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arXiv:quant-ph/0311170 · Physical Review A

Improving the performance of probabilistic programmable quantum processors

Mark S. Hillery, Mário Ziman, Vladimír Bužek

Published Apr 16, 2004• 19 citations
arXivHTMLPDF

Abstract

We present a systematic analysis of how one can improve performance of probabilistic programmable quantum processors. We generalize a simple Vidal-Masanes-Cirac processor that realizes U(1) rotations on a qubit with the phase of the rotation encoded in a state of the program register. We show how the probability of success of the probabilistic processor can be enhanced by using the processor in loops. In addition we show that the same strategy can be utilized for a probabilistic implementation of nonunitary transformations on qubits. In addition, we show that an arbitrary SU(2) transformation of qubits can be encoded in program state of a universal programmable probabilistic quantum processor. The probability of success of this processor can be enhanced by a systematic correction of errors via conditional loops. Finally, we show that all our results can be generalized also for qudits. In particular, we show how to implement SU(N) rotations of qudits via programmable quantum processor and how the performance of the processor can be enhanced when it is used in loops.

DOI
10.1103/physreva.69.042311

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We present a systematic analysis of how one can improve performance of probabilistic programmable quantum processors. We generalize a simple Vidal-Masanes-Cirac processor that realizes U(1) rotations on a qubit with the phase of the rotation encoded in a state of the program register. We show how the probability of success of the probabilistic processor can be enhanced by using the processor in loops. In addition we show that the same strategy can be utilized for a probabilistic implementation of nonunitary transformations on qubits. In addition, we show that an arbitrary SU(2) transformation of qubits can be encoded in program state of a universal programmable probabilistic quantum processor. The probability of success of this processor can be enhanced by a systematic correction of errors via conditional loops. Finally, we show that all our results can be generalized also for qudits. In particular, we show how to implement SU(N) rotations of qudits via programmable quantum processor and how the performance of the processor can be enhanced when it is used in loops.
Abstract We analyze how to improve performance of probabilistic programmable quantum processors. We show how the probability of success of the probabilistic processor can be enhanced by using the processor in loops. In addition, we show that an arbitrary SU(2) transformations of qubits can be encoded in program state of a universal programmable probabilistic quantum processor. The probability of success of this processor can be enhanced by a systematic correction of errors via conditional loops. Finally, we show that all our results can be generalized also for qudits.
We analyze probabilistic realizations of programmable quantum processors that allow us to realize unitary operations on qubits as well as on qudits. Programmable processors are composed of two inputs — the data register and the program register. In the input state of the program register information about the operation that is supposed to be performed on the data is encoded. At the output of the probabilistic processor a measurement over the program register is performed. An intrinsic property of probabilistic processors is that they sometimes fail, but we know when this happens. We present a complete analysis of two processors: (1) The so-called [Formula: see text] processor that is based on a simple controlled-NOT gate. (2) The so-called [Formula: see text] processor that utilizes the quantum-information-distributor circuit.
We present a probabilistic quantum processor for qudits on a single qudit of dimension N. The processor itself is represented by a fixed array of gates. The input of the processor consists of two registers. In the program register the set of instructions (program) is encoded. This program is applied to the data register. The processor can perform any operation on a single qudit of dimension N with a certain probability. For a general unitary operation, the probability is 1/N2,{1/N}^{2},1/N2, but for more restricted sets of operators the probability can be higher. In fact, this probability can be independent of the dimension of the qudit Hilbert space of the qudit under some conditions.
In the first part of the work, the equivalence of quantum deterministic and probabilistic processors was investigated. A programmable quantum processor is a device able to transform input data states in a desired way. Deterministic equivalence as well as three types of probabilistic equivalences - strong, weak, and structural - were defined. Necessary and sufficient conditions for deterministic and structural equivalence of unitarily related processors were discovered. Equivalence of deterministic SWAP processor for two-dimensional data and two-dimensional program space was completely solved. It was found that spans of operators of structurally equivalent processors are identical. Relations between types of individual equivalences were also examined. In the second part, robustness of probabilistic storing and retrieval device (PSAR), originally optimized for implementing a phase gate, to noise was examined - specifically to depolarization and phase damping. In the case of a depolarizing channel mixed with a unitary channel, the device implements noisy channel with the probability that decreases with an increasing number of times the given channel is applied. In the case of the phase damping channel, the device implements noisy channel with the same probability as the original PSAR device optimized for phase gate. Concrete implementations - through the Vidal-Masanes-Cirac scheme and virtual qudit - were examined. Vidal-Masanes-Cirac gives the same result for both noisy channels which is better than the result from PSAR. Implementation through virtual qudit for depolarization yields worse probability of successful measurement than Vidal-Masanes-Cirac. However, it is still better than the probability for PSAR. Probability of successful measurement obtained for phase damping implemented through virtual qudit is the same as for Vidal-Masanes-Cirac and PSAR.
We analyze and compare the optimality of approximate and probabilistic universal programmable quantum processors. We define several characteristics how to quantify the optimality and we study in detail performance of three types of programmable quantum processors based on (1) the C-NOT gate, (2) the SWAP operation, and (3) the model of the quantum information distributor - the QID processor. We show under which conditions the measurement assisted QID processor is optimal. We also investigate optimality of the so-called U-processors and we also compare the optimal approximative implementation of U(1) qubit rotations with the known probabilistic implementation as introduced by Vidal, Masanes and Cirac [ {\\em Phys. Rev. Lett.} {\\bf 88}, 047905 (2002)].
We examine the execution of general U(1)\mathrm{U}(1)U(1) transformations on programmable quantum processors. We show that, with only the minimal assumption of availability of copies of the 111-qubit program state, the apparent advantage of existing schemes proposed by G. Vidal et al. [Phys. Rev. Lett. 88, 047905 (2002)] and M. Hillery et al. [Phys. Rev. A 65, 022301 (2003)] to execute a general U(1)\mathrm{U}(1)U(1) transformation with greater probability using complex program states appears not to hold.