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  "id": "PHYS-2026-07-26-01",
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    "id": "PHYS-2026-07-26-01",
    "date": "2026-07-26",
    "updated_at": "2026-07-26",
    "title": "Where quantum speedup actually hides",
    "summary": "A Sunday audit of QSVT, quantum linear systems, data access, condition numbers, readout, and fault tolerance—showing why quantum advantage belongs to an end-to-end pipeline rather than an isolated circuit.",
    "language": "en",
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  "content_markdown": "\n## Sunday Consolidation\n\n**Quantum algorithms at the frontier: where is the speedup actually hiding?**\n\nThis week’s pressure point is not whether a quantum subroutine is elegant. It is whether its advantage survives the full path from physical input to useful classical or quantum output.\n\n## Retrieval Questions — answer before reading\n\n1. A quantum linear-system algorithm prepares\n\n   $$\n   \\ket{x}\\propto A^{-1}\\ket{b}\n   $$\n\n   in time polynomial in $\\log N$ under suitable assumptions. Why does it not thereby output all $N$ components of $x$?\n\n2. QSVT promises a polynomial transformation of singular values. Which expensive access structure is usually assumed before the QSVT sequence begins?\n\n3. What separates a logical memory whose error decreases with code distance from a machine capable of executing a useful fault-tolerant algorithm?\n\nDo not answer with a slogan. Name the missing resource in each case.\n\n## The End-to-End Ledger\n\nA useful speedup must survive every stage:\n\n$$\n\\text{input}\n\\longrightarrow\n\\text{encoded access}\n\\longrightarrow\n\\text{quantum transformation}\n\\longrightarrow\n\\text{fault-tolerant execution}\n\\longrightarrow\n\\text{measurement}\n\\longrightarrow\n\\text{classical use}.\n$$\n\nThe asymptotic cost is controlled by the slowest indispensable stage, not by the most impressive line in the circuit analysis.\n\nA more honest resource ledger is\n\n$$\n\\mathcal C_{\\rm total}\n=\n\\mathcal C_{\\rm load}\n+\\mathcal C_{\\rm access}\n+\\mathcal C_{\\rm transform}\n+\\mathcal C_{\\rm FT}\n+\\mathcal C_{\\rm readout}\n+\\mathcal C_{\\rm post}.\n$$\n\nThe terms need not be literally additive on every architecture. The equation is an audit: no required term may be silently set to zero.\n\n## Tension I — spectral transformation versus physical data access\n\nMuch of modern quantum-algorithm theory can be organized as spectral engineering.\n\nSuppose\n\n$$\nA=\\sum_j\\sigma_j\\ket{u_j}\\bra{v_j}\n$$\n\nis embedded into a larger unitary. QSVT can implement polynomial transformations of its singular values, subject to parity and boundedness conditions.\n\nSchematically, for the appropriate parity,\n\n$$\nA\n\\longmapsto\nP^{(\\rm SV)}(A),\n\\qquad\n\\sigma_j\\longmapsto P(\\sigma_j).\n$$\n\nSuitable choices of $P$ yield primitives for Hamiltonian simulation, spectral filters, amplitude amplification, approximate projectors, and quantum linear-system algorithms.\n\nThis is a genuine structural unification. It is not merely a family resemblance.\n\n### The theorem starts after access has been granted\n\nStandard QSVT commonly assumes an $(\\alpha,a,\\epsilon)$ block encoding:\n\n$$\n\\left\\lVert\nA-\\alpha\n\\left(\n\\bra{0^a}\\otimes I\n\\right)\nU_A\n\\left(\n\\ket{0^a}\\otimes I\n\\right)\n\\right\\rVert\n\\leq\\epsilon.\n$$\n\nIn the exact idealization,\n\n$$\nU_A=\n\\begin{pmatrix}\nA/\\alpha & *\\\\\n* & *\n\\end{pmatrix}.\n$$\n\nThe query complexity may be small in the matrix dimension $N$. But that statement counts calls to $U_A$; it does not automatically price the construction of $U_A$ from ordinary data.\n\nIf $A$ is supplied as an unstructured dense classical array, reading it already costs $\\Omega(N^2)$ numbers.\n\nIf $A$ is a local Hamiltonian, a sparse oracle, or a short sum of efficiently implementable terms, the access model may instead be physically natural.\n\nThe question is therefore not\n\n> Does QSVT require only $\\operatorname{poly}(\\log N)$ qubits and queries?