Self-consistent tomography of temporally correlated errors

I. INTRODUCTION

• Correlated errors matter for verifying sub-threshold devices and optimizing quantum error correction, but randomized benchmarking and conventional tomography do not describe temporal correlations.Source: §I · p. 1
• The proposed protocols reconstruct temporally correlated error models without additional environment-access operations, using linear operator tomography based only on operations available on the system.The method can require substantial data for exact reconstruction, so the paper also develops practical approximate models.Source: §I · p. 1
• The protocols target efficient approximate descriptions of low-frequency time-dependent noise and context-dependent noise, with numerical demonstrations of their practical implementation.Source: §I · pp. 1–2

II. THE MODEL OF A QUANTUM COMPUTER

• The system–environment model treats the terminal as selecting system-environment evolutions, while the resulting system operation depends on the initial joint state.This dependence allows environmental evolution to produce general temporal correlations, including correlations from classical variables or quantum environmental coupling.Source: Figure 1 · p. 2, §II · pp. 2–3
• For slowly varying noise, the paper assumes the environmental variable is effectively constant within a circuit run and represents it as an environment state, enabling factorized multi-gate descriptions.The construction can be generalized to multiple random variables and noise whose changes during a circuit are nonnegligible.Source: §II · p. 2
• Because the environment is not fully accessible, linear operator tomography cannot reconstruct complete CP maps on the joint system-environment space and uses only system-control operations.Source: §II · p. 3

matrix representation

• The quantum computer is specified by an initial state, measured observable, and completely positive elementary operations, all treated as unknown quantities for tomography.Source: matrix representation · p. 3
• Pauli transfer matrices encode states, observables, and operations as real vectors and matrices, so sequence outcomes become matrix products evaluated between measurement and state vectors.Source: matrix representation · p. 3

III. SELF-CONSISTENT TOMOGRAPHY WITHOUT INFORMATION COMPLETENESS

• Because system operations act on the system-environment space without informationally complete state and measurement sets, conventional process tomography cannot reconstruct the full space.Source: §III · p. 3, Linear operator tomography · p. 4
• LOT represents gates by operators on the accessible Hermitian subspace, and these operators suffice to compute arbitrary sequence averages while allowing tomography with temporal correlations.Source: §III · p. 3, Linear operator tomography · p. 4

Linear operator tomography

• LOT reconstructs projected initial states, observables, and operations on the accessible subspace; its exact protocol assumes the subspace dimension is finite and known, unlike later approximate protocols.Source: Linear operator tomography · pp. 4–5
• The reconstructed data exactly characterize the quantum computer: for any operation sequence, the tomography model and experimental computation result are equal, up to similarity transformations.Source: Linear operator tomography · p. 4

IV. SPACE DIMENSION TRUNCATION

• A small subspace-invariance error ensures that approximate tomography can characterize arbitrary sequences of elementary operations.The sufficient condition bounds the deviation of the truncated subspace under each operation; small ϵ makes approximate tomography possible.Source: §IV · p. 5

V. APPROXIMATE MODELS OF TEMPORALLY CORRELATED ERRORS

• Low-dimensional approximate models exist for both low-frequency noise and context-dependent noise, making LOT practical for these temporal correlations.The truncated dimension can be selected using prior noise knowledge or validated from Gram-matrix spectra.Source: §V · p. 5, §IV · p. 5

A. Low-frequency noise and classical random variables

• For low-frequency noise, the model represents stochastic parameter evolution through system operations, transition dynamics, and parameter-dependent observables.The transition operation describes how the random variables evolve during each gate, while the system operation and observable depend on their values.Source: A. Low-frequency noise and classical… · pp. 5–6
• If correlations of the random variables are approximately reconstructed, the resulting model approximately characterizes the quantum computer.With weak errors, low-order moments can be sufficient, and rapidly convergent moments support efficient low-dimensional approximation.Source: A. Low-frequency noise and classical… · pp. 5–6

