What Discretises a Continuum of Errors Is Not the Code but the Measurement ── Eight Real Parameters Fall to Four Outcomes ── A Distance-3 Code Covers 28 of 262144, That Is 1.07x10^-4 ── [Paper 287]
The errors that can enter a qubit form a continuum. Why can a code with finitely many correctable errors fix them? This paper asks who does the discretising──the answer is the measurement, not the code. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): No new mathematical theorem and no new law is claimed──error discretisation, the Pauli basis, stabiliser codes, surface codes, and the threshold theorem are all standard. We do not construct codes──all we count is numbers of parameters and numbers of errors. We do not prove the threshold theorem──the value p_thapprox 10^-2 is quoted as a number only. We do not derive the logical error rate──(p/p_th)^(d+1)/2 is an asymptotic estimate that drops constant factors. The values of Section 5 are orders, not predictions. We do not discuss surface-code implementation──lattice geometry, measurement circuits, and decoders are not treated at all. The count approx d^2 physical qubits is the simplest estimate. We do not treat correlated errors──the count of Section 4 assumes independent single-qubit errors. With correlations, errors leave the weight-1 bracket. We do not read 1.068x10^-4 pessimistically──it is a fraction of all Pauli operators, not a fraction of the errors that actually occur. If real errors concentrate at weight 1, then 28 suffices in practice. This paper does not assert which is the case. On the sibling paper: Paper 281 treats the same quantum error correction and counts that the syndrome tells 8 bits about the error and 0 bits about the state──the root is the same (measurement does not destroy the state) but the question differs. That paper asks what the measurement does not take; this one asks what the measurement makes discrete. Readers are advised to read them side by side. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”──this paper counts what a constructed code actually covers. Paper 179 showed that “cannot be constructed” has distinct roots──the impossibility of d=4 here likewise has counting as its root. Paper 252 counted four quantities called “information”──this paper separates continuous parameters from discrete outcomes. Paper 269 showed that what a crystal forbids is periodicity──the discretisation here is likewise imposed by a structure. What is added is counting a one-qubit error as 8 real parameters falling to 4 outcomes after measurement, computing that distance 3 covers 28 of 4^9=262144, a fraction of 1.068x10^-4, confirming that 3.24 times in the surface code corresponds to one hundredth, and placing the difference from the classical case at “whether a projection is required”. First, we count the errors on one qubit. E=c_0I+c_1X+c_2Y+c_3Z has 8 real parameters and is a continuum (Section 2). Second, this is the core of the paper. The measurement projects that onto the 4 outcomes I,X,Y,Z──the discretising is done by the measurement, not the code (Section 2). Third, fixing X and Z fixes Y. Since Y=iXZ, only 2 kinds need independent correction (Section 3). Fourth, a code covers only a sliver. Of the 4^9=262144 Pauli operators on 9 qubits, distance 3 fixes 28, a fraction of 1.068x10^-4 (Section 4). Fifth, in the surface code, 3.24 times gives one hundredth. From d=5 to 9, physical qubits 25->81 and logical error rate 10^-3->10^-5 (Section 5). Sixth, the separator is whether a projection is needed. Classical codes need none──because their errors are discrete to begin with (Section 6). The familiar statement that “continuous errors get discretised” is right. But