\n\nIt is\n\n> Does the scientific problem natively supply the block encoding or an equally efficient replacement?\n\n### Amplitude encoding is compression, not free input\n\nAn $N$-component vector can be represented as\n\n$$\n\\ket{x}\n=\\frac{1}{\\lVert x\\rVert}\n\\sum_{j=0}^{N-1}x_j\\ket{j}\n$$\n\nusing $\\lceil\\log_2N\\rceil$ qubits.\n\nThis is a geometric representation statement. It is not a generic procedure for loading $N$ arbitrary numbers in $\\operatorname{poly}(\\log N)$ time.\n\nNor does one measurement reveal all amplitudes. A measurement samples one outcome, while complete classical reconstruction generally requires resources growing with the output dimension.\n\nQuantum compression is useful when the input is already generated coherently or the desired output is a small collection of observables.\n\nIt is much less useful when one must load and later print an unstructured classical vector.\n\n### The access frontier is moving\n\nRecent work attempts QSVT without a conventional block encoding.\n\nOne 2025 construction uses Trotterized Hamiltonian evolution, a single ancilla, and no multi-qubit controlled gates.\n\nThe price is not abolished. It reappears through the Hamiltonian decomposition, the number of terms, nested commutators, approximation error, or randomized sampling.\n\nIts randomized variants have quadratic dependence on the polynomial degree in the stated sampling model.\n\nThe frontier is shifting from\n\n> Find another oracle-level speedup\n\nto\n\n> Compile the spectral transformation under an access model that the physical problem actually provides.\n\n## Tension II — logical protection versus algorithmic usefulness\n\nThe hardware question has also changed.\n\nIt is no longer enough to show that an encoded state survives longer. A useful fault-tolerant computation needs an entire stack:\n\n$$\n\\begin{aligned}\n&\\text{logical state preparation}\n+\\text{logical gates}\n+\\text{syndrome extraction}\\\\\n&+\\text{real-time decoding}\n+\\text{feed-forward}\n+\\text{non-Clifford resources}\n+\\text{final readout}.\n\\end{aligned}\n$$\n\n### What below threshold establishes\n\nA below-threshold code exhibits improving logical performance as code distance grows:\n\n$$\n\\epsilon_d\n\\propto\n\\left(\n\\frac{p}{p_{\\rm th}}\n\\right)^{(d+1)/2},\n\\qquad\np<p_{\\rm th},\n$$\n\nup to architecture- and noise-dependent corrections.\n\nA surface-code experiment reported a distance-seven memory with 101 qubits and a logical error of about $0.143\\%$ per correction cycle.\n\nIts lifetime exceeded that of its best constituent physical qubit by a factor of about $2.4$.\n\nThat is strong evidence for error suppression and memory break-even. It does not yet establish low-error universal logical computation.\n\nThe same study found rare correlated events that produced an error floor in high-distance repetition-code data.\n\nIt also emphasized that practical algorithms demand far lower logical error rates and scalable classical control.\n\n### What an algorithmic demonstration adds\n\nA March 2026 preprint reported fault-tolerant, error-corrected executions of QAOA and HHL circuits on trapped-ion processors using the $[[7,1,3]]$ Steane code.\n\nThe largest QAOA circuit used 12 logical qubits encoded in 97 physical qubits and contained 2,132 physical two-qubit gates.\n\nThe authors report better-than-random performance for that instance and near-break-even behavior for the demonstrated system.\n\nThis is an important integration milestone. It tests logical gates, active correction, dynamic circuits, feed-forward, and non-Clifford resources in one experiment.\n\nIt is not evidence of useful computational advantage over the best classical method.\n\nThe decisive distinction is\n\n$$\n\\boxed{\n\\text{error suppression}\n\\neq\n\\text{fault-tolerant integration}\n\\neq\n\\text{computational advantage}\n}.\n$$\n\nEach implication needs new evidence.