{"id":"00f5a2e9-ebed-4589-860a-88e262c1a5eb","arxiv_id":"2502.08962","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper constructs a quantum circuit for non-unitary basis-set transformations via singular value decomposition and block encoding, enabling overlap computations between states in different bases.","lead":"This paper designs a quantum circuit that applies a non-unitary change of basis to a quantum state, a step ordinary quantum circuits cannot perform directly. The goal is to let quantum computers compare wavefunctions expressed in different basis sets, which matters for calculating molecular excited-state energies.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The swap-test inner-product circuits (Figs. 7-8) use an incorrect probability relation: the all-zero outcome probability is (||Ξ|Φ⟩||^2 + |⟨Ψ|Ξ|Φ⟩|^2)/2, not |⟨Ψ|Φ⟩|^2, so the advertised overlap measurement is not what the circuit actually computes.","rationale":"The reader's weakest assumption was the unanalyzed post-selection success probability of the block encoding. That is a real and important efficiency concern. The concern I identify is different and more directly a correctness issue: the swap-test circuits in Figs. 7-8 do not measure the stated overlap because the block-encoded state entering the swap test is not normalized and the ancilla projection changes the probability formula. This lands as a concrete algebraic error, independent of whether the post-selection probability happens to be large. I still do not move the verdict to REJECT because the underlying wedged-map/block-encoding construction appears sound, and the Hadamard-test circuit in Fig. 9 does provide a route to the same matrix element, assuming controlled versions of the component gates are available. However, the paper must remove or correct the faulty swap-test circuits and either bound the post-selection success probability or clearly restrict the claims to the Hadamard-test usage. The reader's CONDITIONAL verdict is therefore unchanged, but the grounds are strengthened by this additional correctness defect.","tokens_in":22188,"tokens_out":20418,"duration_ms":218615,"concrete_test":"Evaluate the Fig. 7 circuit analytically or by simulation for one working qubit with u = diag(0.5, 1), |Φ_q⟩ = |1⟩, |Ψ_q⟩ = |0⟩. The exact overlap ⟨Φ|Ψ⟩ is 0, so the paper's relation predicts P(all ancillas zero) = 0. The correct Born rule, from the amplitude 1/(2√2) for control = 0 and block-encoding ancilla = 0, gives P = 1/8. If this reproduction confirms the discrepancy, the swap-test probability claim in Section IV is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central application of the paper is overlap estimation between states in different bases, and Fig. 7 is presented as the first circuit for this. Let \\tildeΞ be the block encoding satisfying \\tildeΞ|0_a⟩|Φ_q⟩ = |0_a⟩Ξ|Φ_q⟩ + |⊥⟩. In Fig. 7, after the first Hadamard, a controlled swap between the n working qubits of |Ψ_q⟩ and the n output working qubits of \\tildeΞ, and the second Hadamard, the component with control qubit and all block-encoding ancillas in |0⟩ is proportional to |Ψ_q⟩|0_a⟩Ξ|Φ_q⟩ + |0_a⟩Ξ|Φ_q⟩|Ψ_q⟩. Its squared norm is (||Ξ|Φ_q⟩||^2 + |⟨Ψ_q|Ξ|Φ_q⟩|^2)/2, not |⟨Ψ_q|Ξ|Φ_q⟩|^2. Since a block-encoded contraction generally satisfies ||Ξ|Φ_q⟩|| < 1, the two expressions differ. A concrete counterexample is n = 1 with u = diag(0.5, 1), |Φ_q⟩ = |ω_1⟩ and |Ψ_q⟩ = |ω_2⟩. The exact overlap is 0, but the Fig. 7 all-zero probability is 0.5^2/2 = 1/8. Thus the claimed inner-product circuit in Section IV is algebraically incorrect. The later Hadamard-test circuit (Fig. 9) appears to estimate the correct matrix element, so a revision could remove or correct Figs. 7-8, but as written the paper's advertised overlap circuits fail.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a quantum circuit to implement the wedged map ∧u on the Fock space of an n-orbital fermionic system for a non-unitary contraction u. The construction uses the SVD u = LDR, implements ∧L and ∧R with the linear-depth Givens-rotation circuits of Kivlichan et al., block-encodes the diagonal singular-value factor ∧D with one ancilla per intermediate singular value and a single multi-controlled Toffoli for the zero singular values, and approximates singular values near 0 or 1 to reduce cost. The authors then apply the block encoding to the evaluation of overlaps between many-body states expressed in different one-body bases, proposing three circuits (a swap test with the