what discretises is the measurement, not the code. A one-qubit error has 8 real parameters and forms a continuum, yet the syndrome measurement projects it onto the 4 outcomes I,X,Y,Z──what remains is a single Pauli error, and undoing it suffices. Since Y=iXZ, only 2 kinds need independent correction. Meanwhile the reach of a code is narrow──of the 262144 Pauli errors on 9 qubits, distance 3 fixes 28, a mere 1.068x10^-4. It works because the assumption that errors are sparse is doing the work. And the price is polynomial──multiplying the physical qubits by 3.24 divides the logical error rate by 100. One thing separates them──whether the errors are discrete to begin with, or a projection is required. A classical code needs no projection. In a quantum code, measuring is a precondition of correcting, and the separate fact that measuring does not destroy the state is what makes it possible. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 量子ビットに入る誤りは連続無限にある。それを有限個の符号で訂正できるのは、なぜか。本稿が問うのは、離散化しているのは誰かである──答は、符号ではなく測定である。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない──誤りの離散化、パウリ基底、安定符号、表面符号、閾値定理は、いずれも標準的である。符号を作らない──数えるのはパラメータの個数と、誤りの個数だけである。閾値定理を証明しない──p_thapprox 10^-2 という値は数値として引くのみである。論理誤り率の式を導出しない──(p/p_th)^(d+1)/2 は漸近形の目安であり、定数因子を無視している。第5節の値は「桁」であって予言値ではない。表面符号の実装を論じない──格子の形も、測定回路も、デコーダも一切扱わない。物理量子ビット数 approx d^2 は最も単純な見積りである。相関誤りを扱わない──第4節の数え上げは独立な単一量子ビット誤りを仮定している。相関があれば重み 1 の枠から外れる。1.068x10^-4 を悲観的に読まない──これは全パウリ演算子に対する割合であって、実際に起きる誤りに対する割合ではない。起きやすい誤りが重み 1 に集中していれば実用上は十分である。本稿はどちらが実際かを主張しない。兄弟編について:論文281 は同じ量子誤り訂正を扱い、シンドロームが誤りについて 8 ビット・状態について 0 ビット語ることを数えた──根は同じ(測定が状態を壊さない)だが、問いが違う。あちらは測定が何を取らないか、本稿は測定が何を離散にするかである。読者は両方を並べて読むことを勧める。既刊との関係:論文163 は「良い符号が在る」と「これが良い符号だ」を分けた──本稿は構成された符号が実際に何個の誤りを覆うかを数える。論文179 は「構成できない」に別根があると示した──本稿の d=4 不可も数え上げが根である。論文252 は「情報量」が四つあると数えた──本稿は連続なパラメータと離散な結果を分ける。論文269 は結晶が禁じるのが周期性だと示した──本稿の離散化も構造が課すものである。加えたのは一量子ビットの誤りが実パラメータ 8 つで、測定後に 4 通りになると数えたこと、4^9=262144 に対して距離 3 が覆うのが 28 個・1.068x10^-4 だと計算したこと、表面符号で 3.24 倍が 100 分の一に対応すると確かめたこと、古典との差を「射影が要るかどうか」に置いたことである。 第一に、一量子ビットの誤りを数える。 E=c_0I+c_1X+c_2Y+c_3Z には実パラメータが 8 つあり、連続無限である(第2節)。 第二に、これが本稿の芯である。測定がそれを I,X,Y,Z の 4 通りに射影する──離散にしているのは測定であって、符号ではない(第2節)。 第三に、X と Z を直せば Y も直る。 Y=iXZ だから、独立に直すべきは 2 種類だけである(第3節)。 第四に、符号が直せるのはごく一部である。9 量子ビットのパウリ演算子 4^9=262144 個のうち、距離 3 が直すのは 28 個、割合は 1.068x10^-4(第4節)。 第五に、表面符号では 3.24 倍で 100 分の一である。 d=5->9、物理量子ビット 25->81 で論理誤り率 10^-3->10^-5(第5節)。 第六に、分離子は「射影があるか」である。古典符号には射影が要らない──誤りが最初から離散だからである(第6節)。 量子誤り訂正でよく言われる「連続な誤りが離散化される」は正しい。ただし離散化しているのは符号ではなく、測定である。一量子ビットの誤りには実パラメータが 8 つあり、連続無限であるが、シンドローム測定が I,X,Y,Z の 4 通りに射影する──そのあとに残るのは一つのパウリ誤りであり、打ち消せばよい。 Y=iXZ なので、独立に直すべきは 2 種類である。一方で、符号が覆う範囲は狭い──9 量子ビットのパウリ誤り 262144 通りのうち、距離 3 が直せるのは 28 通り、1.068x10^-4 にすぎない。それでも動くのは誤りが疎だという仮定が効いているからである。そして代償は多項式で済む──物理量子ビットを 3.24 倍にすると、論理誤り率が 100 分の一になる。分けるものは一つ──誤りが最初から離散か、射影を要するか。古典符号に射影は要らない。量子符号では、測ることが訂正の前提であり、測ることが状態を壊さないという別の事実が、それを可能にしている。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
The errors that can enter a qubit form a continuum. Why can a code with finitely many correctable errors fix them? This paper asks who does the discretising──the answer is the measurement, not the code. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): No new mathematical theorem and no new law is claimed──error discretisation, the Pauli basis, stabiliser codes, surface codes, and the threshold theorem are all standard. We do not construct codes──all we count is numbers of parameters and numbers of errors. We do not prove the threshold theorem──the value p_thapprox 10^-2 is quoted as a number only. We do not derive the logical error rate──(p/p_th)^(d+1)/2 is an asymptotic estimate that drops constant factors. The values of Section 5 are orders, not predictions. We do not discuss surface-code implementation──lattice geometry, measurement circuits, and decoders are not treated at all. The count approx d^2 physical qubits is the simplest estimate. We do not treat correlated errors──the count of Section 4 assumes independent single-qubit errors. With correlations, errors leave the weight-1 bracket. We do not read 1.068x10^-4 pessimistically──it is a fraction of all Pauli operators, not a fraction of the errors that actually occur. If real errors concentrate at weight 1, then 28 suffices in practice. This paper does not assert which is the case. On the sibling paper: Paper 281 treats the same quantum error correction and counts that the syndrome tells 8 bits about the error and 0 bits about the state──the root is the same (measurement does not destroy the state) but the question differs. That paper asks what the measurement does not take; this one asks what the measurement makes discrete. Readers are advised to read them side by side. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”──this paper counts what a constructed code actually covers. Paper 179 showed that “cannot be constructed” has distinct roots──the impossibility of d=4 here likewise has counting as its root. Paper 252 counted four quantities called “information”──this paper separates continuous parameters from discrete outcomes. Paper 269 showed that what a crystal forbids is periodicity──the discretisation here is likewise imposed by a structure. What is added is counting a one-qubit error as 8 real parameters falling to 4 outcomes after measurement, computing that distance 3 covers 28 of 4^9=262144, a fraction of 1.068x10^-4, confirming that 3.24 times in the surface code corresponds to one hundredth, and placing the difference from the classical case at “whether a projection is required”. First, we count the errors on one qubit. E=c_0I+c_1X+c_2Y+c_3Z has 8 real parameters and is a continuum (Section 2). Second, this is the core of the paper. The measurement projects that onto the 4 outcomes I,X,Y,Z──the discretising is done by the measurement, not the code (Section 2). Third, fixing X and Z fixes Y. Since Y=iXZ, only 2 kinds need independent correction (Section 3). Fourth, a code covers only a sliver. Of the 4^9=262144 Pauli operators on 9 qubits, distance 3 fixes 28, a fraction of 1.068x10^-4 (Section 4). Fifth, in the surface code, 3.24 times gives one hundredth. From d=5 to 9, physical qubits 25->81 and logical error rate 10^-3->10^-5 (Section 5). Sixth, the separator is whether a projection is needed. Classical codes need none──because their errors are discrete to begin with (Section 6). The familiar statement that “continuous errors get discretised” is right. But what discretises is the measurement, not the code. A one-qubit error has 8 real parameters and forms a continuum, yet the syndrome measurement projects it onto the 4 outcomes I,X,Y,Z──what remains is a single Pauli error, and undoing it suffices. Since Y=iXZ, only 2 kinds need independent correction. Meanwhile the reach of a code is narrow──of the 262144 Pauli errors on 9 qubits, distance 3 fixes 28, a mere 1.068x10^-4. It works because the assumption that errors are sparse is doing the work. And the price is polynomial──multiplying the physical qubits by 3.24 divides the logical error rate by 100. One thing separates them──whether the errors are discrete to begin with, or a projection is required. A classical code needs no projection. In a quantum code, measuring is a precondition of correcting, and the separate fact that measuring does not destroy the state is what makes it possible. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 量子ビットに入る誤りは連続無限にある。