layout: true <slide-template title="Quantum Computing" subtitle="Quantum Logic Gates"></slide-template> --- class: title-slide, center, middle <title-slide titletype="Lecture" title="Quantum Computing" modulenumber="(CS5070)" subtitle="Quantum Logic Gates"></title-slide> --- # Operations via .darkblue[Quantum Logic Gates] .reference[Quantum Logic gates in <a href="https://qiskit.org/documentation/apidoc/circuit_library.html" target="">Qiskit</a>, <a href="https://www.quantum-inspire.com/kbase/cqasm-qubit-gate-operations/" target="">cQASM</a>, <a href="https://github.com/eth-sri/silq/blob/master/library/prelude-nocheck.slq" target="">Silq</a> and <a href="https://docs.microsoft.com/en-us/qsharp/api/qsharp/microsoft.quantum.intrinsic" target="">Microsoft Q#</a>] .more-more-condensed[ - Operation of a .darkblue[quantum logic gate is defined by] - .darkblue[its matrix $U$], - which .darkblue[transforms] a .darkblue[quantum state $|\psi_1\rangle$] via matrix multiplication - .darkblue[into] the resulting .darkblue[quantum state $|\psi_2\rangle$:<br/>$U|\psi_1\rangle = |\psi_2\rangle$] - .darkblue[Authors use] sometimes .darkblue[different names] for the same gates - .darkblue[Most common gates] are introduced .darkblue[on the following slides]... - .darkblue[References], e.g.: - Colin P. Williams (2011). .darkblue[Explorations in Quantum Computing]. Springer. ISBN 978-1-84628-887-6. - Nielsen, Michael A.; Chuang, Isaac (2010). .darkblue[Quantum Computation and Quantum Information]. Cambridge: Cambridge University Press. ISBN 978-1-10700-217-3. - Yanofsky, Noson S.; Mannucci, Mirco (2013). .darkblue[Quantum computing for computer scientists]. Cambridge University Press. ISBN 978-0-521-87996-5. ] --- # .darkblue[Identity gate] .more-more-condensed[ - also known as .darkblue[Pauli I]-gate - .darkblue[defined by] the .darkblue[identity matrix] - for 1 qubit: <img src="../../img/qc/gates/identity.svg" align="middle" width="150px"></img> - for $n$ qubits: <img src="../../img/qc/gates/identityN.svg" align="middle" width="250px"></img> - .darkblue[quantum states remain] the same:<br/><img src="../../img/qc/gates/identityCalculation.svg" align="middle" class="fullwidth"></img> - $I$ behaves like a NOP (no operation) and can be .darkblue[represented as bare wire] in quantum circuits, .darkblue[or not shown at all] ] --- # .darkblue[Rotation around $x$-axis] .more-more-condensed.no-margin.no-padding[ - .darkblue[quantum logic gates for 1 qubit: often rotation around one axis] - Relatively general quantum logic gate: rotation around a specified angle $\theta$: <table> </tbody> <tr> <td style="vertical-align: top;" width="45%"> <div class="textcenter"> <img src="../../img/qc/bloch/BlochSphereNaked.svg" align="top" width="100%"></img> </div> </td> <td style="vertical-align: top;" class="small-font"> Rotation operator for rotation around $x$-axis:<br/> <img src="../../img/qc/bloch/RotationGateFormula.svg" align="top" width="500em"></img><br/>e.g. .bold[Pauli_$x$-Gate]:<br/> <table class="result small-font tablesmallpadding small-font"> <tbody> <tr> <th>Rotation Matrix</th> <td>.small-font[$X = NOT = \left[\\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array}\right]$]</td> </tr> <tr> <th>Quantum Circuit</th> <td><img src="../../img/qc/bloch/NOT.svg" align="top" width="150em"></img><br/> Alternatively: <img src="../../img/qc/bloch/XGate.svg" align="top" width="150em"></img> </td> </tr> <tr> <th>Table of in- & outputs:</th> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font" class="tablepadding"> <tbody> <tr> <td style="background-color:#d9ffb3">$In$</td> <td style="background-color:#ffb3d1">$Out$</td> </tr> <tr> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr> <td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td> <td