Spatio-Temporal Structure of an Effective Qubit Hamiltonian (v4.7)

Diurnal Phase 𝜙(𝑡) and Thermal Lag 𝛿𝑘 as a Slow Driver

Status: Theoretical Version 4.7

DOI: 10.5281/zenodo.18047530

Abstract

Standard effective Hamiltonian models for superconducting qubits often treat the laboratory environment as stationary over calibration horizons. In practice, slow, repeatable modulations correlated with a daily cycle may appear through thermal, mechanical, or infrastructure-driven pathways. We propose an operational extension: a diurnal phase coordinate 𝜙(𝑡) ∈ [0, 1) (a 𝑈 (1) phase on S1) and a qubit-specific thermal lag 𝛿𝑘 capturing spatial phase delays across a chip or package. We express the resulting slow driver as an effective 𝔰𝔲(2) term added to each qubit Hamiltonian, and we outline a deterministic feedforward correction strategy based on model fitting and time-dependent control-frame updates. The framework is a control/identification model and does not propose new physics.

1. Introduction

In superconducting platforms, qubit parameters are influenced by device physics, control electronics, and slow environmental variations. Motivated by the empirical possibility of repeatable daily modulation patterns, we introduce a minimal phase-based regressor and a spatial lag parameter for each qubit.

The key conceptual separation is:

Phase as a coordinate (canonical): a normalized cyclic phase 𝜙(𝑡) ∈ [0, 1).
HS representations (display languages): degrees HS◦_deg and HS segment index are representations of the same phase point for human/UI use.

2. Phase conventions and HS representations

2.1 Canonical daily phase

Let the daily cycle period be 𝑇_cycle = 86,400 s and let 𝑡₀ denote a fixed epoch (protocol convention). Define the normalized daily phase

𝜙(𝑡) := ((𝑡 − 𝑡₀) / 𝑇_cycle) mod 1 ∈ [0, 1) ≅ S1.

We also use the radian phase 𝜃(𝑡) := 2𝜋𝜙(𝑡) ∈ [0, 2𝜋) when convenient.

2.2 HS as equivalent representations

The same phase point can be displayed as:

𝐻𝑆◦_deg(𝑡) := 360◦ · 𝜙(𝑡) ∈ [0◦, 360◦),

𝐻𝑆₁₂(𝑡) := 12 · 𝜙(𝑡) ∈ [0, 12),

𝐻𝑆_idx(𝑡) := ⌊12𝜙(𝑡)⌋ + 1 ∈ {1, . . . , 12}.

Important precision note: 𝐻𝑆_idx is a quantized representation and does not uniquely determine 𝜙. Trigonometric modeling must use 𝜙 (or 𝜃), not 𝐻𝑆_idx.

3. Spatio-temporal effective Hamiltonian

For each qubit 𝑘, we write an operational effective model:

H^(𝑘)_eff(𝑡) = H^(𝑘)₀ + 𝛿H^(𝑘)_slow(𝑡),

where H^(𝑘)₀ is the standard calibrated effective qubit Hamiltonian (including control terms as appropriate), and 𝛿H^(𝑘)_slow(𝑡) captures slow modulation correlated with the daily phase.

3.1 Thermal lag parameter 𝛿𝑘 (definition and meaning)

We model a qubit-specific phase delay 𝛿𝑘 ∈ R (radians), interpreted as a spatially induced lag between the canonical daily phase 𝜃(𝑡) = 2𝜋𝜙(𝑡) and the observed peak/trough of a drift observable for qubit 𝑘.

Operational estimation of 𝛿𝑘: Let 𝑦𝑘(𝑡) be a measured drift observable for qubit 𝑘 (e.g. frequency detuning estimate, phase error proxy, or a calibration residual) sampled over time with associated phase 𝜙(𝑡). Fit the first-harmonic model:

𝑦𝑘(𝑡) ≈ 𝑎𝑘 cos(2𝜋𝜙(𝑡)) + 𝑏𝑘 sin(2𝜋𝜙(𝑡)) + 𝑐𝑘.

