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ノイズがある量子振幅推定法の解析と実験 (本文)

田中, 智樹 慶應義塾大学

2022.03.23

概要

次世代のコンピュータの一つとして, 昨今, 量子コンピュータに注目が集まっている. 歴史的には, 1982 年に Richard Phillips Feynman が量子コンピュータの可能性を指摘し [1], 1985 年に David Deutsch が量子コン ピュータを提案した [2]. そして, 1992 年には David Deutsch と Richard Jozsa によって, Deutsch–Jozsa の アルゴリズム [3] が提案され, 既存の決定論的古典アルゴリズムより早い量子アルゴリズムが発見された. そ して, 1994 年には Peter Williston Shor が Shor のアルゴリズムを提案した [4]. これは素因数分解や離散対 数問題を多項式時間で解けるアルゴリズムの提案である. 素因数分解の困難性は, 現在, RSA 暗号 [5] や楕円 曲線 [6, 7] を始めとした様々な暗号方式の計算量的安全性の根拠になっているものであり, 非常に大きな影 響があることから, 量子コンピューティングに注目が集まるきっかけの一つとなっている. その後, Andrew Steane[8] や A. R. Calderbank ら [9] によって量子誤り訂正が, Grover によってデータベース探索を効率よく 行う Grover のアルゴリズム [10] が, Brassard らによって量子状態に埋め込まれた振幅を効率よく推定する振 幅推定手法 [11] など, 様々な量子アルゴリズムが提案されている [12, 13, 14].

他方, ハードウェアの開発も進み, 1999 年には, 中村らによって超電導の量子ビットの実現がなされた [15]. そして, 直近 2016 年には, IBM 社がクラウドベースで超電導量子コンピュータの提供 [16, 17] を始め, 並行し て, Google 社 [18], rigetti 社 [19] も超電導量子コンピュータを開発を進めており, Google 社からは量子超越 [18] に関する発表がされている. 更に, 超電導だけでなく, Honeywell 社 [20] や IonQ 社 [21] 等からはイオン トラップ方式など, 様々な方式での実装が行われている.
しかしながら, これらのデバイスは, まだノイズがあり, 量子ビットの数も限られている量子コンピュータで あり, Preskill は NISQ(Noisy Intermediate-Scale Quantum)[22] と名付けている. 前述の量子コンピュータ は, まだ数十量子ビット程度であったが, 2021 年 11 月には IBM 社から 100 を超える量子ビットを持つ量子コ ンピュータの発表がされ, 日進月歩で発展している. 今後, 大規模化が進み, FTQC(Fault Tolerant Quantum Computer)[23, 24, 25, 26, 27, 28, 29, 30], つまり誤り訂正 [8, 9, 31, 32] された量子コンピュータの実現を 目指し開発が進んでいくことが期待されるが, 一方で, 現状のノイズがある量子コンピュータを使うことも重要 なテーマの一つである. 将来, 誤り訂正ができるようになり, 論理量子ビットを扱う状況であったとしても, 物 理量子ビットではノイズがあり, ノイズに対しての解析が必要なことは明らかである.

前述の通り, 量子コンピュータは, 現状のコンピュータより優位なアルゴリズムがあることが知られいてお り, 主なものを表 1.1 にまとめる. Shor のアルゴリズム [4] や Kitaev による位相推定アルゴリズム [33], 連立 方程式を高速に解く HHL(Harrow-Hassidim-Lloyd) アルゴリズム [34] は指数加速が, Grover のアルゴリズム [10] や振幅推定アルゴリズム [11] は二乗加速することが知られている. これらは, 現状知られている古典的な 計算に対しての計算量での優位性が保証されている一方で, これらのアルゴリズムを使うためには, 相応のハードウェアリソースつまり, 量子ビットや量子ゲート操作が必要となるため, 現在のノイズのある量子コンピュー タで実行することは簡単ではないと考えられている. そこで, 本論文では, 特にノイズのある量子コンピュータ で振幅推定アルゴリズムについて述べる.

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