Accurate measurement of an intended conditional phase does not by itself place a multiqubit phase gate in its declared generalized-gate equivalence class. We cast the diagnosis as structured Hamiltonian learning of the diagonal successful block in a known basis: Ramsey measurements on a connected graph reconstruct a target-relative eigenphase map; a Walsh transform then yields the per-cycle stroboscopic error generator in a selected logarithm branch, whose branch-dependent coefficients alone cannot decide global equivalence. For diagonal Pauli-Z dressing of weight at most m we derive an exact branch-independent criterion from multiplicative Boolean-Mobius invariants, converted at finite shot counts into a pass/fail/unresolved decision under prespecified tolerances and simultaneous confidence, separate from model rejection. For shared-control C^mZ gates a circular target-face contrast isolates each intended conditional phase, yet a support theorem exhibits forbidden interactions invisible to every target face once multiple targets or external spectators are present. We derive a Cramer-Rao bound for arbitrary integer phase contrasts, recovering the 4^m attempted-circuit scaling of a uniform (m+1)-body face, and quantify degradation from visibility loss, heralded survival, readout confusion, and correlated noise, while systematic bias triggers model rejection. Seeded simulations implement the 4M-setting multiplexed acquisition end to end: for M=3,4,5 at a common budget its mean phase-map RMSE is 0.689-0.794 times a separately compiled statewise scan, and a blind-interaction test yields 300/300 global-audit rejections versus 1/300 for target faces. These results estimate the error generator of a diagonal successful block under a specified acquisition model; they are not unknown-basis Hamiltonian identification, arbitrary-channel tomography, or hardware certification.
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The paper asks when measured conditional phases establish that a multiqubit diagonal gate belongs to a specified equivalence class, rather than merely implementing its intended control–target factors. It assumes a successful computational block diagonal in a known basis and specifies which diagonal phase dressings are allowed.
Conditional Ramsey measurements on a connected graph reconstruct target-relative eigenphase differences. A Walsh transform gives a commuting stroboscopic error generator within a selected logarithm branch. Separately, multiplicative Boolean–Möbius invariants provide an exact, branch-independent criterion: equivalence modulo diagonal interactions of weight at most $m$ holds iff every invariant of degree at least $m+1$ equals one. Finite-shot decisions use prespecified tolerances and simultaneous Gaussian confidence intervals, returning pass, fail, or unresolved after model checks.
Support identities show that shared-control target-face contrasts miss forbidden interactions whenever multiple targets or external spectators are present. For uniform independent gate/reference acquisition, a local Cramér–Rao bound gives $\operatorname{Var}(\widehat\delta)\ge 4^{m+1}/(Ns_q q^2V_q^2)$. Joint spectator readout reduces compiled settings to $4M$ per depth, not total-shot complexity. Count-level simulations for $M=3,4,5$ yield phase-map RMSE ratios of $0.689$–$0.794$ against a statewise comparator; a blind-interaction test produces $300/300$ global-audit detections versus $1/300$ target-face detections.
The results are analytical identities and protocol-specific bounds supported by seeded simulations, not hardware certification. Limitations include exponential full-map size, branch-local generator identification, asymptotic statistical guarantees, incomplete coverage of model violations, and primarily same-model control-update validation. Unknown-basis identification and arbitrary-channel tomography are outside scope.
Some signs of AI-generated textIncrementalStrong assumptions
Error found: The abstract's simultaneous-confidence finite-shot claim needs the body's local-Gaussian, asymptotic qualification; no nonasymptotic coverage guarantee is established.
The abstract overstates systematic-bias detection: a bias consistent with a diagonal phase field and linear depth accumulation can evade the stated model checks and remain a non-averaging estimation error.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
Accurate measurement of an intended conditional phase does not by itself place a multiqubit phase gate in its declared generalized-gate equivalence class. We cast the diagnosis as structured Hamiltonian learning of the diagonal successful block in a known basis: Ramsey measurements on a connected graph reconstruct a target-relative eigenphase map; a Walsh transform then yields the per-cycle stroboscopic error generator in a selected logarithm branch, whose branch-dependent coefficients alone cannot decide global equivalence. For diagonal Pauli-Z dressing of weight at most m we derive an exact branch-independent criterion from multiplicative Boolean-Mobius invariants, converted at finite shot counts into a pass/fail/unresolved decision under prespecified tolerances and simultaneous confidence, separate from model rejection. For shared-control C^mZ gates a circular target-face contrast isolates each intended conditional phase, yet a support theorem exhibits forbidden interactions invisible to every target face once multiple targets or external spectators are present. We derive a Cramer-Rao bound for arbitrary integer phase contrasts, recovering the 4^m attempted-circuit scaling of a uniform (m+1)-body face, and quantify degradation from visibility loss, heralded survival, readout confusion, and correlated noise, while systematic bias triggers model rejection. Seeded simulations implement the 4M-setting multiplexed acquisition end to end: for M=3,4,5 at a common budget its mean phase-map RMSE is 0.689-0.794 times a separately compiled statewise scan, and a blind-interaction test yields 300/300 global-audit rejections versus 1/300 for target faces. These results estimate the error generator of a diagonal successful block under a specified acquisition model; they are not unknown-basis Hamiltonian identification, arbitrary-channel tomography, or hardware certification.