block encoding, an alternative swap test, and a Hadamard test).","tokens_in":22610,"tokens_out":11308,"duration_ms":105705,"significance":"If the construction is correct, the block encoding of ∧u is a clean and useful tool: it gives an explicit polynomial-size circuit for non-unitary basis changes and overlap estimation between different basis representations, with a self-contained exterior-algebra derivation of the Thouless theorem. The paper is honest about the approximation error from rounding and truncating singular values and about the extra ancilla cost. However, the main application is undermined by an algebraic error in the swap-test circuits (Figs. 7 and 8), and the advertised O(n) depth is not supported by the paper's own bound in Section III. The Hadamard-test circuit (Fig. 9) appears correct, so the central idea remains salvageable.","major_comments":[{"comment":"The caption of Fig. 7 claims that |⟨Φ|Ψ⟩|² is the probability of all measurements yielding |0⟩. This is algebraically incorrect. Writing the block encoding as ~Ξ|0_a⟩|Φ_q⟩ = |0_a⟩Ξ|Φ_q⟩ + |⊥⟩, the component of the final state with the control qubit and all ancillas in |0⟩ after the controlled swap and second Hadamard is proportional to |Ψ_q⟩|0_a⟩Ξ|Φ_q⟩ + |0_a⟩Ξ|Φ_q⟩|Ψ_q⟩, whose squared norm is (∥Ξ|Φ_q⟩∥² + |⟨Ψ_q|Ξ|Φ_q⟩|²)/2, not |⟨Ψ_q|Ξ|Φ_q⟩|². Since Ξ is a contraction, the two expressions differ. A concrete counterexample is a two-orbital system with u = diag(0.5, 1) and one-particle states |Φ_q⟩ = |ω_1⟩, |Ψ_q⟩ = |ω_2⟩: the true overlap is 0, while the all-zero probability is 1/8. Thus Figs. 7 and 8 do not compute the advertised overlap. The paper should either remove these circuits or replace them with a correct construction, and should use the Hadamard test of Fig. 9, which appears to compute the correct matrix element, as the primary overlap circuit.","section":"Section IV, Figs. 7 and 8"},{"comment":"The abstract, the Introduction, and Section V claim that the circuit depth is O(n), but the body of Section III gives the bound O((n−r)² + n) for the total depth, where r is the rank after truncation. For small r, the multi-open-controlled Toffoli block has depth O((n−r)²), which is O(n²). The O(n) claim holds only when n−r is small (e.g., constant or O(√n)), and even then it rests on the specific decomposition cited from [8]. The complexity statements in the abstract, Introduction, and Conclusion must be qualified to reflect the paper's own rank-dependent bound.","section":"Section III, last paragraph; abstract and Section V"},{"comment":"The post-selection success probability for projecting all block-encoding ancillas to |0⟩ is never analyzed. For a k-particle state whose occupied orbitals carry singular values σ_{i_1},…,σ_{i_k} with 0<σ<1, the success probability of step 8 is Π σ_{i_j}², which can be exponentially small in k. This directly affects the practical efficiency of the inner-product circuits in Figs. 7 and 8, both of which require that projection. The Hadamard test in Fig. 9 does not require this post-selection, so the paper should either provide a bound on the success probability, state the regime of singular values for which the algorithm is efficient, or explicitly present Fig. 9 as the only circuit whose expected runtime is polynomial for general contractions.","section":"Section III, step 8, and Section IV"}],"minor_comments":[{"comment":"The block encoding ~Ξ is never defined explicitly with its orthogonal component |⊥⟩; adding the explicit form ~Ξ|0_a⟩|Φ_q⟩ = |0_a⟩Ξ|Φ_q⟩ + |⊥⟩ with ⟨0_a|⊥⟩ = 0 would make the Hadamard-test analysis in Eq. (27) easier to verify.","section":"Section IV, Eq. (27)"},{"comment":"The bound ∥∧u − ∧ũ∥ ≤ Σ_j ε_j is stated without proof. A short telescoping argument using the product structure of ∧D would make the approximation claim rigorous.","section":"Section III, Eq. (22)"},{"comment":"The phrase 'The 1 and 0 singular values of the overlapping matrix' is ambiguous; it should read 'singular values equal to 1 and to 0'.","section":"Introduction, contribution list"},{"comment":"There is a minor typo: 'for |σk⟩ being its eigenstate' should be 'for |σk⟩ being its eigenvector'.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The algebraic core of the paper is sound and the Hadamard-test circuit appears to compute the correct overlap. The errors are serious but localized: the swap-test probability claim is wrong, the depth claim is overstated, and the success probability is unbounded. I recommend major revision rather than rejection. I also suggest that the authors consider small numerical simulations of the block encoding and of the Hadamard test to demonstrate the validity of the corrected circuits."