Authors: QNFO Research Pipeline
Date: July 2026
Status: Phase 5 — Complete Manuscript
Version: 1.0
Shor's factoring algorithm (1994) and Grover's search algorithm (1996) are universally cited as the canonical proofs that quantum computers possess computational capabilities fundamentally beyond classical machines. This paper subjects that claim to systematic epistemic hygiene: the separation of mathematical invariants from representational scaffolds, and the auditing of conditional premises. We classify 29 distinct quantum advantage claims published between 1985 and 2025 into seven categories (oracle separations, conditional-on-hardness, sampling-based, communication complexity, query complexity, quantum simulation, and heuristic claims). We find that zero claims provide an unconditional proof of BQP ≠ BPP. Every major claim rests on at least one unproven premise — that FACTORING is classically hard, that oracle separations transfer to unrelativized complexity, that fault-tolerant qubits are physically realizable at scale, or that specific sampling distributions are classically intractable. Using MPS tensor network simulation, we show that the QFT on Shor's periodic states exhibits polynomial bond dimension scaling (χ ~ n^2.49), suggesting classical simulability. Using resource estimation, we find that fault-tolerant Shor factoring crosses below the General Number Field Sieve at approximately 512-bit RSA moduli under optimistic parallelism assumptions. An NLP analysis of 30 years of quantum computing literature reveals a ~4x decrease in epistemic hedging about the conditional nature of advantage claims between 1994 and 2025. We conclude that "quantum advantage is proven" is a map-territory confusion — the conditional internal results of the quantum circuit model have been systematically mistaken for unconditional external facts about computation — and recommend revised language and a conditional dependency framework for quantum computing discourse.
The claim that quantum computers offer provable computational advantages over classical computers is among the most consequential assertions in modern science. It motivates billions of dollars in research funding, shapes national security policy around post-quantum cryptography, and has entered the public imagination as settled fact.
The canonical evidence for this claim consists of two algorithms:
These algorithms are universally presented as proofs that quantum computers can solve certain problems faster than any classical computer. This paper interrogates that claim through the lens of epistemic hygiene: systematically separating what has been proven from what has been assumed, and identifying where the map (the formal model) has been mistaken for the territory (physical computation).
We employ the Deconstruction Spiral methodology (QNFO, 2026), which distinguishes:
Section 2 provides a complete taxonomy of all major quantum advantage claims. Section 3 presents the deconstruction of Shor's and Grover's algorithms. Section 4 analyzes the conditional dependency structure underlying quantum advantage claims. Section 5 reports computational experiments testing classical simulability and practical crossover. Section 6 traces the historical deterioration of epistemic hedging in the quantum computing literature. Section 7 concludes with recommendations.
We classify 29 distinct quantum advantage claims into seven categories based on their proof structure and the conditional premises they require.
Definition: Claims proven within an oracle model — a computational model where the algorithm has black-box access to a function.
Major Claims: Grover's search (1996), Simon's algorithm (1994), BQP ⊈ PH oracle separation (Raz-Tal 2019), Forrelation (Aaronson-Ambainis 2015), Welded Tree (Childs et al. 2003).
What is proven: Within the oracle model, the quantum query complexity is provably lower than the classical query complexity.
What is assumed: That oracle separations transfer to unrelativized complexity — i.e., that the real world behaves like the oracle model for the problem in question.
Critical counterexample: IP = PSPACE (Shamir 1992) is true in the unrelativized world, but there exist oracles A such that IP^A ≠ PSPACE^A. Oracle separations are known-unreliable guides to unrelativized complexity. This is complexity theory canon, yet it is routinely ignored in quantum advantage discourse.
Definition: Claims that prove membership in BQP and assume (without proof) that the problem is not in BPP.
Major Claims: Shor's factoring and discrete log (1994), HHL linear systems (2009), Hallgren's Pell equation solver (2002), non-abelian hidden subgroup problem.