それを有限個の符号で訂正できるのは、なぜか。本稿が問うのは、離散化しているのは誰かである──答は、符号ではなく測定である。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない──誤りの離散化、パウリ基底、安定符号、表面符号、閾値定理は、いずれも標準的である。符号を作らない──数えるのはパラメータの個数と、誤りの個数だけである。閾値定理を証明しない──p_thapprox 10^-2 という値は数値として引くのみである。論理誤り率の式を導出しない──(p/p_th)^(d+1)/2 は漸近形の目安であり、定数因子を無視している。第5節の値は「桁」であって予言値ではない。表面符号の実装を論じない──格子の形も、測定回路も、デコーダも一切扱わない。物理量子ビット数 approx d^2 は最も単純な見積りである。相関誤りを扱わない──第4節の数え上げは独立な単一量子ビット誤りを仮定している。相関があれば重み 1 の枠から外れる。1.068x10^-4 を悲観的に読まない──これは全パウリ演算子に対する割合であって、実際に起きる誤りに対する割合ではない。起きやすい誤りが重み 1 に集中していれば実用上は十分である。本稿はどちらが実際かを主張しない。兄弟編について:論文281 は同じ量子誤り訂正を扱い、シンドロームが誤りについて 8 ビット・状態について 0 ビット語ることを数えた──根は同じ(測定が状態を壊さない)だが、問いが違う。あちらは測定が何を取らないか、本稿は測定が何を離散にするかである。読者は両方を並べて読むことを勧める。既刊との関係:論文163 は「良い符号が在る」と「これが良い符号だ」を分けた──本稿は構成された符号が実際に何個の誤りを覆うかを数える。論文179 は「構成できない」に別根があると示した──本稿の d=4 不可も数え上げが根である。論文252 は「情報量」が四つあると数えた──本稿は連続なパラメータと離散な結果を分ける。論文269 は結晶が禁じるのが周期性だと示した──本稿の離散化も構造が課すものである。加えたのは一量子ビットの誤りが実パラメータ 8 つで、測定後に 4 通りになると数えたこと、4^9=262144 に対して距離 3 が覆うのが 28 個・1.068x10^-4 だと計算したこと、表面符号で 3.24 倍が 100 分の一に対応すると確かめたこと、古典との差を「射影が要るかどうか」に置いたことである。 第一に、一量子ビットの誤りを数える。 E=c_0I+c_1X+c_2Y+c_3Z には実パラメータが 8 つあり、連続無限である(第2節)。 第二に、これが本稿の芯である。測定がそれを I,X,Y,Z の 4 通りに射影する──離散にしているのは測定であって、符号ではない(第2節)。 第三に、X と Z を直せば Y も直る。 Y=iXZ だから、独立に直すべきは 2 種類だけである(第3節)。 第四に、符号が直せるのはごく一部である。9 量子ビットのパウリ演算子 4^9=262144 個のうち、距離 3 が直すのは 28 個、割合は 1.068x10^-4(第4節)。 第五に、表面符号では 3.24 倍で 100 分の一である。 d=5->9、物理量子ビット 25->81 で論理誤り率 10^-3->10^-5(第5節)。 第六に、分離子は「射影があるか」である。古典符号には射影が要らない──誤りが最初から離散だからである(第6節)。 量子誤り訂正でよく言われる「連続な誤りが離散化される」は正しい。ただし離散化しているのは符号ではなく、測定である。一量子ビットの誤りには実パラメータが 8 つあり、連続無限であるが、シンドローム測定が I,X,Y,Z の 4 通りに射影する──そのあとに残るのは一つのパウリ誤りであり、打ち消せばよい。 Y=iXZ なので、独立に直すべきは 2 種類である。一方で、符号が覆う範囲は狭い──9 量子ビットのパウリ誤り 262144 通りのうち、距離 3 が直せるのは 28 通り、1.068x10^-4 にすぎない。それでも動くのは誤りが疎だという仮定が効いているからである。そして代償は多項式で済む──物理量子ビットを 3.24 倍にすると、論理誤り率が 100 分の一になる。分けるものは一つ──誤りが最初から離散か、射影を要するか。古典符号に射影は要らない。量子符号では、測ることが訂正の前提であり、測ることが状態を壊さないという別の事実が、それを可能にしている。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
A quantum state cannot be copied (the no-cloning theorem). Yet quantum error correction works. To correct one must know the error; to know it one must measure; and measuring destroys the state──so it would seem. This paper asks why these two are compatible──the answer is that what is measured is not the state but the error. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): No new mathematical theorem and no new law is claimed──the no-cloning theorem, Shor’s 9-qubit code, the 5-qubit code, BB84, the 25% of intercept-resend, and the 11% of Shor--Preskill are all standard. We do not construct codes──all we count is the number of syndromes and the number of errors to be corrected. We do not prove the no-cloning theorem──we merely quote it. We give no security proof──the 11.0028% of Shor--Preskill is quoted as a number only, and we do not enter the proof. We do not discuss real quantum key distribution──loss, dark counts, multi-photon components, and device attacks are not treated at all. The 25% of this paper is a value for ideal intercept-resend only. We do not claim the best attack──intercept-resend is not the best attack. A cleverer attack obtains the same information at a lower error rate. 25% is a number for “how conspicuous a naive attack is”. We do not treat finite-key effects──the 0.75^73 of Section 5 is the miss probability under full interception, not a complete assessment of statistical security. We do not discuss quantum computation──neither Shor’s factoring nor the threshold theorem is treated. Relation to earlier papers: Paper 163 separated “a good code exists” from “here is a good code”──this paper stands on the construction side and counts the syndromes