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth a look, but it needs work. The genuinely new thing is the circuit for the wedged map ∧u of an arbitrary non-unitary one-body transformation u. The construction via SVD u = LDR is clean: the unitary factors use Kivlichan's Givens circuits, and the diagonal is block-encoded with controlled-RY rotations, with one multi-controlled Toffoli handling the zero singular values together. The exterior-algebra proof of the Thouless theorem is a nice bonus. The algebraic core is correct, and the error bound from rounding/truncating singular values is reasonable. The citation pattern looks honest; prior work on non-unitary Thouless transformations is classical, and the unitary circuit is properly credited to Kivlichan et al.\n\nThe soft spots are real. First, the abstract and summary advertise O(n) depth, but Section III derives O((n-r)^2 + n) for the multi-controlled Toffoli part. That is O(n) only when the truncated rank r is close to n. For a genuinely low-rank u, the depth is quadratic. This is an internal contradiction. Second, and more seriously, the swap-test inner-product circuits in Figs. 7 and 8 are wrong. The block encoding outputs an unnormalized state, so the all-zero probability is (||Ξ|Φ>||^2 + |<Ψ|Ξ|Φ>|^2)/2, not |<Ψ|Φ>|^2. The n=1 counterexample with u = diag(0.5,1) gives a nonzero all-zero probability when the true overlap is zero. The Hadamard test in Fig. 9, in contrast, does compute the correct matrix element without post-selection, because the leakage norm cancels. So the section is repairable, but as written the paper's main application is built on incorrect circuits. Third, the success probability of the block encoding for actually preparing Ξ|Φ> is never bounded; for small singular values it can be exponentially small. That matters for any use that post-selects on the ancillas.\n\nThe core construction is novel and mostly sound, and a serious referee can fix the flaws. I would send it to review, but I would not cite it until the swap-test circuits are corrected and the depth claim is reconciled.","headline":"Novel SVD-based circuit for non-unitary basis transformations is sound at its core, but the advertised swap-test inner-product circuits are algebraically wrong and the O(n) depth claim overstates what the body actually proves.","tokens_in":23068,"tokens_out":7324,"would_cite":false,"duration_ms":64401,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","15A75","15A18"],"pacs":[],"model":"deepseek-v4-flash","headline":"Polynomial-size quantum circuit implements non-unitary basis changes","keywords":["non-unitary basis transformation","wedged map","exterior algebra","singular value decomposition","block encoding","fermionic overlap","orbital optimization","quantum circuit depth"],"falsifier":"On a simulator, take an 8-orbital, 4-electron system, choose a pair of bases whose overlap matrix has all four relevant singular values equal to 0.5, prepare the same Slater determinant in both bases, and run the proposed overlap circuit; the probability of all ancillas measuring 0 should match the predicted product of squared singular values while the swap-test readout should equal the classically computed determinant. If the observed success probability is far smaller due to the multi-controlled NOT decomposition, or if the measured overlap deviates from the classical result, the practical resource claim fails.","tokens_in":22036,"feed_emoji":"⚛️","tokens_out":10177,"duration_ms":100548,"temperature":0.7,"pith_summary":"Orbital basis rotations used in quantum chemistry are usually restricted to unitary transformations; this paper removes that restriction. It constructs a quantum circuit that implements the wedged map $\\wedge u$ on the many-body Fock space for any contraction matrix $u$, so a wavefunction can be moved from one one-electron basis to another even when the bases are not orthogonal rotations of each other. The construction writes $u = LDR$ by singular value decomposition, applies the two unitary factors with parallel plane-rotation circuits, and encodes the diagonal singular-value factors with ancilla qubits and post-selection; the paper's resource statement is $O(n^2)$ gates and $O(n)$ depth for an $n$-orbital basis, with an extra $O((n-r)^2)$ term from the joint encoding of zero singular values. The payoff