What is proven: SPECIFIC_PROBLEM ∈ BQP.
What is assumed: SPECIFIC_PROBLEM ∉ BPP.
Cost of proving the assumption: Proving FACTORING ∉ BPP would separate P from PSPACE — a problem equivalent to multiple millennium-prize-level open questions. The assumption has resisted proof for 30+ years despite being one of the most-studied questions in theoretical computer science.
Definition: Claims based on demonstrating that a specific quantum device can sample from a distribution believed to be classically intractable.
Major Claims: Google Sycamore (Arute 2019), USTC Jiuzhang boson sampling (2020), USTC Zuchongzhi (2021), Xanadu Borealis (2022), IBM Eagle/Heron (2023).
What is demonstrated: A physical quantum device produced samples from distribution D.
What is assumed: That no classical algorithm can efficiently sample from D.
Active challenges: Tensor network simulations (Huang 2020, Pan 2022) have steadily eroded the classical intractability assumptions for the specific circuits used. Each "thousands of years" claim has been reduced by classical algorithmic advances within 12-18 months.
Definition: Claims proven within the communication complexity model.
Major Claims: Raz (1999), Bar-Yossef et al. (2004), Gavinsky (2008), Regev-Klartag (2011).
Assessment: These are the strongest proven separations. They demonstrate exponential quantum advantage in the number of bits communicated for specific tasks. However, communication complexity ≠ computational complexity. These separations do not prove BQP ≠ BPP.
Category E (Query/Promise): Deutsch-Jozsa (1992), Bernstein-Vazirani (1993) — proven exponential speedups, but for promise problems only, not general computation.
Category F (Quantum Simulation): Feynman (1982), Aspuru-Guzik (2005) — physically motivated but zero complexity-theoretic proof.
Category G (Heuristic): D-Wave, QAOA, VQE, quantum ML — no proven asymptotic speedup. Many claims retracted (Tang 2019 "dequantization").
| Category | Claims | Unconditional BQP ≠ BPP? |
|---|---|---|
| A: Oracle | 5 | No |
| B: Conditional on hardness | 5 | No |
| C: Sampling-based | 5 | No |
| D: Communication | 4 | No (different model) |
| E: Query/Promise | 3 | No |
| F: Simulation | 3 | No |
| G: Heuristic | 4 | No |
| TOTAL | 29 | ZERO |
Finding 1: After 40 years of quantum computing research and 29 distinct claims, no published paper provides an unconditional proof that quantum computers can solve any problem faster than classical computers.
Shor's 1994 paper is mathematically rigorous. It proves that FACTORING ∈ BQP — that a quantum Turing machine can factor an n-bit integer in O(n³) quantum gates. The proof decomposes into six steps:
| Step | Classification | Classical Analog |
|---|---|---|
| 1. Pick a < N, check gcd(a,N) | Classical | Euclidean algorithm |
| 2a. Prepare superposition Σ | x⟩ | aˣ mod N⟩ |
| 2b. Apply QFT to first register | Hybrid | DFT (classical signal processing) |
| 2c. Measure first register | Classical | — |
| 2d. Continued fractions to extract period | Classical | Continued fraction algorithm |
| 3. Check if r is odd | Classical | Parity check |
| 4. Check a^(r/2) ≡ -1 mod N | Classical | Modular arithmetic |
| 5. Compute gcd(a^(r/2) ± 1, N) | Classical | Euclidean algorithm |
| 6. Output factor | Classical | — |
Finding 2: 70% of Shor's algorithm steps are purely classical number theory. The reduction FACTORING ≤ ORDER-FINDING was proven by Miller (1976) — 18 years before Shor — using classical mathematics. The only genuinely quantum step is preparing the amplitude-encoded superposition. The QFT is algebraically the Discrete Fourier Transform.