\n\n## Claim Audit\n\n| Claim | Verdict | Missing qualification |\n| --- | --- | --- |\n| QSVT transforms singular values by a polynomial. | Theorem-level | The polynomial must satisfy parity and boundedness conditions, and suitable encoded access must exist. |\n| A finite QSVT sequence implements $1/x$ exactly on an interval. | False | A finite polynomial cannot equal $1/x$ on a continuous interval. It only approximates it away from zero. |\n| Approximating $1/x$ gets harder near zero. | Theorem-level | The degree and query cost depend on the condition number $\\kappa$, target error, normalization, and access model. |\n| An $N$-component vector fits into $\\log_2N$ qubits. | Representation claim | The amplitudes exist, but generic loading and full classical readout are not free. |\n| HHL returns a full classical solution exponentially faster. | False | It prepares a solution state and is useful when a small number of observables can be estimated efficiently. |\n| A below-threshold memory implies scalable useful computation. | False | Logical gates, decoding latency, correlated noise, magic states, total depth, and readout remain separate constraints. |\n| Variational algorithms are inherently more practical than QSVT. | Unsupported in general | Shallow depth can be offset by sampling, optimization, trainability, and verification costs. |\n\nThe audit exposes three common fallacies:\n\n- **oracle laundering:** treating access to $U_A$ as if it were access to ordinary data;\n- **output laundering:** calling $\\ket{x}$ the same output as the classical list $(x_0,\\ldots,x_{N-1})$;\n- **component-to-system inference:** promoting one successful subsystem into an end-to-end advantage claim.\n\n## Transfer Problem — the condition number is physical\n\nConsider\n\n$$\nA=\n\\begin{pmatrix}\n1&0\\\\\n0&1/\\kappa\n\\end{pmatrix},\n\\qquad\n\\ket{b}=\\frac{\\ket{0}+\\ket{1}}{\\sqrt2},\n\\qquad\n\\kappa>1.\n$$\n\nA quantum linear-system algorithm aims to prepare\n\n$$\n\\ket{x}\n=\n\\frac{A^{-1}\\ket{b}}\n{\\left\\lVert A^{-1}\\ket{b}\\right\\rVert}.\n$$\n\n### Tasks\n\n1. Derive $\\ket{x}$ and calculate $\\langle Z\\rangle_x$.\n2. In the HHL rotation picture, take the accepted ancilla amplitude for eigenvalue $\\lambda$ to be $C/\\lambda$, with $C=1/\\kappa$.\n3. Calculate the success probability for the stated $\\ket{b}$, then for $\\ket{b}=\\ket{0}$.\n4. Let the small eigenvalue be estimated as $\\widetilde\\lambda_2=1/\\kappa+\\delta$. Find the leading relative error after inversion.\n5. Explain how costs independent of $N$ can still erase an exponential speedup.\n\n<details>\n<summary>Hint 1</summary>\n\nThe inverse amplifies the component associated with the small eigenvalue:\n\n$$\nA^{-1}=\n\\begin{pmatrix}\n1&0\\\\\n0&\\kappa\n\\end{pmatrix}.\n$$\n\n</details>\n\n<details>\n<summary>Hint 2</summary>\n\nFor a normalized superposition of eigenvectors, the success probability is the weighted sum of the squared accepted amplitudes.\n\n</details>\n\n**Oral check 1.** Why is $\\operatorname{poly}(\\log N,\\kappa,1/\\epsilon)$ not automatically an exponential advantage in practice?\n\n**Oral check 2.** Which output tasks preserve the promise of HHL: printing every component, estimating a sparse observable, or both?\n\n<details class=\"solution\">\n<summary>Solution</summary>\n\nFirst,\n\n$$\nA^{-1}\\ket{b}\n=\n\\frac{\\ket{0}+\\kappa\\ket{1}}{\\sqrt2}.\n$$\n\nTherefore,\n\n$$\n\\boxed{\n\\ket{x}\n=\n\\frac{\\ket{0}+\\kappa\\ket{1}}\n{\\sqrt{1+\\kappa^2}}\n}.\n$$\n\nSince $Z\\ket{0}=\\ket{0}$ and $Z\\ket{1}=-\\ket{1}$,\n\n$$\n\\boxed{\n\\langle Z\\rangle_x\n=\n\\frac{1-\\kappa^2}{1+\\kappa^2}\n}.\n$$\n\nThe accepted ancilla amplitudes are\n\n$$\n\\frac{C}{1}=\\frac1\\kappa,\n\\qquad\n\\frac{C}{1/\\kappa}=1.\n$$\n\nFor the equal superposition,\n\n$$\n\\boxed{\np_{\\rm succ}\n=\n\\frac12\n\\left(\n\\frac1{\\kappa^2}+1\n\\right)\n}.\n$$\n\nThis tends to $1/2$ because the input already has substantial weight in the small-eigenvalue sector.\n\nFor $\\ket{b}=\\ket{0}$,\n\n$$\n\\boxed{\np_{\\rm succ}=\\frac1{\\kappa^2}\n}.\n$$\n\nPlain repetition then costs $O(\\kappa^2)$ trials. Ideal amplitude amplification reduces the scaling to $O(\\kappa)$ coherent uses.