Hybrid Real-Imaginary Time Evolution for Low-Depth Hamiltonian Simulation in Quantum Optimization

TL;DR

The paper tackles the inefficiencies of counterdiabatic (CD) driving in complex quantum optimization, where long evolution times reduce CD effectiveness and inflate circuit depth. It introduces HAVQDS, a hybrid real-imaginary time evolution framework that combines adaptive real-time dynamics (AVQDS) with variational imaginary-time filtering to suppress excitations without adding quantum gates. In Sherrington-Kirkpatrick benchmarks up to $n=14$ qubits, HAVQDS achieves higher final approximation ratios than both adiabatic and CD approaches while reducing CNOT counts by $1$–$2$ orders of magnitude, with a resource footprint that scales favorably as $O(n^2)$. This CD-free methodology offers a practical route to high-fidelity quantum optimization on NISQ devices and can be extended to other challenging optimization problems.

Abstract

Counterdiabatic (CD) driving is a powerful technique for accelerating adiabatic quantum computing. However, it becomes self-limiting in complex optimizations like the Sherrington-Kirkpatrick model: long evolution times $T$ needed to traverse crossings force the CD strength to scale as $1/T$, causing it to vanish before convergence and wasting the quantum resources invested in its implementation. We break this trade-off with a Hybrid adaptive variational quantum dynamics simulation (HAVQDS). HAVQDS combines adaptive real-time evolution for circuit compression with imaginary-time steps that suppress excitations at no extra gate cost. For the SK model (6--14 qubits), HAVQDS achieves higher approximation ratios than adiabatic or CD approaches, while reducing CNOT counts by 1--2 orders of magnitude, enabling high-fidelity quantum optimization.

Figures (6)

• Figure 1: Approximation Ratio $r$ vs Total Time $T$ for AD and CD Evolution Schemes. The approximation ratio $r$ is plotted as a function of total time $T$ for AD and CD evolution strategies, for system sizes of $n=8$ and $n=10$ qubits. Data points represent the mean approximation ratio, with error bars indicating one standard deviation. The convergence of CD and AD performance at larger $T$ demonstrates the efficacy trade-off of the CD approach.
• Figure 2: Left panels: Instantaneous approximation ratio $r(s)$ for AD (top) and CD (bottom) as a function of the dimensionless time $s=t/T$. Different colors correspond to annealing times $T=1, T=5$, and $T=10$ (purple, blue, orange). The decline in performance for $s > 0.4$ is evident for both protocols at longer $T$, highlighting the inability of the weakened CD term to prevent non-adiabatic transitions. Right panel: Instantaneous energy levels $E_0$ to $E_4$ (from bottom to top) of the Hamiltonian as a function of $s$. Shaded regions indicate standard deviation over 10 samples. Between $s=0.4$ and $s=0.8$, the energy gap narrows or even closes. All data in both panels are for an 8-qubit system.
• Figure 3: Performance and efficiency of evolution schemes. (a) Approximation error $1-r$ (log scale) versus total time $T$. (b) CNOT gate count (log scale) versus $T$. Results are for a 10-qubit SK model, with error bars indicating standard deviation over multiple instances. HAVQDS achieves a lower approximation error than both AD and CD Trotterized approaches, while requiring significantly fewer CNOT gates.
• Figure 4: Instantaneous approximation ratio across system sizes. Each subplot corresponds to a different qubit number $n$ (6 to 14). The horizontal axis is the dimensionless time $s = t/T$ ($T=1$), and the vertical axis is the instantaneous approximation ratio $r(s)$. Each curve represents an independent instance of the SK model. HAVQDS maintains high fidelity throughout the evolution, especially in the region of avoided crossings ($s \approx 0.4$--$0.8$).
• Figure 5: Imaginary-time iterations versus rescaled time. Each subplot shows the number of imaginary-time steps per real-time step as a function of $s = t/T$ for a system size $n$ (6 to 14). The shaded area represents the variance across 10 independent samples. The required number of steps peaks in the critical region of avoided crossings.