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Estimation of coherent error sources from stabilizer measurements

Davide Orsucci, Markus Tiersch, Hans J. Briegel

arXiv:1512.07083v1quant-ph

TL;DR

The paper examines whether stabilizer measurements can reveal local unitary errors on graph states, despite correlations among the outcomes. It analyzes reconstruction under axis-aligned field assumptions and finds that recoverability depends on graph connectivity, while small imperfections only mildly affect precision.

  • Problem

    Because stabilizer outcomes are correlated, whether their measurements reveal all local unitary error parameters is a nontrivial reconstruction question.

    Source: §X · p. 14

  • Method

    The paper uses stabilizer-measurement information to infer site-dependent unitary errors, analyzing axis-aligned fields and polynomial solving methods.

    Source: §III · p. 2, §VI · p. 4, §X · p. 14

  • Results

    Reconstructing x- and y-axis rotations requires full rank of A and A + 1, respectively, whereas z-axis rotations can always be reconstructed; small imperfections only mildly affect precision.

    Source: §X · pp. 14–15

  • Takeaways & Limitations

    The authors assess reconstructability for chains, square lattices in arbitrary dimensions, and GHZ states represented as fully connected or star graphs.

    Source: §X · p. 14

Abstract

from arXiv · show

In the context of measurement-based quantum computation a way of maintaining the coherence of a graph state is to measure its stabilizer operators. Aside from performing quantum error correction, it is possible to exploit the information gained from these measurements to characterize and then counteract a coherent source of errors; that is, to determine all the parameters of an error channel that applies a fixed - but unknown - unitary operation to the physical qubits. Such a channel is generated, e.g., by local stray fields that act on the qubits. We study the case in which each qubit of a given graph state may see a different error channel and we focus on channels given by a rotation on the Bloch sphere around either the x, y or z axis, for which analytical results can be given in a compact form. The possibility of reconstructing the channels at all qubits depends non-trivially on the topology of the graph state. We prove via perturbation methods that the reconstruction process is robust and supplement the analytic results with numerical evidence.

I. INTRODUCTION

The paper asks whether stabilizer syndromes alone can characterize local unitary errors, and proposes automated statistical analysis of those measurements. It frames the setting as graph-state preparation for MBQC under fixed, site-dependent unitary errors.

I. INTRODUCTION

  • Automated classical analysis of syndrome measurements is proposed to detect and characterize noise across qubits without adding sensors.The motivation is on-the-fly calibration for increasingly integrated devices whose components may be difficult to access after assembly.Source: §I · p. 1

II. PRELIMINARY CONCEPTS AND NOTATION

  • The setting uses graph states, whose stabilizer correlators can be measured together to prepare a unique resource state for MBQC.The paper studies the restrictive QEC case in which only one state is protected.Source: §I · p. 1, §II · p. 2

III. ERROR MODEL

  • The error model is a fixed, site-dependent local unitary, concretely motivated by static stray fields causing Larmor precession.The results are intended to apply beyond the specified physical implementation when local unitary operations dominate; other decoherence can make reconstruction harder without qualitatively changing the results.Source: §III · pp. 2–3

IV. EFFECT OF THE UNITARY ERROR ON STABILIZER MEASUREMENTS

The paper derives how local unitary errors affect stabilizer statistics, then restricts reconstruction to promised settings with known Cartesian field direction. It gives methods for solving the resulting equations, while noting that recoverability depends on graph connectivity.

IV. EFFECT OF THE UNITARY ERROR ON STABILIZER MEASUREMENTS

  • The protocol prepares a graph state, lets errors act for Δt, measures correlators, and corrects the resulting state, assuming these operations are fast and nearly error-free.The initial analysis assumes the stray field is the only error source.Source: §IV · p. 3

V. PROMISE SETTINGS

  • Estimating 3N field parameters from N probability differences is underdetermined, so promise settings fix a known global Cartesian direction and reduce the unknowns to N intensities.The settings consider fields aligned with the x, y, or z axis.Source: §V · pp. 3–4

VI. SOLVING METHODS

  • Under these promises, the N polynomial equations can be solved efficiently, but graph connectivity can prevent recovery of every field intensity.The z-field case directly yields the cosine of each field intensity from measured rates.Source: §VI · p. 4

A. Complex logarithms

The solving method takes complex logarithms of the field equations to obtain a linear system over values modulo 2πi. A modified elimination procedure handles its multiple solutions and distinguishes singular from nonsingular cases.