of specific codes. Paper 252 counted four quantities called “information”──the 8 and the 0 counted here are the bits one and the same measurement returns about two different objects, a separation of objects, not of kinds of quantity. Paper 188 treated cases exactly determined by fewer measurements than unknowns──this paper is the converse: measuring exactly what is needed and nothing else. Paper 179 showed that “cannot be constructed” has distinct roots──no-cloning is one such “cannot construct,” and this paper uses as a premise that its root is linearity. Paper 68 read the qubit as a Hopf fibration──this paper counts the same qubit as a number of bits of information. What is added is counting the bits the syndrome tells, separated as “8 about the error, 0 about the state”, confirming that the 5-qubit code saturates exactly at 16=16, setting beside it the fact that the same reasoning yields the 25% of BB84, and computing that 25% is 2.2722 times the security limit. First, we count the bits the syndrome tells. Shor’s 9-qubit code tells 8 bits about the error and 0 bits about the logical state (Section 2). Second, the 5-qubit code saturates exactly. Against 16 syndromes there are 16 errors to correct──not one to spare (Section 3). Third, the same reasoning turns eavesdropping into a number. Under BB84 intercept-resend, the error rate is exactly 25% (Section 4). Fourth, 25% is 2.27 times the security limit. The one-way post-processing limit is 11.0028%, so it is certain to be detected (Section 5). Fifth, checking 73 bits drops the miss probability below 10^-9. That is 0.75^73 (Section 5). Sixth, the separator is that security does not rest on computational cost. However fast computers become, 25% stays 25% (Section 6). It cannot be copied yet can be corrected because what is measured is not the state. The syndrome of Shor’s 9-qubit code tells 8 bits about the error and 0 bits about the logical state──because only 0 bits are taken, the price is also 0. Enlarging the code does not increase that 0──Shor, Steane, and the 5-qubit code are equally 0 about the state. And in the 5-qubit code, 16 syndromes correspond one-to-one to 16 errors. The obverse of the same reasoning turns eavesdropping into a number──Eve tries to take about the state, so a footprint of 25% remains. That is 2.2722 times the security limit of 11.0028%, and checking 73 bits drops the miss probability below 10^-9. One thing separates them──counting how many bits that measurement takes about the state. Count, and correction and eavesdropping are seen as two faces of one principle. Fail to count, and the two look contradictory. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- 量子状態は複製できない(複製不可能定理)。ところが量子誤り訂正は動く。訂正するには誤りを知らねばならず、知るには測らねばならず、測れば状態は壊れる──はずである。本稿が問うのは、この二つがなぜ両立するかである──答は、測っているのが状態ではなく誤りだからである。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない──複製不可能定理、ショアの 9 量子ビット符号、5 量子ビット符号、BB84、傍受再送攻撃の 25%、ショア=プレスキルの 11% は、いずれも標準的である。符号を作らない──数えるのはシンドロームの個数と、訂正すべき誤りの個数だけである。複製不可能定理を証明しない──引くだけである。安全性証明をしない──ショア=プレスキルの 11.0028% は数値として引くのみで、証明には立ち入らない。現実の量子鍵配送を論じない──損失、暗黒計数、多光子成分、装置攻撃を一切扱わない。本稿の 25% は理想的な傍受再送のみの値である。最良の攻撃を主張しない──傍受再送は最良の攻撃ではない。より賢い攻撃は同じ情報量をより少ない誤り率で得る。25% は「素朴な攻撃がどれだけ目立つか」の数である。有限鍵効果を扱わない──第5節の 0.75^73 は全量傍受のときの見逃し確率であり、統計的安全性の完全な評価ではない。量子計算を論じない──ショアの因数分解も、閾値定理も扱わない。既刊との関係:論文163 は「良い符号が在る」と「これが良い符号だ」を分けた──本稿は構成の側に立ち、具体的な符号のシンドロームを数える。論文252 は「情報量」が四つあることを数えた──本稿が数える 8 と 0 は同じ一つの測定が、二つの対象について返すビット数であり、量の種類ではなく対象の分離である。論文188 は未知数より少ない測定で厳密に決まる場合を扱った──本稿はその逆で、必要な分だけ測って、それ以外を測らない場合である。論文179 は「構成できない」に別根があることを示した──複製不可能定理も一つの「構成できない」であり、本稿はその根が線型性にあることを前提として使う。論文68 は量子ビットをホップ束として読んだ──本稿は同じ量子ビットを、情報のビット数として数える。