---
layout: page
title: H ゲート
description: 重ね合わせ状態の作成
---

<p>
  H ゲートはフランスの数学者、ジャック・サロモン・アダマール (Jacques Salomon Hadamard, 1865 - 1963)
  の名前を冠したゲートで、別名アダマールゲートとも呼びます{% include margin_note.html
  content="英語では「ハダマードゲート (Hadamard
  gate)」と言います。日本語の「アダマールゲート」は、フランス語の元々の発音 (Hadamard の最初の h
  と最後の d を発音しない) に近いカタカナ表記です。" %}。
</p>

<p>
  H ゲートはシンプルな重ね合わせ状態を作ります。 {% include ket.html label="0" %} または {% include
  ket.html label="1" %} に作用させると、次のように大きさの等しい重ねあわせ状態を作ります。
</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
        id="circle-notation-H0-before"
        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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation
        id="circle-notation-H0-after"
        data-size="xl"
        data-show-kets
      ></circle-notation>
    </div>

    <div class="flex flex-row items-center space-x-1">
      <circle-notation
        id="circle-notation-H1-before"
        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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation
        id="circle-notation-H1-after"
        data-size="xl"
        data-show-kets
      ></circle-notation>
    </div>
  </div>

  <script type="module">
    import { Complex } from './application.js'

    let circleNotationH0before = document.getElementById('circle-notation-H0-before')
    let circleNotationH1before = document.getElementById('circle-notation-H1-before')
    let circleNotationH0after = document.getElementById('circle-notation-H0-after')
    let circleNotationH1after = document.getElementById('circle-notation-H1-after')

    circleNotationH0before.setAmplitudes({
      0: 1,
      1: 0,
    })
    circleNotationH0after.setAmplitudes({
      0: new Complex(Math.sqrt(0.5), 0),
      1: new Complex(Math.sqrt(0.5), 0),
    })
    circleNotationH1before.setAmplitudes({
      0: 0,
      1: 1,
    })
    circleNotationH1after.setAmplitudes({
      0: new Complex(Math.sqrt(0.5), 0),
      1: new Complex(-Math.sqrt(0.5), 0),
    })
  </script>
</div>

<p>
  {% include ket.html label="1" %} に対して H ゲートを適用したときのみ、{% include ket.html
  label="1" %} 位相が <code>π</code> 変化していることに注意してください。{% include ket.html
  label="0" %} に対する H ゲート操作は、位相の回転を含みません。
</p>

<p>
  H ゲートは QPU
  独自の命令です。量子ビットだけが持つ重ね合わせ状態を作るという性質を持つので、従来の論理演算には H
  ゲートに対応するものはありません。
</p>

<h2>ハンズオン</h2>

<p>
  H ゲートを {% include ket.html label="0" %} と {% include ket.html label="1" %}
  に適用して、重ね合わせ状態が作られることを確認しましょう。 {% include ket.html label="0" %}
  は最初から回路にセットされていますが、{% include ket.html label="1" %}
  を作るにはどうしたらいいでしょうか？ (ヒント: X ゲートを使います)
</p>

<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 space-x-2 rounded-xl border-2 border-b-4 border-gray-200 bg-white px-4 py-5"
      >
        <palette-dropzone>
          <h-gate></h-gate>
        </palette-dropzone>
        <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>

<h2>逆演算</h2>

<p>
  H ゲートを続けて 2 回適用するとどうなるでしょうか? 直感的には分かりづらいですが、そんな時には Qni
  で試してみましょう! 実際に H ゲートを 2 個連続して置くと、{% include ket.html label="0" %} も {%
  include ket.html label="1" %} もそれぞれ元の状態に戻ります。 つまり、H ゲートも X
  ゲートと同様にそれ自身を打ち消す効果を持っているので、それ自身の逆演算です。
</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 id="circle-notation-H0-a" 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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation id="circle-notation-H0-b" 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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation id="circle-notation-H0-c" data-size="xl" data-show-kets></circle-notation>
    </div>