A. Complex logarithms

  • Taking complex logarithms converts the product equations into linear equations because field cosines and probability differences can lie in [−1, 1].The logarithm is treated on the complex plane modulo 2πi, where multiplication becomes addition.Source: A. Complex logarithms · p. 4, B. Linear equations · p. 4

B. Linear equations

  • The logarithmic equations reduce to a linear system involving the graph’s adjacency matrix As and the vector of logarithmic field values.The adjacency matrix records which vertex pairs are connected by graph edges.Source: B. Linear equations · p. 4, C. Multiple solutions · p. 5

C. Multiple solutions

  • A modified Gauss-elimination procedure triangularizes the integer matrix while preserving the multiple solutions required by division modulo 2πi.The resulting form is the Hermite normal form, with an integer unimodular matrix P and upper-triangular Q.Source: C. Multiple solutions · p. 5
  • When As is nonsingular, the system has |det As| complex solutions; when As is singular, it is inconsistent or underdetermined.The solutions v yield the field parameters βa by element-wise exponentiation.Source: C. Multiple solutions · p. 5

D. Constraints on the solutions

The solution constraints reduce the allowed complex solutions to a discrete set, with their number governed by the even diagonal entries of Q; physical-range constraints further depend on the measured data. For linear chains, whether fields can be reconstructed depends on field axis, boundary conditions, and chain length.

D. Constraints on the solutions

  • The number of allowed solutions is 2^μ, where μ counts the even diagonal entries of Q; requiring βa and Δpa to lie in [−1, +1] further restricts solutions based on the collected data.The resulting number of acceptable solutions cannot be determined from the adjacency matrix alone.Source: D. Constraints on the solutions · pp. 5–6

VII. SOLUTION FOR PARTICULAR CLASSES OF GRAPH STATES

  • The analysis treats x-, y-, or z-directed fields on graph states, proceeding from linear chains to square lattices and then to quantum-repeater and QEC applications.The stated goal is to determine whether each graph permits complete reconstruction of the fields at every site.Source: §VII · p. 6

A. Linear chains

  • For open chains, X-fields are underdetermined when m is even and Y-fields when m ≡ 2 (mod 3); otherwise the equations have exactly one complex solution.For closed chains, X-fields are underdetermined at m = 1 or m ≡ 0 (mod 4), and Y-fields at m = 2 or m ≡ 0 (mod 3); otherwise the number of complex solutions is |det(Aclose_s)|.Source: A. Linear chains · p. 6
  • For closed-chain X fields, odd m with determinant 2 gives two real solutions, while the reported determinant-4 case has two diagonal entries of 2 in Q and four admissible solutions.The determinant-4 case is stated for m ≡ 1 (mod 2) in the supplied passage.Source: A. Linear chains · p. 6

B. Square lattices

For square lattices, the authors derive adjacency-matrix eigenvalues for planar, cylindrical, and toroidal boundary conditions and use zero eigenvalues to determine whether fields are reconstructible. The method extends to higher dimensions, and the Y-field eigenvalues are the X-field eigenvalues shifted by one.

B. Square lattices

  • Zero eigenvalues of a lattice adjacency matrix signal that not all field information can be reconstructed; their number, the rank defect, measures the amount unreconstructible.The section obtains eigenvalues in closed form to test for zero eigenvalues.Source: B. Square lattices · p. 7
  • The lattice spectra are obtained using tridiagonal Toeplitz and symmetric circulant matrix formulas, including the case of commuting diagonalisable matrix coefficients.The section applies these spectral methods to block-Toeplitz adjacency matrices.Source: B. Square lattices · pp. 7–8
  • The authors derive adjacency-matrix eigenvalues for planar, cylindrical, and toroidal square lattices, then identify reconstructible cases by checking those spectra for zeros.The lattices use different combinations of open and closed boundaries.Source: B. Square lattices · pp. 7–8
  • For any adjacency matrix A, the eigenvalues of Ay equal those of Ax plus one; Figure 1 lists square-lattice graph states up to 20×20 for which the error channel can be reconstructed.Source: B. Square lattices · p. 8

C. Cubic lattices in arbitrary dimension

The square-lattice spectral procedure extends to cubic lattices in arbitrary dimension, with boundary conditions chosen independently across dimensions.