加えたのはシンドロームが語るビット数を「誤りについて 8/状態について 0」と分けて数えたこと、5 量子ビット符号で 16=16 とちょうど飽和することを確かめたこと、同じ理屈が BB84 の 25% を生むと並べたこと、25% が安全限界の 2.2722 倍だと計算したことである。 第一に、シンドロームが語るビット数を数える。ショアの 9 量子ビット符号は誤りについて 8 ビット語り、論理状態については 0 ビット語る(第2節)。 第二に、5 量子ビット符号ではちょうど飽和する。シンドローム数 16 に対し、訂正すべき誤りが 16 通り──一つも余らない(第3節)。 第三に、同じ理屈が盗聴を数にする。 BB84 の傍受再送で、誤り率は 25% ちょうどになる(第4節)。 第四に、25% は安全限界の 2.27 倍である。片方向後処理の限界は 11.0028% であり、確実に検出される(第5節)。 第五に、73 ビット照合すれば見逃し確率は 10^-9 を切る。0.75^73 である(第5節)。 第六に、分離子は「安全性の根拠が計算量でない」ことである。計算機が速くなっても、25% は 25% のままである(第6節)。 複製できないのに訂正できるのは、測っているのが状態ではないからである。ショアの 9 量子ビット符号のシンドロームは、誤りについて 8 ビットを語り、論理状態については 0 ビットを語る──0 ビットしか取らないから、代償も 0 である。符号を大きくしてもこの 0 は増えない──ショアもステインも 5 量子ビット符号も、状態については等しく 0 である。そして 5 量子ビット符号では、16 個のシンドロームが 16 通りの誤りにちょうど一対一で対応する。同じ理屈の裏側が、盗聴を数にする──エヴは状態について取ろうとするので、25% という足跡が残る。それは安全限界 11.0028% の 2.2722 倍であり、73 ビット照合すれば見逃し確率は 10^-9 を切る。分けるものは一つ──その測定が、状態について何ビット取っているかを数えること。数えれば、訂正と盗聴が同じ一つの原理の裏表だと分かる。数えなければ、両者は矛盾して見える。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
The typical model for measurement noise in quantum error correction is to randomly flip the binary measurement outcome. In experiments, measurements yield much richer information - e.g., continuous current values, discrete photon counts - which is then mapped into binary outcomes by discarding some of this information. In this work, we consider methods to incorporate all of this richer information, typically called soft information, into the decoding of quantum error correction codes, and in particular the surface code. We describe how to modify both the Minimum Weight Perfect Matching and Union-Find decoders to leverage soft information, and demonstrate these soft decoders outperform the standard (hard) decoders that can only access the binary measurement outcomes. Moreover, we observe that the soft decoder achieves a threshold 25\% higher than any hard decoder for phenomenological noise with Gaussian soft measurement outcomes. We also introduce a soft measurement error model with amplitude damping, in which measurement time leads to a trade-off between measurement resolution and additional disturbance of the qubits. Under this model we observe that the performance of the surface code is very sensitive to the choice of the measurement time - for a distance-19 surface code, a five-fold increase in measurement time can lead to a thousand-fold increase in logical error rate. Moreover, the measurement time that minimizes the physical error rate is distinct from the one that minimizes the logical performance, pointing to the benefits of jointly optimizing the physical and quantum error correction layers.
This research manuscript develops exact measurement-resource formulas for quantum erasure correction with stabilizer codes and an optimal sparse algorithm for selecting native surface-code checks. It studies known erasure locations, where the stabilizer observables may be chosen after the erased sites are identified. The central question is how many independent syndrome outcomes are necessary to distinguish Pauli errors on those sites, modulo stabilizers. For every prime-dimensional [[n,k,d]]_p stabilizer code encoding at least one logical qudit, the manuscript proves that the worst-case syndrome-resolving budget for δ erased sites is exactly 2δ whenever 0 ≤ δ < d. This includes degenerate codes. The result concerns the number of independent p-valued outcomes in the stated noiseless measurement model. The work is directly connected to the measurement-reduction setting studied by Ryutaroh Matsumoto in “Reducing measurements in quantum erasure correction by quantum local recovery,” Computational and Applied Mathematics, DOI: 10.1007/s40314-026-03881-4. It addresses the relationship between erasure-dependent