The paper asks when measured conditional phases establish that a multiqubit diagonal gate belongs to a specified equivalence class, rather than merely implementing its intended control–target factors. It assumes a successful computational block diagonal in a known basis and specifies which diagonal phase dressings are allowed.
Conditional Ramsey measurements on a connected graph reconstruct target-relative eigenphase differences. A Walsh transform gives a commuting stroboscopic error generator within a selected logarithm branch. Separately, multiplicative Boolean–Möbius invariants provide an exact, branch-independent criterion: equivalence modulo diagonal interactions of weight at most m holds iff every invariant of degree at least m+1 equals one. Finite-shot decisions use prespecified tolerances and simultaneous Gaussian confidence intervals, returning pass, fail, or unresolved after model checks.
Support identities show that shared-control target-face contrasts miss forbidden interactions whenever multiple targets or external spectators are present. For uniform independent gate/reference acquisition, a local Cramér–Rao bound gives Var(δ)≥4m+1/(Nsqq2Vq2). Joint spectator readout reduces compiled settings to 4M per depth, not total-shot complexity. Count-level simulations for M=3,4,5 yield phase-map RMSE ratios of 0.689–0.794 against a statewise comparator; a blind-interaction test produces 300/300 global-audit detections versus 1/300 target-face detections.
The results are analytical identities and protocol-specific bounds supported by seeded simulations, not hardware certification. Limitations include exponential full-map size, branch-local generator identification, asymptotic statistical guarantees, incomplete coverage of model violations, and primarily same-model control-update validation. Unknown-basis identification and arbitrary-channel tomography are outside scope.
We study Hamiltonian learning for k-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown s-sparse Hamiltonian with fixed locality k and ∥H∥op≤Λ, our algorithm recovers its Pauli coefficients to ℓ2 error ε in total evolution time O(s/ε), matching the best known scaling while allowing a near-maximal step size Θ(1/Λ). As an independent contribution, we strengthen the evolution-time lower bound to Ωk(s1−1/(2k)/ε), placing our algorithm within a factor O(s1/(2k)) of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a n overhead in ℓ2 learning time for n qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.
Recent progress in quantum physics and quantum technologies is driving quantum computing from the noisy intermediate-scale (NISQ) era toward fault-tolerant operation. High-precision control of two-qubit gates is among the most critical requirements in this transition and hinges on accurate two-qubit calibration. For controlled-phase and CZ-style operations, the residual conditional phase (the nonlocal ZZ-type deviation after local compensation) is weakly resolved at leading order in average infidelity and randomized benchmarking, and repeated Ramsey amplification does not reliably isolate it from ordinary target detuning, SPAM errors, and contrast loss in long sequences. We introduce branch-resolved Pauli-block spectroscopy to estimate the per-cycle residual ZZ-rotation angle theta_c with its sign, from which the controlled-phase residual follows by a fixed convention. The protocol repeats a fixed probe for N cycles, measures the closed Pauli block IX, IY, ZX, and ZY, and forms branch coherences C+ and C- conditioned on the control qubit; theta_c splits the two branch phase slopes in opposite directions, while local target phase beta_c shifts them together. An echoed-cycle variant suppresses removable local terms while preserving the nonlocal contribution. Numerical simulations with injected theta_c, detuning, damping, and SPAM confirm unbiased signed readout where scalar-sector alternatives fail and distinguish opposite-sign errors at equal infidelity. On one superconducting cloud qubit-coupler pair, a pulse-level calibration closed loop shows near-linear injection, preserved branch contrast, and tracking of the native residual conditional phase through one iteration. The approach yields a low-overhead, signed per-cycle estimate of residual conditional phase that standard fidelity benchmarks underresolve at leading order.
