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Measurement-dependent reduction of open dynamics in the quantum phase estimation algorithm: the visible subspace and the damped Fej\'er kernel
Marina O. Gordeichuk, Oleg M. Kiselev
TL;DR
QPE output under open counting-register dynamics depends on which Liouvillian modes survive the full Fourier measurement, whose probabilities are determined by cyclic sums. The paper constructs the minimal observable reduction and a finite-time Gramian framework for exact and approximate reduction. It establishes explicit spectral structure and dimension bounds, while numerical calculations check the kernel and residual-based error behavior.
Problem
The relevant modes of open dynamics for the full QPE output distribution are not identified by a single Ramsey contrast, since QPE depends on cyclic combinations of several coherence harmonics.
Source: §1 · pp. 1–2Method
The paper constructs the measurement-dependent Krylov space of QPE observables, computes it through cyclic spectral classes for diagonal generators, and uses a finite-time visibility Gramian and residual estimates for approximate reduction.
Source: §1 · p. 2, §8 · p. 19Results
The dimension of the QPE-visible space lies between M and M^2 − M + 1 for matrix-unit-diagonal generators, and the calculations report agreement among the full propagator, Krylov reduction, and analytic kernel within accuracy of order 10^-14.
Source: §8 · p. 19Takeaways & Limitations
The QPE distribution distinguishes random-unitary channels with the same Ramsey contrast, and its outcome probabilities can be represented through the visible observable space.
Source: §1 · p. 2, §8 · p. 19
Abstract
from arXiv · showhide
For the quantum phase estimation (QPE) algorithm we study the minimal space of observables that is sufficient to predict the entire output distribution under open dynamics of the counting register. Building on the general Krylov theory of reduction, we compute the structure of this space explicitly for the full Fourier measurement. The outcome probabilities are determined by cyclic sums of the state matrix. If the adjoint generator is diagonal on the matrix units, the spectral classes inside the residue classes of the index difference modulo the register dimension $M$ give a constructive basis, an exact dimension formula, and attainable bounds: the dimension of the visible space lies between $M$ and $M^2-M+1$. The class includes independent qubit dephasing, and an analytically solvable reference model of Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) type yields a damped Fej\'er kernel; two random-unitary channels with the same Ramsey contrast show that higher harmonics are distinguishable in the QPE distribution. For approximate reduction we use a finite-time Gramian and an a posteriori residual estimate, with a quadratic error order for a reducing unperturbed subspace and a compatible initial state. Numerical calculations check the consistency of the representations and the behaviour of the error under non-Toeplitz perturbations.
1 Introduction
The paper asks which open-dynamics modes affect the full QPE output distribution and constructs the minimal observable reduction for that measurement. It makes the reduction explicit for matrix-unit-diagonal generators and relates it to prior reduction and stability work.
1 Introduction
- The full QPE measurement depends on cyclic sums of density-matrix entries, and dynamics can make initially invisible state differences visible.The paper therefore uses the invariant closure of the measurement effects to determine probabilities at all times.Source: §1 · p. 2
- Two random-unitary channels with the same Ramsey contrast can have different QPE distributions because QPE probes cyclic combinations of several coherence harmonics.A Ramsey experiment probes the first harmonic, whereas QPE depends on g1 through gM−1.Source: §1 · p. 2
- For generators diagonal on matrix units, cyclic spectral classes give a basis and exact dimension for the minimal observable space of the full QPE POVM.The paper also obtains an exact reduction to cyclic sums of diagonals and a Toeplitz specialization.Source: §1 · p. 2
- The work extends prior QPE stability analysis for input-unitary perturbations to open GKLS dynamics and the observable part of the generator.It retains the full QPE POVM and factors its output statistics through a minimal observable space.Source: §1.1 · p. 3
1.1 Relation to previous work and novelty
- The finite-time Gramian links exact rank with approximate visibility, while residual-based estimates bound approximate-reduction errors in outcome probabilities.The paper reports a quadratic perturbation law for new modes and a special quadratic estimate under stated compatibility conditions.Source: §1 · p. 2, §1.1 · p. 3
2 QPE as a fixed measurement projection
QPE is represented as a fixed measurement on the counting register after the inverse Fourier transform. This representation separates the open evolution, coherent detuning, and classical readout channel.
2 QPE as a fixed measurement projection
- The counting register has dimension M = 2^n, and its measurement outcomes encode phase estimates k/M.The work register stores the system to which U is applied.Source: §2 · p. 3
- Moving the inverse QFT onto the measurement operators gives the QPE effects used to compute outcome probabilities from the pre-transform register state.The effects form a POVM of orthogonal rank-one projectors.Source: §2 · p. 3
- For the ideal counting-register state, substitution yields the Fejér kernel, while the operational probability formula separates open evolution, coherent detuning, and classical readout.The three components act before the inverse QFT, through phase factors, and after quantum measurement, respectively.Source: §2 · p. 3
3 Minimal space of visible Liouvillian modes
The section defines the minimal invariant observable space that exactly carries measurement probabilities, then develops finite-time Gramian tools for quantifying and controlling reductions. It also gives spectral and perturbative conditions that characterize visible dynamics and error.