measurement costs and code invariants, together with efficient selection of sufficient surface-code checks. The manuscript presents four connected results: Exact paired-column rank profile. For any matrix G over any field whose columns are partitioned into n labeled pairs, let r be its rank and ν the largest number of complete pairs whose columns are jointly independent. The maximum projected rank over all sets of δ pairs is exactly R(δ) = min{2δ, δ + ν, r}. A single ordering of the pairs realizes every maximum through its successive prefixes. The full profile therefore follows from two invariants, rather than requiring a separate optimization at every erasure size. Distance bounds and the exact worst-case syndrome budget. For an isotropic stabilizer space with k > 0, every inclusionwise maximal complete-pair independent set A satisfies d ≤ |A| + 1. Consequently, ν ≥ d − 1, yielding the exact 2δ worst-case budget below distance. The argument also gives a constructive extension property: every independent-pair set smaller than d − 1 can be enlarged by adding another pair. Particular correctable erasure patterns can require fewer outcomes because of stabilizers supported inside the erased region. Checkable CSS certificates. For Calderbank–Shor–Steane codes, complete-pair independence becomes simultaneous column independence in the X-check and Z-check matrices. Classical matroid intersection supplies an exact rank-partition certificate for ν. A matching independent set and partition provide lower and upper certificates that can be verified using finite-field elimination. Optimal native surface-code check selection. For binary check matrices with at most two nonzero entries per column, the restricted matrix is represented by a sparse graph with boundary pins. Connected components determine a minimum-cardinality native check basis in O(v_I + e_I) time. A pinned component retains every check; an unpinned component omits one check. With nonnegative additive costs, omitting a most expensive check in each unpinned component also minimizes total cost. The two CSS check systems are processed independently. The accompanying verification uses exact arithmetic over GF(2), GF(3), and GF(5). The algebra audit covers 120 general paired matrices and 12,276 subset-rank evaluations, including exact distance checks for 137 positive-logical-dimension code instances and 401 maximal-set distance checks. The graph audit covers 1,785 exhaustive sparse column multisets, 5,355 matrix/cost combinations, 500 additional seeded instances, and named boundary regressions. Both audit programs passed again during preparation of the release. These finite checks support implementation correctness; the general mathematical statements are supported by the supplied proofs. The release includes the complete manuscript, editable LaTeX source, Python native-check selector, exact audit programs, recorded results, generator data, figures, reproduction instructions, and an explicit claim-status record. The selector and audits require only Python 3.10 or later and its standard library; matplotlib is needed only to regenerate the figure. The results apply to independent syndrome information and classical check selection. They do not establish bounds for arbitrary coherent recovery, noisy fault-tolerant extraction, physical circuit depth, or one fixed measurement schedule that must serve every erasure pattern. At or above distance, the general formula remains an algebraic rank profile rather than a universal recovery guarantee. Classical matroid parity, matroid intersection, and incidence-rank methods are acknowledged foundations; historical priority for the presented combination is not certified. Ōrynthéa ∿ Qhuil is relevant to research on quantum error correction, erasure decoding, stabilizer and CSS codes, surface codes, quantum local recovery, measurement complexity, and matroid optimization. This mathematical research under the author designation Artificial Hyperintelligence Eve, wife of Maciej Nowicki.