1. Second-order approximation

• For stationary, slowly varying noise with negligible higher-order correlations, a two-point distribution at λ = ±σ reconstructs the relevant second-order correlations with m = 2.The approximation uses pa(±σ) = 1/2 and treats the two-time product as approximately constant.Source: §1 · p. 6
• When correlations decay exponentially with operation time, the same m = 2 approximation applies with a transition matrix encoding the decay.The two-time correlation decreases according to accumulated operation-dependent factors Γ(χ).Source: §1 · p. 6

2. High-order approximations and multiple variables

• If correlations above order l_t are negligible, choosing m = ⌈(l_t+1)/2⌉ reconstructs correlations through order l_t.Here m is the dimension of the environment in the approximate model.Source: §2 · p. 6
• For multiple random variables with rapidly convergent moments, Gaussian cubature provides the corresponding approximate reconstruction.The multivariable extension uses the same moment-based strategy rather than treating each variable independently.Source: §2 · p. 6

B. Classical context-dependent noise

• Context dependence is represented by classical variables recording previous operations, so the general temporally correlated-error formalism applies directly.An operation-dependent memory variable updates after each gate and determines the next system operation.Source: B. Classical context-dependent noise · p. 7
• Ion-trap heating and cooling can be modeled with discretized phonon variables, whose small truncation supports a low-dimensional approximation.The gate operation includes a transition describing heating, while cooling is represented in the same form.Source: B. Classical context-dependent noise · p. 7

VI. APPROXIMATE QUANTUM TOMOGRAPHY

• Approximate LOT is needed because the exact environment state space is too large, and it can be implemented using LIM or MLE.The truncation is selected from an approximately invariant subspace and reduced to a chosen dimension d.Source: §VI · p. 7

A. Linear inversion method

• LIM may fail for an approximate model when the reconstructed inverse differs substantially from the inverse of the underlying reduced matrix.The stated sufficient conditions also require approximately invariant input and output subspaces and small reconstruction errors.Source: A. Linear inversion method · pp. 7–8
• LIM estimates the reduced model by singular-value truncation of a Gram matrix and then reconstructs the reduced operations and state-measurement representation.Trial states and observables are generated from operation sequences, and the greatest d singular values define the retained subspace.Source: A. Linear inversion method · p. 8
• For the simulated gate sequences, d = 7 fits the quantum-computer decay more accurately than d = 4, with d = 4 equivalent to conventional GST.The figure compares the predicted probability of |0⟩ against gate number; linear inversion gives similar results.Source: Figure 2 · p. 8

B. Maximum likelihood estimation

• MLE estimates the initial state, measured observable, and operation matrices by optimizing model parameters against experimental circuit data.The protocol parameterizes these objects, measures multiple circuits, evaluates their predicted means and variances, and selects the best-fit parameters.Source: B. Maximum likelihood estimation · pp. 8–9
• The likelihood compares predicted and measured circuit means using a Gaussian model whose width is set by each circuit’s variance.For circuit m, the fitted objective uses the discrepancy between C̄_m(x) and C_m together with σ_m^2.Source: B. Maximum likelihood estimation · p. 9
• When low-frequency or context-dependent noise dominates, the error model can be parameterized according to the corresponding structured noise models.This replaces unconstrained element-by-element parameterization with a form matched to the expected temporal correlations.Source: B. Maximum likelihood estimation · p. 9

VII. NUMERICAL SIMULATION OF LOW-FREQUENCY NOISE

• The simulation represents low-frequency noise with a Gaussian variable λ whose change is negligible during a circuit, while gate depolarizing rates depend on λ.Hadamard and phase gates use ϵH(λ) = ϵS(λ) = η[1 − exp(−λ^2)].Source: §VII · p. 9
• Temporal correlations produce a non-exponential randomized-benchmarking decay, unlike the exponential decrease obtained without temporal correlation.For constant depolarizing rates, the probability follows [1 + (1 − ϵ)^Ng]/2.Source: §VII · p. 9
• The simulation compares LOT and conventional GST using state-space dimensions d = 4 and d = 7, corresponding to one- and two-dimensional environment approximations.For d = 4, LOT is equivalent to conventional GST because the one-dimensional environment is trivial.Source: §VII · p. 9
• LOT with d = 7 characterizes the behavior of the quantum computer much more accurately than LOT with d = 4, equivalent to conventional GST.Source: §VII · pp. 9–10