3 Minimal space of visible Liouvillian modes
- The visible space is the Krylov span generated by applying the adjoint generator repeatedly to the POVM effects, and restricting to it preserves each outcome probability.The probability trajectories are finite combinations of exponential-polynomial terms determined by the visible restriction's eigenvalues and Jordan blocks.Source: §3 · p. 4
- Universal minimality means this space is the unique smallest linear carrier representing every outcome probability for all initial density matrices and times.The result relies on requiring a common carrier for all states, rather than a restricted family of initial states.Source: §3 · pp. 4–5
3.1 Finite-time Gramian and effective visibility
- The finite-time visibility Gramian contains the chosen POVM effects; its range equals the exact visible space, and its leading eigenspaces give optimal spectral truncations.The spectral projector onto the top q eigenvectors maximizes captured Gramian weight among rank-q projectors.Source: §3.1 · pp. 5–6
- The Gramian spectrum defines an effective dimension that equals the exact dimension at zero tolerance and weights modes by their visibility over the chosen time horizon.This provides a finite-time scale in addition to algebraic rank.Source: §3.1 · p. 6
- For a small generator perturbation, the total weight of newly visible modes is quadratic in perturbation size under the corollary's stated setup.For an invariant unperturbed space, the general approximate-probability error bound is of order O(|η|) at fixed time.Source: §3.1 · pp. 6, 8
- An a posteriori estimate uses the invariance residual to bound each approximate probability when the trial space contains all measurement effects.The bound applies to the probabilities rather than only to a projected state trajectory.Source: §3.1 · p. 7
- For a reducing unperturbed subspace, a compatible initial state and measurement yield a quadratic error estimate through first-order cancellation.The stated conditions include G0P = PG0, Pρ0 = ρ0, and PEk = Ek.Source: §3.1 · pp. 8–9
- Visible oscillations require a visible eigenvalue with nonzero imaginary part; state differences orthogonal to the observable space change no outcome probability at any time.Visible oscillations decay when the eigenvalue's real part is negative and retain amplitude when it is zero.Source: §3.1 · p. 9
4 Exact QPE-visible projection
The full QPE probability vector is related by a discrete Fourier transform to cyclic sums of the counting-register matrix. Under population-preserving entrywise-multiplication noise, the outcome law is expressed through coherence-transfer coefficients and depends on cyclic combinations of several harmonics.
4 Exact QPE-visible projection
- The full QPE probability vector and cyclic sums of the counting-register matrix are related by a discrete Fourier transform.These cyclic sums capture the information in the matrix that determines the measurement outcomes.Source: §4 · p. 10
- Dynamical invisibility is characterized by orthogonality to the entire QPE-visible space.This is distinct from instantaneous invisibility, which is characterized through cyclic sums.Source: §4 · p. 10
4.1 Representation via coherence transfer
- For population-preserving entrywise-multiplication noise, the coherence-transfer matrix has C_jj = 1, and complete positivity is equivalent to C being positive semidefinite.The exact QPE law is expressed through the individual coherence-transfer coefficients.Source: §4.1 · pp. 10–11
- For M = 4, two random-unitary phase channels with common g1 = 1/2 at ω = 0 have coincident single-step Ramsey contrasts, yet their QPE distributions differ in supports and outcome weights.Source: §4.1 · p. 11
- The QPE distribution depends on cyclic combinations of several coherence-transfer harmonics, not only the first harmonic.The exact outcome law is expressed through the individual coherence-transfer coefficients.Source: §4.1 · p. 11
- At fixed M and ω, two Toeplitz sequences give the same QPE distribution exactly when all h_r coincide.For nonnegative g_d = a_d, the kernel is even and maximal at k/M = ω; a factor e^(2πidδ) shifts its centre to ω + δ.Source: §4.1 · p. 11
- Algorithmic phase error is represented by a coherent shift δ, Markov decoherence changes the moduli a_d, and readout error acts on the finished probability vector.Source: §4.1 · p. 11
5 Damped Fejér kernel and the visible coherence length
For generators diagonal on matrix units, the QPE-visible space is built from spectral classes within cyclic index-difference classes. The section applies this structure to dephasing, an analytically solvable damped-kernel model, and preparation and readout errors.