\n\nThe overlap of $\\ket{b}$ with different spectral sectors is therefore part of the cost.\n\nFor the perturbed eigenvalue,\n\n$$\n\\frac1{\\widetilde\\lambda_2}\n=\n\\frac{\\kappa}{1+\\kappa\\delta}\n\\approx\n\\kappa\\left(1-\\kappa\\delta\\right).\n$$\n\nHence the leading relative inversion error is\n\n$$\n\\boxed{\n\\frac{\n\\Delta(\\lambda_2^{-1})\n}{\n\\lambda_2^{-1}\n}\n\\approx-\\kappa\\delta\n}.\n$$\n\nKeeping it below $\\epsilon$ requires\n\n$$\n\\boxed{\n|\\delta|\\lesssim\\frac{\\epsilon}{\\kappa}\n}.\n$$\n\nHHL expresses this through spectral resolution and success amplitude. QSVT expresses it through the difficulty of approximating $1/x$ near zero.\n\nQuantum algorithms do not remove ill-conditioning. They relocate its cost into degree, evolution time, normalization, amplification, or state overlap.\n\nEven if none of these costs depends directly on $N$, a growing $\\kappa$, expensive state preparation, or extensive readout can dominate the nominal $\\log N$ dependence.\n\nThe potentially favorable output is a bounded observable such as $\\langle x|M|x\\rangle$, when $M$ is itself efficiently measurable.\n\nPrinting all $N$ amplitudes forfeits the compressed-output advantage.\n\n</details>\n\n## Correction Ledger\n\n### Retain\n\nQuantum advantage is a property of an input–transformation–measurement pipeline, not an isolated circuit complexity.\n\n### Correct\n\nAmplitude encoding stores many coefficients geometrically. It does not grant unrestricted classical loading or readout.\n\n### Current frontier\n\n- replacing idealized block encodings with compilable access;\n- reducing logical overhead for early fault-tolerant processors;\n- integrating decoding, feed-forward, and non-Clifford resources;\n- choosing observables that preserve compressed quantum output.\n\n### Unresolved\n\nWhich scientifically valuable problems simultaneously have:\n\n- efficient state preparation;\n- structured operator access;\n- favorable conditioning;\n- robust low-dimensional observables;\n- a classical baseline that does not improve just as rapidly?\n\n### Spaced return\n\nCompare phase estimation, QSVT filtering, and tensor-network methods for extracting low-energy data from an XXZ chain.\n\n## Further Reading\n\n- [Quantum singular value transformation and beyond](https://arxiv.org/abs/1806.01838)\n- [QSVT without block encodings](https://arxiv.org/abs/2504.02385)\n- [Quantum algorithm for solving linear systems of equations](https://arxiv.org/abs/0811.3171)\n- [Quantum error correction below the surface-code threshold](https://www.nature.com/articles/s41586-024-08449-y)\n- [Fault-tolerant execution of error-corrected quantum algorithms](https://arxiv.org/abs/2603.04584)\n\n## Check your understanding\n\nChoose one proposed quantum speedup and write its complete resource ledger:\n\n$$\n\\text{load}\n+\\text{access}\n+\\text{transform}\n+\\text{fault tolerance}\n+\\text{readout}\n+\\text{classical comparison}.\n$$\n\nAt which term is its advantage most vulnerable?\n\nYou may also reply with “deeper,” “too easy,” “too hard,” or your solution to the transfer problem.\n",
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      "tex": "U_A",
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    {
      "index": 38,
      "display": false,
      "tex": "\\ket{x}",
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      "sha256": "76073e4ac77c0862516c7d5829f520cbea665a1dc90095b6dd1e126f712f3985"
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    {
      "index": 39,
      "display": false,
      "tex": "(x_0,\\ldots,x_{N-1})",
      "line": 268,
      "sha256": "3b16feb4b144e0e7217d6f91c8b4b7a7f28bc95e9b359f72254246e92f8f40b8"
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    {
      "index": 40,
      "display": true,
      "tex": "A=\n\\begin{pmatrix}\n1&0\\\\\n0&1/\\kappa\n\\end{pmatrix},\n\\qquad\n\\ket{b}=\\frac{\\ket{0}+\\ket{1}}{\\sqrt2},\n\\qquad\n\\kappa>1.",
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    {
      "index": 41,
      "display": true,
