{ "schema_version": 1, "commit": "ab8f277c5704b7831c7a5b5ecddce61c760b4ac8", "policy": "Projected from existing chapter/result and implementation metadata, not a newly scored source-coverage percentage. A compiled prerequisite does not close a route.", "chapters": [ { "id": "linear-algebra", "title": "Vectors, matrices, and basis states", "track": "Shared foundations", "summary": "Fix the finite matrix model, pointwise equality, dimensions, and resource records used by every later certificate.", "results": [ { "declaration": "QuantumBlockEncoding.Matrix.PointwiseEq", "title": "Pointwise matrix equality", "math": "A = B \\;\\Longleftrightarrow\\; \\forall i\\,j,\\; A_{ij}=B_{ij}.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.Matrix" ], "route_closures": [ "QuantumBlockEncoding.Matrix.PointwiseEq" ] }, { "declaration": "QuantumBlockEncoding.Resource", "title": "Executable resource records", "math": "r=(q_{\\mathrm{anc}},q_{\\mathrm{tot}},d,n_{1q},n_{2q}).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared reusable route.", "dependencies": [ "QuantumBlockEncoding.gridSize" ], "route_closures": [ "QuantumBlockEncoding.ExecutableResourceCertificate.resource_eq_program_resource", "QuantumBlockEncoding.ExecutableResourceCertificate.cost_gateCount", "QuantumBlockEncoding.ExecutableResourceCertificate.cost_depth" ] } ] }, { "id": "circuit-semantics", "title": "Gates, unitaries, and circuit meaning", "track": "Shared foundations", "summary": "Separate gate syntax from matrix evaluation and make register order an explicit part of the semantic boundary.", "results": [ { "declaration": "QuantumBlockEncoding.evalGateMatrices", "title": "Circuit matrix evaluation", "math": "\\mathrm{Eval}([g_1,\\ldots,g_m])=G_m\\cdots G_1.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.GateMatrix", "QuantumBlockEncoding.Matrix" ], "route_closures": [ "QuantumBlockEncoding.evalGateMatrices" ] }, { "declaration": 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"Pauli X prepares the one state", "math": "X|0\\rangle=|1\\rangle,\\qquad X|1\\rangle=|0\\rangle.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.TextbookStatePreparation.pauliX_unitary", "QuantumBlockEncoding.TextbookStatePreparation.oneTarget_normalized" ], "route_closures": [ "QuantumBlockEncoding.TextbookStatePreparation.pauliXCertificate_prepares_one" ] }, { "declaration": "QuantumBlockEncoding.StatePreparationCandidate.preparesTarget", "title": "The state-preparation contract", "math": "U\\lvert 0^n\\rangle=\\lvert\\psi\\rangle.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared reusable route.", "dependencies": [ "QuantumBlockEncoding.StatePreparationTarget", "QuantumBlockEncoding.StatePreparationCandidate" ], "route_closures": [ "QuantumBlockEncoding.StatePreparationCandidate.certify", "QuantumBlockEncoding.StatePreparationCandidate.certify_firstColumn" ] } ] }, { "id": "state-preparation-certificates", "title": "Reading and reusing a preparation certificate", "track": "State preparation", "summary": "Connect the ket equation to a matrix column, package the proof, and reuse exact preparation where an approximate interface is expected.", "results": [ { "declaration": "QuantumBlockEncoding.ConcreteSemantics.firstColumnMatches_iff_applyVec_zeroKet", "title": "The ket equation is a first-column statement", "math": "\\operatorname{column}_0(U)=\\psi\\iff U\\lvert0^n\\rangle=\\lvert\\psi\\rangle.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared reusable route.", "dependencies": [ "QuantumBlockEncoding.FirstColumnMatches", "QuantumBlockEncoding.ConcreteSemantics.applyVec_zeroKet" ], "route_closures": [ "QuantumBlockEncoding.ConcreteSemantics.ComplexStatePreparationCertificate.ofFirstColumn", "QuantumBlockEncoding.ConcreteSemantics.ComplexStatePreparationCertificate.verifiedOfFirstColumn", "QuantumBlockEncoding.ConcreteSemantics.ComplexStatePreparationCertificate.verifiedOfFirstColumn_preparesTarget" ] }, { "declaration": "QuantumBlockEncoding.VerifiedStatePreparation.firstColumn", "title": "A verified preparer exposes its amplitudes", "math": "U_{i,0}=\\psi_i.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.VerifiedStatePreparation" ], "route_closures": [ "QuantumBlockEncoding.VerifiedStatePreparation.firstColumn" ] }, { "declaration": "QuantumBlockEncoding.VerifiedStatePreparation.asZeroErrorApprox", "title": "Exact preparation can enter an approximate interface", "math": "\\varepsilon=0.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.VerifiedStatePreparation" ], "route_closures": [ "QuantumBlockEncoding.VerifiedStatePreparation.asZeroErrorApprox" ] } ] }, { "id": "block-encoding", "title": "The block-encoding contract", "track": "Block encoding", "summary": "State the block contract with normalization, explicit register layout, and a verifier-facing certificate record.", "results": [ { "declaration": "QuantumBlockEncoding.OperatorBlockEncodingCandidate.cost", "title": "Candidate cost is derived from its layout", "math": "c(U)=(a,q,d,n_{1q},n_{2q},\\ldots).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared reusable route.", "dependencies": [ "QuantumBlockEncoding.OperatorBlockEncodingCandidate", "QuantumBlockEncoding.RegisterLayout", "QuantumBlockEncoding.BlockEncodingCost" ], "route_closures": [ "QuantumBlockEncoding.OperatorBlockEncodingCandidate.cost_eq_fromLayoutAndResource", "QuantumBlockEncoding.OperatorBlockEncodingCandidate.certify" ] }, { "declaration": 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2^n\\rangle|i\\rangle.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared semantic-plus-finite-compiler route. The paper's arbitrary-size one-qubit/CNOT upper bound remains a separate general compiler theorem.", "dependencies": [ "QuantumBlockEncoding.BandedSparseAccess.modularSumEquiv", "QuantumBlockEncoding.BandedSparseAccess.accessMatrix_unitary", "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3Program_eval" ], "route_closures": [ "QuantumBlockEncoding.BandedSparseAccess.accessMatrix_unitary", "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3_cleanAction", "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3Program_eval", "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3Program_unitary", "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3Program_oracleCalls_eq_zero" ] }, { "declaration": "QuantumBlockEncoding.BlockEncodingClassics.partialPermutationCertificate", "title": "Partial-permutation certificate", "math": "A_{ij}\\in\\{0,1\\},\\quad \\text{at most one supported entry per routed index}.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.BlockEncodingClassics.ExactCleanBlock" ], "route_closures": [ "QuantumBlockEncoding.BlockEncodingClassics.partialPermutationCertificate" ] }, { "declaration": "QuantumBlockEncoding.BlockEncodingClassics.productExactCleanBlockCertificate", "title": "Product closure for exact clean blocks", "math": "\\operatorname{block}(U_B U_A)=BA.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.BlockEncodingClassics.ExactCleanBlock", "QuantumBlockEncoding.BlockEncodingClassics.matrix_mul_congr_pointwise" ], "route_closures": [ "QuantumBlockEncoding.BlockEncodingClassics.productExactCleanBlockCertificate" ] }, { "declaration": "QuantumBlockEncoding.BlockEncodingClassics.QSVTConsumerContract", "title": "QSVT consumer boundary", "math": "U_A\\leadsto p^{(\\mathrm{SV})}(A/\\alpha).