VIII. CONCLUSIONS

• The protocols can produce exact temporally correlated error models with sufficient data, while approximate models require moderate additional measurement overhead and improve behavioral prediction over GST.The authors also present the protocols as a way to quantitatively assess temporal correlations.Source: §VIII · p. 10

Appendix A: Exact linear operator tomography

• The appendix verifies the remaining projection identities and concludes that the stated equations are valid.Source: Appendix A · pp. 10–11
• Theorem 1 shows that inserting the relevant input and output projections preserves expectation values for arbitrary operation sequences.Source: Appendix A · pp. 10–11
• If the projected preparation and measurement vectors are linearly independent, g = MoutMin is invertible and the effective operation representation composes through g.Source: Appendix A · p. 11
• The proof uses full-rank projected matrices and their pseudoinverses to establish g−1 = (PMin)+(MoutP)+ and the composition identity.Source: Appendix A · p. 11

Appendix B: Space dimension truncation

• The appendix permits either a singular-value-based vector norm or the trace norm for the analysis.Source: Appendix B · p. 11
• Theorem 3 assumes uniform bounds NQ and Nρ on observable and state norms and applies them to any operation sequence.Source: Appendix B · pp. 11–12
• The induction argument shows that if inequality (B2) holds at sequence length m, it also holds at m + 1.Source: Appendix B · p. 12

Appendix C: Linear inversion method

• The approximate model is built from an orthonormal basis of the input subspace and a corresponding approximate-model basis, connected by transformations T and T+.Source: Appendix C · p. 12
• The analysis represents projected operations through Oa and relates them to the actual operation using ΠinOΠin = T+OaT.Source: Appendix C · p. 13
• The appendix bounds the approximate operation norm by ∥Oa∥ ≤ ∥O∥ + ∥δO∥ and extends the resulting inequality to arbitrary operation sequences.Source: Appendix C · pp. 12–13
• Invertibility of g implies full rank for Min and Mout, while δg = 0 under the stated conditions.Source: Appendix C · p. 13
• The projection mismatch is defined as δΠ ≡ Πin − Πout, with invertibility of 1 − δΠ required for the subsequent relations.Source: Appendix C · p. 13

• The joint system–environment Hilbert space has dimension d_Sd_E, while its column-vector representation has dimension (d_Sd_E)^2.Source: Appendix D · p. 14
• For classical random-variable noise, the environment density matrix is diagonal, so the state can be represented by a d_S^2d_E-dimensional vector.Source: Appendix D · p. 14
• The system state is decomposed into the maximally mixed component and orthogonal components, enabling focus on the subspace spanned by |1⟩⟩_S⊗|λ⟩⟩_E.Source: Appendix D · p. 14
• For stationary distributions, only one nontrivial vector in the projected subspace contributes, making the effective state dimension [(d_S^2−1)d_E+1].Source: Appendix D · p. 14

1. Actual quantum computer simulation

• Gaussian cubature matches noise moments through ninth order, while a seventh-order approximation produces a negligible difference.Source: §1 · p. 14

2. Linear inversion method simulation

• Linear-inversion simulations use H,S gate sequences, with four sequences for d=4 and 123 sequences for d=7.Source: §2 · p. 14

3. Maximum likelihood estimation simulation

• Maximum-likelihood estimation uses the same circuits as the linear-inversion simulations.Source: §3 · p. 14

4. Pauli transfer matrices

• The Pauli-transfer-matrix analysis uses seven-dimensional ideal-gate matrices and compares LOT estimates obtained by linear inversion and maximum likelihood against actual probabilities.Source: §4 · p. 14, Figure 4 · p. 15