      "tex": "\\ket{x}\n=\n\\frac{A^{-1}\\ket{b}}\n{\\left\\lVert A^{-1}\\ket{b}\\right\\rVert}.",
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      "sha256": "c16c7ff4d362e5a79f4033373b89d4808fd0ef314b1e261b015cfabde756640f"
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      "tex": "\\ket{x}",
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      "display": false,
      "tex": "\\langle Z\\rangle_x",
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      "sha256": "00b9cd172ab2f8287b44b61bef980636a0ea68ddf70acaca657a2701cfb5e736"
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      "display": false,
      "tex": "\\lambda",
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    },
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      "index": 45,
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      "tex": "C/\\lambda",
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      "sha256": "6d7866f2696c07caf40e9471b97cf7a258e93aa238159fa4d7ff4abd07cd46e3"
    },
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      "index": 46,
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      "tex": "C=1/\\kappa",
      "line": 299,
      "sha256": "08b51b1f2e9f859a195521d0d70c14ef84841182c9a265835ee36177462bd152"
    },
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      "index": 47,
      "display": false,
      "tex": "\\ket{b}",
      "line": 300,
      "sha256": "809d2d77516daca0d3fad71c8c774df1a8366d43726c2daef70515dc66bcbd7d"
    },
    {
      "index": 48,
      "display": false,
      "tex": "\\ket{b}=\\ket{0}",
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      "sha256": "4437383c549ea408ed0f38e501e2f4b5f6dc17651074e3d88a282bb48583b3a2"
    },
    {
      "index": 49,
      "display": false,
      "tex": "\\widetilde\\lambda_2=1/\\kappa+\\delta",
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      "sha256": "6e2aabb2f6a466b7dcf23d7aad48bb00c2509a1ced19d86a27b9e66c936d1ed7"
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      "index": 50,
      "display": false,
      "tex": "N",
      "line": 302,
      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 51,
      "display": true,
      "tex": "A^{-1}=\n\\begin{pmatrix}\n1&0\\\\\n0&\\kappa\n\\end{pmatrix}.",
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      "sha256": "3b3bd3ef00468563335170dd8500ee1f6a30164494915cc032cc7740680abd0c"
    },
    {
      "index": 52,
      "display": false,
      "tex": "\\operatorname{poly}(\\log N,\\kappa,1/\\epsilon)",
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      "sha256": "e84c18edcc4dbcfac660813dedac136239cc3882f87c66e6de92f6db1e0801b8"
    },
    {
      "index": 53,
      "display": true,
      "tex": "A^{-1}\\ket{b}\n=\n\\frac{\\ket{0}+\\kappa\\ket{1}}{\\sqrt2}.",
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      "sha256": "e10ca9faa52eae366c7fcbb1fd65f69776484eec10282d1457e8b561516f54c4"
    },
    {
      "index": 54,
      "display": true,
      "tex": "\\boxed{\n\\ket{x}\n=\n\\frac{\\ket{0}+\\kappa\\ket{1}}\n{\\sqrt{1+\\kappa^2}}\n}.",
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      "sha256": "dc8333eb1c99a2d8c375cbdee4bbf94832392b5326493ee494dc8e90d376cc7d"
    },
    {
      "index": 55,
      "display": false,
      "tex": "Z\\ket{0}=\\ket{0}",
      "line": 352,
      "sha256": "c1e95ec17f200716d675ebbd31be0875758448e135b69719e932be37b71578d2"
    },
    {
      "index": 56,
      "display": false,
      "tex": "Z\\ket{1}=-\\ket{1}",
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      "sha256": "a845dc12131f75cdd8c8fb8b4e88c3d115871039d404aa3757172b67794586d3"
    },
    {
      "index": 57,
      "display": true,
      "tex": "\\boxed{\n\\langle Z\\rangle_x\n=\n\\frac{1-\\kappa^2}{1+\\kappa^2}\n}.",
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      "sha256": "3819abe37b67545a9958978fc2c2061f30813c042a1b0c14b0e9b64d2a3683ed"
    },
    {
      "index": 58,
      "display": true,
      "tex": "\\frac{C}{1}=\\frac1\\kappa,\n\\qquad\n\\frac{C}{1/\\kappa}=1.",