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the declared reusable route.", "dependencies": [ "QuantumBlockEncoding.BlockEncodingClassics.HermitianDilationContract" ], "route_closures": [ "QuantumBlockEncoding.BlockEncodingClassics.QSVTConsumerContract.identity", "QuantumBlockEncoding.BlockEncodingClassics.QSVTConsumerContract.identity_output" ] } ] }, { "id": "certified-cases", "title": "Certified block-encoding cases", "track": "Block encoding", "summary": "Distinguish completed local certificates from paper-facing contract models and historical experiments.", "results": [ { "declaration": "QuantumBlockEncoding.coldE1Candidate_blockProjection", "title": "BE Case 1 block projection", "math": "\\Pi U_{\\mathrm{E1}}\\Pi^\\dagger=A_{\\mathrm{E1}}/\\alpha.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.coldE1CandidateImage_permutation_certificate", "QuantumBlockEncoding.BlockEncodingClassics.partialPermutationCertificate" ], "route_closures": [ "QuantumBlockEncoding.coldE1Candidate_blockProjection" ] }, { "declaration": "QuantumBlockEncoding.CubicDiagonalOracle.cubicDiagonalHouseholderExactBEContract_complete", "title": "BE Case 2 exact Householder completion", "math": "\\Pi U_{\\mathrm{cubic}}\\Pi^\\dagger=D_{\\mathrm{cubic}}/\\alpha.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.CubicDiagonalOracle.linearDiagonalHouseholderInputBEContract_complete", "QuantumBlockEncoding.CubicDiagonalOracle.cubicDiagonalRationalCompletion_backendSupport" ], "route_closures": [ "QuantumBlockEncoding.CubicDiagonalOracle.cubicDiagonalHouseholderExactBEContract_complete" ] }, { "declaration": "QuantumBlockEncoding.GHL2025.Hamiltonian.theorem4_source_lcu_route_closed", "title": "GHL Theorem 4: source-audited LCU composition to the Hamiltonian", "math": "A=\\sum_kA_k,\\quad A^\\dagger=\\sum_kA_k^\\dagger,\\quad H=S_1\\otimes x_\\xi+S_2\\otimes I_\\xi.", "local_status": "Compiled", "route_status": "Compiled", "missing": "The source-audited Theorem-4 composition is closed. Uniform arbitrary-width primitive compilation of all Theorem-3 source oracles and their concrete gate/resource realization remains a separate compiler frontier.", "dependencies": [ "QuantumBlockEncoding.GHL2025.Hamiltonian.adjoint_sumTerms", "QuantumBlockEncoding.Robin.ComplexLCU.prepareAmplitudeSelectUnprepare_unitary", "QuantumBlockEncoding.Robin.ComplexLCU.prepareAmplitudeSelectUnprepare_cleanEntry", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq29PrintedClean_ne_S1", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq29PhaseBalancedClean_eq_S1", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq30Clean_eq_S2" ], "route_closures": [ "QuantumBlockEncoding.GHL2025.Hamiltonian.adjoint_sumTerms", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq29PrintedClean_lowerRight", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq29PrintedClean_ne_S1", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq29PhaseBalancedClean_eq_S1", "QuantumBlockEncoding.GHL2025.Hamiltonian.eq30Clean_eq_S2", "QuantumBlockEncoding.GHL2025.Hamiltonian.OneDimCompositionCertificate.H_eq_S1_tensor_xXi_add_S2_tensor_I", "QuantumBlockEncoding.GHL2025.Hamiltonian.oneDimHamiltonianClaim_normalization_closed", "QuantumBlockEncoding.GHL2025.Hamiltonian.oneDimHamiltonianClaim_layout_closed", "QuantumBlockEncoding.GHL2025.Hamiltonian.oneDimHamiltonianResource_pureAncilla_closed", "QuantumBlockEncoding.GHL2025.Hamiltonian.theorem4_source_lcu_route_closed" ] }, { "declaration": "QuantumBlockEncoding.Examples.RobinHeat.oneTermRobinGamma3BoundaryUnitaryEntry_ne_backendFold_n3", "title": "Robin raw-fold route is rejected", "math": "U_{00}\\ne\\sum_{s=0}^{6} B_s.", "local_status": "Compiled", "route_status": "Compiled", "missing": "The rejected branch is closed. Fixed-N8 source circuits and the GHL Theorem-4 A-to-H composition are compiled; arbitrary-width primitive compilers for the general source oracles remain separately scoped.", "dependencies": [ "QuantumBlockEncoding.Examples.RobinHeat.oneTermRobinGamma3BoundaryBackendExpansionStatement_not_n3", "QuantumBlockEncoding.Examples.RobinHeat.oneTermRobinGamma3BoundaryBackendExpansionStatement_equivUnitaryEntryFold_n3" ], "route_closures": [ "QuantumBlockEncoding.Examples.RobinHeat.oneTermRobinGamma3BoundaryUnitaryEntry_ne_backendFold_n3" ] } ] }, { "id": "resources-and-exports", "title": "Candidate comparison and executable exports", "track": "System and evidence", "summary": "Keep formal validity lexicographically ahead of resource quality, then expose accepted candidates to executable tooling.", "results": [ { "declaration": "QuantumBlockEncoding.PromiseGateOptimization.controlledConjugation_matrix", "title": "Control only the middle of a conjugation", "math": "C(V^\\dagger U V)=(I\\otimes V^\\dagger)\\,C(U)\\,(I\\otimes V).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.PromiseGateOptimization.controlledConjugation_equiv", "QuantumBlockEncoding.Robin.ComplexLCU.equivPermutationMatrix_mul" ], "route_closures": [ "QuantumBlockEncoding.PromiseGateOptimization.controlledConjugation_matrix" ] }, { "declaration": "QuantumBlockEncoding.PromiseGateOptimization.dirtyControlledInvolution_action", "title": "Replace a clean flag by a dirty flag", "math": "U^2=I\\Longrightarrow(c,b,|\\psi\\rangle)\\mapsto(c,b,U^{[c]}|\\psi\\rangle).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.PromiseGateOptimization.dirtyControlledInvolutionEquiv", "QuantumBlockEncoding.PromiseGateOptimization.dirtyControlledInvolution_unitary" ], "route_closures": [ "QuantumBlockEncoding.PromiseGateOptimization.dirtyControlledInvolution_action" ] }, { "declaration": "QuantumBlockEncoding.BlockEncodingCost.betterThan", "title": "Lexicographic candidate comparison", "math": "c_1\\prec_{\\mathrm{lex}}c_2.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None for this local declaration.", "dependencies": [ "QuantumBlockEncoding.BlockEncodingCost" ], "route_closures": [ "QuantumBlockEncoding.BlockEncodingCost.betterThan" ] }, { "declaration": "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveVerifiedBlockEncoding", "title": "Finite cubic amplitude oracle", "math": "\\Pi_0U_{\\mathrm{cubic},2}\\Pi_0^\\dagger=\\operatorname{diag}(0,(1/4)^3,(2/4)^3,(3/4)^3).", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the fixed n=2 primitive route. Scalable arithmetic and general QSVT phase synthesis remain separately scoped research routes.", "dependencies": [ "QuantumBlockEncoding.compileUniformlyControlledRy_eval_controlledRyBlockMatrix", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveProgram_cleanEntry", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveFlatUnitary_unitary" ], "route_closures": [ "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveProgram_cleanEntry", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveFlatUnitary_unitary", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveFlatUnitary_cleanBlock", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveVerifiedBlockEncoding", "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2Primitive_oracleCalls_eq_zero" ] } ] }, { "id": "automation-and-roadmap", "title": "Search control, evidence, and open work", "track": "System and evidence", "summary": "Show how typed stages, agent layers, candidate populations, proof gates, and explicit open