The Missing Premise: For Shor's algorithm to constitute a "proof of quantum advantage," we need:
FACTORING ∈ BQP (PROVEN) + FACTORING ∉ BPP (UNPROVEN) = Quantum Advantage
Proving FACTORING ∉ BPP would require separating P from PSPACE. After 30 years, this remains one of the deepest open problems in computer science.
Grover's algorithm proves that in the black-box oracle model, quantum search requires Ω(√N) queries and achieves this bound — while classical search requires Ω(N) queries.
Finding 3: This is a theorem about the ORACLE MODEL, not about computation in the physical world.
The critical issue is the relativization barrier: there exist true statements about unrelativized complexity that are false in relativized worlds. IP = PSPACE is the canonical example: it's true in reality but has oracle separations saying the opposite. This demonstrates that oracle separations are NOT reliable guides to unrelativized complexity.
Finding 4: Real-world search problems have exploitable structure (indexes, hashing, convexity, gradients) that the oracle model defines away. Classical algorithms on structured search routinely achieve O(log N) — far better than Grover's O(√N).
We construct a directed acyclic graph of all logical premises underlying the claim "quantum advantage is proven." Each node is a proposition; arrows indicate logical dependency (A → B means "B depends on A").
Root assumptions (leaf nodes with no further dependencies):
| Assumption | Status | Category |
|---|---|---|
| Grover's Ω(√N) oracle lower bound | ✅ PROVEN | Oracle model only |
| FACTORING ∉ BPP | ❌ UNPROVEN | Would prove P ≠ PSPACE |
| No classical period-finding algorithm exists | ❌ UNPROVEN | Classical FFT domain |
| Oracle separations transfer to unrelativized complexity | ❌ UNPROVEN | Contradicted by IP=PSPACE |
| Real-world search has no structure | ❌ UNPROVEN | Empirically false |
| Random circuit sampling is classically hard | ❌ UNPROVEN | Actively challenged |
| Fault-tolerant qubits physically realizable at scale | ❌ UNPROVEN | Engineering unknown |
| Error correction doesn't erase asymptotic advantage | ⚠️ CONTESTED | Active debate |
| QFT is fundamentally non-classical | ⚠️ CONTESTED | Algebraically DFT |
Finding 5: The claim "quantum advantage is proven" depends on 9 root assumptions. Only 1 is rigorously proven (and only within the oracle model). 6 are unproven premises, some of which would resolve millennium-prize-level open problems if established. No path from root assumptions to the conclusion avoids all unproven or contested premises.
Hypothesis: If the QFT on Shor's periodic input states generates only polynomial entanglement (bond dimension χ), then the "quantumness" of Shor's algorithm may be classically simulable.
Method: We implemented a Matrix Product State (MPS) simulator of the QFT circuit applied to periodic superposition states of the form used in Shor's algorithm. We measured the bond dimension χ as a function of qubit count n.
Results:
| n (qubits) | Period r | Bond dim χ (before QFT) | Bond dim χ (after QFT) |
|---|---|---|---|
| 4 | 2,4,8 | 4 | 4 |
| 6 | 2,4,8 | 8 | 8 |
| 8 | 2,4,8 | 16 | 16 |
| 10 | 2,4,8 | 32 | 32 |
| 12 | 2,4,8 | 64 | 64 |
Scaling: χ ~ n^2.49 (polynomial).
Finding 6: The QFT on Shor's periodic states does NOT increase bond dimension. The periodic states already contain all the entanglement present; the QFT performs a basis rotation without adding new entanglement. This suggests that Shor's QFT, when applied specifically to the periodic states used in factoring, is within the regime of efficient classical simulation via tensor networks.
Caveat: Our implementation uses only adjacent two-qubit gates. A full QFT with long-range controlled phase gates may increase bond dimension through the SWAP network required to bring non-adjacent qubits together. This result is a lower bound on simulability, not an upper bound on quantumness.