Define

𝐴𝑘 := √(𝑎²𝑘 + 𝑏²𝑘), 𝛿𝑘 := atan2(𝑏𝑘, 𝑎𝑘).

Then 𝐴𝑘 is the fitted modulation amplitude and 𝛿𝑘 is the fitted phase lag.

Remark 3.1: Higher harmonics may be included if data support them. The first harmonic is a minimal, interpretable starting point.

3.2 Slow driver as an effective 𝔰𝔲(2) field

A generic single-qubit Hermitian driver can be written as:

𝛿H^(𝑘)_slow(𝑡) = (ℏ/2) (Δ𝜔𝑘(𝑡)𝜎𝑧 + 𝜖𝑘(𝑡)𝜎𝑥 + 𝛾𝑘(𝑡)𝜎𝑦),

where (Δ𝜔𝑘, 𝜖𝑘, 𝛾𝑘) are real-valued slow coefficients.

Minimal diurnal model with thermal lag. A simple choice consistent with the definition of 𝛿𝑘 is:

Δ𝜔𝑘(𝑡) = 𝐴𝑘 cos(2𝜋𝜙(𝑡) − 𝛿𝑘) + 𝑏𝑘,

𝜖𝑘(𝑡) = 𝐴′𝑘 sin(2𝜋𝜙(𝑡) − 𝛿𝑘),

𝛾𝑘(𝑡) = Γ𝑘(𝑡),

where 𝐴′𝑘 allows a (possibly smaller) transverse component and Γ𝑘(𝑡) can capture other slow drifts.

Why not only 𝜎𝑧?: If the dominant effect is frequency drift, setting 𝜖𝑘 = 𝛾𝑘 = 0 may be sufficient. Nonzero transverse terms can model effective axis drift or cross-couplings in an operational way, but should be justified empirically.

4. Formal consistency

4.1 Hermiticity and unitary evolution

All coefficients are real-valued, hence H^(𝑘)_eff(𝑡) is Hermitian and generates unitary evolution.

4.2 Algebraic closure

The driver remains within the span of Pauli operators, i.e. within 𝔰𝔲(2) for a single qubit:

[𝜎𝑖, 𝜎𝑗] = 2𝑖𝜀𝑖𝑗𝑘𝜎𝑘.

5. Deterministic correction as a control-plane strategy

5.1 Time-ordered correction operator (formal)

If one defines a correction evolution generated by the slow term, the formal expression is

𝑈^(𝑘)_corr(𝑡) = T exp(−(𝑖/ℏ) ∫^𝑡₀ 𝛿H^(𝑘)_slow(𝜏)𝑑𝜏).

5.2 Practical implementation viewpoint

In practice, deterministic correction is implemented as feedforward updates to control parameters, e.g.:

adjust drive frequency by −Δ𝜔𝑘(𝑡) (compensate detuning),
update virtual 𝑍-frames / phases to track predictable drifts,
schedule recalibration or parameter refresh at chosen phase windows.

The slow driver is treated as an exogenous, low-frequency context feature.

6. Illustrative drift plot

Illustrative plot (not experimental validation). It shows a diurnal drift pattern with a qubit-specific lag and a large reduction in residual drift after model-based correction.

Figure 1: Illustrative diurnal drift proxy showing a lagged peak (as if induced by a thermal delay), and a large residual reduction under model-based feedforward correction.

7. Future work

Validate 𝛿𝑘 estimation on measured drift observables and compare first-harmonic vs multi-harmonic models.
Couple spatial structure (chip regions) to fitted lags 𝛿𝑘 and amplitudes 𝐴𝑘 to test thermal/structural hypotheses.
Extend from single-qubit terms to multi-qubit couplings if data show correlated drifts in coupling strengths.
Integrate phase-window scheduling and Q-Address metadata so that correction refresh is coordinated deterministically.

Data Availability Statement

This conceptual foundation and all versions are available under the Concept DOI: 10.5281/zenodo.17882161. This specific manuscript version is archived under: 10.5281/zenodo.18047530.

References

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