problems coordinate without overstating evidence.", "results": [ { "declaration": "QuantumBlockEncoding.threeLayerAgentContracts", "title": "Three-layer agent contracts", "math": "\\text{upper}\\rightarrow\\text{middle}\\rightarrow\\text{lower}\\rightarrow\\text{reviewer}\\rightarrow\\text{accepted}.", "local_status": "Compiled", "route_status": "Compiled", "missing": "None within the typed handoff and acceptance route. Running external models remains engineering evidence rather than a Lean theorem.", "dependencies": [ "QuantumBlockEncoding.AutomationStage", "QuantumBlockEncoding.AutomationTask", "QuantumBlockEncoding.ThreeLayerTrace" ], "route_closures": [ "QuantumBlockEncoding.threeLayerCanonicalTrace_allValid", "QuantumBlockEncoding.threeLayerCanonicalTrace_reachesAccepted", "QuantumBlockEncoding.threeLayerAccepted_requiresLeanGate", "QuantumBlockEncoding.threeLayerAccepted_requiresReviewerApproval", "QuantumBlockEncoding.threeLayerFailedGateTrace_notAccepted" ] }, { "declaration": "QuantumBlockEncoding.openProblems", "title": "Open problems are first-class data", "math": "\\mathcal O=[o_1,\\ldots,o_7],\\qquad \\operatorname{Nodup}(\\operatorname{id}(\\mathcal O)).", "local_status": "Compiled", "route_status": "Compiled", "missing": "The registry route is closed; its seven mathematical problems remain intentionally open and are not presented as solved theorems.", "dependencies": [ "QuantumBlockEncoding.OpenProblem", "QuantumBlockEncoding.openProblemIds" ], "route_closures": [ "QuantumBlockEncoding.openProblems_count", "QuantumBlockEncoding.openProblemIds_nodup", "QuantumBlockEncoding.openProblems_all_actionable", "QuantumBlockEncoding.openProblemRegistry_compiled" ] } ] } ], "paper_frontier": { "schemaVersion": 1, "featured": [ { "slug": "ghl2025-robin", "title": "Quantum Framework for Simulating Linear PDEs with Robin Boundary Conditions", "authors": "Nikita Guseynov, Xiajie Huang, Nana Liu", "year": 2025, "url": "https://arxiv.org/abs/2506.20478", "category": "blockEncoding", "sourceAnchors": [ "Eq. (9)", "Theorem 3", "Theorem 4", "Fig. 4" ], "status": "reproduced-fixed-benchmark", "route": "papers/ghl2025-robin/", "summary": "Theorem 3/4 are mapped to explicit Lean scope. The fixed N=8 Robin benchmark closes source normal forms, an exact XOR four-slot winner, and same-tier resource comparisons; the arbitrary-width primitive compiler remains a separate frontier." } ], "queue": [ { "key": "mottonen-2005-state-preparation", "title": "Transformation of quantum states using uniformly controlled rotations", "authors": "Mikko Möttönen, Juha J. Vartiainen, Ville Bergholm, Martti M. Salomaa", "year": 2005, "url": "https://arxiv.org/abs/quant-ph/0407010", "category": "statePreparation", "sourceAnchors": [ "Eq. (6)", "Eq. (7)", "Eq. (8)", "Fig. 3" ], "sourceStatement": "Eq. (6) eliminates one qubit by a uniformly controlled y rotation; Eq. (7) composes those eliminations recursively; Eq. (8) specifies the required angles; Fig. 3 displays the resulting state-preparation architecture.", "status": "finite-benchmark-formalized", "route": "papers/mottonen-2005-state-preparation/", "exampleCase": "mottonen-dense-state-preparation", "formalized": "An exact dense two-qubit target, full unitary completion, typed root-RY plus one-control-UCRY circuit, exact clean-input state action, and the circuit-derived resource tuple (5 gates, depth 4, no auxiliary qubits, no unresolved oracle calls).", "leanRoots": [ "QuantumBlockEncoding.StatePreparationBenchmarks.mottonenDenseTarget_normalized", "QuantumBlockEncoding.StatePreparationBenchmarks.mottonenDenseMatrix_unitary", "QuantumBlockEncoding.StatePreparationBenchmarks.mottonenDensePrimitive_prepares_target", "QuantumBlockEncoding.StatePreparationBenchmarks.mottonenDenseVerified_cost" ], "remaining": "General n-qubit state-to-state synthesis, phase layer, analytic rotation-angle construction at arbitrary width, and the paper-wide CNOT/one-qubit rotation count theorem." }, { "key": "grover-rudolph-2002", "title": "Creating superpositions that correspond to efficiently integrable probability distributions", "authors": "Lov Grover, Terry Rudolph", "year": 2002, "url": "https://arxiv.org/abs/quant-ph/0208112", "category": "statePreparation", "sourceAnchors": [ "Eq. (1)", "Eq. (3)", "Eq. (5)", "Eq. (6)" ], "sourceStatement": "Eq. (1) is the target probability-amplitude state; Eq. (3) gives the recursive probability refinement; Eq. (5)–(6) turn the coherently computed split angle into a controlled rotation.", "status": "structured-benchmark-formalized", "route": "papers/grover-rudolph-2002/", "exampleCase": "grover-rudolph-product-state-preparation", "formalized": "An exact two-bit product distribution, a typed generic binary-tree circuit and typed factorized circuit preparing the same target, and a Lean-certified resource improvement from (5,4,0,0) to (2,1,0,0).", "leanRoots": [ "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphProductTarget_normalized", "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphFactorized_prepares_target", "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphTree_prepares_target", "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphTreeVerified_cost", "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphFactorizedVerified_cost", "QuantumBlockEncoding.StatePreparationBenchmarks.groverRudolphFactorized_betterThan_tree" ], "remaining": "The general efficiently-integrable recursive probability-loading theorem and an end-to-end arithmetic/integration oracle compiler." }, { "key": "li-luo-sparse-state-2025", "title": "Nearly Optimal Circuit Size for Sparse Quantum State Preparation", "authors": "Lvzhou Li, Jingquan Luo", "year": 2025, "url": "https://arxiv.org/abs/2406.16142", "category": "statePreparation", "sourceAnchors": [ "Eq. (1)", "Eq. (2)", "Theorem 1" ], "sourceStatement": "Eq. (1) records a d-sparse state through its nonzero amplitudes and basis labels; Eq. (2) is the exact state-preparation unitary contract with clean ancillas; Theorem 1 is the asymptotic circuit-size statement that remains outside the finite ASPBE witness.", "status": "finite-sparse-benchmark-formalized", "route": "papers/li-luo-sparse-state-2025/", "exampleCase": "sparse-three-state-preparation", "formalized": "An exact n=3, d=3 witness of Eq. (2), a typed pruned UCRY route, a same-target typed dense zero-fill baseline, and a Lean-certified resource improvement from (15,13,0,0) to (5,4,0,0).", "leanRoots": [ "QuantumBlockEncoding.StatePreparationBenchmarks.sparseThreeTarget_normalized", "QuantumBlockEncoding.StatePreparationBenchmarks.sparseThreeMatrix_unitary", "QuantumBlockEncoding.StatePreparationBenchmarks.sparsePruned_prepares_target", "QuantumBlockEncoding.StatePreparationBenchmarks.sparseDenseTree_prepares_target", "QuantumBlockEncoding.StatePreparationBenchmarks.sparseDenseTreeVerified_cost", "QuantumBlockEncoding.StatePreparationBenchmarks.sparsePrunedVerified_cost", "QuantumBlockEncoding.StatePreparationBenchmarks.sparsePruned_betterThan_denseTree" ], "remaining": "The asymptotic sparse synthesis constructions, ancilla/circuit-size tradeoffs, and the matching lower bounds of Theorem 1." }, { "key": "low-kliuchnikov-schaeffer-2018", "title": "Trading T gates for dirty qubits in state preparation and unitary synthesis", "authors": "Guang Hao Low, Vadym