5 Damped Fejér kernel and the visible coherence length
- Each residue class contains M matrix units, and for each nonzero residue class 1 ≤ |Λr| ≤ M; the lower and upper bounds are attained at G = 0 and by a purely Hamiltonian generator, respectively.Source: §5 · p. 13
- Coincident frequencies in different cyclic classes must be counted separately: the M = 4 example has five generator eigenvalues but eight visible directions.The spectral classes are formed separately within each residue class, even when frequencies coincide across classes.Source: §5 · p. 13
- For independent qubit dephasing with n = 2 and equal positive rates, the listed spectral classes give dim KQPE = 6.The ordinary diagonal with index difference one includes pairs at Hamming distances one and two, with distinct damping rates.Source: §5 · p. 14
- Distinct damping rates on the two ordinary diagonals merged by a cyclic sum add a visible direction, while the visible space remains within the Toeplitz matrices.When the rates differ for every nonzero residue, the visible dimension is 2M − 1.Source: §5 · pp. 14–15
- The reference GKLS model uses threshold operators to give an analytically solvable counting-register evolution and an exact damped Fejér-kernel expression.The section identifies α as integrated damping per unit distance between time branches and δ as coherent phase detuning.Source: §5 · pp. 12, 16
5.1 Preparation and classical readout
- Preparation errors change the initial state, whereas classical readout errors act on the final probability vector.The readout transformation is specified by a stochastic matrix.Source: §5.1 · p. 17
6 Computational verification and numerical applications
The calculations check agreement among the full, reduced, and analytic models, then test how a non-Toeplitz perturbation affects the visible reduction. For the compatible initial state, the probability error is approximately quadratic in perturbation strength and remains below the a posteriori bound.
6 Computational verification and numerical applications
- For M = 4, 8, 16, 32, 64, the full, reduced, and analytic models agree within about 10^-14, and the Krylov algorithm recovers dimension 2M − 1.Independent tests also confirm the dimension formula, intermediate degeneracies, the upper bound M^2 − M + 1, and the two random-unitary distributions.Source: §6 · p. 17
6.1 Stability beyond the Toeplitz model
- The test adds inhomogeneous mixing and amplitude relaxation to the Toeplitz-model generator, then compares full dynamics with a projection onto the fixed space T_M.The approximate model is A_η = PG_ηP; Table 1 compares the maximum outcome-probability error with the a posteriori estimate at t = 1.Source: §6.1 · p. 17, Table 1 · p. 18
- The probability error stays below the a posteriori bound and scales approximately quadratically with perturbation strength for M = 8, 16, 32.Across 10^-5 ≤ η ≤ 10^-1 the error decreases with M for the chosen normalization; Figure 1 shows quadratic error versus linear bound, and Table 1 compares actual error with the estimate.Source: §6.1 · p. 18, Figure 1 · p. 18, Table 1 · p. 18
- The universal bound exceeds the error by approximately 63, 113, and 186 times, while the compatible state’s first-order cancellation yields the special estimate.The universal O(|η|) guarantee applies to arbitrary initial states of a given purity, whereas the compatible-state error is O(η^2).Source: §6.1 · p. 18
7 Discrete and time-dependent dynamics
The paper extends the observable-space construction to discrete channels and time-dependent generators. The resulting visible space is built by backward propagation of measurement effects.
7 Discrete and time-dependent dynamics
- For a sequence of quantum channels, the discrete analogue of the continuous construction is a backward space.The supplied passage introduces this construction but does not specify its further structure.Source: §7 · p. 18
- For a time-dependent generator, the visible space is generated by solutions of the adjoint evolution with final conditions set by the measurement effects.The evolution is given by ˙E_k(t) = −L(t)†E_k(t), with E_k(T) = E_k.Source: §7 · p. 19
8 Conclusion
The conclusion identifies the minimal QPE-visible observable space and its structure when the adjoint generator is diagonal on matrix units. It also summarizes the Gramian and residual-based approximate reduction, alongside the model and perturbation results.
8 Conclusion
- When the adjoint generator is diagonal on matrix units, cyclic spectral classes give a basis and exact dimension between M and M^2 − M + 1 for the minimal QPE-visible space.The basis groups spectral classes within residues of j − ℓ modulo M.Source: §8 · p. 19
- The finite-time Gramian has range equal to the visible space, while its leading eigenspaces provide spectral truncations and the residual controls probability errors.The weight of perturbed modes outside the exact unperturbed visible space is O(η^2). Two random-unitary models with common Ramsey contrast but different QPE distributions distinguish higher harmonics.Source: §8 · p. 19
- The GKLS reference model yields a damped Fejér kernel; calculations agree within about 10^-14, and non-Toeplitz perturbations show approximately quadratic error at M = 8, 16, 32.The stated conclusion is that QPE-visible modes determine the output distribution and the residual estimate bounds reduction error.Source: §8 · p. 19
Funding
The paper reports no external funding.
Funding
- The research received no external funding.Source: Funding · p. 19
Data and code availability
Ancillary files provide scripts, result tables, and reproduction instructions for deterministic numerical checks of the paper’s mathematical models.
Data and code availability
- Ancillary files include scripts, result tables, and instructions for reproducing deterministic numerical checks of the stated mathematical models.The checks cover full GKLS dynamics, Krylov reduction, the analytic kernel, spectral degeneracies, a Ramsey counterexample, and non-Toeplitz perturbations.Source: Data and code availability · p. 19