Device-independent certification of a quantum gate requires the input and output tests to identify the same reference qubits. Self-testing the output Choi state alone does not guarantee this consistency. We develop a robust certification scheme for Boolean-phase gates, a broad family of computational-basis diagonal gates specified by Boolean functions. Boolean derivatives convert the target-dependent phase information into classical signs that can be evaluated from local measurement outcomes. This leads to Bell tests built from CHSH blocks using two binary measurements per party and no entangling measurements. At maximal violation, the tests self-test the normalized Choi state and the measured observables. Away from the maximum, they give an explicit affine lower bound on the extracted-state squared fidelity that is uniform over all Boolean functions and valid in arbitrary local dimensions. We then combine an identity test and a gate-output test in an independent-source network in which the reference devices use the same physical observables, obtaining a closed-form Choi-fidelity bound for an effective n-qubit channel. For CCZ, a joint six-party analysis gives a stronger robustness bound without changing the Bell expression or measurement settings. The construction shows how the algebraic structure of a gate can shift target-dependent information from quantum measurement design to classical processing of local outcomes.
We present a variational algorithm for learning an unknown quantum unitary from time-series observable measurements, with no structural assumption about the target. The core separation: a hardware-efficient parametrised circuit learns the evolution operator U via observable matching; Hamiltonian identification follows as classical post-processing via matrix logarithm, when the target happens to be exp(-iH*tau).
Three experiments establish the method's generality. First, a noiseless proof of correctness with exact gradients (L-BFGS-B) achieves MSE 1.61e-14 and recovers all Hamiltonian coefficients to six decimal places. Second, a gate-learning experiment fits CNOT, iSWAP, and a Haar-random SU(4) element -- none generated by any fixed Hamiltonian -- all to process fidelity 1.000000, confirming the method does not rely on Trotterisation structure. Third, quantum deployment via SPSA-Adam under Qiskit Aer depolarising noise (p1=0.001, p2=0.01, Nshots=1024) recovers all three Ising Hamiltonian terms with errors below 8%.
The optimiser, SPSA-Adam, combines SPSA's hardware-efficient two-point gradient estimation with Adam's adaptive moment updates. A four-stage moment-warm curriculum progressively extends the training horizon, converting a global non-convex problem into a sequence of well-posed local ones.
We construct a source-matched reference calibration for distinguishing the coherent law h = 0 from the specified added generator h = J/25 in a finite quantum-record model. A complete-carrier transfer bound connects an accessible reference contrast to a delayed target contrast while retaining the original drift and actual law. On reference coverage and measured admission, restricting both hypotheses by the same observed reference band yields a directed target-set separation of at least 0.0071230619 whenever both compatible classes are nonempty. Rival worlds may have different admitted nuisance parameters. The constants are uniform over independently known finite lengths m >= 65, with tag 65, access prefix 1:32 and target deadline 25.02/J. Finite reference preparation and work/source reset satisfy the declared response budgets. Reset acts on the actual joint input and consumed supply, with the retained reference R exactly inert. It does not assign a universal reduced channel to correlated supplies. Under almost-sure conditional resource and supply bounds, the comparison extends to history-dependent acquisition. A prespecified rule, including abstention below the calibration threshold, has each-law unconditional wrong-declaration probability at most 2 exp(-6) + exp(-85/16) < 0.01. This is neither a posterior probability nor an error bound conditional on admission. The statistical step uses standard bounded-martingale concentration. Global target states and specified auxiliary supplies remain resources. Reset requires additional globally conditioned controls and has very large finite costs. The result is a conditional theorem for the two declared laws; it does not establish practical implementation or a unique diagnosis of fundamental collapse. A supporting Lean 4 companion contains 85 theorem declarations and 30 supporting definitions across Calibration, Operations and Risk. It provides checked scalar, sufficient-bound, resource-arithmetic and finite-history risk consequences under explicit premises. The companion has an executed Linux preflight and accepted returned macOS compilation, named-type and transitive-axiom evidence. The quantum constructions and conditional moment, filtration, tower and tail derivations remain written proofs or external premises; this is not an end-to-end formalization of the physical theorem chain. The original checked companion targeted author v0.4 and is carried forward unchanged, with its scope and execution chronology preserved.
Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting n-qubit system on a graph G of effective dimension D. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous C1 approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb--Robinson estimates proved here, and a certified effective approximation/interpolation degree meff, the protocol reconstructs every coefficient uniformly on a constant interval with error ε and success probability 1−δ using O(ε−2poly(meff)log(nδ−1)) samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.