Kliuchnikov, Luke Schaeffer", "year": 2018, "url": "https://arxiv.org/abs/1812.00954", "category": "statePreparation", "sourceAnchors": [ "Eq. (2)", "Eq. (5)", "Table 2", "Fig. 1(c,d)" ], "sourceStatement": "Eq. (2) states the arbitrary target-state problem; Eq. (5) gives the coherent data-lookup interface; Table 2 and Fig. 1(c,d) expose the SelectSwap T-count/space tradeoff used by the finite arithmetic witness.", "status": "resource-lemma-formalized", "route": "papers/low-kliuchnikov-schaeffer-2018/", "formalized": "The clean-qubit SelectSwap T-count formula at the repository arithmetic tier, including N=16, b=1 values 72 at lambda=1 and 48 at lambda=4, plus the strict finite comparison 48<72.", "leanRoots": [ "QuantumBlockEncoding.StatePreparationBenchmarks.selectSwapCleanTCount_16_1_1", "QuantumBlockEncoding.StatePreparationBenchmarks.selectSwapCleanTCount_16_1_4", "QuantumBlockEncoding.StatePreparationBenchmarks.selectSwapCleanTCount_lambda4_better_lambda1" ], "remaining": "Approximate Clifford+T state preparation, coherent lookup/SelectSwap semantics, dirty-qubit correctness, error accounting, and the full asymptotic optimality theorem." }, { "key": "sun-tian-yang-yuan-zhang-2021", "title": "Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis", "authors": "Xiaoming Sun, Guojing Tian, Shuai Yang, Pei Yuan, Shengyu Zhang", "year": 2021, "url": "https://arxiv.org/abs/2108.06150", "category": "statePreparation", "sourceAnchors": [], "status": "queued", "formalized": "Not yet reproduced in Lean.", "remaining": "The ancilla-sensitive state-preparation construction, depth/size upper bounds, and the matching optimal-depth statements across the claimed parameter regimes." }, { "key": "plesch-brukner-2011", "title": "Quantum-state preparation with universal gate decompositions", "authors": "Martin Plesch, Časlav Brukner", "year": 2011, "url": "https://arxiv.org/abs/1003.5760", "category": "statePreparation", "sourceAnchors": [], "status": "queued", "formalized": "Not yet reproduced in Lean.", "remaining": "Universal state-preparation decomposition and its CNOT/depth counts, including the four-qubit benchmark." }, { "key": "shende-bullock-markov-2006", "title": "Synthesis of Quantum Logic Circuits", "authors": "Vivek V. Shende, Stephen S. Bullock, Igor L. Markov", "year": 2006, "url": "https://arxiv.org/abs/quant-ph/0406176", "category": "statePreparation", "sourceAnchors": [], "status": "queued", "formalized": "The local uniformly-controlled-RY compiler provides a related reusable primitive, but this paper is not claimed reproduced.", "remaining": "Paper-faithful quantum-multiplexor/state-initialization synthesis, its CNOT complexity theorem, and the lower-bound comparison." } ] }, "implementation_frontier": [ { "goal": "Prepare the one-qubit plus state", "contract": "H|0\\rangle=(|0\\rangle+|1\\rangle)/\\sqrt2", "obligation": "Normalization, Mathlib unitarity, state action, and logical cost", "declaration": "QuantumBlockEncoding.TextbookStatePreparation.hadamardCertificate_prepares_plus", "dependencies": "hadamard_unitary; plusTarget_normalized; hadamard_prepares_plus", "status": "Compiled", "missing": "None for this textbook route", "chapter": "state-preparation" }, { "goal": "Prepare the one-qubit basis state one", "contract": "X|0\\rangle=|1\\rangle", "obligation": "Permutation unitarity, state action, and logical cost", "declaration": "QuantumBlockEncoding.TextbookStatePreparation.pauliXCertificate_prepares_one", "dependencies": "pauliX_unitary; oneTarget_normalized; pauliX_prepares_one", "status": "Compiled", "missing": "None for this textbook route", "chapter": "state-preparation" }, { "goal": "Prepare a finite target state", "contract": "U|0^n\\rangle=|\\psi\\rangle", "obligation": "Normalization, unitarity, and first-column equality supplied to the promotion constructor", "declaration": "QuantumBlockEncoding.StatePreparationCandidate.preparesTarget", "dependencies": "StatePreparationCandidate.certify; textbook Pauli-X and Hadamard witnesses", "status": "Compiled", "missing": "None for the reusable promotion route; each new target family supplies its own proof terms", "chapter": "state-preparation" }, { "goal": "Consume a verified prepared state", "contract": "U_{i0}=\\psi_i", "obligation": "First-column projection", "declaration": "QuantumBlockEncoding.VerifiedStatePreparation.firstColumn", "dependencies": "VerifiedStatePreparation", "status": "Compiled", "missing": "None locally", "chapter": "state-preparation-certificates" }, { "goal": "Read first-column evidence as state action", "contract": "\\operatorname{column}_0(U)=\\psi\\iff U|0^n\\rangle=|\\psi\\rangle", "obligation": "Finite basis-ket matrix-action bridge", "declaration": "QuantumBlockEncoding.ConcreteSemantics.firstColumnMatches_iff_applyVec_zeroKet", "dependencies": "FirstColumnMatches; applyVec_zeroKet", "status": "Compiled", "missing": "Candidate normalization and unitarity remain separate", "chapter": "state-preparation-certificates" }, { "goal": "Evaluate a gate list", "contract": "\\mathrm{Eval}(C)=G_m\\cdots G_1", "obligation": "Fixed fold and register order", "declaration": "QuantumBlockEncoding.evalGateMatrices", "dependencies": "GateMatrix; Matrix", "status": "Compiled", "missing": "None locally", "chapter": "circuit-semantics" }, { "goal": "Extract a block from a circuit", "contract": "\\Pi U\\Pi^\\dagger=A/\\alpha", "obligation": "Typed circuit semantics, selected projection equality, and finite identity witness", "declaration": "QuantumBlockEncoding.CircuitMatrixSemantics.blockExtractionTarget", "dependencies": "CertifiedCircuitBlockExtraction; teachingIdentityBlockExtraction", "status": "Compiled", "missing": "None for the reusable extraction route; each concrete circuit supplies its unitary and entry proof", "chapter": "circuit-semantics" }, { "goal": "Move between flat and product-register block views", "contract": "\\Pi_s U\\Pi_s^\\dagger=\\operatorname{cleanBlockProduct}(s,U)", "obligation": "Shared register-order projection equality", "declaration": "QuantumBlockEncoding.ConcreteSemantics.signalSystemBlockProjection_eq_cleanBlockProduct", "dependencies": "signalSystemBlockProjection; cleanBlockProduct", "status": "Compiled", "missing": "Candidate-level block and unitarity proofs remain separate", "chapter": "circuit-semantics" }, { "goal": "Package an exact block encoding", "contract": "\\Pi U\\Pi^\\dagger=A/\\alpha", "obligation": "Candidate unitarity and exact block identity supplied to the promotion constructor", "declaration": "QuantumBlockEncoding.VerifiedOperatorBlockEncoding", "dependencies": "OperatorBlockEncodingCandidate.certify; certified finite cases", "status": "Compiled", "missing": "None for the reusable promotion route; concrete candidate fields remain route-specific inputs", "chapter": "block-encoding" }, { "goal": "Reuse an exact certificate approximately", "contract": "\\|A-\\alpha\\Pi U\\Pi^\\dagger\\|=0", "obligation": "Zero-error conversion", "declaration": "QuantumBlockEncoding.VerifiedOperatorBlockEncoding.asZeroErrorApprox", "dependencies": "VerifiedOperatorBlockEncoding", "status": "Compiled", "missing": "Positive-error analyses are route-specific", "chapter": "block-encoding" }, { "goal": "Certify a partial permutation", "contract": "\\operatorname{block}(U)=A", "obligation": "Finite routed-entry