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arXiv:quant-ph/0008011 · Physical Review A

Universal manipulation of a single qubit

Lucién Hardy, David D. Song

Published Feb 8, 2001• 15 citations
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Abstract

We find the optimal universal way of manipulating a single qubit, $|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,$ such that $(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).$ Such optimal transformations fall into two classes. For $0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,$ the optimal map is the identity and the fidelity varies monotonically from 1 (for $\ensuremath{\alpha}=0)$ to $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2).$ For $\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},$ the optimal map is the universal-NOT gate and the fidelity varies monotonically from $\frac{1}{2}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}/2)$ to $\frac{2}{3}$ (for $\ensuremath{\alpha}=\ensuremath{\pi}).$ The fidelity $\frac{2}{3}$ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of $\ensuremath{\alpha}$ the fidelity is lower than $\frac{2}{3}.$ For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.

DOI
10.1103/physreva.63.032304

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We find the optimal universal way of manipulating a single qubit, ∣\ensuremathψ(\ensuremathϑ,\ensuremathφ)〉,|\ensuremath{\psi}(\ensuremath{\vartheta},\ensuremath{\varphi})〉,∣\ensuremathψ(\ensuremathϑ,\ensuremathφ)〉, such that (\ensuremathϑ,\ensuremathφ)\ensuremath→(\ensuremathϑ\ensuremath−\ensuremathα,\ensuremathφ\ensuremath−\ensuremathβ).(\ensuremath{\vartheta},\ensuremath{\varphi})\ensuremath{\rightarrow}(\ensuremath{\vartheta}\ensuremath{-}\ensuremath{\alpha},\ensuremath{\varphi}\ensuremath{-}\ensuremath{\beta}).(\ensuremathϑ,\ensuremathφ)\ensuremath→(\ensuremathϑ\ensuremath−\ensuremathα,\ensuremathφ\ensuremath−\ensuremathβ). Such optimal transformations fall into two classes. For 0< \ensuremathα< \ensuremathπ/2,0<~\ensuremath{\alpha}<~\ensuremath{\pi}/2,0< \ensuremathα< \ensuremathπ/2, the optimal map is the identity and the fidelity varies monotonically from 1 (for \ensuremathα=0)\ensuremath{\alpha}=0)\ensuremathα=0) to 12\frac{1}{2}21​ (for \ensuremathα=\ensuremathπ/2).\ensuremath{\alpha}=\ensuremath{\pi}/2).\ensuremathα=\ensuremathπ/2). For \ensuremathπ/2< \ensuremathα< \ensuremathπ,\ensuremath{\pi}/2<~\ensuremath{\alpha}<~\ensuremath{\pi},\ensuremathπ/2< \ensuremathα< \ensuremathπ, the optimal map is the universal-NOT gate and the fidelity varies monotonically from 12\frac{1}{2}21​ (for \ensuremathα=\ensuremathπ/2)\ensuremath{\alpha}=\ensuremath{\pi}/2)\ensuremathα=\ensuremathπ/2) to 23\frac{2}{3}32​ (for \ensuremathα=\ensuremathπ).\ensuremath{\alpha}=\ensuremath{\pi}).\ensuremathα=\ensuremathπ). The fidelity 23\frac{2}{3}32​ is equal to the fidelity of measurement. It is therefore rather surprising that for some values of \ensuremathα\ensuremath{\alpha}\ensuremathα the fidelity is lower than 23.\frac{2}{3}.32​. For instance, a universal square root of NOT operation is more difficult to approximate than the universal NOT gate itself.
It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F= (N+1)/(N+2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation.
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It is known that a computationally universal gate set {H,CCZ}\{H,CCZ\}{H,CCZ} can be transformed to a strictly universal one {H,Λ(S)}\{H, Λ(S)\}{H,Λ(S)} using one maximally imaginary state ∣+i⟩|+i \rangle∣+i⟩ and non-imaginary ancillary qubits. We succeed this transformation with a shorter circuit that eliminates non-imaginary ancillary qubits. We further extend this to the continuous gate-set setting, showing that any multi-qubit unitary can be exactly generated by real single-qubit unitary gates, CCZCCZCCZ gates and ∣0⟩∣+i⟩|0 \rangle |+i \rangle∣0⟩∣+i⟩.