View source for Cirac–Zoller controlled-NOT gate

{{Short description|Quantum logic gate}}
{{Quantum mechanics}}

The '''Cirac–Zoller controlled-NOT gate''' is an implementation of the [[Controlled NOT gate|controlled-NOT]] (CNOT) [[quantum logic gate]] using [[ion trap|cold trapped ions]] that was proposed by [[Juan Ignacio Cirac Sasturain|Ignacio Cirac]] and [[Peter Zoller]] in 1995 and represents the central ingredient of the Cirac–Zoller proposal for a [[Trapped ion quantum computer|trapped-ion quantum computer]].<ref>{{Cite journal|last1=Cirac|first1=J. I.|last2=Zoller|first2=P.|date=1995-05-15 |title=Quantum Computations with Cold Trapped Ions|journal=Physical Review Letters|volume=74 |issue=20|pages=4091–4094 |doi=10.1103/PhysRevLett.74.4091|pmid=10058410|bibcode=1995PhRvL..74.4091C}}</ref> The key idea of the Cirac–Zoller proposal is to mediate the interaction between the two qubits through the joint motion of the complete chain of trapped ions. 

The quantum CNOT gate acts on two [[qubit]]s and can [[Quantum entanglement|entangle]] them. It forms part of the standard [[Quantum logic gate#Universal quantum gates|universal set of gates]],<ref name="NC00">{{cite book |author1=M. A. Nielsen |author2=I. L. Chuang |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2000}}</ref> meaning that any gate ([[Unitary transformation (quantum mechanics)|unitary transformation]]) on the <math>N</math>-qubit [[Hilbert space]] can be approximated to arbitrary precision by a sequence of gates from the universal set. 

The Cirac–Zoller gate was experimentally first realized in 2003 (in slightly modified form) at the [[University of Innsbruck]], Austria by [[Ferdinand Schmidt-Kaler]] and coworkers in the group of [[Rainer Blatt]] using two calcium ions.<ref>{{Cite journal |last1=Schmidt-Kaler |first1=Ferdinand |last2=Häffner |first2=Hartmut |last3=Riebe |first3=Mark |last4=Gulde |first4=Stephan |last5=Lancaster |first5=Gavin P. T. |last6=Deuschle |first6=Thomas |last7=Becher |first7=Christoph |last8=Roos |first8=Christian F. |last9=Eschner |first9=Jürgen |last10=Blatt |first10=Rainer |date=2003-03-27 |title=Realization of the Cirac–Zoller controlled-NOT quantum gate |journal=Nature |language=en |volume=422 |issue=6930 |pages=408–411 |bibcode=2003Natur.422..408S |doi=10.1038/nature01494 |issn=0028-0836 |pmid=12660777 |s2cid=4401898}}</ref>

== Procedure ==
The qubits on which the Cirac–Zoller gate operates are represented by two internal states, ground state and excited state (called in the following ''g'' and ''e'') of trapped ions. An additional auxiliary excited state ''a'' is used to implement the gate. Due to their mutual [[Coulomb interaction|Coulomb repulsion]] the ions line up in a linear chain. The ions are [[laser cooling|cooled]] to their collective [[ground state]], so that the quantization of the motion of the chain becomes relevant. The proposal assumes that each ion can be individually addressed by [[laser]] pulses. Both the transitions "<math>g\to e</math>" and "<math>g\to a</math>" can be driven by choosing different laser [[Photon polarization|polarizations]]. For each transition, one can distinguish two kinds of such pulses. Those on resonance with the transition and those that are [[Laser detuning|detuned]] from the respective transition by an energy difference that corresponds to the energy of a single [[phonon|quantum of motion]] of the ion chain. The former are called ''direct'' pulses, the latter ''sideband'' pulses. The proposal uses ''red'' sideband pulses (that have less energy than corresponds to the direct transition). 

