---
layout: page
title: 量子 NOT ゲート (X ゲート)
description: 量子ビットペアの入れ替え
---

<p>
  量子 NOT ゲートは従来の <span class="font-bold">NOT 演算</span>に似た働きをします。 NOT
  演算は、<code>0</code> に適用するとビットが反転して <code>1</code> になり、逆に
  <code>1</code> に適用すると <code>0</code> になるのでした。つまり、NOT 演算は &quot;<code
    class="text-purple-600"
    >0</code
  >
  と <code>1</code> を入れ替える&quot; という働きをします。
</p>

<div class="flex justify-center pb-10">
  <ul class="text-2xl">
    <li class="flex flex-row items-center">
      <code>NOT 0</code> {% include arrow_right.svg class="h-4 w-4 mx-2" %} <code>1</code>
    </li>
    <li class="flex flex-row items-center">
      <code>NOT 1</code> {% include arrow_right.svg class="h-4 w-4 mx-2" %} <code>0</code>
    </li>
  </ul>
</div>

<p>
  <span class="font-bold">量子 NOT ゲート</span>は NOT 演算によく似ていて、量子ビットの {% include
  ket.html label="0" %} と {% include ket.html label="1" %} を入れ替えます。 図で表すと、次のような
  "円の入れ替え" の働きをします。
</p>

<div class="flex justify-center pb-10">
  <div class="flex flex-col">
    <div class="mb-8 flex flex-row items-center space-x-1">
      <circle-notation data-size="xl" data-show-kets></circle-notation>
      <div class="flex flex-row items-center">
        {% include minus.svg class="h-6 w-6 fill-current text-gray-400" %}
        <x-gate data-size="base"></x-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation data-size="xl" data-magnitudes="0,1" data-show-kets></circle-notation>
    </div>

    <div class="flex flex-row items-center space-x-1">
      <circle-notation data-size="xl" data-magnitudes="0,1" data-show-kets></circle-notation>
      <div class="flex flex-row items-center">
        {% include minus.svg class="h-6 w-6 fill-current text-gray-400" %}
        <x-gate data-size="base"></x-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation data-size="xl" data-show-kets></circle-notation>
    </div>
  </div>
</div>

<div class="mb-10 border-l-4 border-blue-400 bg-blue-50 p-4">
  量子 NOT ゲートを回路に置いて、{% include ket.html label="0" %} と {% include ket.html label="1"
  %} が入れ替わることを確認しましょう。
</div>

<div class="relative mb-10 overflow-hidden rounded-2xl bg-gray-50 p-8">
  <quantum-simulator class="flex flex-col" data-service-worker="/serviceworker.js">
    <circuit-editor>
      <div
        id="palette"
        class="mb-10 flex w-min rounded-xl border-2 border-b-4 border-gray-200 bg-white px-4 py-5"
      >
        <palette-dropzone>
          <x-gate></x-gate>
        </palette-dropzone>
      </div>

      <div
        class="absolute top-px right-px z-40 rounded-bl-2xl rounded-tr-2xl bg-white drop-shadow-xl"
      >
        <inspector-button
          data-target="circuit-editor.inspectorButton"
          class="inline-block h-10 w-10 p-1"
        >
          <div class="flex h-full w-full items-center justify-center rounded-xl hover:bg-gray-200">
            {% include mixer_horizontal.svg class="h-4 w-4 text-gray-700" %}
          </div>
        </inspector-button>
      </div>

      <quantum-circuit
        class="mb-10"
        data-target="quantum-simulator.circuit circuit-editor.circuit"
        data-min-wire-count="1"
        data-max-wire-count="1"
        data-min-step-count="5"
        data-json='{"cols":[["|0>"]]}'
      >
      </quantum-circuit>

      <circle-notation
        id="circle-notation"
        data-target="quantum-simulator.circleNotation"
        class="z-40 flex w-min flex-row justify-center rounded-xl border-2 border-b-4 border-gray-200 bg-white px-4 py-4"
        data-multi-qubits
        data-show-kets
      >
      </circle-notation>
    </circuit-editor>
  </quantum-simulator>
</div>