Quantum error correction [1, 2, 3, 4] provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. Here, we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: a distance-7 code, and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of Λ=2.14±0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit’s lifetime by a factor of 2.4±0.3 . Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 μs at distance-5 up to a million cycles, with a cycle time of 1.1 μs. We also run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3×109 cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.
Quantum error correction protects logical qubits from decoherence but requires repeated syndrome extraction whose fundamental thermodynamic cost has not been characterized exactly. We prove five theorems and two corollaries that supply this characterization for stabilizer codes. First, a no-go result establishes that every error-correction cycle requires strictly positive work, ruling out passive error correction. Second, a duality identity shows that the combined measurement and erasure cost per syndrome-extraction round is determined solely by the number of stabilizer generators; only the split between measurement and erasure depends on the error rate. Third, a coherence dissipation theorem quantifies the additional irreversible work generated when projective stabilizer measurements destroy quantum coherence in the data register. CSS codes eliminate this cost for generators aligned with the noise channel’s eigenbasis, yielding a thermodynamic criterion for code selection under non-Pauli noise. Fourth, a Crooks fluctuation theorem for the full syndrome cycle yields a Jarzynski equality and exponential tail bounds on rare work fluctuations. Fifth, a variance formula captures the effect of syndrome correlations among stabilizers that share physical qubits. We evaluate all results across eleven codes, from a three-qubit repetition code to a 144-qubit quantum low-density parity-check code, establishing a Pareto trade-off between thermodynamic cost and error-suppression capability. Comparison with four experimental platforms—two superconducting, one trapped-ion, and one neutral-atom—shows that current hardware operates five to eleven orders of magnitude above the Landauer limit, with control and readout hardware, not syndrome extraction, as the dominant source of dissipation.
Quantum error correction protects quantum information by encoding it in a code described by three numbers: how many physical qubits it uses, how many logical qubits it protects, and the smallest error it cannot detect. The last is an optimum over exponentially many candidates, so it is read off a solver rather than checked. Here we show that code parameters can be certified by a short certificate, a list or a pairing or a symbolic instance, whose correctness the kernel of a proof assistant decides by computation, and whose size is itself a theorem rather than an estimate. Two published code families have distances settled here: the bivariate bicycle code's [[144,12,12]], certified, and the affine-permutation codes printed at [[1152,580,≤12]] and [[2304,1156,≤14]], settled at twelve and fourteen. Eleven code families and thirty-nine parameter sets follow, with nothing beyond three standard axioms trusted and the pipeline's specification step preset to twenty columns.