style="background-color:#ffb3d1">$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$</td> </tr> <tr> <td style="background-color:#d9ffb3">$\frac{3\cdot i}{5}|0\rangle+\frac{4}{5}|1\rangle$</td> <td style="background-color:#ffb3d1">$\frac{4}{5}|0\rangle+\frac{3\cdot i}{5}|1\rangle$</td> </tr> </tbody> </table> </td> </tr> </tbody> </table> </td> </tr> </tbody> </table> ] --- # .darkblue[Pauli$_\\{x,y,z\\}$-Gates] .reference[deutsch: ¹ selbst-invers ² anti-kommutativ] .more-more-condensed[ - .darkblue[Pauli gates ($X$,$Y$,$Z$): rotation around the x, y and z axes] of the Bloch sphere .darkblue[by $\pi$] radians:<img src="../../img/qc/gates/PauliGates.svg" align="top" style="width:60%"></img> - .darkblue[Circuit] representation:<img src="../../img/qc/gates/PauliGatesCircuit.svg" align="top" style="width:65%"></img> - .darkblue[Pauli matrices are involutory]¹ (i.e. .darkblue[square] of Pauli matrix .darkblue[is identity matrix]): .darkblue[$I^2=X^2=Y^2=Z^2=-iXYZ=I$] - .darkblue[Pauli matrices are anti-commute]², e.g., .darkblue[$ZX=iY=-XZ$] - .darkblue[Pauli$_x$-gate] (also called .darkblue[bit-flip]) - .darkblue[quantum equivalent of] the .darkblue[NOT] gate for classical computers - .darkblue[Pauli$_y$-gate] - maps .darkblue[$|0\rangle$ to $i|1\rangle$ and $|1\rangle$ to $-i|0\rangle$] - .darkblue[Pauli$_z$-gate] (also called .darkblue[phase-flip]) - leaves the basis state .darkblue[$|0\rangle$ unchanged and] maps .darkblue[$|1\rangle$ to $-|1\rangle$] ] --- # .darkblue[Controlled NOT (CNOT/CX)]-Gate .more-more-condensed[ - ".em[.darkblue[If] the .darkblue[control bit] is set, .darkblue[then] it .darkblue[flips] the .darkblue[target bit].]" <table class="result small-font tablesmallpadding small-font"> <tbody> <tr> <th>Quantum Circuit</th> <th>Table of in- & outputs</th> <th>Rotation Matrix $R$</th> </tr> <tr> <td><img src="../../img/qc/bloch/CNOT.svg" align="top" width="220em"></img></td> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font" class="tablepadding"> <tbody> <tr> <td colspan="2" style="background-color:#d9ffb3"><b>Inputs</b></td> <td style="background-color:#ffb3d1"><b>Output</b> </td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>Control $C$</b></td> <td style="background-color:#d9ffb3;width:3em;"><b>Target $T_{before}$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>Target $T_{after}$</b></td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;">$|0\rangle$</td> <td style="background-color:#d9ffb3;width:3em;">$|0\rangle$</td> <td style="background-color:#ffb3d1;width:3em;">$|0\rangle$</td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;">$|0\rangle$</td> <td style="background-color:#d9ffb3;width:3em;">$|1\rangle$</td> <td style="background-color:#ffb3d1;width:3em;">$|1\rangle$</td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;">$|1\rangle$</td> <td style="background-color:#d9ffb3;width:3em;">$|0\rangle$</td> <td style="background-color:#ffb3d1;width:3em;">$|1\rangle$</td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;">$|1\rangle$</td> <td style="background-color:#d9ffb3;width:3em;">$|1\rangle$</td> <td style="background-color:#ffb3d1;width:3em;">$|0\rangle$</td> </tr> <tr> <td colspan="3" style="background-color:#ffb3d1;width:3em;">$R\cdot\frac{1}{\sqrt{2}}(|01\rangle+|11\rangle) = \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$</td> </tr> </tbody> </table> </td> <td> $\left[\\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ \\end{array}\right]$<br/> </td> </tr> </tbody> </table> - .darkblue[reversible] gate: 2 applications of CNOT retrieves the original input - .darkblue[Classical analog] of the CNOT gate .darkblue[is] a .darkblue[reversible XOR] gate (i.e., they have analogous in- & and outputs for $\\{|0\rangle, |1\rangle\\}$ inputs) - $|a,b\rangle \mapsto |a,a\oplus b\rangle$, where $\oplus$ is XOR ] --- # .darkblue[Controlled-$U$ ($CU$)-Gate] .more-more-condensed[ - .darkblue["If the control bit is set, then the $U$ transformation is applied."]