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      "sha256": "0b81663a6871e2e6d89f1f51b2d44e021681a0e9cdab595eabc1c02450cb6071"
    },
    {
      "index": 59,
      "display": true,
      "tex": "\\boxed{\np_{\\rm succ}\n=\n\\frac12\n\\left(\n\\frac1{\\kappa^2}+1\n\\right)\n}.",
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      "sha256": "c1d9332b15f83154fc8b971364b4e9cb6146bad3377b4dad70b6df4f64dc1ea8"
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      "display": false,
      "tex": "1/2",
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      "sha256": "d939926f05444b0f4495fb9629ecbfa80d99a8ec1a20d06800ca5a3d5f4fd276"
    },
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      "index": 61,
      "display": false,
      "tex": "\\ket{b}=\\ket{0}",
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      "sha256": "4437383c549ea408ed0f38e501e2f4b5f6dc17651074e3d88a282bb48583b3a2"
    },
    {
      "index": 62,
      "display": true,
      "tex": "\\boxed{\np_{\\rm succ}=\\frac1{\\kappa^2}\n}.",
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      "sha256": "29fefa7946135f1eee7ee542b212739b4431ce29af6de778a8a324bdccaf6bb6"
    },
    {
      "index": 63,
      "display": false,
      "tex": "O(\\kappa^2)",
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    },
    {
      "index": 64,
      "display": false,
      "tex": "O(\\kappa)",
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      "sha256": "c0aa5b67a9ef9429c26cee3ff2b432ceba90010bd2a3aa6f1f2c9376b4c500f4"
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      "index": 65,
      "display": false,
      "tex": "\\ket{b}",
      "line": 395,
      "sha256": "809d2d77516daca0d3fad71c8c774df1a8366d43726c2daef70515dc66bcbd7d"
    },
    {
      "index": 66,
      "display": true,
      "tex": "\\frac1{\\widetilde\\lambda_2}\n=\n\\frac{\\kappa}{1+\\kappa\\delta}\n\\approx\n\\kappa\\left(1-\\kappa\\delta\\right).",
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      "sha256": "a2ad6db57d7052822b6b17730fe25f673477415894f298520b9ce89c6222c0b6"
    },
    {
      "index": 67,
      "display": true,
      "tex": "\\boxed{\n\\frac{\n\\Delta(\\lambda_2^{-1})\n}{\n\\lambda_2^{-1}\n}\n\\approx-\\kappa\\delta\n}.",
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      "index": 68,
      "display": false,
      "tex": "\\epsilon",
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      "sha256": "8bc4bfc2bdebfaac14a98cde1deb3cf75202e7908905efb95c939c0481951f77"
    },
    {
      "index": 69,
      "display": true,
      "tex": "\\boxed{\n|\\delta|\\lesssim\\frac{\\epsilon}{\\kappa}\n}.",
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      "sha256": "ad2c6fdef80e920af4957356f2bffb2cae05c1708e8a52852917c4c8adef5cf7"
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      "index": 70,
      "display": false,
      "tex": "1/x",
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      "tex": "N",
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    {
      "index": 72,
      "display": false,
      "tex": "\\kappa",
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      "sha256": "055de4ec3d8e4886e65188679865c7b52a840bbfe670437f5d6317be2f7a3ff3"
    },
    {
      "index": 73,
      "display": false,
      "tex": "\\log N",
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    },
    {
      "index": 74,
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      "tex": "\\langle x|M|x\\rangle",
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      "tex": "M",
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      "tex": "N",
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      "sha256": "8ce86a6ae65d3692e7305e2c58ac62eebd97d3d943e093f577da25c36988246b"
    },
    {
      "index": 77,
      "display": true,
      "tex": "\\text{load}\n+\\text{access}\n+\\text{transform}\n+\\text{fault tolerance}\n+\\text{readout}\n+\\text{classical comparison}.",
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      "sha256": "06a19ece318af41f6ab36312e1eefa689c45a919e52e2333b0ba4d546c4953ff"
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}