The Cirac–Zoller gate between two qubits represented by ions A and B is then realized in a three-step process: 
# A red sideband pulse is directed onto ion A. Length and strength of the pulse are chosen such that it realizes the following transformation: if initially ion A is in state ''e'' and the ion chain in the ground state, then at the end of the pulse the ion is in state ''g'' and the chain in its first excited state 1. Conversely, ''g'' is mapped to ''e'' if initially the chain was in its first excited state:<br> <math> |g_A,1\rangle \mapsto -i |e_A,0\rangle</math> and <math>|e_A,0\rangle \mapsto -i|g_A,1\rangle</math>, all other states are unaffected. <ref>Certain states with more motional excitations (such as, say <math>|g_A,2\rangle</math>) would be affected, but are never populated throughout the protocol, since ground state cooling is assumed.</ref><br>  This transformation is referred to as a ''<math>\pi</math>-pulse''.
# A <math>2\pi</math>-pulse is applied to the ion B on red sideband of the "<math>g\to a</math>"-transition: this changes the phase of the ion B if it is in the state ''g'' and the chain is in the first excited state: <math> |g_B,1\rangle \mapsto - |g_B,1\rangle</math>, all other states are unaffected. Given initial cooling and the design of the first step this means that the phase of ion B is flipped only if ion A originally was in the state ''e''.
# Another red sideband <math>\pi</math>-pulse on ion A completes the two-qubit gate. This returns ion A and the motion of the chain to its initial state. 

[[File:Diagram_of_the_Cirac-Zoller_CNOT_gate.svg|thumb|400px|Diagram of the three-pulse sequence used to implement the gate between qubits A and B. The pulses are as follows: (1): A π-pulse is applied to ion A. (2): A 2π-pulse is applied to ion B. (3): Another π-pulse is applied to ion A. ]]
In total, the three pulses realize the following transformation on the two-qubit subspace in the motional ground state:
:<math>\begin{matrix}
&\xrightarrow{(1)}&&\xrightarrow{(2)}&&\xrightarrow{(3)}\\
|gg0\rangle && |gg0\rangle && |gg0\rangle && |gg0\rangle\\
|ge0\rangle && |ge0\rangle && |ge0\rangle && |ge0\rangle\\
|eg0\rangle && -i |gg1\rangle && i |gg1\rangle && |eg0\rangle\\
|ee0\rangle && -i |ge1\rangle && -i |ge1\rangle && - |ee0\rangle
\end{matrix},</math>
that is, the state ''ee'' acquires a phase <math>-1</math> the other three states are unaffected. This transformation is called a ''controlled-phase'' or ''[[Quantum_logic_gate#Controlled_gates|controlled-Z]]'' gate (<math>U_{CZ}</math>), since the first qubit controls whether a phase flip (which corresponds to applying the [[Pauli matrix|Pauli <math>\sigma_z</math> matrix]]) is applied to the second qubit. It can be turned into the CNOT gate by applying a single-qubit gate, the [[Hadamard gate]] to the ion B before and after the application of <math>U_{CZ}</math>:
:<math>U_{\mathrm{CNOT}} = (1\otimes H) U_{CZ} (1\otimes H).</math>

The central theoretical realization, on which the above steps and much of the subsequent theoretical progress in trapped-ion quantum computation is based, is that the ion chain driven by red sideband pulses realizes the [[Jaynes–Cummings model]] for the two-level system formed by ''g'' and ''e'' and one of the [[normal mode]]s of the chain.<ref>Namely the one whose frequency is resonant with the detuning of the laser pulse.</ref> To achieve this, it is necessary that the light interacting with the ions can change their motional state. This requires [[Raman transition]]s. To suppress transitions in which more than one quantum of motion is transferred, one has to work in the [[Lamb Dicke regime]] where the wavelength of the light used is large compared to the size of the wave packet of the trapped ion. In this regime, the coupling strength is reduced and leads to a relatively slow gate.

== See also ==

* [[Mølmer–Sørensen gate]]

== References ==
{{Reflist}}

{{Quantum computing}}

{{DEFAULTSORT:Cirac-Zoller controlled-NOT gate}}
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