<br/><img src="../../img/qc/gates/UGate.svg" align="top" style="width:80%"></img> - .darkblue[If $U=X$: "controlled-$X$" ($CX$),<br/>if $U=Y$: "controlled-$Y$" ($CY$),<br/>if $U=Z$: "controlled-$Z$" ($CZ$)] - .darkblue[Circuit] representations: <img src="../../img/qc/gates/UGates.svg" align="top" style="width:80%"></img> ] --- # .darkblue[Hadamard] (.darkblue[H])-Gate .more-more-condensed[ - unitary transformation that .darkblue[maps qubit operations in z-axis to the x-axis and vice versa] - represents a rotation of $\pi$ about the axis $({\hat {x}}+{\hat {z}})/{\sqrt {2}}$ at the Bloch sphere - For example, $HZH=X$, $H{\sqrt {X}} H={\sqrt {Z}}=S$, and $HR_z(\theta)H=R_x(\theta)$ - maps the basis states .darkblue[$|0\rangle \mapsto {\frac {|0\rangle +|1\rangle }{\sqrt {2}}}$ and $|1\rangle \mapsto {\frac {|0\rangle -|1\rangle }{\sqrt {2}}}$] - i.e., it .darkblue[creates a superposition] if given a basis state - <img src="../../img/qc/gates/H.svg" align="top" style="width:40%"></img> - $H^2=I$, $R_y(\pi /2)Z=H$ and $XR_y(\pi /2)=H$ - Controlled H gate: $CU$-Gate with $U=H$ ] --- # .darkblue[Hadamard] transformation .darkblue[on] the .darkblue[Bloch Sphere for $|0\rangle$] .more-more-condensed[ <img src="../../img/qc/gates/Hadamard0Bloch.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Hadamard] transformation .darkblue[on] the .darkblue[Bloch Sphere for $|1\rangle$] .more-more-condensed[ <img src="../../img/qc/gates/Hadamard1Bloch.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Bell States via Entanglement] 1/4 .reference[<a href="https://doi.org/10.1126/science.aan3211" target="_blank">Yin et al., "Satellite-based entanglement distribution over 1200 kilometers". Quantum Optics. 356: 1140–1144, 2017</a>] .more-more-condensed[ <table class="result small-font tablesmallpadding"> <tbody> <tr> <th>Entanglement</th> <th>Quantum Circuit</th> <th>Table of in- & outputs</th> </tr> <tr> <th>Correlated</th> <td><img src="../../img/qc/bloch/CorrelatedBellState.svg" align="top" width="160em"></img></td> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font"> <tbody> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$A$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$B$</b></td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$|0\rangle$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$|0\rangle$</b></td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$|1\rangle$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$|1\rangle$</b></td> </tr> <tr> <td colspan="3" style="background-color:#ffb3d1;width:3em;" style="text-align="right"><b>$\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$</b></td> </tr> </tbody> </table> </td> </tr> <tr> <th>Anti-Correlated</th> <td><img src="../../img/qc/bloch/AntiCorrelatedBellState.svg" align="top" width="160em"></img></td> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font"> <tbody> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$A$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$B$</b></td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$|0\rangle$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$|1\rangle$</b></td> </tr> <tr> <td style="background-color:#d9ffb3;width:3em;"><b>$|1\rangle$</b></td> <td style="background-color:#ffb3d1;width:3em;"><b>$|0\rangle$</b></td> </tr> <tr> <td colspan="3" style="background-color:#ffb3d1;width:3em;" style="text-align="right"><b>$\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$</b></td> </tr> </tbody> </table> </td> </tr> </tbody> </table> - Even