proof", "declaration": "QuantumBlockEncoding.BlockEncodingClassics.partialPermutationCertificate", "dependencies": "ExactCleanBlock", "status": "Compiled", "missing": "None locally", "chapter": "classic-routes" }, { "goal": "Compose exact block encodings", "contract": "\\operatorname{block}(U_BU_A)=BA", "obligation": "Projected product equality", "declaration": "QuantumBlockEncoding.BlockEncodingClassics.productExactCleanBlockCertificate", "dependencies": "ExactCleanBlock; matrix_mul_congr_pointwise", "status": "Compiled", "missing": "Register compatibility for each application", "chapter": "classic-routes" }, { "goal": "Close BE Case 1", "contract": "\\Pi U_{\\mathrm{E1}}\\Pi^\\dagger=A_{\\mathrm{E1}}/\\alpha", "obligation": "Transfer-operator projection", "declaration": "QuantumBlockEncoding.coldE1Candidate_blockProjection", "dependencies": "coldE1CandidateImage_permutation_certificate", "status": "Compiled", "missing": "None locally", "chapter": "certified-cases" }, { "goal": "Close BE Case 2", "contract": "\\Pi U_{\\mathrm{cubic}}\\Pi^\\dagger=D/\\alpha", "obligation": "Exact rational Householder completion", "declaration": "QuantumBlockEncoding.CubicDiagonalOracle.cubicDiagonalHouseholderExactBEContract_complete", "dependencies": "Householder input; rational backend support", "status": "Compiled", "missing": "None for the selected exact route", "chapter": "certified-cases" }, { "goal": "Feed a certified clean block to a typed polynomial consumer", "contract": "U_D\\leadsto p^{(\\mathrm{SV})}(D/\\alpha)", "obligation": "Source certificate, side conditions, and a finite degree-one identity realization", "declaration": "QuantumBlockEncoding.BlockEncodingClassics.QSVTConsumerContract", "dependencies": "QSVTConsumerContract.identity; teachingIdentityQSVTConsumer", "status": "Compiled", "missing": "None for the typed identity consumer; general phase synthesis and approximation checking remain roadmap items", "chapter": "classic-routes" }, { "goal": "Compile a finite banded sparse-access witness", "contract": "|s\\rangle|i\\rangle|0\\rangle\\mapsto|s\\oplus3+i\\bmod8\\rangle|i\\rangle|0\\rangle", "obligation": "Primitive refinement, clean workspace, unitarity, and zero oracle calls", "declaration": "QuantumBlockEncoding.BandedSparseAccess.primitiveAccess3Program_eval", "dependencies": "modularAdd3ReversibleProgram; compileReversibleProgram_eval", "status": "Compiled", "missing": "Arbitrary-width source gate upper bounds remain a separate compiler theorem", "chapter": "classic-routes" }, { "goal": "Compile a finite cubic diagonal amplitude oracle", "contract": "\\Pi_0U\\Pi_0^\\dagger=\\operatorname{diag}((j/4)^3)", "obligation": "Exact multiplexed RY semantics, clean projection, and verified BE packaging", "declaration": "QuantumBlockEncoding.CubicDiagonalOracle.cubicN2PrimitiveVerifiedBlockEncoding", "dependencies": "UniformlyControlledRy; PrimitiveBasisLE; exact RY bridge", "status": "Compiled", "missing": "Scalable arithmetic and general QSVT phase synthesis are separate routes", "chapter": "resources-and-exports" }, { "goal": "Verify three-layer harness promotion", "contract": "\\text{upper}\\to\\text{middle}\\to\\text{lower}\\to\\text{reviewer}\\to\\text{accepted}", "obligation": "Logged artifacts, valid role order, Lean gate, and reviewer approval", "declaration": "QuantumBlockEncoding.threeLayerCanonicalTrace_reachesAccepted", "dependencies": "ThreeLayerHandoff.validFlag; ThreeLayerTrace.finalPhase", "status": "Compiled", "missing": "External model execution remains engineering evidence", "chapter": "automation-and-roadmap" }, { "goal": "Audit the open-problem registry", "contract": "|\\mathcal O|=7\\land\\operatorname{Nodup}(\\operatorname{id}(\\mathcal O))", "obligation": "Unique identifiers and nonempty actionable fields", "declaration": "QuantumBlockEncoding.openProblemRegistry_compiled", "dependencies": "openProblemIds_nodup; openProblems_all_actionable", "status": "Compiled", "missing": "The registered mathematical problems intentionally remain open", "chapter": "automation-and-roadmap" }, { "goal": "Compose the GHL one-dimensional Hamiltonian", "contract": "A=\\sum_kA_k,\\ A^\\dagger=\\sum_kA_k^\\dagger,\\ H=S_1\\otimes x_\\xi+S_2\\otimes I_\\xi", "obligation": "Finite sum/adjoint bridge, Hermitian S1/S2 split, and final Hamiltonian composition", "declaration": "QuantumBlockEncoding.GHL2025.Hamiltonian.theorem4_source_lcu_route_closed", "dependencies": "adjoint_sumTerms; verified ComplexLCU kernel; eq29PrintedClean_ne_S1; eq29PhaseBalancedClean_eq_S1; eq30Clean_eq_S2", "status": "Compiled", "missing": "None for the source-audited Theorem-4 composition; arbitrary-width primitive one-term gate/resource compilation is a separate frontier", "chapter": "certified-cases" }, { "goal": "Close the historical Robin raw-fold branch", "contract": "U_{00}\\ne\\sum_s B_s", "obligation": "Convert the raw fold to the refuted backend expansion", "declaration": "QuantumBlockEncoding.Examples.RobinHeat.oneTermRobinGamma3BoundaryUnitaryEntry_ne_backendFold_n3", "dependencies": "Equivalence bridge; all-one counterexample", "status": "Compiled", "missing": "None for the rejected branch; cited oracle implementations remain experimental", "chapter": "certified-cases" } ], "historical_roadmap": [ { "title": "State-preparation contracts and first-column consumer", "status": "Compiled" }, { "title": "Textbook Pauli X and Hadamard certificates", "status": "Compiled" }, { "title": "Circuit syntax to matrix semantics", "status": "Compiled" }, { "title": "Reusable exact block-encoding routes", "status": "Compiled" }, { "title": "Finite three-bit primitive banded sparse access", "status": "Compiled" }, { "title": "BE Case 1 transfer-operator certificate", "status": "Compiled" }, { "title": "BE Case 2 exact Householder certificate", "status": "Compiled" }, { "title": "Finite two-qubit cubic primitive amplitude oracle", "status": "Compiled" }, { "title": "Degree-one QSVT identity consumer realization", "status": "Compiled" }, { "title": "Three-layer controller trace and registry audit", "status": "Compiled" }, { "title": "Fixed-N8 Robin T3 reproduction and evolved winner", "status": "Compiled" }, { "title": "Arbitrary-width banded-access source resource compiler", "status": "Planned" }, { "title": "General QSVT phase synthesis and approximation checker", "status": "Planned" }, { "title": "GHL Theorem 4 A-to-H Hamiltonian composition", "status": "Compiled" }, { "title": "Arbitrary-width GHL one-term primitive resource compiler", "status": "Planned" }, { "title": "Historical Robin H-free raw-fold rejection", "status": "Compiled" } ], "research_routes": [ { "id": "spw-structured", "title": "High-dimensional structured function states", "priority": 1, "status": "research-target", "setting_id": "setting:structured-tt-supplied-cores", "formula": "|f\\rangle=\\frac{1}{\\|f\\|_{2,N}}\\sum_{\\mathbf j\\in[N]^D}f(\\mathbf x_{\\mathbf j})|\\mathbf j\\rangle,\\quad N=2^n", "goal": "Prove a constructive cost and state-error theorem for a precisely specified low-rank, sparse-frequency, mixed-smoothness or compositional class; do not equate these assumptions.", "motivation": "Multivariate PDE data, localized scientific functions and conditional distributions need more than generic amplitude loading.", "input_model": "An explicit formula-to-core supplier