal quantum computation in decoherence free subsystems and subspaces.

DOI
10.48550/arxiv.1303.0297

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Any single-qubit unitary operation or quantum gate can be considered a rotation. Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes, and up to three rotation steps. Here we show that, if the rotation axes can be tuned arbitrarily in a fixed plane, then two rotation steps are sufficient for implementing a single-qubit gate, and one rotation step is sufficient for implementing a state transformation. The results are relevant for "exchange-only" logical qubits encoded in three-spin blocks, which are important for universal quantum computation in decoherence free subsystems and subspaces.
Typical experimental implementations of single-qubit gates involve two or three fixed rotation axes and up to three rotation steps. In this work, we prove that if the rotation axes can be tuned arbitrarily in a fixed plane, then two rotation steps are sufficient for implementing a single-qubit gate, and one rotation step is sufficient for implementing a state transformation. As concrete examples, we demonstrate two-step single-qubit gate implementations in two different physical qubit systems: (i) a transmon superconducting qubit coupled to an external microwave drive, such as a transmission line; and (ii) a quantum-dot based exchange-only qubit encoded in a three-spin block. These results provide a significant speedup for many common gate implementations, such as Rabi oscillations with phase control.
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.
Universal gate sets for quantum computation, when single and two qubit operations are accessible, include both Hermitian and non-Hermitian gates. Here we utilize the fact that any single-qubit operator may be implemented as two Hermitian gates, and thus a purely Hermitian universal set is possible. This implementation can be used to prepare high fidelity single-qubit states in the presence of amplitude errors, and helps to achieve a high fidelity single-qubit gate decomposition using four Hermitian gates. An implementational convenience can be that non-identity single-qubit Hermitian gates are equivalent to π rotations up to a global phase. We show that a gate set comprised of π rotations about two fixed axes, along with the CNOT gate, is universal for quantum computation. Moreover, we show that two π rotations can transform the axis of any multi-controlled unitary, a special case being a single CNOT sufficing for any controlled π rotation. These gates simplify the process of circuit compilation in view of their Hermitian nature. We exemplify by designing efficient circuits for a variety of controlled gates, and achieving a CNOT count reduction for the four-controlled Toffoli gate in LNN-restricted qubit connectivity.
In recent investigations, Ozawa showed that the angular momentum conservation laws led to a precision limit on quantum computing in single-spin qubit systems and suggested the importance of encoding of a logical qubit by multiple-spin qubits for circumventing this difficulty. Subsequently, Lidar pointed out that the universal encoding on decoherence-free (noiseless) subspaces or subsystems can be used effectively for this purpose. In this paper, we investigate the three-qubit decoherence-free subsystem proposed by Knill, Laflamme, and Viola to construct error-free quantum logic gates in the serial mode satisfying the angular-momentum conservation law. This paper shows that the class of quantum gates defined by the 19-gate sequence with the repetition and correlation symmetries precisely coincides with the class of controlled-UUU operations up to local equivalence. As a result, any controlled rotation, frequently used in quantum Fourier transforms, can be performed by the 26-gate sequence, the same as the controlled-NOT operation.
The Toffoli gate is an important universal quantum gate, and will alongside the Clifford gates be available in future fault-tolerant quantum computing hardware. Many quantum algorithms rely on performing arbitrarily small single-qubit rotations for their function, and these rotations may also be used to construct any unitary from a limited (but universal) gate set. How to carry out such rotations is then of significant interest. In this work, we evaluate the performance of a recently proposed single-qubit rotation algorithm using the Clifford plus Toffoli gate set by implementation of a one-shot version on both a real and a simulated quantum computer. We test the algorithm under various simulated noise levels using a per-qubit depolarizing error noise model and examine how the probabilities and process fidelities are affected. We then conduct live runs and find that the results reasonably match the simulated results. We also attempt to model the hardware noise by combining a number of noise models, matching the results to results of the live runs to approximate the hardware noise. Our results suggest that the algorithm will perform well for up to 1 % noise under the noise models we chose. We further posit the use of our algorithm as a benchmark for quantum processing units, given that it has a low complexity that is easy to fine-tune in small steps. We provide details for how to do this.
Optimal implementation of quantum gates is crucial for realization of quantum computation. We slightly modify the Khaneja-Glaser decomposition (KGD) for n-qubits and give a new Cartan subalgbra in the second step of the decomposition. Based on this modified KGD, we investigate the realization of three-qubit logic gate and obtain the result that a general three-qubit quantum logic gate can be implemented using at most 73 one-qubit gates rotations with respect to the y and z axes and 26 CNOT gates.