if the .darkblue[entangled qubits] are .darkblue[at different locations], they .darkblue[are still entangled] (phenomena appears to happen .darkblue[instantaneously ignoring speed of light]: open question in physics) - As of 2017 experimentally verified for distances of up to 1200 kilometers - .darkblue[Succeeding operations on entangled qubits] do not change the state of the other entangled qubits if the operation is not a measurement ] --- # .darkblue[Bell States via Entanglement] 2/4 .reference[<a href="https://doi.org/10.1103/PhysRevX.13.021031" target="_blank">[C+23]</a>] <img src="../../img/qc/gates/CorrelatedBellStateWithHadamard.svg" align="top" class="fullwidth"></img> .more-more-condensed[ - .darkblue[How many atoms] can be .darkblue[entangled?] - .darkblue[Experiment] [C+23] .darkblue[with 2] spatially separated Bose-Einstein .darkblue[condensates, each containing about 700] rubidium .darkblue[atoms] - Entanglement between the condensates results in <span class="darkblue">strong correlations of their collective spins</span> ] --- # .darkblue[Bell States via Entanglement] 3/4 .more-more-condensed.small-font22[ - .darkblue[Difference to cloning] in the sense of no-cloning theorem - .darkblue[If one qubit is measured, the other collapses to one of $|0\rangle$ and $|1\rangle$] according to the entanglement and succeeding operations, too. - .darkblue[No-cloning] theorem .darkblue[refers to] that it is not possible to .darkblue[get an independent qubit by copying the superposition] of another, which could be independently measured without that the other qubit collapses to $|0\rangle$ or $|1\rangle$ - As computer scientists, I would formulate it in the following way (physicists may forgive me...):<br/>.em.bold.darkblue[Some kind of cloning via entanglement and independent succeeding operations is possible, but once one of the (entangled) qubits have been measured, all (entangled) qubits collapse to $|0\rangle$ or $|1\rangle$ with having the same effect as measuring all (entangled) qubits at the same time.] ] --- # .darkblue[Bell States via Entanglement] 4/4 .more-more-condensed[ - Verifying entanglement of quantum states:<br/>.bold.darkblue[None of the Bell states can be broken into (tensor) products!] - e.g., for 2 qubits: $\left[\begin{array}{c} x \\\\ y \end{array}\right] \otimes \left[\begin{array}{c} w \\\\ z \end{array}\right] = \left[\begin{array}{c} x\cdot w \\\\ x\cdot z \\\\ y\cdot w \\\\ y\cdot z \end{array}\right]$ - $\left[\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\\\ 0 \end{array}\right] \Rightarrow y=z=0, x\cdot w = 1$, i.e., .bold.darkred[no entanglement]! - $\left[\begin{array}{c} 0 \\\\ 1 \\\\ 1 \\\\ 0 \end{array}\right] \Rightarrow (x=0\vee w=0)\wedge x\neq 0 \wedge w \neq 0$ .bold.darkred[↯], i.e., .bold.darkred[entanglement]! ] --- # .darkblue[Phase shift gates] .reference[^* deutsch: Dies entspricht dem Zeichnen eines horizontalen Kreises (einer Breitenlinie) auf der Blochkugel] .more-more-condensed[ - map the basis states .darkblue[$|0\rangle \mapsto |0\rangle$ and $|1\rangle \mapsto e^{i\varphi }|1\rangle$] - .darkblue[probability of] measuring a .darkblue[$|0\rangle$ or $|1\rangle$ is unchanged] - .darkblue[modify the phase of the quantum state]. This is equivalent to tracing a horizontal circle (a line of latitude) on the Bloch sphere^* by $\varphi$ radians - <img src="../../img/qc/gates/PhaseGateMatrix.svg" align="top" style="width:30%"></img>, where $\varphi$ is the phase shift with the period $2\pi$ - <img src="../../img/qc/gates/PhaseGateMatrices.svg" align="top" class="fullwidth"></img> - Controlled phase shift gates: $CU$-Gate with $U=P(\varphi)$ ] --- # .darkblue[Swap]-Gate (.darkblue[$SWAP$]) .more-more-condensed[ - swaps two qubits:<br/><img src="../../img/qc/gates/SWAP.