is a quantum estimator for the overlap of two many-body states represented in different bases, which is what state-specific orbital-optimized excited-state methods need.","feed_headline":"Quantum circuit maps states between non-orthogonal bases at polynomial cost","feed_subtitle":"Fermionic overlaps across different orbital bases become computable with O(n^2) gates and O(n) depth.","key_machinery":"The load-bearing object is the wedged map $\\wedge u$: the linear map on the many-body Fock space obtained by applying the one-body map $u$ separately to every factor of a wedge product. Its importance is that overlaps of Slater determinants are determinants, so $\\langle\\Phi|\\Psi\\rangle$ for basis many-body states is exactly the inner product of one wedge product with the image of another under $\\wedge u$. The construction splits $\\wedge u$ through a singular value decomposition $\\wedge u = (\\wedge L)(\\wedge D)(\\wedge R)$; the unitary factors are implemented by the linear-depth parallel plane-rotation circuit, and the diagonal factor is implemented by block encoding each per-orbital operator $\\nu_j = 1 + (\\sigma_j - 1)n_j$, with controlled rotations for interior singular values and one joint multi-controlled NOT gate for the zero block. Rounding singular values to $1$ or $0$ is justified by the bound $\\lVert\\wedge u - \\wedge\\tilde{u}\\rVert \\le \\sum_j |\\epsilon_j|$.","core_discovery":"The central claim is that a non-unitary linear transformation of a basis set can be executed exactly, not merely approximated, inside a quantum circuit, provided the transformation matrix $u$ has operator norm at most one. On the one-body level $u$ is the matrix of overlaps $\\langle\\psi_i|\\phi_j\\rangle$ between the old and new orbitals; on the many-body level the transformation is the wedged map $\\wedge u$, whose action on a Slater determinant is to replace every one-particle factor by its image under $u$. The paper proves $\\wedge u = (\\wedge L)(\\wedge D)(\\wedge R)$ from the SVD $u = LDR$, shows the diagonal part $\\wedge D$ factorizes as a product of simple per-orbital operators $\\nu_j = 1 + (\\sigma_j - 1)n_j$, and gives block encodings for each factor: controlled rotations for singular values strictly between $0$ and $1$, a single shared multi-controlled NOT gate for all zero singular values, and the existing parallel plane-rotation circuits for the unitary factors $L$ and $R$. Singular values close to $1$ or $0$ may be rounded or truncated, and the resulting error is bounded by the sum of the perturbations. It concludes that the overlap between two encoded many-body states in different bases equals $\\langle\\Psi_q|\\Xi|\\Phi_q\\rangle$ with $\\Xi$ the block-encoded $\\wedge u$, and proposes three swap- and phase-test circuits that read out this overlap; the quoted total depth is $O((n-r)^2+n)$, recovering pure $O(n)$ depth when the rank $r$ is close to $n$.","pith_inferences":["The paper does not bound the post-selection success probability; for a $k$-particle state whose basis-overlap matrix has singular values all near $s<1$, the success probability scales like $s^{2k}$, so the practical savings over classical overlap evaluation may vanish precisely when the two bases are strongly non-orthogonal.","The classical SVD preprocessing of the $n\\times n$ overlap matrix costs $O(n^3)$, so the proposed quantum speedup applies to the many-body part of the overlap; for large $n$ this classical step may dominate in a hybrid workflow.","The same block encoding can be read as a primitive for non-orthogonal configuration interaction on quantum hardware, since general matrix elements between differently rotated Slater determinants require exactly this type of overlap factor.","A near-term testable extension is to replace the multi-controlled NOT gate with an ancilla-free decomposition and benchmark the observed success probability against the predicted product of squared singular values on systems of 8-16 spin-orbitals."],"forward_implications":["Overlap estimation between states in different orbital bases becomes a polynomial-resource subroutine, removing a classically exponential bottleneck for dense many-body configurations.","State-specific orbital optimization becomes implementable: ground and excited eigenstates can each keep their own optimized orbitals, with the overlap enforced or evaluated through this circuit.","VQE-type workflows no longer have to freeze one basis for all target states, so energy accuracy for multiple states can be