Temporal Phase-Aware Scheduling and Correction for Variational Quantum Algorithms

1. Introduction

1.1 The Barren Plateau Problem

The barren plateau phenomenon fundamentally limits the scalability of variational quantum algorithms [1]. For a parameterized quantum circuit 𝑈 (𝜽) with cost function L(𝜽) = ⟨0|𝑈† (𝜽) 𝑂 𝑈 (𝜽)|0⟩, the variance of the gradient vanishes exponentially with system size:

Var[𝜕L/𝜕𝜃𝑘] ∼ exp(−𝛼𝑛)

where 𝑛 is the number of qubits and 𝛼 > 0 depends on circuit structure [2]. This is a mathematical property of high-dimensional Hilbert spaces arising from concentration of measure, and cannot be eliminated by error mitigation alone.

1.2 Decomposition of Gradient Variance

The observed gradient variance on real hardware can be decomposed as:

Var_observed = exp(−𝛼𝑛) × exp(−𝛽𝐿𝜀₀) × exp(−𝛾𝐿𝜀_dyn(𝑡))

1.3 Key Hypothesis

Hypothesis 1.1 (Diurnal Noise Component). The dynamic error component 𝜀_dyn(𝑡) on superconducting quantum processors contains a predictable, periodic component correlated with the local daily cycle:

𝜀_dyn(𝑡) = 𝐴 · 𝑓(𝜙(𝑡)) + 𝜉(𝑡)

where 𝐴 is the diurnal modulation amplitude, 𝜙(𝑡) ∈ [0, 1) is the normalized daily phase, and 𝜉(𝑡) is residual stochastic noise. If Hypothesis 1.1 holds with 𝐴 ≳ 0.5%, the dynamic component becomes exploitable through prediction and correction.

1.4 Scope and Claims

2. The HS◦ Framework

2.1 Temporal Phase Coordinate

Definition 2.1 (Canonical Daily Phase). Let 𝑇_cycle = 86400 s be the daily period and 𝑡₀ a reference epoch. The normalized daily phase is:

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

The radian phase is 𝜃(𝑡) := 2𝜋𝜙(𝑡) ∈ [0, 2𝜋).

2.2 HS Representations

Definition 2.2 (Equivalent HS Representations). The same phase point admits multiple representations:

HS◦(𝑡) := 360◦ × 𝜙(𝑡) ∈ [0◦, 360◦)
HS₁₂(𝑡) := 12 × 𝜙(𝑡) ∈ [0, 12)
HS_idx(𝑡) := ⌊12𝜙(𝑡)⌋ + 1 ∈ {1, . . . , 12}

Remark 2.1 (Precision). HS_idx is quantized and does not uniquely determine 𝜙. All correction formulas must use continuous representations (𝜙, HS◦, or HS₁₂).

2.3 Error Rate Model

Definition 2.3 (Diurnal Error Model). For qubit 𝑘, the effective gate error rate is:

𝜀_𝑘(𝑡) = 𝜀⁽⁰⁾_𝑘 + 𝐴_𝑘/2 (1 − cos(2𝜋𝜙(𝑡) − 𝛿_𝑘))

where: 𝜀⁽⁰⁾_𝑘 ≥ 0 is the baseline error (calibration), 𝐴_𝑘 ≥ 0 is the diurnal modulation amplitude, and 𝛿_𝑘 ∈ [0, 2𝜋) is the thermal phase lag. This model satisfies:

𝜀_min^𝑘 = 𝜀⁽⁰⁾_𝑘 (at 𝜙 = 𝛿_𝑘/(2𝜋), night)
𝜀_max^𝑘 = 𝜀⁽⁰⁾_𝑘 + 𝐴_𝑘 (at 𝜙 = 𝛿_𝑘/(2𝜋) + 0.5, noon + lag)

2.4 Thermal Lag Parameter

Definition 2.4 (Thermal Lag Estimation). Given time-series measurements 𝑦_𝑘(𝑡) of a drift observable for qubit 𝑘, fit the first-harmonic model:

𝑦_𝑘(𝑡) ≈ 𝑎_𝑘cos(2𝜋𝜙(𝑡)) + 𝑏_𝑘sin(2𝜋𝜙(𝑡)) + 𝑐_𝑘

Then: 𝐴_𝑘 := √(𝑎_𝑘² + 𝑏_𝑘²), 𝛿_𝑘 := atan2(𝑏_𝑘, 𝑎_𝑘). The thermal lag 𝛿_𝑘 captures the delay between external forcing (solar heating) and qubit response. Typical values are 1–3 hours (𝛿_𝑘 ∈ [0.26, 0.79] rad) for well-isolated cryostats.

3. Effective Hamiltonian Formulation

3.1 Slow Driver Model

For each qubit 𝑘, the effective Hamiltonian is:

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

where H^(𝑘)₀ is the calibrated qubit Hamiltonian.

Definition 3.1 (Diurnal Slow Driver). The slow driver is an 𝔰𝔲(2) perturbation:

𝛿H^(𝑘)_slow(𝑡) = (ℏ/2) [Δ𝜔_𝑘(𝑡)𝜎_z + 𝜖_𝑘(𝑡)𝜎_x + 𝛾_𝑘(𝑡)𝜎_y]

For frequency drift dominance, we use the minimal model:

Δ𝜔_𝑘(𝑡) = A^(𝜔)_𝑘sin(2𝜋𝜙(𝑡) − 𝛿_𝑘), 𝜖_𝑘 = 𝛾_𝑘 = 0

3.2 Hermiticity and Unitarity

Proposition 3.1. The slow driver with real coefficients is Hermitian, hence generates unitary evolution.

4. Feed-Forward Correction Protocol

4.1 Correction Unitary

The correction operator cancels the predicted slow driver:

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

where T denotes time-ordering.

4.2 Practical Implementation

For the frequency-drift model, correction is implemented as a drive frequency adjustment:

𝜔^(𝑘)_drive(𝑡) = 𝜔^(𝑘)₀ − 𝐴^(𝜔)_𝑘sin(2𝜋𝜙(𝑡) − 𝛿_𝑘)

The accumulated phase correction is:

𝜃^(𝑘)_corr(𝑡) = −∫^𝑡₀Δ𝜔_𝑘(𝜏)𝑑𝜏 = 𝐴^(𝜔)_𝑘(𝑇_cycle/2𝜋)[cos(2𝜋𝜙(𝑡) − 𝛿_𝑘) − cos(−𝛿_𝑘)]

4.3 Corrected Error Model

With correction efficiency 𝜂 ∈ [0, 1]:

𝜀^corr_𝑘(𝑡) = 𝜀⁽⁰⁾_𝑘 + (1 − 𝜂)(𝜀_𝑘(𝑡) − 𝜀⁽⁰⁾_𝑘)

For 𝜂 = 0.85, residual diurnal modulation is reduced to 15% of original amplitude.

5. Resource-Theoretic Formulation

5.1 U(1) Asymmetry Framework

The phase coordinate 𝜙(𝑡) defines a U(1) action. Within the asymmetry resource theory [3, 4], temporal alignment is quantified by the asymmetry monotone.

Definition 5.1 (HS◦ Asymmetry Monotone).

𝐴_HS(𝜌) := 𝑆(T(𝜌)) − 𝑆(𝜌) = min_𝜎∈Free 𝐷(𝜌∥𝜎)

where 𝑆(𝜌) = −Tr(𝜌 log 𝜌) is von Neumann entropy, T is the U(1) twirling channel, and 𝐷 is relative entropy.

5.2 Consumption Under Phase Diffusion

Theorem 5.1 (Entropy Production Identity [7]). For a phase-diffusion channel N(𝜌) = ∫ 𝑝(𝛼)𝑈(𝛼)𝜌𝑈(𝛼)†𝑑𝛼:

Δ𝐴_HS := 𝐴_HS(𝜌) − 𝐴_HS(N(𝜌)) = 𝑆(N(𝜌)) − 𝑆(𝜌) ≥ 0

Corollary 5.1. For pure input 𝜌 = |𝜓⟩⟨𝜓|:

Δ𝐴_HS = 𝑆(N(𝜌)) = 𝐻(𝑝)

where 𝐻(𝑝) is Shannon entropy of the residual distribution.