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Square root of Swap]-Gate (.darkblue[$\sqrt{SWAP}$]) .reference[^* <a href="https://doi.org/10.1103%2FPhysRevResearch.3.013113" target="_blank">J.Nemirovsky, Y.Sagi, Fast universal two-qubit gate for neutral fermionic atoms in optical tweezers, Physical Review Research, 3(1), 2021</a>] .more-more-condensed[ - .darkblue[arises naturally] in systems that exploit exchange interaction^* - .darkblue[universal] gate such that any many-qubit gate can be constructed from only $\sqrt{SWAP}$ and single qubit gates - .darkblue[not maximally entangling] (i.e., more than one application of it is required to produce a Bell state) - .darkblue[performs half-way of a two-qubit swap]:<br/>$\sqrt{SWAP}=\left[\\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\ 0 & \\frac{1}{2}(1+i) & \\frac{1}{2}(1-i) & 0 \\\\ 0 & \\frac{1}{2}(1-i) & \\frac{1}{2}(1+i) & 0 \\\\ 0 & 0 & 0 & 1 \\\\ \\end{array}\right]$ - .darkblue[Circuit] Representation: <img src="../../img/qc/gates/RootOfSwap.svg" align="top" style="width:2em"></img> ] --- # .darkblue[Toffoli]-Gate .small-font22[(.darkblue[controlled-controlled-not (CCNOT)/Deutsch gate $D(\frac{\pi}{2})$])] .more-more-condensed.no-margin[ - ".em[.darkblue[If] the .darkblue[first two (qu)bits] are set, .darkblue[then] the Toffoli gate .darkblue[flips] the .darkblue[third (qu)bit].]", i.e., it maps .small-font[$(C_1, C_2, T)$] to .small-font[$(C_1, C_2, T\ XOR\ (C_1\ \wedge\ C_2))$] - .darkblue[reversible] gate - .darkblue[universal]: for classical computation and for quantum computation when combined with the single qubit Hadamard gate <table class="result small-font tablesmallpadding small-font"> <tbody> <tr> <th>Quantum Circuit</th> <th>Table of in- & outputs</th> <th>Rotation Matrix $R$</th> </tr> <tr> <td><img src="../../img/qc/bloch/CCNOT.svg" align="top" width="200em"></img></td> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font"> <tbody> <tr align="center"> <td style="background-color:#d9ffb3" class="bold">$C_1$</td> <td style="background-color:#d9ffb3" class="bold">$C_2$</td> <td style="background-color:#d9ffb3" class="bold">$T_{before}$</td> <td style="background-color:#ffb3d1" class="bold">$T_{after}$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}\cdot(|0\rangle+|1\rangle)$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> </tbody> </table> </td> <td class="smaller-font"> $\left[\\begin{array}{cccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ \\end{array}\right]$<br/> </td> </tr> </tbody> </table> ] --- # .darkblue[Fredkin]-Gate (.darkblue[$CSWAP/CS$]) .more-more-condensed[ - 3-bit gate that performs a .darkblue[controlled swap] - .darkblue[universal for classical] computation <table class="result small-font tablesmallpadding small-font"> <tbody> <tr> <th>Quantum Circuit</th> <th>Table of in- & outputs</th> <th>Rotation Matrix $R$</th> </tr> <tr> <td><img src="../../img/qc/gates/Fredkin.svg" align="top" width="150em"></img></td> <td> <table style="text-align:center;line-height:0.9;" class="smaller-font"> <tbody> <tr align="center"> <td style="background-color:#d9ffb3" class="bold">$C$</td> <td style="background-color:#d9ffb3" class="bold">$I_1$</td> <td style="background-color:#d9ffb3" class="bold">$I_2$</td> <td style="background-color:#ffb3d1" class="bold">$O_1$</td> <td style="background-color:#ffb3d1" class="bold">$O_2$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> <td style="background-color:#ffb3d1">$|1\rangle$</td> </tr> <tr align="center"> <td style="background-color:#d9ffb3">$|1\rangle$</td> <td style="background-color:#d9ffb3">$|0\rangle$</td> <td