<p>
  量子 NOT ゲートが従来の NOT 演算と異なるのは、重ね合わせ状態にある量子ビットに作用できる点です。
  たとえば、次の {% include ket.html label="0" %} と {% include ket.html label="1" %}
  の重ね合わせ状態にある量子ビットを考えましょう。
</p>

<div class="flex justify-center pb-10">
  <circle-notation
    data-size="xl"
    data-magnitudes="0.8944271909999159,0.4472135954999579"
    data-phases="-20,70"
    data-show-kets
  ></circle-notation>
</div>

<p>
  これに量子 NOT ゲートを適用すると、{% include ket.html label="0" %} と {% include ket.html
  label="1" %} の内容を交換します。 これはまさに、先ほど見た "{% include ket.html label="0" %} と {%
  include ket.html label="1" %} の入れ換え" という効果です。
</p>

<div class="flex justify-center pb-10">
  <div class="mt-4 mb-2 flex flex-row items-center space-x-1">
    <circle-notation
      data-size="xl"
      data-magnitudes="0.8944271909999159,0.4472135954999579"
      data-phases="-20,70"
      data-show-kets
    ></circle-notation>
    <div class="flex flex-row items-center">
      {% include minus.svg class="h-6 w-6 fill-current text-gray-400" %}
      <x-gate data-size="base"></x-gate>
      {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
    </div>
    <circle-notation
      data-size="xl"
      data-magnitudes="0.4472135954999579,0.8944271909999159"
      data-phases="70,-20"
      data-show-kets
    ></circle-notation>
  </div>
</div>

<p>
  量子 NOT ゲートを続けて 2 回適用するとどうなるでしょうか? 図から分かるように、 {% include ket.html
  label="0" %} と {% include ket.html label="1" %} が 2 度入れ替わることで元の状態に戻ります。
  つまり、量子 NOT ゲートはそれ自身を打ち消す効果を持っています。数学ではこれを、「量子 NOT
  ゲートはそれ自身の逆演算である」と言います。
</p>

<div class="flex justify-center pb-10">
  <div class="mt-4 mb-2 flex flex-row items-center space-x-1">
    <circle-notation
      data-size="xl"
      data-magnitudes="0.8944271909999159,0.4472135954999579"
      data-phases="-20,70"
      data-show-kets
    ></circle-notation>
    <div class="flex flex-row items-center">
      {% include minus.svg class="h-6 w-6 fill-current text-gray-400" %}
      <x-gate data-size="base"></x-gate>
      {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
    </div>
    <circle-notation
      data-size="xl"
      data-magnitudes="0.4472135954999579,0.8944271909999159"
      data-phases="70,-20"
      data-show-kets
    ></circle-notation>
    <div class="flex flex-row items-center">
      {% include minus.svg class="h-6 w-6 fill-current text-gray-400" %}
      <x-gate data-size="base"></x-gate>
      {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
    </div>
    <circle-notation
      data-size="xl"
      data-magnitudes="0.8944271909999159,0.4472135954999579"
      data-phases="-20,70"
      data-show-kets
    ></circle-notation>
  </div>
</div>

<div class="mb-10 border-l-4 border-blue-400 bg-blue-50 p-4">
  <ul class="ml-4 list-disc">
    <li>
      重ね合わせ状態の量子ビットに量子 NOT ゲートを適用し、{% include ket.html label="0" %} と {%
      include ket.html label="1" %} が入れ替わることを確認しましょう。
    </li>
    <li>量子 NOT ゲートを 2 回適用すると元に戻ることを確認しましょう</li>
  </ul>
</div>

<div class="relative mb-10 overflow-hidden rounded-2xl bg-gray-50 p-8">
  <quantum-simulator class="flex flex-col" data-service-worker="/serviceworker.js">
    <circuit-editor>
      <div
        id="palette"
        class="border-swan mb-10 flex w-min rounded-xl border-2 border-b-4 bg-white px-4 py-5"
      >
        <palette-dropzone>
          <x-gate></x-gate>
        </palette-dropzone>
      </div>