or a charged sparse coefficient oracle; grid, complex phase and a nonzero norm are part of the input.", "assumptions": [ "D and n specified; rank r and degree q are certified, not numerical fit labels", "Core/coefficients and normalizer are computably supplied", "Approximation error includes model compression, grid, arithmetic and primitive synthesis" ], "target_bound": "A candidate target is poly(D,r,q,n,log(1/epsilon)) cost under stated bit-length and stability promises; this bound is not claimed here.", "known_boundary": "Generic tensor-product degree-q expansions have (q+1)^D coefficients; smoothness alone does not remove dimensionality. Hermite's one-dimensional exact-real compiler is a substrate, not this multivariate theorem.", "families": [ "family:bounded-memory", "family:gram-normalization", "family:local-isometry", "family:charged-access" ], "source_ids": [ "holmes-matsuura-2020", "multivariate-2025", "qkan-2026" ], "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec" ], "steps": [ { "id": "class-contract", "target": "Choose one function class and prove a uniform rank/degree certificate.", "acceptance": "A source-faithful class predicate and formula-to-core action theorem, including phases and supports." }, { "id": "error-supplier", "target": "Bound normalized-state error from core approximation and arithmetic.", "acceptance": "An explicit norm lower bound and a theorem for the exact requested vector norm; no uncharged condition number." }, { "id": "compiler", "target": "Compose the class supplier with the existing clean TT compiler.", "acceptance": "Primitive-list semantics, workspace cleanup, and every cost coordinate under the same model." } ], "next": "Start from the Hermite corridor; add a tensor-product or shallow coupled class with a proved rank bound before claiming general high dimension.", "lower_bound": { "status": "model-definition-pending", "task": "Separate arbitrary-vector counting barriers from lower bounds for the selected structured class. Specify which class parameters enter the hard family.", "comparison_key": "setting:structured-tt-supplied-cores" }, "benchmarks": [ "Product Gaussian with explicitly bounded widths", "Weakly coupled Gaussian with a proved rank estimate", "Hermite-smoothed PDE auxiliary profile" ] }, { "id": "spw-envelope", "title": "Preconditioned function preparation with an explicit envelope", "priority": 1, "status": "research-target", "setting_id": "setting:coherent-envelope-ratio", "formula": "\\kappa_{\\rm env}=\\frac{C\\|g\\|_2}{\\|f\\|_2},\\quad p_{\\rm succ}=\\kappa_{\\rm env}^{-2},\\quad C\\ge\\max_{g_j\\ne0}|f_j/g_j|", "goal": "Construct g for one useful localized function class so that reference preparation, ratio implementation and amplification are all controlled.", "motivation": "Uniform-state postselection can have exponentially small filling fraction in dimension; a reference should match localization without hiding an equally hard preparation problem.", "input_model": "A clean U_g and U_g-dagger with charged construction; coherent access to a bounded complex ratio; classical or certified scalar C.", "assumptions": [ "support(f) subset support(g); define the ratio as zero off support(g)", "f and g nonzero and C>0", "A constructive domination certificate, not an empirical maximum on a few grid points" ], "target_bound": "For a selected class, prove kappa_env and reference/ratio costs polynomial in declared class parameters; add a model-matched obstruction or lower bound.", "known_boundary": "Coherent rejection and reweighting are prior art. Uniform g recovers kappa_env=1/F2 where F2=||f||2/(sqrt(M)||f||infinity). A small ratio on average does not imply a valid pointwise envelope.", "families": [ "family:envelope-reweight", "family:bounded-memory", "family:spectral-filter", "family:charged-access" ], "source_ids": [ "quantum-rejection-2013", "mcardle-gilyen-berta-2026", "multivariate-2025" ], "lean_refs": [], "steps": [ { "id": "success-identity", "target": "Formalize support-aware coherent reweighting and its exact success probability.", "acceptance": "All amplitudes including zero-reference entries and complex phases agree; normalizer nonzero." }, { "id": "construct-envelope", "target": "Build a certified reference for a restricted localized class.", "acceptance": "A concrete supplier for g, a global domination proof and kappa bound." }, { "id": "compose-cost", "target": "Add robust ratio approximation and amplification.", "acceptance": "State error and actual query/primitive costs, including U_g inversion and repeated calls." } ], "next": "First close the support-aware probability identity, then study anisotropic product envelopes and a controlled perturbation class.", "lower_bound": { "status": "source-audit-pending", "task": "Audit the quantum rejection-sampling query lower bound and prove whether its oracle assumptions match this structured envelope model before transferring it.", "comparison_key": "setting:coherent-envelope-ratio" }, "benchmarks": [ "Localized Gaussian profiles", "Boundary-layer initial data", "Piecewise payoff functions with declared phase convention" ] }, { "id": "spw-no-qram", "title": "Data loading without a free QRAM assumption", "priority": 2, "status": "research-target", "setting_id": "setting:explicit-data-charged-access", "formula": "T_{\\rm total}=T_{\\rm build}+Q\\,T_{\\rm query}+T_{\\rm update}+T_{\\rm readout}", "goal": "Identify succinct data classes for which explicit access construction preserves the intended algorithmic advantage.", "motivation": "An algorithm with few abstract queries can still be expensive when the query unitary is built from real data.", "input_model": "Classical stored arrays, QROM, arithmetic formulas and physical QRAM are separate settings with explicit initialization, updates and memory costs.", "assumptions": [ "Fix data bit length, update frequency and reuse count", "Charge controlled/inverse queries when the algorithm uses them", "Compare against a classical algorithm with the same access assumptions" ], "target_bound": "A full cost theorem for one succinct data representation, not universal polylogarithmic loading of an arbitrary array.", "known_boundary": "General sparse-access and block-encoding constructions can require near-linear cost in matrix dimension. Sparsity does not make all entries free.", "families": [ "family:charged-access", "family:prepare-select", "family:matrix-contract" ], "source_ids": [ "zhang-yuan-2024", "yuan-zhang-2023" ], "lean_refs": [], "steps": [ { "id": "access-contract", "target": "Define the exact classical/quantum access interface.", "acceptance": "Separate data construction, coherent calls, memory and update costs." }, { "id": "succinct-supplier", "target": "Compile one formula, sparse-index or repeated-block supplier.", "acceptance": "Basis-action theorem and count of actual access primitives." }, { "id": "application", "target": "Reinsert access costs into a downstream algorithm.", "acceptance": "A comparison using the same input/output model and total cost, not query count alone." } ], "next": "Reuse existing sparse/LCU routes only after auditing their oracle contracts and compiler frontiers.", "lower_bound": { "status": "source-audit-pending", "task": "Port a pinned circuit-capacity/access lower bound with precision and ancilla assumptions