style="background-color:#d9ffb3">$\frac{1}{\sqrt{2}}\cdot(|0\rangle+|1\rangle)$</td> <td style="background-color:#ffb3d1">$\frac{1}{\sqrt{2}}\cdot(|0\rangle+|1\rangle)$</td> <td style="background-color:#ffb3d1">$|0\rangle$</td> </tr> </tbody> </table> </td> <td class="smaller-font"> $\left[\\begin{array}{cccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\ \\end{array}\right]$<br/> </td> </tr> </tbody> </table> ] --- # .darkblue[Measurement] (also called observation) .more-more-condensed[ <table class="result tablepadding"> <tbody> <tr> <th colspan="2">Input</th> <th rowspan="2">Circuit<br/>Represen-<br/>tation</th> <th colspan="2">Output</th> </tr> <tr> <th>Information Unit</th> <th>Line</th> <th>Line</th> <th>Information Unit</th> </tr> <tr> <td class="textcenter">Qubit</td> <td class="textcenter">Single</td> <td class="textcenter"><img src="../../img/qc/gates/Measurement.svg" align="top" width="150em"></img></td> <td class="textcenter">Double</td> <td class="textcenter">Classical Bit</td> </tr> </tbody> </table> - .darkblue[irreversible and] therefore .darkblue[not] a .darkblue[quantum gate], because it assigns the observed quantum state to a single value ] --- # .darkblue[Measurement] - Probabilities for basis state collapses <img src="../../img/qc/gates/MeasurementProbabilities.svg" align="top" class="fullwidth"></img> --- # Set of .darkblue[Universal Quantum Gates] .more-more-condensed[ - .darkblue[is any set of gates to which any operation can be reduced] - In other words: .darkblue[any other unitary operation can be expressed as a finite sequence of gates from this set] - .darkblue[Examples] of universal quantum gates sets - Rotation operators $R_x(\theta), R_y(\theta), R_z(\theta)$, the phase shift gate $P(\varphi), CNOT$ - $CNOT, H, S$ and $T$ gates - Two-gate set of universal quantum gates: Toffoli and Hadamard gates - ... ] --- # .darkblue[Unitary Inversion $^\dagger$ of Gates] 1/2 .reference[^* <a href="https://arxiv.org/abs/1504.05155" target="_blank">S. Aaronson, D. Grier, L. Schaeffer. The Classification of Reversible Bit Operations. arXiv:1504.05155, 2015</a>] .more-more-condensed[ - .darkblue[All quantum logical gates are reversible $\Rightarrow$ any composition of multiple gates is also reversible] - Series and parallel .darkblue[combinations of unitary matrices are] also .darkblue[unitary matrices $\Rightarrow$ Inversing] all algorithms and functions containing only gates .darkblue[is possible] - .darkblue[Inversion not possible for initialization, measurement, I/O and spontaneous decoherence] - $U$ is a unitary matrix .darkblue[$\Rightarrow U^\dagger U=UU^\dagger=I$ and $U^\dagger=U^\{-1\}$] - The .darkblue[dagger $\dagger$ denotes] the .darkblue[conjugate transpose] (also called the .darkblue[Hermitian adjoint].darkgray[, deutsch: adjungierte/hermitesch transponierte/transponiert-konjugierte Matrix]) - .darkblue[used in uncomputation]^* for cleaning up temporary effects on ancilla (i.e., auxiliary) bits so that they can be re-used ] --- # .darkblue[Unitary Inversion $^\dagger$ of Gates] 2/2 .reference[deutsch: ¹ selbst-adjungierter Operator ² schiefhermetische Matrix] .more-more-condensed[ - .darkblue[Hermitian (also called self-adjoint operators¹): Gates that are their own unitary inverses] like Hadamard ($H$) and the Pauli gates ($I, X, Y, Z$) - .darkblue[Skew-Hermitian² (also called adjoint operators): Gates that are not their own unitary inverses] in general like the phase shift ($S, T, P, CPHASE$) - .darkblue[$(UV)^\dagger=V^\dagger U^\dagger, (A_1 \cdots A_m)^\dagger=A_m^\dagger\cdots A_1^\dagger$] - .darkblue[$(U\otimes V)^\dagger=U^\dagger\otimes V^\dagger, (A_1\otimes\cdots\otimes A_m)^\dagger=A_1^\dagger\otimes\cdots\otimes A_m^\dagger$] Example:<br/> <img src="../../img/qc/gates/InverseOfCorrelatedBellStateWithHadamard.