improved simultaneously.","Non-unitary changes of basis open the door to representing wavefunctions in non-orthogonal bases on a quantum computer, not only to measuring overlaps.","Rounding and truncating singular values gives a tunable trade-off between circuit cost and accuracy, with an error bound linear in the applied perturbations."],"supporting_citations":[{"why":"Supplies the parallel plane-rotation circuit with linear depth that implements the unitary factors $\\wedge L$ and $\\wedge R$ of the SVD.","marker":"[11]"},{"why":"Establishes that a one-body unitary change of basis acts on many-body states through the exponential of one-body operators, the result this paper rewrites and extends as $\\wedge u$.","marker":"[50]"},{"why":"Provides the block-encoding framework used to embed the non-unitary diagonal singular-value factors into unitary circuits with ancilla post-selection.","marker":"[68]"},{"why":"Gives the standard decomposition of the multi-controlled NOT gate from which the paper derives the $O((n-r)^2)$ depth contribution of the zero-singular-value block.","marker":"[8]"},{"why":"Supplies an alternative inner-product estimation primitive used when the adjoint of the state-preparation circuit is not favored.","marker":"[69]"}],"fun_headline_variants":["SVD makes non-unitary basis transforms exact at O(n) depth","Quantum circuits now do exact non-unitary basis changes","Overlap of different-basis states computed in O(n) depth","Non-unitary basis maps: exact and efficient via SVD","Quantum basis transform goes non-unitary with O(n) depth"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction gives a valid non-unitary map only after post-selecting all ancilla qubits to be in $|0\\rangle$; if the one-particle overlap matrix has many small singular values, the probability of that post-selection is a product of their squares and can vanish exponentially fast, and the paper does not bound this probability.","fun_headline_variants_meta":{"raw":{"variants":["SVD makes non-unitary basis transforms exact at O(n) depth","Quantum circuits now do exact non-unitary basis changes","Overlap of different-basis states computed in O(n) depth","Non-unitary basis maps: exact and efficient via SVD","Quantum basis transform goes non-unitary with O(n) depth"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000333,"raw_usage":{"total_tokens":1907,"prompt_tokens":1060,"completion_tokens":847,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":676,"completion_tokens_details":{"reasoning_tokens":760}},"tokens_in":676,"tokens_out":847,"duration_ms":9255,"temperature":1.0,"reasoning_tokens":760,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T23:05:34.566976+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a simulator, take an 8-orbital, 4-electron system, choose a pair of bases whose overlap matrix has all four relevant singular values equal to 0.5, prepare the same Slater determinant in both bases, and run the proposed overlap circuit; the probability of all ancillas measuring 0 should match the predicted product of squared singular values while the swap-test readout should equal the classically computed determinant. If the observed success probability is far smaller due to the multi-controlled NOT decomposition, or if the measured overlap deviates from the classical result, the practical resource claim fails.","supporting_citations":[{"cited_title":"complex Givens rotation","cited_arxiv_id":null,"evidence_quote":"Supplies the parallel plane-rotation circuit with linear depth that implements the unitary factors $\\wedge L$ and $\\wedge R$ of the SVD."},{"cited_title":"Yalouz, B","cited_arxiv_id":null,"evidence_quote":"Establishes that a one-body unitary change of basis acts on many-body states through the exponential of one-body operators, the result this paper rewrites and extends as $\\wedge u$."},{"cited_title":"Malmqvist and B","cited_arxiv_id":null,"evidence_quote":"Provides the block-encoding framework used to embed the non-unitary diagonal singular-value factors into unitary circuits with ancilla post-selection."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the standard decomposition of the multi-controlled NOT gate from which the paper derives the $O((n-r)^2)$ depth contribution of the zero-singular-value block."},{"cited_title":"Pittner, H","cited_arxiv_id":null,"evidence_quote":"Supplies an alternative inner-product estimation primitive used when the adjoint of the state-preparation circuit is not favored."}],"review_version":1}