intact.", "comparison_key": "setting:explicit-data-charged-access" }, "benchmarks": [ "Repeated structured blocks", "Sparse matrices with formulaic index/value access", "Amortized repeated-query workloads" ] }, { "id": "spw-controlled", "title": "Coherent parameter-dependent families", "priority": 1, "status": "research-target", "setting_id": "setting:joint-parameter-data-supplier", "formula": "U:\\sum_a c_a|a\\rangle|0\\rangle\\longmapsto\\sum_a c_a|a\\rangle|f_a\\rangle", "goal": "Produce one coherent template for jointly structured parameters and data with a uniform error bound.", "motivation": "Parameterized PDEs, source histories and Bayesian conditional states require superpositions of inputs, not separate classically compiled circuits.", "input_model": "Parameter a is a quantum register; one shared joint coefficient/core supplier is provided with a uniform norm bound.", "assumptions": [ "Relative phases are fixed across a", "Norms stay bounded away from zero on the promised parameter domain", "Branch selection and numerical rank changes are controlled coherently or avoided" ], "target_bound": "Polynomial dependence on the joint structural parameters and parameter bit length, rather than a table of independent preparations.", "known_boundary": "A family of individually easy states need not have an easy coherent supplier. Per-parameter correctness up to phase is insufficient.", "families": [ "family:state-action", "family:bounded-memory", "family:charged-access" ], "source_ids": [ "yuan-zhang-2023", "qkan-2026" ], "lean_refs": [], "steps": [ { "id": "phase-contract", "target": "Define the coherent family contract including phases and clean workspace.", "acceptance": "One matrix action for arbitrary parameter superpositions." }, { "id": "joint-structure", "target": "Derive a joint tensor/polynomial representation.", "acceptance": "Uniform rank/degree and nonzero normalization proof over all allowed parameters." }, { "id": "uniform-error", "target": "Compile with uniform error and cost.", "acceptance": "Operator-action error on the whole parameter-data subspace, not only sampled parameter values." } ], "next": "Use an affine parameterized Hermite or Gaussian family as a small exact model before time-dependent PDE source histories.", "lower_bound": { "status": "model-definition-pending", "task": "Keep arbitrary CQSP lower bounds separate from the restricted joint-structure promise.", "comparison_key": "setting:joint-parameter-data-supplier" }, "benchmarks": [ "Width-dependent Gaussians", "Parameter-dependent Hermite smoothing", "Time-dependent source terms" ] }, { "id": "spw-ground", "title": "Useful trial states for strong-correlation ground-state preparation", "priority": 3, "status": "research-target", "setting_id": "setting:ground-filter-overlap-gap", "formula": "\\gamma=\\|\\Pi_0|\\psi_{\\rm trial}\\rangle\\|,\\qquad \\Delta=E_1-E_0", "goal": "For a specified physical family, construct a cheap trial-state supplier with a proved useful overlap or target-subspace guarantee.", "motivation": "Near-optimal filtering is only useful when trial-state and spectral promises are available.", "input_model": "Hamiltonian access, a charged trial-state circuit and explicit energy/gap information; target may be a ground subspace rather than a unique vector.", "assumptions": [ "Hamiltonian family and locality stated", "Overlap convention and gap promise stated", "Trial-state optimization and evaluation costs included" ], "target_bound": "A class-specific overlap and total preparation theorem; no unconditional removal of inverse-overlap or inverse-gap barriers.", "known_boundary": "Lin–Tong provides algorithms and lower bounds under overlap/gap assumptions. Strong correlation does not itself certify an MPS rank or overlap.", "families": [ "family:spectral-filter", "family:bounded-memory", "family:charged-access" ], "source_ids": [ "lin-tong-2020", "chemistry-initial-2024" ], "lean_refs": [], "steps": [ { "id": "filter-baseline", "target": "Formalize one pinned filtering theorem with exact promises.", "acceptance": "Gap, overlap, energy information and success convention match the source." }, { "id": "trial-supplier", "target": "Prove an overlap bound for a concrete trial family.", "acceptance": "The trial circuit and overlap certificate are both explicit." }, { "id": "compose", "target": "Compose trial preparation and filtering.", "acceptance": "All repeated preparations, oracle calls, error and workspace are charged." } ], "next": "Begin with a controlled family with a rigorous trial-state overlap, not arbitrary strongly correlated molecules.", "lower_bound": { "status": "external-baseline", "task": "Retain search/approximate-counting barriers from the pinned source and test whether extra structure changes their applicability.", "comparison_key": "setting:ground-filter-overlap-gap" }, "benchmarks": [ "Gapped one-dimensional families with a supplied rank certificate", "Small selected-CI/MPS trial models with explicit overlap" ] }, { "id": "spw-gibbs", "title": "Low-temperature Gibbs preparation with quantitative mixing", "priority": 3, "status": "research-target", "setting_id": "setting:local-kms-generator", "formula": "\\rho_\\beta=\\frac{e^{-\\beta H}}{\\operatorname{Tr}(e^{-\\beta H})},\\quad \\|\\mathcal E_t(\\rho)-\\rho_\\beta\\|_1\\le\\epsilon", "goal": "For a restricted noncommuting local Hamiltonian family, prove a mixing certificate and implemented-channel error bound.", "motivation": "An invariant Gibbs state does not guarantee that an implementable sampler reaches it quickly at low temperature.", "input_model": "Specified Hamiltonian and KMS generator with charged simulation access, temperature, initial state and convergence norm.", "assumptions": [ "Primitivity/ergodicity or invariant-subspace restriction stated", "Spectral or functional-inequality mixing certificate supplied", "Generator implementation and statistical output contract specified" ], "target_bound": "A model-specific polynomial mixing and total-cost theorem in the promised temperature regime.", "known_boundary": "KMS detailed balance and stationarity are not polynomial mixing. Classical sampling inequalities cannot be transferred to quantum operators without a hypothesis map.", "families": [ "family:mixing-certificate", "family:charged-access", "family:matrix-contract" ], "source_ids": [ "ding-li-lin-2025" ], "lean_refs": [], "steps": [ { "id": "invariance", "target": "Port a finite-dimensional invariant-state identity.", "acceptance": "Exact generator, Gibbs normalizer and adjoint convention." }, { "id": "mixing", "target": "Identify and prove a coercivity or mixing condition for one class.", "acceptance": "An explicit rate with temperature/system-size dependence; no assumed conclusion." }, { "id": "channel-error", "target": "Compose mixing with generator-simulation error.", "acceptance": "A trace-norm output theorem and total implementation budget." } ], "next": "Use a finite-dimensional warm-up and then separate the high-temperature baseline from the intended low-temperature regime.", "lower_bound": { "status": "model-definition-pending", "task": "State the oracle and Hamiltonian class before importing hardness or mixing obstructions.", "comparison_key": "setting:local-kms-generator" }, "benchmarks": [ "Commuting warm-up", "Small