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Circuit composition] - Serially Wired Gates .more-more-condensed[ - .darkblue[When] gate .darkblue[$B$ is put after] gate .darkblue[$A$ in] a .darkblue[series] circuit.darkblue[, then the effect] of the two gates .darkblue[can be described as a single gate $C=B\cdot A$] - Example:<br/><img src="../../img/qc/gates/SeriesYX.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Exponents of Quantum Gates] .more-more-condensed[ - .darkblue[All quantum gates are unitary matrices] - .darkblue[Positive integer exponents are equivalent to sequences of serially wired gates] - e.g., $X^3=X\cdot X\cdot X$ - .darkblue[Real exponents is] a .darkblue[generalization of] the .darkblue[series] circuit - .darkblue[All real exponents of unitary matrices are] also .darkblue[unitary matrices/quantum gates] - e.g., $X^\pi$ and $\sqrt{X}=X^\frac{1}{2}$ are valid quantum gates - .darkblue[$U^0=I$] for any unitary matrix $U$ - .darkblue[$U^{-n}=(U^{n})^{\dagger}$] - e.g., $T^{-1}=T^{\dagger}$ and $T^{-2}=(T^{2})^{\dagger }=S^{\dagger }$ ] --- # .darkblue[Circuit composition] - Parallel Gates .more-more-condensed[ - .darkblue[The tensor product] (or Kronecker product) .darkblue[of two quantum gates is the gate that is equal to the two gates in parallel]<br/>Example:<br/> <img src="../../img/qc/gates/ParallelYX.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Circuit composition] - Parallel Hadamard .reference[^* deutsch: gleichmäßige Überlagerung] .more-more-condensed[ - .darkblue[Parallel application of Hadamard gates on $n$ qubits (all in basis state $|0\rangle$):]<br/> <img src="../../img/qc/gates/ParallelH.svg" align="top" class="fullwidth"></img> - This state is a .darkblue[uniform superposition]^*: - If not measured, it is a .darkblue[quantum state with equal probability amplitude] $\frac {1}{\sqrt {2^{n}}}$ for each of its possible states - Measuring this state results in a .darkblue[random number] between $|0\rangle$ and $|2^{n}-1\rangle$ - .darkblue[Often first step of quantum computing algorithms like Grover's search] ] --- # .darkblue[Computational Complexity of Simulating] Quantum Computers .reference[^* <a href="https://doi.org/10.1145%2F509907.509932" target="_blank">R. Raz, On the complexity of matrix product. STOC, 2002</a>] .more-more-condensed[ - .darkblue[Time complexity for multiplying two $k\times k$-matrices is at least $\Omega(k^{2}\log k)$] on a classical machine^* - .darkblue[For $n$-qubits-gate: $k=2^n$<br/>$\Rightarrow$ intractable to simulate] large entangled quantum systems using classical computers - .darkblue[Efficient simulation of subsets of the gates], such as the Clifford gates, or combinations of $X$, $CNOT$, Toffoli - .darkblue[State vector of a quantum register is $2^n$ complex entries<br/>$\Rightarrow$] Storing the probability amplitudes as a list of floating point values is .darkblue[not tractable for large $n$] ] --- # .darkblue[More Quantum Logic Gates...] .more-more-condensed[ - .darkblue[Ising coupling] gates - implemented natively in some trapped-ion quantum computers - <a href="https://doi.org/10.1038/nature18648" target="_blank" class="smaller-font">Debnath et al. Demonstration of a small programmable quantum computer with atomic qubits. Nature 536, 63–66,2016</a> - .darkblue[Imaginary swap] ($iSWAP$) and it root version $\sqrt{iSWAP}$ - for systems with Ising like interactions - <a href="https://doi.org/10.1103%2FPhysRevResearch.2.033097" target="_blank" class="smaller-font">S.E. Rasmussen, N.T. Zinner. Simple implementation of high fidelity controlled- i swap gates and quantum circuit exponentiation of non-Hermitian gates. Physical Review Research. 