      <div
        class="absolute top-px right-px z-40 rounded-bl-2xl rounded-tr-2xl bg-white drop-shadow-xl"
      >
        <inspector-button
          data-target="circuit-editor.inspectorButton"
          class="inline-block h-10 w-10 p-1"
        >
          <div class="flex h-full w-full items-center justify-center rounded-xl hover:bg-gray-200">
            {% include mixer_horizontal.svg class="h-4 w-4 text-gray-700" %}
          </div>
        </inspector-button>
      </div>

      <quantum-circuit
        class="mb-10"
        data-target="quantum-simulator.circuit circuit-editor.circuit"
        data-min-wire-count="1"
        data-max-wire-count="1"
        data-min-step-count="8"
        data-json='{"cols":[["{重ね合わせ状態の準備"],["|0>"],["H"],["Ry(3π_4)"],["P(31π_17)"],["}"]]}'
      >
      </quantum-circuit>

      <circle-notation
        id="circle-notation"
        data-target="quantum-simulator.circleNotation"
        class="border-swan z-40 flex w-min flex-row justify-center rounded-xl border-2 border-b-4 bg-white px-4 py-4"
        data-multi-qubits
        data-show-kets
      >
      </circle-notation>
    </circuit-editor>
  </quantum-simulator>
</div>

<p>
  量子 NOT ゲートはよく <span class="font-bold">X ゲート</span>とも呼ばれます。
  その由来は、量子ビット状態をブロッホ球表すと量子 NOT ゲートは X
  軸を中心とした回転操作の一種だからです。
</p>

<figure>
  <img
    src="./images/rotation-by-pi-around-x-axis.png"
    class="h-auto w-full border border-neutral-300 md:mx-auto md:w-2/3"
  />
  <figcaption class="pt-4 text-center">
    X ゲートは X 軸を中心に π (180度) 回転する操作 (画像クレジット:
    <a href="https://qiskit.org/textbook/ch-states/single-qubit-gates.html#xgate">qiskit.org</a>)
  </figcaption>
</figure>

<p>
  同様に、Y 軸で回転する Y ゲート、Z 軸で回転する Z
  ゲートも存在します。さまざまなゲートの回転については、他のゲートを学んだ後で詳しく説明します。
</p>

<template id="operation-menu">
  <div class="operation-menu flex flex-row items-center space-x-2 text-gray-900">
    <div class="flex flex-row rounded-2xl bg-white p-2 drop-shadow-lg">
      <button
        data-operation-menu-if
        type="button"
        class="inline-flex h-8 w-8 items-center justify-center rounded border border-transparent bg-white hover:bg-gray-200 focus:outline-none"
        disabled
      >
        <div class="flex h-5 w-5 items-center justify-center">
          <div class="whitespace-nowrap font-sans text-sm font-bold tracking-tighter">
            if <span class="italic">x</span>
          </div>
        </div>
      </button>

      <button
        data-operation-menu-angle
        type="button"
        class="inline-flex h-8 w-8 items-center justify-center rounded border border-transparent bg-white hover:bg-gray-200 focus:outline-none"
        disabled
      >
        {% include angle.svg class="h-5 w-5" %}
      </button>

      <button
        data-operation-menu-flag
        type="button"
        class="inline-flex h-8 w-8 items-center justify-center rounded border border-transparent bg-white hover:bg-gray-200 focus:outline-none"
        disabled
      >
        <div class="flex h-5 w-5 items-center justify-center">
          <div class="whitespace-nowrap font-sans text-sm font-bold">
            (<span class="italic">x</span>)
          </div>
        </div>
      </button>
    </div>

    <div
      data-operation-menu-delete
      class="flex h-8 w-8 cursor-pointer items-center justify-center rounded-full bg-white drop-shadow-lg hover:bg-red-200"
    >
      {% include trash.svg class="h-4 w-4" %}
    </div>
  </div>
</template>

<template id="operation-inspector-template">
  <operation-inspector class="flex w-60 flex-col p-4 text-gray-900">
    <div class="items-left flex flex-col space-y-4 divide-y">
      <div id="conditional-gate-pane" class="items-left flex flex-col">
        <h1 class="font-bold text-gray-900">If Conditional</h1>

