Digitized counterdiabatic quantum critical dynamics

1. Problem Statement

The paper addresses the challenge of topological defect formation during fast quenches across quantum phase transitions (QPTs).

• The Kibble-Zurek Mechanism (KZM): In slow, adiabatic driving, the density of defects (e.g., domain walls or kinks) scales as a power law of the driving rate. However, this scaling breaks down in fast quenches (non-adiabatic regimes), where the defect density saturates to a plateau independent of the quench rate.
• Adiabatic Quantum Computing Limitations: While adiabatic evolution is ideal for quantum optimization and state preparation, the time required to maintain adiabaticity often exceeds the coherence times of current Noisy Intermediate-Scale Quantum (NISQ) devices.
• The Gap: While Counterdiabatic (CD) driving (a "shortcut to adiabaticity") has been theoretically proposed to suppress excitations by adding auxiliary terms to the Hamiltonian, its effectiveness in suppressing topological defects in many-body systems during fast quenches had not been experimentally verified, particularly on large-scale quantum hardware.

2. Methodology

The authors employed a hybrid approach combining theoretical derivation, numerical simulation, and experimental execution on cloud-based superconducting quantum processors.

A. Theoretical Framework

• Model: They utilized the Transverse-Field Ising Model (TFIM) in 1D and 2D geometries, driving the system from a paramagnetic phase to a ferromagnetic phase.
• Counterdiabatic Protocol:
  • The total Hamiltonian is $H(\lambda) = H_0(\lambda) + \dot{\lambda}A_\lambda$, where $H_0$ is the standard annealing Hamiltonian and $A_\lambda$ is the Adiabatic Gauge Potential (AGP).
  • Since exact AGP calculation is intractable for large systems, they used a Nested-Commutator (NC) expansion to approximate $A_\lambda$.
  • They derived the first-order approximation: $A^1_\lambda = 2gJ\alpha_1(\lambda) \sum_{\langle i,j \rangle} (\hat{Y}_i \hat{Z}_j + \hat{Z}_i \hat{Y}_j)$.
  • Analytic solutions for defect statistics were derived for the 1D case in the fast-quench limit ($T \to 0$).

B. Experimental Implementation

• Hardware: Experiments were conducted on IBM Quantum processors (specifically ibm_fez and ibm_marrakesh), utilizing up to 156 qubits with a heavy-hexagonal coupling map.
• Digitization: The continuous time evolution was digitized using the first-order Suzuki-Trotter decomposition.
  • The evolution operator $U(T)$ was decomposed into $M$ steps of size $\delta t$.
  • The circuit included gates for the initial Hamiltonian ($H_i$), final Hamiltonian ($H_f$), and the CD term.
• Geometries Tested:
  1. 1D chain ($N=100$).
  2. 2D Heavy-hexagonal lattice ($N=156$).
  3. 3-leg ladder ($N=45$).
  4. 2D Square lattice ($6 \times 6$).
• Error Mitigation: The team utilized graph coloring for gate parallelization and tested various transpilation strategies (e.g., As-Late-As-Possible scheduling, Dynamical Decoupling). They found that standard gates with no specific scheduling performed best in their specific noise environment.

C. Metrics

• Defect Density ($\kappa_1$): Measured as the expectation value of kinks in magnetization: $\langle \hat{n}_{def} \rangle = \frac{1}{2N_e} \sum_{\langle i,j \rangle} \langle 1 - \hat{Z}_i \hat{Z}_j \rangle$.
• Higher-Order Cumulants: They analyzed the variance ($\kappa_2$) and skewness ($\kappa_3$) of the defect distribution to characterize the full statistical behavior, not just the mean.
• Reference: Results were compared against digitized annealing (without CD) and theoretical benchmarks (Exact solutions for 1D, Matrix Product State (MPS) simulations for 2D).

3. Key Contributions

1. First Experimental Demonstration: This is the first experimental evidence that digitized CD protocols successfully suppress topological defect formation in fast quenches across quantum phase transitions on NISQ hardware.
2. Scaling Breakdown Verification: The study experimentally confirms the breakdown of KZM scaling in fast quenches, showing the defect density plateau, and demonstrates that CD can lower this plateau.
3. Analytic Derivation: The authors derived an exact analytic solution for the defect number distribution in the 1D fast-quench limit, providing a rigorous benchmark for the experimental data.
4. Statistical Characterization: Beyond mean defect density, the paper provides a comprehensive analysis of the full distribution (variance and skewness), revealing geometry-dependent asymmetries in defect formation.

4. Key Results

• Defect Suppression: In the fast-quench regime, the digitized CD protocol reduced the formation of topological defects by up to 48% compared to standard digitized annealing.
• 1D Chain Results:
  • Experimental data matched theoretical predictions closely for short evolution times.
  • Without CD, the defect density plateaued near the initial state value ($\kappa_1 \approx 0.5$).
  • With CD, the plateau was significantly lowered ($\kappa_1 \approx 0.22$ in the $T \to 0$ limit).
  • The distribution shifted from Gaussian (no CD) to a slightly positively skewed distribution (with CD).
• 2D and Complex Geometries:
  • Heavy-hexagonal lattice: Showed clear separation between CD and non-CD defect densities, with CD reducing the mean defect density.
  • Square Lattice: Showed a reduction in mean defects but exhibited negative skewness (asymmetric distribution with a tail below the mean), unlike the 1D and heavy-hexagonal cases. This highlights a qualitative difference in non-equilibrium dynamics based on lattice connectivity.
• Noise and Deviations:
  • Experimental data deviated from theoretical/MPS predictions at longer evolution times ($T \gtrsim 1/J$) due to hardware noise.
  • In the fast-quench limit (short times), the CD protocol's benefit was robust against noise, whereas standard annealing failed to suppress defects.
• Cumulant Behavior:
  • The skewness ($\kappa_3$) showed non-monotonic behavior and strong dependence on geometry, indicating that the shape of the defect distribution is a sensitive probe of the underlying dynamics.

5. Significance

• Quantum Optimization: The results suggest that CD protocols can significantly improve the fidelity of final states in quantum optimization problems (like QAOA) where fast annealing is necessary due to coherence time limits.
• Material Design: The ability to control defect formation in fast quenches is crucial for simulating material properties and phase transitions on current quantum hardware.
• Beyond KZM: The work extends the understanding of non-equilibrium critical phenomena, proving that "shortcuts to adiabaticity" are viable even in the deep non-adiabatic regime where traditional scaling laws fail.
• NISQ Utility: It demonstrates that current noisy quantum processors (up to 156 qubits) can perform meaningful many-body simulations that are difficult to replicate classically (especially for 2D geometries), provided that error-mitigating strategies like CD are employed.

In conclusion, the paper establishes digitized counterdiabatic driving as a practical and effective tool for controlling quantum critical dynamics on real-world quantum hardware, offering a pathway to high-fidelity quantum state preparation and optimization despite the constraints of NISQ devices.