2(3), 2020</a> - <a href="https://doi.org/10.1103%2FPhysRevA.67.032301" target="_blank" class="smaller-font">N. Schuch, J. Siewert. Natural two-qubit gate for quantum computation using the XY interaction. Physical Review A. 67(3), 2003</a> - <a href="https://doi.org/10.1103%2FPhysRevA.94.062304" target="_blank" class="smaller-font">P.-L. Dallaire-Demers, F.K. Wilhelm. Quantum gates and architecture for the quantum simulation of the Fermi-Hubbard model. Physical Review A. 94 (6), 2016</a> - .darkblue[Deutsch] gate (named after physicist David Deutsch) - some proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms - <a href="https://doi.org/10.1103%2FPhysRevApplied.9.051001" target="_blank" class="smaller-font">X.-F. Shi. Deutsch, Toffoli, and cnot Gates via Rydberg Blockade of Neutral Atoms. Physical Review Applied. 9(5), 2018</a> - .darkblue.bold[...] ] --- # COVID-19 Variant of the .darkblue[El Farol Bar Problem] for simplicity reasons for 4 friends .reference[<a href="https://en.wikipedia.org/wiki/El_Farol_Bar_problem" target="_blank">El Farol Bar problem</a>] .more-more-condensed[ - .darkblue[4 friends want to go to] a .darkblue[restaurant] for lunch. - However, because of COVID-19 confinements, .darkblue[only 3 guests are allowed] to sit at a table of a restaurant. - .darkblue[If all friends go] to this restaurant.darkblue[, then no one will go inside] because of solidarity, because one of them would stay hungry if 3 of them go into the restaurant. - .darkblue[If 3 go to restaurant, then these 3 have most fun compared to] only .darkblue[1, 2 or none of them go to restaurant]. - The friends do not want to decide which of them goes to the restaurant. Therefore, .darkblue[they independently decide at home without communication whether or not to go to] the .darkblue[restaurant]. ] -- <span style="line-height:1"> .small-font.more-more-condensed.margin-top.round-border.blue-border[ After some trials the friends are not satisfied with the procedure. One of the friends studied quantum information theory and proposes: .bold.darkblue["Let's decide it by a quantum computer!"] ] </span> --- # COVID-19 Variant of the .darkblue[El Farol Bar Problem] for simplicity reasons for 4 friends .more-more-condensed.no-margin[ - .darkblue[$k$-th friend gets the $k$-th qubit. If the $k$-th qubit is set, then the $k$-th friend stays at home] and the other friends go to restaurant - Hence the quantum circuit should deliver .darkblue[one of $|0001\rangle, |0010\rangle, |0100\rangle, |1000\rangle$ with equal probability<br/>$\rightsquigarrow$ (General) W-State (for $n$ qubits)] ] -- .more-more-condensed.no-margin[ - .darkblue[Use of two ancilla qubits allowed!] ] -- .more-more-condensed.no-margin[ - .darkblue[Map ancilla state $\frac{|00\rangle+|01\rangle+|10\rangle+|11\rangle}{2}$ to $\frac{|0001\rangle+|0010\rangle+|0100\rangle+|1000\rangle}{2}$] ] -- .reference[<a href="https://web.wpi.edu/Pubs/E-project/Available/E-project-051620-220950/unrestricted/Constructions_and_Applications_of_W-States.pdf" target="_blank">James McClung, Constructions and Applications of W-States, Bachelor Thesis, Worcester Polytechnic Institute, 2020</a>] .more-more-condensed.no-margin[ <img src="../../img/qc/gates/WState.svg" align="top" class="fullwidth"></img> ] --- # .darkblue[Summary and Conclusions] .more-more-condensed[ - .darkblue[Quantum Logic Gates] - Rotation: Pauli ($I, X, Y, Z$), $R_x, R_y, R_z, H, P(\varphi)$ - $SWAP, \sqrt{SWAP}$ - Controlled Gates: $CX, CU, CCNOT, CSWAP$ - Measurement - ... - .darkblue[Bell States via Entanglement] - .darkblue[Universal Quantum Logic Gates] - .darkblue[Unitary Inversion] - .darkblue[Circuit] - .darkblue[Composition] - .darkblue[Simulation] ]