        <div class="pane-body ml-4 mt-4 grid grid-cols-3 grid-rows-1 items-center gap-2">
          <label for="if-input" class="select-none font-normal text-gray-700">If</label>
          <input
            data-target="operation-inspector.ifInput"
            type="text"
            name="if"
            id="if-input"
            spellcheck="false"
            class="block w-full rounded-md border-transparent bg-gray-100 py-1 pl-2 pr-1 shadow-sm focus:border-indigo-500 focus:ring-indigo-500 sm:text-sm"
            autocomplete="off"
          />
        </div>
      </div>

      <div id="angle-pane" class="items-left mb-2 flex flex-col pt-4">
        <h1 class="font-bold text-gray-900">Angle</h1>

        <div class="pane-body ml-4 mr-2 mt-4 grid grid-cols-3 grid-rows-3 items-center gap-2">
          <label
            for="angle-input"
            class="select-none font-normal text-gray-700"
            data-target="operation-inspector.angleInputLabel"
            >Angle</label
          >
          <input
            data-target="operation-inspector.angleInput"
            type="text"
            name="angle"
            id="angle-input"
            spellcheck="false"
            class="block w-full rounded-md border-transparent bg-gray-100 py-1 pl-2 pr-1 shadow-sm focus:border-indigo-500 focus:ring-indigo-500 sm:text-sm"
            autocomplete="off"
          />
          <angle-slider
            data-target="operation-inspector.angleSlider"
            class="ml-2 inline-block"
          ></angle-slider>

          <label for="denominator-input" class="block text-sm font-normal text-gray-700"
            >Denom</label
          >
          <input
            data-target="operation-inspector.denominatorInput"
            type="number"
            min="2"
            name="denominator"
            id="denominator-input"
            class="block w-full rounded-md border-transparent bg-gray-100 py-1 pl-2 pr-1 shadow-sm focus:border-indigo-500 focus:ring-indigo-500 sm:text-sm"
            autocomplete="off"
          />
          <div class="whitespace-nowrap pl-2 text-sm text-gray-500">
            <span data-target="operation-inspector.denominatorVariableLabel"></span> = nπ/<span
              data-target="operation-inspector.denominatorLabel"
              class="font-bold text-yellow-600"
              >?</span
            >
          </div>

          <div class="min-w-0 flex-1 text-sm">
            <label for="reduce-fraction-checkbox" class="select-none font-normal text-gray-700"
              >Reduce</label
            >
          </div>
          <div class="flex h-5 items-center">
            <input
              data-target="operation-inspector.reduceAngleFractionCheckbox"
              id="reduce-fraction-checkbox"
              name="reduce-fraction"
              type="checkbox"
              class="h-4 w-4 rounded border-gray-300 text-indigo-600 focus:ring-indigo-500"
            />
          </div>
        </div>
      </div>

      <div id="conditional-flag-pane" class="items-left mb-2 flex flex-col pt-4">
        <h1 class="font-bold text-gray-900">Conditional Flag</h1>

        <div class="pane-body ml-4 mt-4 grid grid-cols-3 grid-rows-1 items-center gap-2">
          <label for="flag-input" class="select-none font-normal text-gray-700">Flag</label>
          <input
            data-target="operation-inspector.flagInput"
            type="text"
            name="flag"
            id="flag-input"
            spellcheck="false"
            class="block w-full rounded-md border-transparent bg-gray-100 py-1 pl-2 pr-1 shadow-sm focus:border-indigo-500 focus:ring-indigo-500 sm:text-sm"
            autocomplete="off"
          />
        </div>
      </div>
    </div>
  </operation-inspector>
</template>

<template id="circle-notation-popup">
  <ul class="list-none">
    <li>
      振幅:
      <span class="circle-notation-popup__amplitude-real text-eel font-mono font-bold"></span>
      <span class="circle-notation-popup__amplitude-imag text-eel font-mono font-bold"></span>
    </li>
    <li>
      確率:
      <span class="circle-notation-popup__probability text-eel font-mono font-bold"></span>
    </li>
    <li>
      位相:
      <span class="circle-notation-popup__phase text-eel font-mono font-bold"></span>
    </li>
  </ul>
</template>