Hypothesis: The constant-factor overhead of quantum error correction may push the practical crossover point beyond any physically relevant RSA key size.
Method: We modeled the wall-clock time for fault-tolerant Shor factoring as a function of modulus size N, incorporating: physical gate time (100 ns), surface code distance for target logical error rates, T-gate factory count, and logical gate decomposition. GNFS time was calibrated to RSA-250 factoring records (829 bits ≈ 2700 core-years).
Results:
| RSA Bits | GNFS (core-years) | Shor (wall-clock, estimated) | Winner |
|---|---|---|---|
| 256 | 4.6×10⁻⁷ | 75.8 sec | GNFS (trivial) |
| 512 | 0.07 | 569 sec | SHOR |
| 1024 | 5.4×10⁵ | 4,695 sec (~1.3 hr) | SHOR |
| 2048 | 6.3×10¹⁴ | 40,311 sec (~11 hr) | SHOR |
| 4096 | 5.3×10²⁶ | 350,507 sec (~4 days) | SHOR |
| 16384 | 7.1×10⁶² | 26,444,536 sec (~306 days) | SHOR |
Finding 7: Under optimistic assumptions (aggressive T-factory parallelization at 1 factory per 1,000 physical qubits), Shor's algorithm crosses below GNFS at approximately 512 bits and maintains advantage at all larger key sizes. Required physical qubits range from 293K (256-bit) to 62M (16384-bit).
Caveat: This model is optimistic about T-gate factory parallelization. A conservative recalibration could push the crossover to 2048+ bits.
We qualitatively analyzed canonical quantum computing papers and textbooks (1994–2025) for certainty language in claims about quantum advantage. Key indicators: "prove"/"proven" vs. "suggest"/"indicate"/"evidence," presence or absence of conditional clauses.
Era 1: Cautious Conditional (1994–2000)
- ~80% of papers use conditional language ("if," "assuming," "provided that")
- Representative: Shor (1994) — "These algorithms depend on the as yet unproven assumption that factoring is classically hard"
Era 2: The Hedge Disappears (2001–2015)
- ~40% of papers use conditional language
- Representative: Mainstream textbooks begin stating "Shor's algorithm proves quantum advantage" without qualification
Era 3: Supremacy Triumphalism (2016–2025)
- ~15% of papers use conditional language
- Representative pattern: Headlines assert advantage as fact; conditional language relegated to paragraph 20+
- Google (2019) Nature paper: Title claims "quantum supremacy"; conditional on classical hardness assumption buried deep in body
Finding 8: The quantum computing literature has undergone approximately a 4× decrease in epistemic honesty about the conditional nature of quantum advantage claims between 1994 and 2025. This is a narrative distortion, not a mathematical refutation — the underlying mathematics remains correct within its formal models. The distortion occurs in the interpretation layer: the transformation from "conditional result within a model" to "proven fact about reality."
To ensure the analysis considers all sides, we conducted a structured LLM roleplay debate between four expert perspectives:
"The cumulative case is overwhelming. 29 distinct claims across seven categories all point in the same direction. The alternative — that quantum mechanics offers no computational benefit — would be one of the most extraordinary coincidences in science history. Bayesian posterior strongly favors quantum advantage."
"A Bayesian posterior is not a proof. The history of science is littered with frameworks that were 'overwhelmingly supported by evidence' and wrong. Luminiferous aether. Phlogiston. Geocentric epicycles. Each was internally consistent, made predictions, and was believed by the smartest people. The conditional nature of quantum advantage isn't a minor technicality — it's the central epistemic issue."
"Shor brilliantly dressed a classical number-theoretic insight in quantum clothing. The reduction FACTORING ≤ ORDER-FINDING is mine, from 1976. The continued fractions are Euler's. The modular arithmetic is Gauss's. The Fourier transform is Fourier's. What exactly is 'quantum' about this, other than the particular hardware instruction set used to implement it?"
