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arXiv:quant-ph/0505202 · Physical Review A

Probabilistic programmable quantum processors with multiple copies of program states

Adam Brazier, Vladimír Bužek, P. L. Knight

Published Mar 9, 2005• 17 citations
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Abstract

We examine the execution of general $\mathrm{U}(1)$ transformations on programmable quantum processors. We show that, with only the minimal assumption of availability of copies of the $1$-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 $\mathrm{U}(1)$ transformation with greater probability using complex program states appears not to hold.

DOI
10.1103/physreva.71.032306

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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.
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 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 present a probabilistic quantum processor for qubits. The processor itself is represented by a fixed array of gates. The input of the processor is constituted by two registers. In the program register the set of instructions (program) is encoded. This program is applied to the data register. The processor we present can perform any operation on a single qubit with a certain probability. If the operation is unitary, the probability is in general 1/4, but for more restricted sets of operators the probability can be higher.
Similar to a classical processor, which is an algorithm for reading a program and executing its instructions on input data, a universal programmable quantum processor is a fixed quantum channel that reads a quantum program ∣ψU⟩\lvertψ_{U}\rangle∣ψU​⟩ that causes the processor to approximately apply an arbitrary unitary UUU to a quantum data register. The present work focuses on a class of simple programmable quantum processors for implementing reflection operators, i.e. U=eiπ∣ψ⟩⟨ψ∣U = e^{i π\lvertψ\rangle\langleψ\rvert}U=eiπ∣ψ⟩⟨ψ∣ for an arbitrary pure state ∣ψ⟩\lvertψ\rangle∣ψ⟩ of finite dimension ddd. Unlike quantum programs that assume query access to UUU, our program takes the form of independent copies of the state to be reflected about ∣ψU⟩=∣ψ⟩⊗n\lvertψ_U\rangle = \lvertψ\rangle^{\otimes n}∣ψU​⟩=∣ψ⟩⊗n. We then identify the worst-case optimal algorithm among all processors of the form trProgram[V(∣ϕ⟩⟨ϕ∣⊗(∣ψ⟩⟨ψ∣)⊗n)V†]\text{tr}_{\text{Program}}[V (\lvertϕ\rangle\langleϕ\rvert \otimes (\lvertψ\rangle\langleψ\rvert)^{\otimes n}) V^\dagger]trProgram​[V(∣ϕ⟩⟨ϕ∣⊗(∣ψ⟩⟨ψ∣)⊗n)V†] where the algorithm VVV is a unitary linear combination of permutations. By generalizing these algorithms to processors for arbitrary-angle rotations, eiα∣ψ⟩⟨ψ∣e^{i α\lvertψ\rangle\langleψ\rvert}eiα∣ψ⟩⟨ψ∣ for α∈Rα\in \mathbb Rα∈R, we give a construction for a universal programmable processor with better scaling in ddd. For programming reflections, we obtain a tight analytical lower bound on the program dimension by bounding the Holevo information of an ensemble of reflections applied to an entangled probe state. The lower bound makes use of a block decomposition of the uniform ensemble of reflected states with respect to irreps of the partially transposed permutation matrix algebra, and two representation-theoretic conjectures based on extensive numerical evidence.
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.