6. Theoretical Improvement Factor

6.1 Derivation

From the variance decomposition, the ratio of corrected to uncorrected variance is:

I := Var_corrected / Var_uncorrected = exp[𝛾𝐿(𝜀_uncorr − 𝜀_corr)]

At peak error (noon), 𝜀_uncorr = 𝜀₀ + 𝐴 and 𝜀_corr = 𝜀₀ + (1 − 𝜂)𝐴, giving:

I = exp(𝛾𝐿𝜂𝐴)

7. Mitigation Strategies

7.1 Strategy 1: Temporal Scheduling

Execute circuits during low-error phases: 𝜙∗ = arg min_𝜙 ¯𝜀(𝜙) ≈ 0 (midnight).

7.2 Strategy 2: HS◦ Correction

Apply feed-forward frequency adjustment per Eq. (17). This enables high-fidelity operation at any time.

7.3 Strategy 3: Quality-Weighted Gradients

Weight gradient samples by estimated quality: ∇L_eff = (∑_𝑖 𝑤(𝜙_𝑖)∇L_𝑖) / (∑_𝑖 𝑤(𝜙_𝑖)), where 𝑤(𝜙) = exp(−𝜆¯𝜀(𝜙)).

10. Discussion

10.1 Physical Mechanisms

Diurnal modulation may arise from: Thermal (Building/cryostat temperature cycles), Mechanical (HVAC, traffic vibrations), Electrical (Power grid load variations), or Operational (Calibration schedules, user load patterns). The thermal lag 𝛿𝑘 arises from heat diffusion through cryostat stages, with typical time constants of 1–3 hours.

10.2 Relation to Fundamental Barren Plateau

This work addresses only the dynamic noise term. The fundamental scaling exp(−𝛼𝑛) remains unchanged. For circuits where exp(−𝛼𝑛) ≫ exp(−𝛾𝐿𝐴), the fundamental BP dominates and HS◦ provides limited benefit. Conversely, when noise-induced suppression is comparable to or larger than fundamental BP, HS◦ correction can meaningfully extend practical circuit depth.

10.3 Implementation Requirements

1. Characterization: Fit (𝐴𝑘, 𝛿𝑘) from 7–14 days of calibration data.
2. Prediction: Compute 𝜙(𝑡) and expected error in real-time.
3. Correction: Apply feed-forward frequency/phase adjustments.
4. Monitoring: Track residual drift, refit periodically.

Computational overhead is negligible (𝜇s per update).

11. Limitations and Caveats

• Simulation only: All results are from numerical simulation, not hardware experiments.
• Amplitude unknown: Actual diurnal amplitude 𝐴 on real hardware has not been systematically characterized.
• Hardware-specific: Parameters (𝐴𝑘, 𝛿𝑘) will vary across processors, cooling systems, and facilities.
• First-harmonic approximation: Higher harmonics may be present; model may require extension.
• Fundamental BP unaffected: Deep circuits with 𝑛 ≫ 20 will still face exponentially small gradients.
• Parameter stability: Seasonal variations and hardware changes require periodic recalibration.
• Interaction with other techniques: Compatibility with ZNE, PEC, dynamical decoupling not analyzed.

12. Conclusion

We have developed the HS◦ framework for temporal phase-aware quantum computing and demonstrated through simulation that:

1. Diurnal noise modulation can cause significant degradation of gradient variance.
2. The HS◦ feed-forward correction can recover a substantial fraction of optimal scheduling performance.
3. The framework integrates naturally with 𝑈(1)-asymmetry resource theory for principled quantification.

The critical next step is experimental characterization of diurnal modulation on real quantum hardware.