noncommuting local families", "Explicit high-temperature comparison baseline" ] }, { "id": "spw-fault-tolerant", "title": "Fault-tolerant resource trade-offs for structured preparation", "priority": 2, "status": "research-target", "setting_id": "setting:structured-clifford-t-connectivity", "formula": "\\mathcal R=(T,\\operatorname{Toffoli},d,q_{\\rm anc},Q_{\\rm oracle},\\epsilon)", "goal": "Compile the exact-real structured route into a finite gate set with a proved error budget and declared connectivity.", "motivation": "A small number of arbitrary real rotations is not a complete fault-tolerant resource estimate.", "input_model": "Finite-bit source data and a fixed Clifford+T gate set, connectivity graph, tolerance and ancilla policy.", "assumptions": [ "Input/separation and conditioning promises exposed", "Each angle approximation and synthesis error budget explicit", "Count or bound the actual emitted primitive list" ], "target_bound": "A source-to-finite-bit theorem and a model-matched resource frontier; do not claim simultaneous global optimality of every coordinate.", "known_boundary": "The Hermite exact-real primitive theorem is already a useful substrate. Full bit complexity and stable angle generation are additional obligations.", "families": [ "family:charged-access", "family:local-isometry", "family:gram-normalization" ], "source_ids": [ "zhang-yuan-2024", "yuan-zhang-2023" ], "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec" ], "steps": [ { "id": "input-bits", "target": "Freeze the finite-bit input and numerical-stability contract.", "acceptance": "All arbitrary-real comparisons and small pivots have explicit handling." }, { "id": "stable-angles", "target": "Bound coefficient, canonicalization and angle errors.", "acceptance": "A normalized-state error theorem tied to supplied bit precision." }, { "id": "logical-compiler", "target": "Compose finite-gate synthesis and connectivity routing.", "acceptance": "T/Toffoli/depth/ancilla accounting plus final state error and cleanup." } ], "next": "Treat the current Hermite bit-complexity boundary as the first local target rather than reopening its exact-real correctness proof.", "lower_bound": { "status": "source-audit-pending", "task": "Compare ancilla/depth trade-offs only in the same finite gate and connectivity model.", "comparison_key": "setting:structured-clifford-t-connectivity" }, "benchmarks": [ "Hermite baseline versus structured route at matched error", "Connectivity-constrained local TT compilation" ] }, { "id": "spw-verification", "title": "Joint design of preparation and experimental verification", "priority": 2, "status": "research-target", "setting_id": "setting:structured-witness-measurements", "formula": "\\Pr[|\\widehat F-F(\\rho,|f\\rangle)|\\le\\epsilon]\\ge1-\\delta", "goal": "Automatically derive a low-sample fidelity witness for a precisely defined structured state class.", "motivation": "Full tomography is not the correct default output task; preparation structure may enable a cheaper witness.", "input_model": "Copies or channels producing rho, a specified measurement family, classical target representation and declared IID/noise assumptions.", "assumptions": [ "Target and implemented noisy state distinguished", "Witness computation and measurement basis-change cost included", "Confidence, bias and non-IID handling not silently omitted" ], "target_bound": "A witness correctness theorem, finite-sample bound and implementation cost for one state family.", "known_boundary": "A Lean ideal-circuit proof does not certify noisy hardware. The user-supplied 2609.08414 reference remains primary-source-unavailable in this ledger.", "families": [ "family:structured-verification", "family:bounded-memory", "family:charged-access" ], "source_ids": [ "butterworth-2026-candidate" ], "lean_refs": [], "steps": [ { "id": "measurement-contract", "target": "Specify the measurement and statistical experiment.", "acceptance": "Outcome law, confidence and noise assumptions are formal objects." }, { "id": "witness", "target": "Derive a structure-specific fidelity witness.", "acceptance": "Unbiasedness or certified one-sided bias and computable observable." }, { "id": "sample-cost", "target": "Prove sample and basis-change complexity.", "acceptance": "End-to-end confidence statement with no uncharged target tomography." } ], "next": "Verify the primary hardware reference first; start formal work from a small exactly analyzable MPS witness.", "lower_bound": { "status": "model-definition-pending", "task": "Distinguish lower bounds for full tomography, fidelity estimation and a promised structured witness.", "comparison_key": "setting:structured-witness-measurements" }, "benchmarks": [ "Product-state witness", "Small exact MPS with prescribed measurement access", "Structured function signals" ] }, { "id": "spw-cvdv", "title": "CV–DV function preparation and non-Gaussian resources", "priority": 2, "status": "research-target", "setting_id": "setting:finite-energy-cv-dv-embedding", "formula": "\\|V_N|f_N\\rangle-|f\\rangle_{L^2}\\|\\le\\epsilon_{\\rm trunc}+\\epsilon_{\\rm grid}+\\epsilon_{\\rm prep}", "goal": "Relate a declared finite-energy oscillator encoding to qubit-grid preparation with a rigorous embedding and separate resource budgets.", "motivation": "Schrödingerisation uses auxiliary profiles in both continuous-variable and qubit implementations, but their resources are not interchangeable.", "input_model": "An explicit isometric embedding V_N, cutoff, grid or Fock truncation, energy/squeezing and non-Gaussian gate model.", "assumptions": [ "Finite-energy/domain conditions and target normalization", "The embedding and quadrature weights are fixed", "CV noise, squeezing and non-Gaussian resources are not counted as free qubits" ], "target_bound": "A model-specific conversion, approximation and preparation theorem, not a universal equivalence of CV and DV costs.", "known_boundary": "The finite Hermite state is a useful first example. No CV–DV equivalence or GKP preparation theorem is currently supplied by this route.", "families": [ "family:harmonic-lift", "family:hermite-bernstein", "family:matrix-contract", "family:charged-access" ], "source_ids": [], "lean_refs": [ "QuantumBlockEncoding.ConstructiveHermitePreparation.prepare_spec" ], "steps": [ { "id": "embedding", "target": "Define and prove an isometric finite-to-continuous embedding.", "acceptance": "Inner products and normalization match the stated quadrature/Fock convention." }, { "id": "truncation", "target": "Prove energy-tail and discretization error for a fixed profile family.", "acceptance": "All cutoff, domain and regularity hypotheses explicit." }, { "id": "resource-map", "target": "Compose preparation and conversion costs.", "acceptance": "Separate DV gates and CV physical resources plus total state error." } ], "next": "Choose a finite-energy Hermite-smoothed profile and one embedding before comparing grid, oscillator or GKP implementations.", "lower_bound": { "status": "source-survey-pending", "task": "First specify the energy and non-Gaussian resource model; no cross-model lower-bound transfer is asserted.", "comparison_key": "setting:finite-energy-cv-dv-embedding" }, "benchmarks": [ "Hermite-smoothed auxiliary profiles", "Gaussian plus controlled non-Gaussian perturbation" ] } ] }