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Physical Review A

Quantum gates generated by rotationally invariant operators in a decoherence-free subsystem

Y. Kawano, Masanao Ozawa

Published Jan 30, 2006• 2 citations
Publisher

Abstract

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-$U$ 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.

DOI
10.1103/physreva.73.012339

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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.
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.
One of the most challenging problems for the realization of a scalable quantum computer is to design a physical device that keeps the error rate for each quantum processing operation low. These errors can originate from the accuracy of quantum manipulation, such as the sweeping of a gate voltage in solid state qubits or the duration of a laser pulse in optical schemes. Errors also result from decoherence, which is often regarded as more crucial in the sense that it is inherent to the quantum system, being fundamentally a consequence of the coupling to the external environment. Grouping small collections of qubits into clusters with symmetries may serve to protect parts of the calculation from decoherence. In this work, we use 4-level cores with a straightforward generalization of discrete rotational symmetry, called ω\omegaω-rotation invariance, to encode pairs of coupled qubits and universal 2-qubit logical gates. We propose a scalable scheme for universal quantum computation where cores play the role of quantum-computational transistors, or \textit{quansistors} for short. Embedding in the environment, initialization and readout are achieved by tunnel-coupling the quansistor to leads. The external leads are explicitly considered and are assumed to be the main source of decoherence. We show that quansistors can be dynamically decoupled from the leads by tuning their internal parameters, giving them the versatility required to act as controllable quantum memory units. With this dynamical decoupling, logical operations within quansistors are also symmetry-protected from unbiased noise in their parameters. We identify technologies that could implement ω\omegaω-rotation invariance. Many of our results can be generalized to higher-level ω\omegaω-rotation-invariant systems, or adapted to clusters with other symmetries.
We investigate how to carry out universal quantum computation deterministically with free electrons in decoherence-free subspace by using polarizing beam splitters, charge detectors, and single-spin rotations. Quantum information in our case is encoded in spin degrees of freedom of the electron-pairs which construct a decoherence-free subspace. We design building blocks for two noncommutable single-logic-qubit gates and a logic controlled phase gate, based on which a universal and scalable quantum information processing robust to dephasing is available in a deterministic way.
We propose a universal gate set for quantum computing with all-to-all connectivity and intrinsic robustness to bit-flip errors based on parity encoding. We show that logical controlled phase gate and R_{z} rotations can be implemented in parity encoding with single-qubit operations. Together with logical R_{x} rotations, implemented via nearest-neighbor controlled-NOT gates and an R_{x} rotation, these form a universal gate set. As the controlled phase gate requires only single-qubit rotations, the proposed scheme has advantages for several cornerstone quantum algorithms, e.g., the quantum Fourier transform. We present a method to switch between different encoding variants via partial on-the-fly encoding and decoding.
One of the most challenging problems for the realization of a scalable quantum computer is to design a physical device that keeps the error rate for each quantum processing operation low. These errors can originate from the accuracy of quantum manipulation, such as the sweeping of a gate voltage in solid state qubits or the duration of a laser pulse in optical schemes. Errors also result from decoherence, which is often regarded as more crucial in the sense that it is inherent to the quantum system, being fundamentally a consequence of the coupling to the external environment. Grouping small collections of qubits into clusters with symmetries may serve to protect parts of the calculation from decoherence. In this work, we use four-level cores with a straightforward generalization of discrete rotational symmetry, called $\ensuremath{\omega}$-rotation invariance, to encode pairs of coupled qubits and universal two-qubit logical gates. We include quantum errors as a main source of decoherence, and show that symmetry makes logical operations particularly resilient to untimely anisotropic qubit rotations. We propose a scalable scheme for universal quantum computation where cores play the role of quantum-computational transistors, or quansistors for short. Initialization and readout are achieved by tunnel-coupling the quansistor to leads. The external leads are explicitly considered and are assumed to be the other main source of decoherence. We show that quansistors can be dynamically decoupled from the leads by tuning their internal parameters, giving them the versatility required to act as controllable quantum memory units. With this dynamical decoupling, logical operations within quansistors are also symmetry-protected from unbiased noise in their parameters. We identify technologies that could implement $\ensuremath{\omega}$-rotation invariance. Many of our results can be generalized to higher-level $\ensuremath{\omega}$-rotation-invariant systems, or adapted to clusters with other symmetries.
In recent investigations, it has been found that conservation laws generally lead to precision limits on quantum computing. Lower bounds of the error probability have been obtained for various logic operations from the commutation relation between the noise operator and the conserved quantity or from the recently developed universal uncertainty principle for the noise-disturbance trade-off in general measurements. However, the problem of obtaining the precision limit to realizing the quantum NOT gate has eluded a solution from these approaches. Here, we develop a method for this problem based on analyzing the trace distance between the output state from the realization under consideration and the one from the ideal gate. Using the mathematical apparatus of orthogonal polynomials, we obtain a general lower bound on the error probability for the realization of the quantum NOT gate in terms of the number of qubits in the control system under conservation of the total angular momentum of the computational qubit plus the control system along the direction used to encode the computational basis. The lower bound turns out to be more stringent than one might expect from previous results. Our method is expected to lead to more accurate estimates for physical realizations of various types of quantum computations under conservation laws and to contribute to related problems such as the accuracy of programmable quantum processors.