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arXiv:1203.0350

Perfect NOT transformation and conjugate transformation

Fengli Yan, Ting Gao, Zhichao Yan

Published Mar 2, 2012• 0 citations
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Abstract

The perfect NOT transformation, probabilistic perfect NOT transformation and conjugate transformation are studied. Perfect NOT transformation criteria on a quantum state set $S$ of a qubit are obtained. Two necessary and sufficient conditions for realizing a perfect NOT transformation on $S$ are derived. When these conditions are not satisfied we discuss a probabilistic perfect NOT transformation (gate). We construct a probabilistic perfect NOT machine (gate) by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate yields perfectly complements of the input states. We prove that one can realize probabilistically the NOT gate of the input states secretly chosen from a certain set $S=\{|Ψ_1>, |Ψ_2>,..., |Ψ_n>\}$ if and only if $|Ψ_1>, |Ψ_2>,...,$ and $|Ψ_n>$ are linearly independent. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.

DOI
10.48550/arxiv.1203.0350

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The perfect NOT transformation, probabilistic perfect NOT transformation and conjugate transformation are studied. Perfect NOT transformation criteria on a quantum state set SSS of a qubit are obtained. Two necessary and sufficient conditions for realizing a perfect NOT transformation on SSS are derived. When these conditions are not satisfied we discuss a probabilistic perfect NOT transformation (gate). We construct a probabilistic perfect NOT machine (gate) by a general unitary-reduction operation. With a postselection of the measurement outcomes, the probabilistic NOT gate yields perfectly complements of the input states. We prove that one can realize probabilistically the NOT gate of the input states secretly chosen from a certain set S={∣Ψ1>,∣Ψ2>,...,∣Ψn>}S=\{|Ψ_1>, |Ψ_2>,..., |Ψ_n>\}S={∣Ψ1​>,∣Ψ2​>,...,∣Ψn​>} if and only if ∣Ψ1>,∣Ψ2>,...,|Ψ_1>, |Ψ_2>,...,∣Ψ1​>,∣Ψ2​>,..., and ∣Ψn>|Ψ_n>∣Ψn​> are linearly independent. We also generalize the probabilistic NOT transformation to the conjugate transformation in the multi-level quantum system. The lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.
Abstract We obtain the perfect NOT transformation criteria, two necessary and sufficient conditions for realizing a perfect NOT transformation on a quantum state set S of a qubit. Furthermore, we discuss a probabilistic perfect NOT transformation when there is no perfect NOT transformation on a state set S. We also generalize the perfect NOT transformation to the conjugate transformation in the multi-level quantum system and a lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.
In [1] Gingrich and Williams extended the standard quantum circuit model of quantum computation to include gates that perform arbitrary linear, yet non-unitary, transformations of their input state. In this paper we use this theory to construct quantum circuits that perform desired non-unitary state transformations of the sort arising in lattice-based quantum search. Such n-qubit non-unitary gates cannot be achieved deterministically, but they can be achieved probabilistically using a single ancilla. Our approach is to use the n-qubit non-unitary gate to design an (n+1)-qubit Hamiltonian that can be seen to induce a suitable conditional dynamics such that whenever the output value of the ancilla qubit is measured and found to be |O>, then the remaining (unmeasured) n qubits will contain the desired non-unitary transform of the n-qubit input state. The scheme is necessarily probabilistic because we cannot force the measurement on the ancilla qubit to return the value we want. Fortunately, however, failed attempts are found to disturb the n-qubit input state very little. This allows us to re-use the result of successive “failed” attempts until the success condition is finally attained, and we give analytic expressions for the success probability and net fidelity in this case. By using our previous method for designing a quantum circuit for an arbitrary n-qubit gate in conjunction with our new probabilistic non-unitary procedure we are able to compute an explicit quantum circuit sufficient to implement an arbitrary linear, yet non-unitary, transformation of an input pure or mixed n-qubit state, using only a single copy of the input state. This allows us to extend the repertoire of computations that may be performed on a quantum computer, and in particular, gives the first explicit construction of the quantum circuits needed to perform some of the key operations arising in quantum lattice search.
The action of a NOT gate on a classical bit results in a change of its value from a 0 to a 1 and vice versa. The action of the classical NOT gate is in principle perfect because with fidelity equal to unity it complements the value of a bit. The action of the quantum NOT gate in a computational basis |0⟩ and |1⟩ is very similar to the action of the classical NOT gate. However, a more general quantum mechanical operation which corresponds to a classical NOT gate would take a qubit in an arbitrary state |Ψ⟩ and produce a qubit in the state |Ψ⊥⟩ orthogonal to |Ψ⟩. This operation is anti-unitary and therefore, cannot be realized exactly. So how well we can do? We find a unitary transformation acting on an input qubit and some auxiliary qubits, which represent degrees of freedom of the quantum NOT gate itself, which approximately realizes the NOT operation on the state of the original qubit. We call this ‘device’ a universal-NOT gate because the size of the error it produces is independent of the input state. We show that an optimal U-NOT gate which has as its input N identical qubits and produces M outputs achieves a fidelity of F = (N + 1)/ (N + 2), which is equal to the fidelity of estimation of the input qubits. We also show that when a priori information about the state of the input qubit is available, the fidelity of a quantum NOT gate can be much better than the fidelity of estimation.
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},|{\ensuremath{\Psi}}_{n}〉}$ can be probabilistically cloned if and only if $|{\ensuremath{\Psi}}_{1}〉,|{\ensuremath{\Psi}}_{2}〉,\dots{},\mathrm{and}|{\ensuremath{\Psi}}_{n}〉$ are linearly independent. We derive the best possible cloning efficiencies. Probabilistic cloning has a close connection with the problem of identification of a set of states, which is a type of n+1n+1n+1 outcome measurement on nnn linearly independent states. The optimal efficiencies for this type of measurement are obtained.