    <div class="flex flex-row items-center space-x-1">
      <circle-notation id="circle-notation-H1-a" 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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation id="circle-notation-H1-b" 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" %}
        <h-gate data-size="base"></h-gate>
        {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
      </div>
      <circle-notation id="circle-notation-H1-c" data-size="xl" data-show-kets></circle-notation>
    </div>
  </div>

  <script type="module">
    import { Complex } from './application.js'

    let circleNotationH0a = document.getElementById('circle-notation-H0-a')
    let circleNotationH0b = document.getElementById('circle-notation-H0-b')
    let circleNotationH0c = document.getElementById('circle-notation-H0-c')
    let circleNotationH1a = document.getElementById('circle-notation-H1-a')
    let circleNotationH1b = document.getElementById('circle-notation-H1-b')
    let circleNotationH1c = document.getElementById('circle-notation-H1-c')

    circleNotationH0a.setAmplitudes({
      0: 1,
      1: 0,
    })
    circleNotationH0b.setAmplitudes({
      0: new Complex(Math.sqrt(0.5), 0),
      1: new Complex(Math.sqrt(0.5), 0),
    })
    circleNotationH0c.setAmplitudes({
      0: 1,
      1: 0,
    })

    circleNotationH1a.setAmplitudes({
      0: 0,
      1: 1,
    })
    circleNotationH1b.setAmplitudes({
      0: new Complex(Math.sqrt(0.5), 0),
      1: new Complex(-Math.sqrt(0.5), 0),
    })
    circleNotationH1c.setAmplitudes({
      0: 0,
      1: 1,
    })
  </script>
</div>

<h2>H ゲートを回転として見る</h2>

<p>
  H ゲートを回転として見ると、H ゲートはブロッホ球の X 軸と Z 軸の中間 (黒い矢印) を軸とした 180
  度の回転となります。 たとえば北極の {% include ket.html label="0" %} 状態に H
  ゲートを適用すると、赤い矢印に沿って X の文字の部分まで回転します。
  もう一度適用すると、青い矢印に沿って 180 度回転し元の位置に戻ります (逆演算)。
</p>

<figure>
  <img
    src="./images/bloch_H0.png"
    class="h-auto w-full border border-neutral-300 md:mx-auto md:w-1/3"
  />
  <figcaption class="pt-4 text-center">
    H ゲートの回転 (画像クレジット:
    <a
      href="https://physics.stackexchange.com/questions/313959/visual-interpretation-on-the-bloch-sphere-when-hadamard-gate-is-applied-twice"
      >physics.stackexchange.com</a
    >)
  </figcaption>
</figure>

<h2>任意の重ね合わせ状態に H ゲートを適用</h2>

<p>
  次のように振幅の大きさと位相の異なる重ね合わせ状態に H
  ゲートを適用した場合、その変化はどのように理解するとよいでしょうか？
</p>

<div class="flex flex-row justify-center space-x-4 pb-10">
  <div class="mb-8 flex flex-row items-center space-x-1">
    <circle-notation
      id="circle-notation-H-superposition-before"
      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" %}
      <h-gate data-size="base"></h-gate>
      {% include arrow_right.svg class="h-6 w-6 fill-current text-gray-400" %}
    </div>
    <circle-notation
      id="circle-notation-H-superposition-after"
      data-size="xl"
      data-show-kets
    ></circle-notation>
  </div>

  <script type="module">
    import { Complex } from './application.js'

    let circleNotationBefore = document.getElementById('circle-notation-H-superposition-before')
    let circleNotationAfter = document.getElementById('circle-notation-H-superposition-after')

    circleNotationBefore.setAmplitudes({
      0: new Complex(-0.38268, 0),
      1: new Complex(0.7855, -0.48636),
    })
    circleNotationAfter.setAmplitudes({
      0: new Complex(0.28483, -0.34391),
      1: new Complex(-0.82603, 0.34391),
    })
  </script>
</div>

<p>
  これを頭の中だけで想像するのは大変なので、さしあたっては Qni に計算をおまかせして OK です。
  実は、円の図だけでおおまかな変化を導き出すこともできるのですが、そのためには H
  ゲートを行列で表したもの (ゲートの行列表現) の知識が必要になります。
  詳しい方法はのちほど紹介します。
</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>