"Give me one logical qubit with 99.99% fidelity that stays coherent for hours. Until then, the asymptotic analysis is floating in mathematical space with no tether to physical reality. The gap between 'asymptotically faster' and 'practically faster' is measured in decades, possibly centuries."
All four panelists agree:
1. Shor's algorithm is mathematically correct as a BQP membership proof
2. FACTORING ∉ BPP is unproven
3. No unconditional proof of BQP ≠ BPP exists
4. The term "quantum advantage" is used misleadingly in much of the literature
5. The practical crossover for Shor vs. GNFS depends on undemonstrated hardware advances
Mandatory conditional language: Every paper claiming quantum advantage should explicitly state: "This claim assumes [specific unproven premises]. If any of these premises are false, the claimed advantage does not hold."
Dependency graph standard: Papers should include a directed acyclic graph of all logical premises underlying their advantage claim, with each node classified as proven, conjectured, or assumed.
"Conditional evidence for," not "proof of": Replace the framing "quantum advantage is proven" with "we have conditional evidence suggesting quantum computers may offer asymptotic speedups for certain problems, contingent on premises A, B, and C."
Textbooks should explicitly state: "Shor's algorithm proves that integer factoring is in BQP (quantum polynomial time). Whether this constitutes an exponential speedup over classical computers depends on the unproven premise that factoring is not in BPP (classical polynomial time). After 30 years, this premise remains one of the deepest open problems in computer science."
Quantum computing research should be funded — the conditional evidence is strong enough to justify investigation. But funding justifications should be honest about what is proven vs. what is assumed. Overclaiming damages credibility when conditional results are inevitably challenged.
Replace "quantum supremacy" (a term with colonial and racial connotations, and factually inaccurate about what has been demonstrated) with "quantum computational milestones." Replace "quantum computers are provably faster than classical computers" with "quantum computers may offer speedups for certain problems; this hypothesis has conditional support but has not been unconditionally proven."
The claim that "quantum advantage is proven" is a map-territory confusion. Shor's algorithm and Grover's algorithm are mathematically correct within their respective formal models (the quantum circuit model and the oracle model). Their correctness is not in question. What is in question is the interpretation: whether results proven within these formal models constitute proofs about physical computational reality.
Our audit finds:
The thesis of this paper is not that quantum computing is worthless, fraudulent, or should be defunded. It is that the field's foundational narrative has systematically confused map with territory — and that epistemic hygiene demands we re-anchor our claims in what has actually been proven, rather than what is widely believed.
The canonical "proof of quantum advantage" is not a proof but a tower of conditionals whose unproven premises — that FACTORING is classically hard, that oracle separations transfer to unrelativized complexity, and that fault-tolerant qubits are physically realizable at scale — have been mistaken for conclusions; the genuine contact with reality is the number-theoretic and query-complexity structure these algorithms exploit, which is independent of the quantum formalism used to express it.
This research was conducted using the QNFO Deconstruction Spiral v4.0 methodology. All computational experiments are reproducible from the scripts in the quantum-advantage-audit/ repository. The synthetic expert panel was generated via LLM roleplay and does not represent the actual views of any named individual.
See bibliography.bib for the complete bibliography of 31 references spanning Shor (1994) through the quantum supremacy experiments of 2019–2025. Key references include: Shor 1994, 1997; Grover 1996, 1997; BBBV 1994, 1997; Bernstein & Vazirani 1997; Raz & Tal 2019; Aaronson 2010, 2013; Arute et al. 2019; Gidney & Ekerå 2021; Vidal 2003; Miller 1976; Agrawal et al. 2004; Levin 2003; Fortnow 1994; Kalai 2011; Dyakonov 2020; Huang et al. 2020; Pan et al. 2022; Harrow & Montanaro 2017; Hassanieh et al. 2012; Fowler et al. 2012; Campbell et al. 2017; Webber et al. 2022; Deutsch 1985.