C. Cubic lattices in arbitrary dimension

  • The procedure generalizes to cubic lattices in d dimensions, with closed boundaries in d1 dimensions and open boundaries in d2 dimensions.The dimensions satisfy d = d1 + d2.Source: C. Cubic lattices in arbitrary dimension · p. 8
  • The adjacency-matrix eigenvalues for these cubic-lattice classes are given by formulas indexed by the lattice size along each dimension.The size along dimension r is denoted mr.Source: C. Cubic lattices in arbitrary dimension · p. 8
  • The square-lattice figure uses each cell to represent an m1×m2 lattice and marks singular adjacency matrices; for cylinders, the open boundary is along rows and the closed boundary along columns.This figure describes two-dimensional square lattices, rather than the general-dimensional extension.Source: Figure 1 · p. 9

D. Application: quantum repeaters

The methods are applied to graph-state resources for quantum repeaters and measurement-based QEC. Recoverability depends on graph representation and topology, with different cases permitting reconstruction of different field axes.

D. Application: quantum repeaters

  • The examined repeater resources include linear chains, GHZ states, and irregular graph states, and the methods apply to many states useful for repeaters and measurement-based QEC.The examples include open and closed chains of lengths three and five, and four- and six-qubit GHZ states.Source: D. Application: quantum repeaters · p. 9
  • Switching between fully connected and star representations makes GHZ stray fields reconstructible for both X and Y fields when N > 2.The fully connected representation permits X-field but not Y-field reconstruction, while the star representation permits Y-field but not X-field reconstruction.Source: Table III · p. 10, D. Application: quantum repeaters · p. 10
  • The six-qubit graph for the Steane 5-qubit QEC code permits reconstruction in both the X-field and Y-field cases.The graph consists of a closed linear chain of five vertices plus a sixth vertex connected to all five.Source: D. Application: quantum repeaters · p. 10
  • In the X-field setting, any pair of vertices with identical neighbour sets makes the adjacency matrix singular and prevents reconstruction.The identical neighbour sets produce equal rows in the adjacency matrix.Source: D. Application: quantum repeaters · p. 10

VIII. ERROR PROPAGATION

The section analyzes how finite measurement precision and field-alignment uncertainty propagate through reconstruction. Its uncertainty estimates depend on the reconstruction matrix and distinguish all complex solutions from physically relevant real-field solutions.

A. Effect of uncertainty in the measured probabilities

  • For finite correlator measurements, the error-rate differences are unbiased estimators of probability differences, with approximately normal, small-variance distributions when M is large.The analysis initially assumes stray fields are perfectly aligned with one of the x, y, or z axes.Source: A. Effect of uncertainty in the… · p. 10
  • When the reconstruction matrix is nonsingular, uncertainty propagates through its inverse; the reconstructed covariance is independent of the solution branch.The analysis gives a perturbation bound in terms of the matrix condition number.Source: A. Effect of uncertainty in the… · p. 11
  • The uncertainty volume around each complex solution shrinks by a factor |det As|, while restricting to physical real-field solutions can reduce total uncertainty volume but never increase it.There are |det As| complex solutions, with an uncertainty region estimated around each valid solution.Source: A. Effect of uncertainty in the… · p. 11

B. Full error analysis

The section examines reconstruction under finite sampling, axis misalignment, and depolarizing noise. Simulations show how these imperfections affect precision across field axes and graph vertices, while perturbation analysis bounds the effect of small deviations from the promised axis.

B. Full error analysis

  • When measurement-rate errors and axis misalignment are small, the perturbed reconstruction remains close to the unperturbed solution, with the distance bound tending to zero as both vanish.The continuity result is established under regularity assumptions.Source: B. Full error analysis · pp. 11–12
  • Reconstruction error is lowest for Z-fields, higher for X-fields, and highest for Y-fields, consistent with the increasing degree of the polynomial systems and estimator cross-correlations.Cross-correlations are maximal in the Y-field case and minimal in the Z-field case.Source: B. Full error analysis · p. 13
  • Estimation precision varies by vertex and depends on the graph’s global connectivity; with uniform disturbance assumptions, vertices a and 11−a have equal average inaccuracy.The symmetry relation applies to the ten-qubit open-ended chain after averaging over field intensities.Source: B. Full error analysis · p. 13
  • For small M, reconstruction error decreases approximately as 1/M, then approaches a non-zero floor from depolarizing noise and axis misalignment.Figure 2 plots errors for X-, Y-, and Z-field cases across vertices of a ten-vertex open chain while varying q, ϵ, or M.Source: Figure 2 · p. 12, B. Full error analysis · p. 13
  • Some vertices are resilient to depolarizing noise: 3, 4, 7, and 8 for X-fields, and 2, 3, 5, 6, 8, and 9 for Y-fields.For these vertices, the reconstruction is almost insensitive to q when q ≲ 0.8; as q approaches 1, finite measurement counts make it unreliable.Source: B. Full error analysis · pp. 12–13

IX. EXCURSUS: EXTENSION OF THE FAMILY OF GRAPH STATES

The paper extends field reconstruction by using stabilizer measurements from graph states defined with different logical Pauli bases, then summarizes how reconstruction depends on graph topology and tolerates small imperfections. The added bases can make the system overdetermined, while the conclusions identify axis-specific rank conditions and robustness boundaries.

IX. EXCURSUS: EXTENSION OF THE FAMILY OF GRAPH STATES

  • The alternative approach gathers more information with different stabilizer-measurement sets, avoiding the assumption that fields point in a known direction.The resulting graph states can preserve the reference state's computational capabilities by redefining the local Pauli basis and adjusting MBQC measurements.Source: §IX · p. 13
  • Using at least four logical Pauli bases yields an overdetermined system that almost certainly has a unique solution and is usually more robust to small uncertainties.Three different bases can in principle reconstruct all stray-field parameters, but may leave multiple physically acceptable solutions.Source: §IX · p. 13
  • The three-qubit example estimates field intensities and directions from stabilizer measurements in four randomly selected logical Pauli bases.The procedure minimizes a cost function using simulated measurements; the displayed estimates are repeated across 10^3 trials.Source: Figure 3 · p. 14, §IX · p. 14

X. CONCLUSIONS

  • Rotations about x are fully reconstructible only when adjacency matrix A has full rank, rotations about y require A + 1 to have full rank, and rotations about z are always reconstructible.The conclusions identify graph connectivity as a non-trivial determinant of complete reconstruction, even with arbitrarily many stabilizer measurements.Source: §X · p. 14
  • The reconstruction analysis covers open and closed linear chains, square lattices in arbitrary dimensions, and GHZ states represented as fully connected or star graphs.Source: §X · p. 14
  • Finite stabilizer sampling, axis misalignment, and depolarizing noise only mildly affect reconstruction precision when the imperfections are small.Source: §X · p. 15

Appendix A: Derivation of the error probability

The appendix derives the probability difference for the two possible stabilizer-measurement outcomes under local unitary field errors. It uses the stabilizer expectation value and Pauli-matrix identities, then writes the field parameter as βa = cos λa.

Appendix A: Derivation of the error probability

  • The probability difference between outcomes κa = 0 and κa = 1 equals the expectation value of stabilizer Ka on the noisy graph state.Source: Appendix A · p. 15
  • The derivation evaluates the identity component of the conjugated stabilizer operator, using vanishing graph-state expectations for nontrivial Pauli terms.The stated vanishing expectations apply to nontrivial graphs without isolated vertices; the X case additionally assumes vertex a has a neighbour.Source: Appendix A · p. 15
  • The resulting expression matches Eq. (6) after defining βa as cos λa.Source: Appendix A · p. 15

Appendix B: Continuity of the reconstruction in presence of small misalignments

The appendix treats small axis misalignments and measurement-rate errors as perturbations to the reconstruction system. Under stated regularity conditions, it bounds the solution’s deviation and shows that this deviation vanishes as both perturbations vanish.

Appendix B: Continuity of the reconstruction in presence of small misalignments

  • The perturbed system adds a misalignment term f_ϵ(x) to the linear system, with ϵ bounding the maximum deviation of rotation axes from Cartesian directions.The measured data enter through ω = w + δw, and the perturbed solution is x = v + δv.Source: Appendix B · p. 16
  • For the vector v obtained by solving the unperturbed linear system, the setup requires r,R>0 with r≤||v||∞≤R and Re(v_a)<0 for every a∈V, and checks a bound on f_ε(v).Source: Appendix B · p. 16
  • The perturbed solution’s distance from the unperturbed solution is at most R∞, which tends to zero as measurement-rate error and misalignment vanish.Source: Appendix B · p. 16
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